hycon-cemacs june 1-2 2006 proceedings · proceedings of hycon & cemacs workshop on automotive...

Network of Excellence Thematic Priority 2

FP6 – IST- 511368

HYCON Hybrid Control: Taming Heterogeneity and Complexity

of Networked Embedded Systems

Starting date: 15 September 2004 Duration: 4 years

Deliverable number D4c.4.2 Title

Proceedings of the HYCON & CEmACS Workshop on Automotive Systems and Control

Work package WP4c Due date Month 21 Actual submission date 14/03/2006 Organisation name(s) of lead contractor for this deliverable

LTH

Author(s)

Rolf Johansson [email protected] Anders.Rantzer [email protected] Alberto Sangiovanni Vincentelli [email protected] Balluchi [email protected]

Nature Report

Revision v1.0 07/09/2006 18:18

Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006) Dissemination Level

PU Public X

PP Restricted to other programme participants (including the Commission Services)

RE Restricted to a group specified by the consortium (including the Commission Services)

CO Confidential, only for members of the consortium (including the Commission Services)

Executive summary

This report contains the Proceedings of the HYCON-CEmACS Workshop on Automotive Applications of Hybrid Systems held in Lund on June 1-2, 2006. The aim of the workshop was identifying challenges and opportunities for hybrid systems in automotive design. In particular, the following topics were discussed: industrial trends and concerns; methodologies, flows and tools; theoretical open problems. The workshop was attended by HYCON partners, CEmACS partners and representatives from automotive companies: DaimlerChrysler, DT Innovations, FIAT, Ford, PSA Peugeot Citroen, Scania, Volvo, Alberto Sangiovanni Vincentelli (PARADES, Univ. of California at Berkeley), gave the invited presentation of the workshop. The presentation of the contributions and the discussion were organized in three sessions devoted to: Hybrid Modeling and Control, Vehicle Dynamics, and Engine Dynamics and Control.

Proceedings of HYCON & CEmACS Workshop on

Automotive Systems and Control

Lund University, Lund, June 1-2, 2006

Lund 2006

HYCON Work package WP4c Automotive

The HYCON (http://www.ist-hycon.org) Network of Excellence is supported by the European Commission within the Sixth Framework Program – Information Society

Technologies (grant FP6-IST-511368).

Preface

This report contains the Proceedings of the HYCON-CEmACS Workshop on Automotive Applications of Hybrid Systems held in Lund on June 1-2, 2006. This workshop is included in the activities of the workpackage WP4c―Automotive control of the HYCON Network of Excellence (http://www.ist-hycon.org), supported by the European Commission within the Sixth Framework Programme―Information Society Technologies (grant FP6- IST-511368, Sep. 2004 - Sep. 2008). The objective of HYCON is to establish a durable community of leading researchers and practitioners who develop and apply the hybrid systems approach to networked embedded control systems. The HYCON network includes 27 partners from 10 countries and six partners in light association with a total of 130 researchers and 113 PhD students. The CEmACS project (http://www.hamilton.ie/cemacs) is a partnership between DaimlerChrysler Research, the Hamilton Institute at NUI Maynooth, Lund University, Glasgow University and SINTEF. The objective of CEmACS is to contribute to a systematic, modular, model-based approach for designing complex automotive control systems. The Specific Target Research Project is aimed at combining research into the theory of multivariable control and nonlinear observers with a selection of novel prototype automotive control applications. Control and observer designs will be evaluated using two real-life benchmark integrated chassis control design applications: (i) vehicle dynamics control for active safety (collision avoidance and roll-over protection), and (ii) multivariable control design for ride and handling using multiple actuators (Generic Prototyping).

Introduction

The HYCON activities are divided in four thematic activities: hybrid system analysis; modeling and simulation of hybrid systems; hybrid system synthesis; and implementation aware control. HYCON partners cooperated in eleven joint programme activities, organized as follows: – Integrating activities • WP1: Creation of the European Institute for Hybrid Systems • WP2: Performance evaluation platform (benchmarking) • WP3: Tool integration – Research activities • WP4a: Energy Management • WP4b: Industrial Controls • WP4c: Automotive Control • WP4d: Multimedia Communication Networks – Spreading the Excellence • WP5: Knowledge Management • WP6: Industrial Bridging – Management activities

• WP7: Management activities • WP8: Assessment, evaluation and quality. The automotive domain, object of the HYCON Work Package WP4c, is certainly one of the most promising application domains for hybrid system techniques. The availability of low-cost computationally-powerful micro-controllers has made it possible to extend the performance and the functionality of the embedded systems used in automotive industry to limits that were unthinkable only a few years ago. Today, most medium-top class cars are equipped with more than 80 embedded controllers that handle different subsystems, such as engine, gear, brakes, suspensions, windows, and dash-board, accounting for more than 30% of the total cost of the car. The increasingly challenging requirements in terms of drivability, safety, emissions and fuel consumption imposed by car manufacturers and regulations call for more powerful design approaches and expose the need for a more structured design flow. The adopted design methodologies have to ensure that the design of the embedded systems achieve the requested specification and meet tight cost and development-time constraints. Hybrid system techniques can provide the basis for a more robust design methodology since they allow the designers to represent and manage the complex combination of time and event-based behaviors as well as the interactions between continuous and discrete phenomena. Hybrid formalisms and methodologies for modeling, analysis, and verification proved to be effective in handling several critical issues of the design flow such as: – Formalization of system specifications; – Plant and environment modeling, including representation of embedded controller inputs and outputs; – Control algorithm design; – Representation of the interaction between multirate discrete–time and event–based asynchronous control loops; – Description of control-flow and data-flow for software implementation; – Validation and verification of control algorithms and their implementations; – Description of the HW/SW implementation requirements. The HYCON work package WP4c - Automotive consists for the first year of the following tasks: – Task 4c.1: Hybrid models for automotive control; – Task 4c.2: Engine and vehicle control during fast transients; – Task 4c.3: Design methodologies for embedded automotive control systems; – Task 4c.4: Dissemination activities. Fourteen HYCON partners contribute to this work package, with a variety of research projects that cover a wide spectrum of automotive applications. Several HYCON partners have established fruitful collaborations with automotive OEMs and Tier-1 companies. The following companies are involved in HYCON research projects: Centro Ricerche Fiat, DaimlerChrysler, Drivetrain Innovation, Ferrari, Ford, Magneti Marelli Powertrain, Piaggio, PSA Peugeot-Citroen, Pirelli, Renault, Scania, TNO Automotive, Toyota, Volvo.

The strong interest demonstrated by industry in this work-package shows that: – The present needs of automotive companies for collaborations with research organizations to address the increasing demand for innovation in the automotive market; – The recognition of the potential of the HYCON research topics to meet the new challenges in the design flow for automotive control systems. In view of the feedback received from industry and reported later in this report, we are confident that these goals remain relevant.

HYCON-CEmACS Workshop on Automotive Systems and Control

The HYCON-CEmACS Workshop on Automotive Applications of Hybrid Systems was hosted by Lund University, Dept Automatic Control, Lund. The aim of the workshop was identification of challenges and opportunities for hybrid systems in automotive design. In particular, the following topics were discussed: industrial trends and concerns; methodologies, flows and tools; theoretical open problems. The workshop was attended by HYCON partners, CEmACS partners and representatives from automotive companies: DaimlerChrysler, DT Innovations, FIAT, Ford, PSA Peugeot Citroen, Scania, Volvo,. Alberto Sangiovanni Vincentelli (PARADES, University of California at Berkeley), gave the invited presentation of the workshop. The presentation of the contributions and the discussion were organized in three sessions devoted to: Hybrid Modeling and Control, Vehicle Dynamics, and Engine Dynamics and Control. Self-evaluation In addition to all rewarding and encouraging remarks from the workshop participants on the organization, industrial participation and intellectual exchange, we enjoyed a workshop atmosphere with high-quality project presentations from industrial and academic partners, open and clear technical communication with attention to commercial, environmental conditions as well as collaboration aspects, thus contributing to European leadership in the automotive area. In view of the feedback received from industry and reported later in this report, we are confident that the HYCON research goals remain relevant. As a follow-up, the workshop organizers are organizing a Special Issue on Automotive Systems and Control in the International Journal of Control containing revised and reviewed research papers included in draft version in this report.

Some photographs are available on our workshop homepage Hybrid Control—European Network Workshop, June 1-2, 2006, http://www.control.lth.se/seminars/HYCON2006/HYCON2006.html#photographs We wish to thank Eva Schildt and other members of our staff for their help in the organization of the workshop. June 2006

Rolf Johansson Anders Rantzer

List of Industrial Participants Pandeli Borodani, Centro Ricerche FIAT, [email protected] Gilberto Burgio, Ford, Aachen, Gilberto Burgio <[email protected]> Dario Castagnoli, ABB, [email protected] Hilding Elmqvist, Dynasim AB , [email protected] Tobias Geyer, GE Global Research, [email protected] Martin Hagström, Swedish Defence Research Agency, [email protected] Johan Hamberg , Swedish Defence Research Agency , [email protected] Hans Hellendoorn, Siemens Netherlands, TU Delft, [email protected] Alf Isaksson , ABB AB, [email protected] Jens Kalkkuhl , DaimlerChrysler AG, [email protected] Karsten-Ulrich Klatt, Bayer Technology Services GmbH, [email protected] Lars Larsen , Danfoss A/S, [email protected] Jorge Mari, GE Global Research, [email protected] Riccardo Minutolo, Thales Italia, <[email protected]> Cédric Nouillant, PSA Peugeot Citroën, [email protected] Loris Schettino, SELEX Communications, [email protected] Alex Serrarens, Drive Train Innovation, [email protected] Petter Strandh, Volvo Powertrain Corp., [email protected] Avshalom Suissa, DaimlerChrysler AG, [email protected] Claus Thybo, Danfoss A/S, [email protected]

Remarks and Feedback from Industrial Participants

The meeting was organized in the framework of the HYCON workpackage WP4 "Automotive control" in close cooperation with the workpackage WP6 "Industrial Bridging" and the HYCON Industrial Advisory Board. Industrial representatives were invited as "discussants" and appointed for each of the four HYCON plenary lectures. The idea was to set aside 15 minutes towards the end of each plenary lecture for discussion, where 1-3 industrial representatives give comments and questions to the plenary speaker regarding the objectives and progress of the work. To support the preparation, the discussants were provided with lecture slides and some additional documentation one week in advance. The discussants approved the Another means of interaction was the co-organization of the HYCON and CEmACS (http://www.hamilton.ie/cemacs/ ) workshops, both projects being including automotive research supported and financed by the EC. Some specific quotations and feedback from industrial participants:

Here are some of my thoughts after attending last week’s workshop in Lund, hope they can give new indications for next years HYCON activities in Vehicle Dynamics. 1. Better consider technologies with shorter market introduction forecast (or

already in the market), like vehicle stability control with active steering, active differentials, etc, and better use market available sensors. Systems with longer forecast for market introduction, moreover using non standard measurements don‘t have a consolidated "controller benchmark case" to show hybrid control / MPC advantages.

2. The integration of active systems in a unique vehicle controller is an interesting benchmark case, to show hybrid control capabilities: this should be more considered in the automotive WP.

3. Show the cases when hybrid derived complexity increase isn‘t well balanced by gain in performance / robustness, too. This is an interesting result like the case where hybrid succeeds.

Again, thanks a lot for the intersting workshop and the perfect organization.

Gilberto Burgio, Global Vehicle Dynamics, Ford Forschungszentrum Aachen

Here are some points I mentioned during the meeting according to my notes and discussions I remembered :

1. Applicative part: The technical presentations show an interest in both theoretical and technical aspects. Today we may say that most control laws use look-up tables to adapt gains with respect to the operating points. Hybrid control is more dedicated to model based control where the hybrid formulation enable to have a trade off between model complexity and gain adaptation (then we may stop copping with tables). I have found with great pleasure a first article dealing with limitations of automobile dampers where most articles do not consider at first the saturation areas and develop linear control laws. The hybrid control shows here its interest of integrating constraints in the design. I may add that a field to be considered is the hybrid engine with internal combustion engine and electric assist motor. The "hybrid" technology meets here the hybrid theory where several "modes" have to be switched to have high performances on the system. This subject is of high interest for automotive industry.

2. Theoretical part: One of the major concern with industrial applications is the robustness of the control laws. Robustness has not been adressed (or too few) during the meeting and should be part of future development. As said with the other automotive manufacturers, one of the key for hybrid control introduction is a comparison with state of art. This comparison should be made not only in the bibliographic part but also in the simulation/experimentations results to see the benefits of hybrid control.

3. Method implementation in a software: The best way to introduce new method in industry is to propose a software implenting the methods. In our field, a hybrid control toolbox could be developed. As discussed with Manfred Morari, such a goal is not an easy task and may take a long time. Anyway, it may propose a scheme of developement and help to classify problem formulations and gives solutions for hybrid control design. Then it will make a bridge with the dedicated Hycon WP for software.

Cédric Nouillant, PSA Peugeot Citroën, DINQ / DRIA / SARA / STEV / EEES -

Mécatronique

Hycon & CEmACS Workshop on Automotive Systems and Control June 1-2, 2006, Lund, Sweden

List of Participants Stefan Almér, Royal institute of Technology (KTH), [email protected] Miguel Alonso, ENST, [email protected] Peter Alriksson, Lund University, [email protected] Karl-Erik Årzén, Lund University LTH, [email protected] Karl Johan Åström, Lund University LTH, [email protected] Mihai Baja, Supelec, [email protected] Andrea Balluchi, PARADES, [email protected] Miroslav Baric,ETH Zürich, [email protected] A. Giovanni Beccuti, ETH Zurich, [email protected] Alberto Bemporad, University of Siena, [email protected] Thomas Besselmann,ETH Zürich, [email protected] Geraint Bevan, University of Glasgow, [email protected] Anders Blomdell, Lund University LTH, [email protected] Carlos Bordons, University of Seville, [email protected] Pandeli Borodani, Centro Ricerche FIAT, [email protected] Rolf Braun,Lund University LTH, [email protected] Gilberto Burgio, Ford, Gilberto Burgio <[email protected]> Dario Castagnoli, ABB, [email protected] Daniele Corona, TU Delft, [email protected] Alex Darlington, Cambridge [email protected] Bart De Schutter, Delft University of Technology, [email protected] Simone del Favero, Università degli studi di Padova, [email protected] Luigi del Re, Johannes Kepler University, Linz, [email protected] Turhan Demiray, ETH Zürich, Power System Laboratory,[email protected] Maria Domenica Di Benedetto, Dipartimento di Ingegneria Elettronica, [email protected] Stefano Di Cairano, Università di Siena, [email protected] Stefano Di Gennaro, University of L'Aquila, [email protected] Lan Anh Dinh Thi,University of Dortmund, [email protected] Camacho Eduardo, University of Seville, [email protected] Johan Eker, Ericsson Mobile Platforms AB, [email protected] Hilding Elmqvist, Dynasim AB, [email protected] Sebastian Engell, Universität Dortmund, [email protected] Giancarlo Ferrari Trecate, INRIA, [email protected] Carlo Fischione, Royal Institute of Technology, [email protected] Daniele Fontanelli, University of Pisa, [email protected] Eloisa Garcia-Canseco, Laboratoire des Signaux et Systèmes (LSS), [email protected] Ather Gattami, Lund University LTH, [email protected] Stephanie Geist, Technische Universität Berlin, [email protected]

Mathieu Gerard, Lund University LTH, [email protected] Tobias Geyer, GE Global Research, [email protected] Gabriela Glanzmann, ETH Zürich, [email protected] Keith Glover, University of Cambridge, [email protected] Henrik Gollee, University of Glasgow, [email protected] Håvard Fjær Grip, SINTEF ICT, Applied Cybernetics, [email protected] Herve Gueguen, Supelec, [email protected] Martin Hagström, FOI, [email protected] Johan Hamberg, FOI, [email protected] Staffan Haugwitz, Lund University LTH, [email protected] Hans Hellendoorn, Siemens Netherlands, TU Delft, [email protected] Toivo Henningsson, Lund University LTH, [email protected] Tore Hägglund, Lund University LTH, [email protected] Alf Isaksson, ABB AB, [email protected] Henrik Jansson, KTH, [email protected] Erik Johannesson, Lund University LTH, [email protected] Karl Henrik Johansson, Royal Institute of Technology, [email protected] Rolf Johansson, Lund University LTH, [email protected] Ulf Jönsson, Royal Institute of Technology, [email protected] Jens Kalkkuhl, DaimlerChrysler AG, [email protected] Maria Karlsson, Lund University LTH, [email protected] Karsten-Ulrich Klatt, Bayer Technology Services GmbH, [email protected] Francoise Lamnabhi-Lagarrigue, Centre National de la Recherche Scientifique, [email protected] Lars Larsen, Danfoss A/S, [email protected] Sture Lindahl, Lund University LTH, [email protected] Klas Malmqvist, Lund University LTH, [email protected] Jorge Mari, GE Global Research, [email protected] Emanuele Mazzi, University of Pisa, [email protected] Riccardo Minutolo, Thales Italia, [email protected] Ralf Mitsching, University of Aachen (RWTH), [email protected] Manfred Morari, ETH Zurich, [email protected] Oskar Nilsson, Lund University LTH, [email protected] Cédric Nouillant, PSA Peugeot Citroën, [email protected] Simon O'Neill, University of Glasgow, [email protected] Romeo Ortega, LSS, Supelec, [email protected] Athanasia Panousopoulou, University of Patras, [email protected] Georgios Papafotiou, ETH Zurich, [email protected] Diego Patino, CRAN-CNRS, [email protected] Joerg Raisch, Technische Universitaet Berlin, [email protected] Anders Rantzer, Lund University LTH, [email protected] Pierre Riedinger, CNRS-CRAN, [email protected] Claudia Rinaldi, University of L'Aquila, [email protected] Anders Robertsson, Lund University LTH, [email protected] Philipp Rostalski, ETH Zürich, [email protected]

Alberto Sangiovanni-Vincentelli, University of California, Berkeley, [email protected] Fortunato Santucci, University of L'Aquila, [email protected] Loris Schettino, SELEX Communications, [email protected] Brad Schofield, Lund University LTH, [email protected] Alex Serrarens, Drivetrain Innovation, [email protected] Iliyana Simeonova, Université Catholique de Louvain, [email protected] Selim Solmaz, Hamilton Institute, [email protected] Christian Sonntag, Universität Dortmund, [email protected] Alberto Speranzon, Royal Institute of Technology, [email protected] Petter Strandh, Volvo Powertrain Corp, [email protected] Avshalom Suissa, DaimlerChrysler AG, [email protected] Claus Thybo, Danfoss A/S, [email protected] Per Tunestål, Lund University LTH, [email protected] Anthony Tzes, University of Patras, [email protected] Arno van der Heijden, Technische Universiteit Eindhoven, [email protected] Andreas Wernrud, Lund University LTH, [email protected] Carl Wilhelmsson, Lund University LTH, [email protected] Carlos Villegas, Hamilton Institute, [email protected] Rafael Wisniewski, Aalborg University, [email protected]

HYCON & CEmACS Workshop onAutomotive Systems and Control

June 1-2, 2006

HYCON-CEmACS Workshop, Lund 2006 ©All Rights Reserved for Authors

This a workshop organized by Lund University, Dept. Automatic Control, LTH on behalf of the European Network ofExcellence HYCON and the European project CEmACS. The aim of the meeting is identifying challenges and opportunitiesfor hybrid systems in automotive design.

Thursday June 1, room M:B

1000 Registration

1045 Welcoming remarksAnders Rantzer, Rolf Johansson

1100 Opening Plenary Session of the General HYCON Workshop.

Highlights from WP4c Automotive ControlAlberto Sangiovanni-Vincentelli

Highlights from WP4d Networked ControlKarl Henrik Johansson

1230 Lunch at Kårhuset, John Ericssons väg 3

1400 Technical Session 1: Hybrid Modeling and Control

Idle speed control - A benchmark for hybrid system research Balluchi, Benvenuti, Sangiovanni-Vincentelli

Coping with the Variability of Hybrid Models for Automotive SoftwareMitsching, Kowalewski

A Hybrid Approach to Modeling, Control and State estimation of Mechanical Systems withBacklashvan Belzen, Rostalski, Bari´c, Morari

Hybrid optimal control of dry clutch engagementvan der Heijden, Serrarens, Camlibel, Nijmeijer

1520 Discussion

1540 Coffee

1600 Tecnical Session 2: Vehicle Dynamics

Nonlinear observer for vehicle lateral velocityImsland, Grip, Böhm, Johansen, Fossen, Kalkkuhl, Suissa

Improved road grade estimation using sensor fusionJansson, Kozica, Sahlholm, Johansson

Development of a controller to perform an automatic lateral emergency collision avoidancemanoeuvre for a passenger carBevan, Gollee, O'Reilly

Design and Probabilistic Validation of a System for Cooperative DrivingGietelink, De Schutter, Verhaegen

A methodology for the design of robust rollover prevention controllers for automotive vehicles:Active steering.Solmaz, Corless, Shorten

1740 Discussion

1800 End of sessions

1830 Bus departure from Ole Römers väg 1 (east side of building) to dinner location at Trollnäs castle

Friday June 2, room M:B

0830 Technical Session 3: Engine Dynamics and Control

Hybrid Modelling and Control of the Common Rail Injection SystemBalluchi, Bicchi, Mazzi, Sangiovanni-Vincentelli1, Serra

Error Feedback Nonlinear Control of Electromagnetic Valves for Camless EnginesDi Gennaro, Castillo Toledo, Di Benedetto

Multilinear identification and model predictive control of a turbocharged Diesel enginedel Re

Model Predictive Control of Magnetically Actuated Mass Spring Dampers for Automotive ApplicationsS. Di Cairano, A. Bemporad, I. Kolmanovsky, D. Hrovat

0950 Discussion

1010

Lund University Combustion Engine Laboratory - Laboratory VisitTunestål, B. Johansson, R. Johansson

Coffee

1100 Closing Plenary Session of the General HYCON Workshop

Highlights from WP4a Energy ManagementManfred Morari

Highlights from WP4b Industrial ControlSebastian Engell

1230 Lunch at Kårhuset, John Ericssons väg 3

1330 Panel Discussion with Members of the Industrial Advisory Board

1500 End of workshop

Time for internal project management meetings in CEMACS and HYCON WP4c

HYCON-CEmACS Workshop, Lund 2006 ©All Rights Reserved for Authors

Idle speed control –

A benchmark for hybrid system research∗

Andrea Balluchi(1), Luca Benvenuti(1,2),Alberto L. Sangiovanni–Vincentelli(1,3)

(1) PARADES, Via di S. Pantaleo, 66, 00186 Roma, Italy.balluchi, [email protected]

(2) DIS, University of Rome La Sapienza, Via Eudossiana 18, 00184 Rome, [email protected]

(3) EECS, Univ. of California at Berkeley, CA 94720, [email protected]

Abstract

The design of engine control systems has been traditionally carried out using a mix ofheuristic techniques validated by simulation and prototyping using approximate mean–value models. However, the ever increasing demands on passengers’ comfort, safety,emissions and fuel consumption imposed by car manufacturers and regulations call formore robust techniques and the use of cycle–accurate models. This urges the use of hybridmethodologies because of the rich combination of time and event-based behaviors. In thisreport, we present the hybrid benchmark problem on “Idle Speed Control” proposed bythe Network of Excellence HYCON. The purpose of the benchmark is to promote theapplication of hybrid system techniques to automotive control problems. In fact, as it isdemonstrated by some recent interesting applications, the introduction of hybrid systemdesign methodologies in the automotive industry is very promising.

1 Introduction

In the automotive industry, increased performance, safety and time-to-market pressure re-quire the use of complex control algorithms with guaranteed properties. Best practices inthis industry are based on extensive experimentation and tuning of parameters for the controlalgorithm and for the engine model. This procedure needs a substantial overhaul to eliminatelong re-design cycles and potential safety problems after the car is introduced in the mar-ket. Using more accurate models and control algorithms with guaranteed properties reducesgreatly the need for extensive experimentation and points to potential problems early in thedesign cycle.In this general scenario, the synthesis of a control strategy for spark–ignition engines at

idle speed is one of the most challenging problems. The goal is to maintain the engine speed∗This work is supported by the Network of Excellence HYCON, E.C. IST-511368.

1

as close as possible to a reference constant engine speed despite load torque disturbances (dueto i.e. the air conditioning system, the steering wheel servo-mechanism) and engagements anddisengagements of the transmission occurring when the driver operates on the clutch. In orderto achieve the best fuel economy, the reference engine speed is chosen at the minimum valuethat yields acceptable combustion and emission quality, and noise, vibration and harshness(NVH) characteristics.A survey on different engine models and control design methodologies for idle control is

given in (Hrovat and Sun, 1997). Both time–domain (e.g. (Butts et al., 1999)) and crank–angle domain (e.g. (Yurkovich and Simpson, 1997)) mean–value models have been proposedin the literature. Mean–value engine models (Hendricks and Vesterholm, 1992) describe thedynamic time development of the mean values of some engine variables and can be usedfor control design, when engine speed fluctuations due to cycle-to-cycle variations in thecombustion process are not considered. More recently, cycle accurate models have beeninvestigated (Shim et al., 1996) in order to reduce the periodic speed variations due to torquefluctuations. Several control algorithm design methodologies have been applied to the idlespeed control problem. A typical approach (see e.g. (Kokotovic and Rhode, 1986)) consistsof using a PID controller for the air loop, a P controller for the spark loop and severalfeedforward compensation schemes which use accessory load and enviromental information.Since the goal is to regulate the engine speed to its reference value, the integral part of the aircontrol loop is the core of this strategy and several efforts were made to tune the PID controlin order to minimize some appropriate cost functions. Other optimization-based methodshave been used: in particular the optimal LQ-based (Abate and Di Nunzio, 1990) controlwas demonstrated to achieve better performances with respect to conventional PID controller,and H∞ methods were applied in order to achieve a more robust control design (Carnevaleand Moschetti, 1993). Controllers based on the µ-synthesis technique were proposed, toensure stability also when large plant perturbations or uncertainties are present (Hrovat andBodenheimer, 1993). Additional improvements are possible when some loads are measurable.In (Butts et al., 1999), an 1 optimal control minimizing the excursion of the engine speedhas been proposed when a bounded load torque accessible to measurement is present. Otherapproaches to the idle speed control design include multivariable control (Onder and Geering,1993) and sliding mode control (Kjergaard et al., 1994).More recently, hybrid system techniques have been applied to the idle speed control

problem. In fact, an accurate model of a four–stroke gasoline engine has a “natural” hybridrepresentation because

• pistons have four modes of operation corresponding to the stroke they are in. Hencetheir behavior can be represented with a finite state model;

• power–train and air dynamics are continuous–time processes.

In addition, these processes interact tightly. In fact, the timing of the transitions betweentwo phases of the pistons is determined by the continuous motion of the power–train, which,in turn, depends on the torque produced by each piston. The hybrid nature of the problemof engine control does not come only from the use of digital control laws, but it is rooted inthe plant to be controlled.

2

In (Balluchi et al., 2000b), a hybrid approach to the design of an idle speed control algo-rithm was first proposed. The problem was formalized as a hybrid game and the correspondingmaximal controller was presented. Formal verification of a PID idle speed controller usinghybrid techniques has been presented in (Balluchi et al., 2002). Hybrid control algorithmshave also been developed using the control–to–facet approach in (Balluchi et al., 2004) andthe command governor technique in (Albertoni et al., 2003).In this report, we present the hybrid benchmark problem on “Idle Speed Control” pro-

posed by the Network of Excellence HYCON. The purpose of the benchmark is to promotethe application of hybrid system techniques to automotive control problems. The documenta-tion and the simulation files related to this benchmark problem are available at the HYCONweb–page www.ist-hycon.org, under “WP2: Performance Evaluation Platform”.The report is organized as follows. In Section 2, a hybrid model of the engine is described.

The idle speed control problem is formulated in Section 3. In Section 4, a more detailed enginehybrid model to be used for control algorithm validation is briefly presented. The validationmodel will be available only as a black–box and will be used to assess the performances ofthe proposed solution to the benchmark.

2 The engine hybrid model

In this section, a hybrid model of a 4–cylinder 4–stroke spark ignition engine equipped withan electronic–throttle is presented. The proposed hybrid model represents accurately thebehavior of the engine during idle speed control. A survey on internal combustion enginesis given in (Heywood, 1988; Stone, 1992). The overall system is composed of four maininteracting blocks, namely the ignition actuators, the intake manifold, the cylinders and thepowertrain (Figure 1).At idle speed, to achieve desired emission performances, the amount of fuel injected in

each cylinder is regulated by a fuel injection controller so that the air and fuel mixture isstoichiometric, i.e. the air-to-fuel ratio is 14.64. Hence, for the purpose of idle speed controllerdesign, we can assume that the engine operates with stoichiometric mixtures and we abstractaway the dynamics of fuel injection.The ignition actuators deliver the sparks sparki to the cylinders with a timing defined by

the desired spark advance angle ϕ. The latter represents the spark ignition control input.When the controller issues a new value ϕ, it emits the synchronization event trigger.The mass of air q loaded in the cylinders depends on the dynamics of the intake manifold.

The manifold pressure p is controlled by a throttle valve powered by an electrical motor; αand α denote, respectively, the throttle valve position and the reference to the throttle valvecontroller. The mass of air q is a function of the pressure p and of the crankshaft revolutionspeed n.The cylinders model describes the mechanims of engine torque generation. The engine

torque T is given by the sum of the torques produced by each cylinder. The latter dependson the mass of air q and the timing of spark ignition. The timing sequence of the four strokesof each cylinder is determined by the motion of the piston between the top and the bottomdead centers (dc), i.e. the piston uppermost and lowermost positions. The position of the

3

T

clutch gear

Driveline

Primary

Secondary

Driveline

Tl

spark1

spark2

spark3

spark4Actuators

Ignition

α, p

Manifold

Intake

Cylinders

θ Powertrain

n

n

Crankshaft

n

.

.

α

θ dc

dc

p

q

trigger

ϕ

Figure 1: Engine hybrid model.

piston is determined by the crankshaft angle θ.Finally, the crankshaft revolution speed n depends on the powertrain dynamics. In idle

speed control it is assumed that the gear is idle, while the clutch can be either open or closed.The powertrain is powered by the balance between the engine torque T and the load torqueTl, due to the auxiliary systems driven by the crankshaft.

2.1 Intake manifold

Manifold pressure dynamics is a continuous-time process controlled by the throttle-valveposition α that changes the effective section of the intake rail of the manifold. Denoting byp the mean–value pressure, the intake manifold dynamics is modelled as (see (Balluchi etal., 1998; Hendricks and Sorenson, 1990)):

α = − 1τthr

(α− α ) (1)

p =RTair

Vpln[ fthr(α)− fcyl(p, n) ] (2)

q =30nfcyl(p, n) (3)

Equation (1) represents the actuation dynamics of the throttle valve, with α being the refer-ence command. The intake manifold pressure dynamics (2) depends on the balance betweenthe air-mass flow through the throttle valve fthr(α) and through the cylinder valves fcyl(p, n),where

fthr(α) = s2 α2 + s1 α (4)

fcyl(p, n) = c0 + c1 p+ c2 n+ c3 p n (5)

Parameters depend on the geometric characteristics of the intake manifold, the physicalcharacteristics of the gas and the atmosphere.

4

2.2 Cylinders.

The torque T produced by the engine is given by the sum of the contributions of the fourcylinders, i.e.

T =4∑

i=1

T i (6)

where T i is the torque generated by the i–th cylinder. The profile of T i depends on the currentstroke of the cylinder, the piston position, the mass of air loaded in the cylinder during theintake stroke, and the spark ignition timing. The 4–stroke engine cycle is composed by thefollowing four phases:

• Intake. The piston goes down from the top dead–center (TDC) to the bottom dead–center (BDC) loading the air–fuel mixture from the intake manifold;

• Compression. The mixture is compressed by the piston during its upward movementfrom the BDC to the TDC;

• Expansion. The combustion takes place pushing down the piston from the TDC to theBDC;

• Exhaust. During its upward movement, from the BDC to the TDC, the piston expelscombustion exhaust gases.

We assume that the engine torque is negligible in the intake and exhaust strokes, i.e. T i = 0,and we model the torque profile as constant and negative in the compression stroke andpiecewise constant and positive in the expansion stroke.Spark ignition must occur at every cycle. Intuitively, it should occur exactly when the

piston reaches the top dead–center of the compression stroke. Since the combustion processtakes non-zero time to complete, the pressure in the cylinder reaches its maximum some timeafter spark ignition. It is then convenient to produce a spark before the piston completes thecompression stroke (positive spark advance), to achieve maximum fuel efficiency. Producinga spark after the piston has completed the compression phase and is in the expansion stroke(negative spark advance) may be used to reduce drastically the value of the torque generatedduring the expansion run. Hence, the spark control input has a very short delay and can beused to reduce torque much faster than using only the throttle valve1. The spark ignitiontime is commonly defined in terms of the spark advance ϕi, which denotes the differencebetween the angle of the crank at the top dead–center between compression and expansionand the one at the time of ignition. It is positive for sparks ignited in the compression stroke,negative otherwise2.

1Because of this property, negative spark advance is very useful to regulate the engine speed in the Idleregion of operation.

2Spark advance has to be bounded both from above and from below to prevent the mix from not burninguniformly thus causing undesired knocking (Franklin and Murphy, 1989; Konig et al., 1989) (upper bound)and from misfiring (Hacohen, 1992; Tang et al., 1994) (lower bound), which causes undesired pollutants. Thesebounds depend on the revolution speed n.

5

.

.

NA

PA

BS

AS

I

spar

ki

ϕi:=

180−

θ

dcT i := Gn(m

i)

spar

ki

ϕi:=

−θ

Ti:=

Ge(m

i )η(−

θ)

dcT i := Ge(m

i)η(ϕi)

dc

dc

mi:=

q

T i:=

Gc (p)

H

dcT i:=

0

E

sparki ∧d

c

ϕi:=

0

Ti :=

Ge(m

i )η(0)

C

Figure 2: Hybrid system describing the behavior of the i–th cylinder.

The air-fuel mixture is loaded in the cylinder during the intake stroke, while the spark isignited when the piston is around the top dead–center between the compression and expan-sion strokes (Hrovat et al., 1998). The delay between mixture intake / ignition and torquegeneration is represented by the hybrid system depicted in Figure 2. The hybrid system hastwo discrete states (I and H corresponding to the intake and exhaust strokes, resp.) andtwo macro–states (C and E corresponding to the compression and expansion strokes, resp.).Since spark ignition may occur either during the compression stroke or during the expansionstroke, the macro–states C and E are splitted as follows:

• BS, denoting Before Spark. The piston is in the compression stroke and no spark hasbeen ignited yet.

• PA, denoting Positive Advance. The piston is in the compression stroke and the sparkhas been ignited.

• NA, denoting Negative Advance. The piston is in the expansion stroke and the sparkhas not been ignited yet.

• AS, denoting After Spark. The piston is in the expansion stroke and the spark has beenignited.

6

S+S− S

dc / mC := q,

TE := Ge(mC )η(ϕ), TC := Gc(p)

dc / mC := q, mE := mC

TE := Gn(mC ), TC := Gc(p)

spark

ϕ := −θ, TE := Ge(mE )η(−θ)

dc ∧ spark

mC := q, ϕ := 0,

TE := Ge(mC)η(0), TC := Gc(p)

spark

ϕ := 180 − θ

Figure 3: Hybrid system describing the behavior of a 4-cylinder engine.

The hybrid system makes a transition either when the piston reaches a dead–center (dc) orwhen the spark is ignited (spark i).The spark advance angle ϕi is evaluated when the spark is ignited (BS → PA, BS → AS,

NA → AS) and expressed in terms of the crankshaft angle θ.The torque T i during the compression stroke is constant and negative and it is evaluated

at the end of the intake stroke (I → BS) as a function of the intake manifold pressure atthat time:

T i = Gc(p) = f0 + f1 p+ f2 p2 . (7)

In case of negative spark advance the mixture is not ignited before the dead–center. Thehybrid system enters the state NA, where the torque T i is positive due to gas expansion. Wemodel T i as constant in NA

T i = Gn(mi) (8)

with amplitude depending on the loaded air mass mi,

Gn(mi) = g0 + g1 mi + g2 (mi)2 . (9)

The hybrid system is in state AS either during the entire expansion stroke, in case of positivespark advance, or just after spark ignition, in case of negative spark advance. In this state,the torque is given by

T i = Ge(mi) η(ϕi) (10)

where

Ge(mi) = h0 + h1 mi + h2 (mi)2 (11)

represents the maximum value of torque achievable by the given stoichiometric mixture withmass of air mi loaded during intake and set at I → BS. In (10), η(ϕ) denotes the ignitionefficiency function. We assume

η(ϕ) = v0 + v1 ϕ+ v2 ϕ2 + v3 ϕ

3 (12)

7

The behavior of a four–cylinder in-line engine can be obtained by composing four cylinderhybrid models as given in Figure 2. However, since at any time each cylinder is in a differentstroke of the engine cycle, the model can be significantly simplified and reduced to a three–state hybrid model, with discrete states S, S+ and S− as depicted in Figure 3. States S,S+ and S− correspond to the following cylinder configurations S = (I,BS,AS,H), S+ =(I, PA,AS,H), S− = (I,BS,NA,H). The engine torque is given by

T = TC + TE (13)

where TC denotes the torque given by the cylinder in the compression stroke and TE thetorque produced by the cylinder in the expansion stroke.In state S, the cylinder in the expansion stroke has received the spark and is generating

torque (AS), and the cylinder in the compression stroke has not yet received the sparkcommand (BS). If spark ignition occurs before the next dead–center, then the cylinder thatis in the compression stroke enters state PA (BS → PA), while the others remain in thesame state. This corresponds to the transition from S to S+. Otherwise, if the dead–center isreached before the spark is ignited, all the cylinders change phase and the transition from S toS− takes place. When the spark is ignited at the dead–center, all the cylinders change phaseand the self–looping transition from S to S is performed. In state S+, the spark commandhas been given for both cylinders in the compression stroke and in the expansion stroke, andat the dead–center all the cylinders change phase so that the transition from S+ to S takesplace. In state S−, both cylinders in the expansion and compression strokes are waiting forthe spark command. No expansion torque is generated in this case. When the spark ignitionis given (necessarily before the next dead–center) the cylinder that is in the expansion strokechanges from NA to AS, and the transition from S− to S takes place. For more details onthe model reduction see (Balluchi et al., 2000a).

2.3 Powertrain

In the idle speed control problem the gear position is fixed in neutral position (idle). Conse-quently, the secondary driveline is disconnected and does not affect the crankshaft dynamics.Due to the actions of the driver on the clutch pedal, the first part of the driveline is eitherconnected or disconnected from the engine (see Figure 1). The dynamics of the crankshaftspeed n is given by the hybrid model depicted in Figure 4, where: the discrete states open andclosed encode the two possible positions of the clutch, the input events on and off representthe driver action on the clutch pedal, and the continuous dynamics are affine.When the clutch is open the primary driveline speed n′ evolves independently from the

crankshaft speed n. Instead, when the clutch is closed, they evolve at the same speed n.The parameters of the continuous dynamics are

ac = −B+Bp

Jc, bc = 1

Jc, ao = − B

Jo, bo = 1

Jo, ap = − Bp

Jc−Jo,

with J0 (Jc) and B (Bc) denoting the inertial momentum and the viscous friction coefficientof the segment of the powertrain from the crankshaft to the clutch (from the crankshaft tothe gear, respectively).

8

on

n′ := n

n = ao n + bo (T − Tl) + eo

n′ = ap n′

off

n := r1 n + r2 n′

n = ac n + bc (T − Tl) + ec

closed open

.

.

Figure 4: Powertrain hybrid model at idle speed.

When the clutch pedal is released and a transition from open to closed occurs, due to theconservation of the driveline angular momentum, the speeds of the shafts are reset to thevalue

n :=Jon+ (Jc − Jo)n′

Jc, (14)

with J0 and Jc denoting the inertial momentum of the segment of the powertrain from thecrankshaft to the clutch and from the crankshaft to the gear, respectively. When the clutchis opened, the primary driveline speed is initialized as follows:

n′ := n. (15)

The evolution of the crankshaft angle in the interval [0, 180] gives the position of thepistons within each stroke. It is described by the simple hybrid model reported in Figure 5.The dynamics is given by the crankshaft speed n. When the crankshaft angle θ reaches thevalue 180, it is reset and the dead–center event dc is emitted.

.

θ = 180 / dc

θ := 0

θ = 6n

.

Figure 5: Crankshaft angle and dead–center events.

9

trigger ∧ ( θ ≤ −ϕmin )

β := ϕτ := −τϕ

trigger ∧ ( θ > −ϕmin )

β := ϕτ := −τϕ

( β ≥ 0 ) ∧ [ ( θ ≥ 180 − β ) ∨ ( θ ≥ −ϕmin ) ]∨

( β < 0 ) ∧ [ ( θ ≥ −β ) ∧ ( θ ≥ −ϕmin ) ]

spark

θ ≥ −β

spark

τ = 0 τ = 0

Synch 1

τ = 0

Charge 1

τ = 1

Charge 2

τ = 1

Synch 2

τ = 0

Wait

τ = 0

Figure 6: Hybrid model of the ignition actuator models.

2.4 Ignition actuators

The ignition actuators introduce a no negligible delay in spark ignition control. This delayis due to the time needed to charge the ignition coils. As a consequence, the desired sparkadvance ϕ has to be set with a sufficiently large advance to allow proper spark actuation.The ignition actuator delay is described by the hybrid model depicted in Figure 6.The system waits in stateWait for the next value of desired spark advance angle ϕ. When

the controller issues a new value ϕ, it emits the event trigger to synchronize with the ignitionactuator that has to read it. It is supposed that the desired spark advance angle ϕ is alwaysissued for each cylinder and that it belongs to the feasible range [ϕmin, ϕmax], with ϕmin < 0and ϕmax > 0.If the new value ϕ is issued when the crankshaft angle θ is lower than or equal to −ϕmin,

then the command is given at the beginning of the expansion stroke (only negative sparkadvance will be feasible) and the system takes the right cycle. Otherwise, if θ > −ϕmin, theneither positive or negative spark advances could be applied and the system takes the leftcycle.After the transition, the system starts charging the ignition coil for a time τϕ, spent either

in state Charge 1 or state Charge 2.When the charging time is elapsed, two cases are in order:

• The crankshaft angle has not reached the desired spark advance yet (either θ < 180−β,for positive spark advance, or θ < −β, for negative spark advance): the system remainsin state Synch 1 / Synch 2 until the desired spark advance is reached. Then, it makesthe transition to state Wait emitting the spark signal.

• The crankshaft angle has already passed the desired spark advance (the command wasissued too late). In this case, no time is spent in states Synch 1 / Synch 2 and thesystem makes the transition to state Wait emitting the spark signal.

10

3 Problem formulation

The purpose of idle speed control is to keep the engine speed as close as possible to a targetvalue n0 when the gear is neutral, preventing the engine to stall. For fuel consumptionminimization, the target value n0 is, ideally, the lowest engine speed for which the enginecan be robustly controlled avoiding engine stall. More precisely, the goal is to maintain thecrankshaft speed n in a specified range, n0 ± ∆n, robustly with respect to two sources ofdisturbances (a continuous and a discrete one):

• The load torque Tl acting on the crankshaft due to auxiliary sub-systems (such as e.g.air conditioning and servo-mechanisms for steering and braking);

• The changes on the crankshaft dynamics due to the motion of the clutch, which maybe operated by the driver.

The control inputs are the desired spark advance angle ϕ and the throttle valve command α.Actuation constraints and dynamics have to be taken into account in the design. Availablesensors provide the following feedbacks: throttle valve position α, manifold pressure p, enginespeed n.The design specification can be formalized as a constrained optimal control problem for

the hybrid model of the engine described in Section 2. The adoption of a hybrid formalismallows us to represent the cyclic behavior of the engine, thus capturing the effect of each sparkcommand on the generated torque, the interaction between the discrete torque generationand the continuous powertrain and air dynamics, and the discrete changes of the powertrainbehavior.For any action of the torque disturbance Tl ∈ [0, Tl max] and any switching of the clutch

state, the controller has to guarantee that the following constraints are satisfied:

C1 : engine speed|n− n0| ≤ ∆n (16)

C2 : throttle angle0 ≤ α ≤ αmax (17)

C3 : spark advance control

ϕmin ≤ ϕ ≤ ϕmax (18)ϕmin ≤ ϕ ≤ ϕmax (19)

The cost function to minimize is the square of the L2–norm of the engine speed error ina transient due to a torque load Tl = Tl max, starting from the steady state point with Tl = 0and assuming that the clutch is open. That is

min ‖n− n0‖2L2=∫ ∞

0(n− n0)2dt . (20)

11

4 Validation

The proposed solution for idle speed control will be validated by simulation using a refine-ment of the engine hybrid model presented in Section 2. The engine hybrid model used forvalidation represents accurately the behavior of the engine and includes the main phenomenathat have been abstracted away in the definition of the model given in Section 2 provided forsynthesis.The refinements introduced in the validation model are the following ones:

• modeling of the continuous evolution of the in–cylinder pressure and the engine torque inthe four strokes of the engine cycle (the latter as a function of the former), see (Heywood,1988; Stone, 1992);

• stochastic modeling of the cycle–to–cycle variations of the engine torque;

• modeling of the clutch as a three state hybrid systems, which includes the clutch slidingmode.

5 Conclusion

The hybrid benchmark problem on “Idle Speed Control” proposed by the Network of Ex-cellence HYCON has been presented. The purpose of the benchmark is to promote theapplication of hybrid system techniques to automotive control problems and demonstrate theeffectiveness of hybrid system methodologies to the automotive industry. The description ofthe benchmark includes: a hybrid model of the engine, formalized system specification anda model for control algorithm validation.

12

Notation

Variables

Intake manifold model

α [o] throttle valve angle

α [o] desired throttle valve angle (air control input)

p [mbar] intake manifold pressure

q [Kg] mass of air loaded by the engine

i–th cylinder model

dc [ ] dead center event

spark i [ ] i–th cylinder spark event

mi [Kg] i–th cylinder loaded air mass

ϕi [o] i–th cylinder spark advance angle

T i [Nm] i–th cylinder generated torque

4–th cylinder engine model

spark [ ] spark event

mC [Kg] air trapped in the cylinder in compression stroke

mE [Kg] air trapped in the cylinder in expansion stroke

ϕ [o] spark advance angle

TC [Nm] torque contribution by the cylinder in compression stroke

TE [Nm] torque contribution by the cylinder in expansion stroke

T [Nm] engine torque

13

Powertrain model

on [ ] clutch open command (discrete disturbance)

off [ ] clutch closed command (discrete disturbance)

n [RPM] crankshaft (engine) speed (controlled output)

n′ [RPM] primary driveline revolution speed

Tl [Nm] load torque on the crankshaft (continuous disturbance)

θ [o] crankshaft angle

Ignition actuator model

ϕ [o] desired spark advance (ignition control input)

Functions

Intake manifold model

fthr(α) [Kgsec ] air mass flow rate through the throttle valve

fcyl(p, n) [Kgsec ] air mass flow rate through the cylinder intake valves

i–th and 4–th cylinder engine models

Gc(p) [Nm] torque in the compression stroke

Gn(m) [Nm] torque due to gas expansion in the power stroke

Ge(m) [Nm] maximum torque in the expansion stroke

η(ϕ) [ ] ignition efficiency

14

References

Abate, M. and V. Di Nunzio (1990). Idle speed control using optimal regulation. Tech. Rep.905008. SAE.

Albertoni, L., A. Balluchi, A. Casavola, C. Gambelli, E. Mosca and A. L. Sangiovanni-Vincentelli (2003). Idle speed control for GDI engines using robust multirate hybridcommand governors. In: Proc. CCA2003, 2003 IEEE Conference on Control Applica-tions. Vol. 1. Istanbul, Turkey. pp. 140–145.

Balluchi, A., F. Di Natale, A. L. Sangiovanni-Vincentelli and J. H. van Schuppen (2004). Syn-thesis for idle speed control of an automotive engine. In: Hybrid Systems: Computationand Control (R. Alur and G. J. Pappas, Eds.). Vol. 2993 of Lecture Notes in ComputerScience. pp. 80–94. Springer-Verlag. Berlin Heidelberg New York.

Balluchi, A., L. Benvenuti, M. D. Di Benedetto and A. L. Sangiovanni-Vincentelli (2002).Idle speed control synthesis using an assume–guarantee approach. In: Nonlinear andHybrid Systems in Automotive Control. pp. 229–243. Springer-Verlag. London, UK.

Balluchi, A., L. Benvenuti, M. D. Di Benedetto, C. Pinello and A. L. Sangiovanni-Vincentelli(2000a). Automotive engine control and hybrid systems: Challenges and opportunities.Proceedings of the IEEE 88(7), 888–912. “Special Issue on Hybrid Systems”.

Balluchi, A., L. Benvenuti, M. D. Di Benedetto, G. M. Miconi, U. Pozzi, T. Villa, H. Wong-Toi and A. L. Sangiovanni-Vincentelli (2000b). Maximal safe set computation for idlespeed control of an automotive engine. In: Hybrid Systems: Computation and Control(Nancy Lynch and Bruce H. Krogh, Eds.). Vol. 1790 of Lecture Notes in ComputerScience. pp. 32–44. Springer-Verlag. New York, U.S.A.

Balluchi, A., M. D. Di Benedetto, C. Pinello and A. L. Sangiovanni-Vincentelli (1998). Ahybrid approach to the fast positive force transient tracking problem in automotive en-gine control. In: Proc. 37th IEEE Conference on Decision and Control. Tampa, Florida,USA. pp. 3226–3231.

Butts, K. R., N. Sivashankar and J. Sun (1999). Application of 1 optimal control to the engineidle speed control problem. IEEE Trans. on Control Systems Technology 7(2), 258–270.

Carnevale, C. and A. Moschetti (1993). Idle speed control with H∞ technique. Tech. Rep.930770. SAE.

Franklin, M. L. and T. E. Murphy (1989). A study of knock and power loss in the automotivespark ignition engine. Tech. Rep. 890161. SAE.

Hacohen, J. (1992). Experimental and theoretical analysis of flame development and misfirephenomena in a spark–ignition engine. Tech. Rep. 920415. SAE.

Hendricks, E. and S. C. Sorenson (1990). Mean value modelling of spark ignition engines.Tech. Rep. 900616. SAE.

15

Hendricks, E. and T. Vesterholm (1992). The analysis of mean value SI engine models. Tech.Rep. 920682. SAE.

Heywood, J. B. (1988). Internal Combustion Engine Fundamental. Mc-Graw Hill.

Hrovat, D. and B. Bodenheimer (1993). Robust automotive idle speed control design basedon µ-synthesis. Proc. IEEE American Control Conference pp. 1778–1783.

Hrovat, D. and J. Sun (1997). Control engineering practice. Models and control methodologiesfor IC engine idle speed control design.

Hrovat, D., D. Colvin and B. K. Powell (1998). Comments on “applications of some newtools to robust stability analysis of spark ignition engine: A case study”. IEEE Trans.on Control Systems Technology 6(3), 435–436.

Kjergaard, L., S. Nielsen, T. Vesterholm and E. Hendricks (1994). Advanced nonlinear engineidle speed control systems. Tech. Rep. 940974. SAE.

Kokotovic, P.K. and D. Rhode (1986). Sensitivity guided design of an idle speed controller.Tech. Rep. 1. Ford Motor Company. PK Research, Urbana, USA.

Konig, G., R. R. Maly, D. Bradley, A.K.C. Lau and C.G.W. Sheppard (1989). Role of exother-mic centres on knock initiation and knock damage. Tech. Rep. 890161. SAE.

Onder, C. H. and H. P. Geering (1993). Model-based multivariable speed and air-to-fuel ratiocontrol of a SI engine. Tech. Rep. 930859. SAE.

Shim, D., J. Park, P. P. Khargonekar and W. B. Ribbens (1996). Reducing automotive enginespeed fluctuation at idle. IEEE Trans. on Control Systems Technology 4(4), 404–410.

Stone, C. R. (1992). Introduction to Internal Combustion Engines. Macmillan. London.

Tang, D.-L., M.-F. Chang and M. C. Sultan (1994). Predictive engine spark timing control.Tech. Rep. 940973. SAE.

Yurkovich, S. and M. Simpson (1997). Crank-angle domain modeling and control for idlespeed. SAE Journal of Engines 106(970027), 34–41.

Zambrano, D., C Bordons, W. Garcia-Gabin and E.F. Camacho (2006). A solar coolingplant: a benchmark for hybrid systems control. submitted to the “2nd IFAC Conferenceon Analysis and Design of Hybrid Systems”, ADHS‘06.

16

Coping with the Variability of Hybrid Models forAutomotive Software

- Draft -Ralf Mitsching, Stefan Kowalewski

Embedded Software Laboratory - Chair ofComputer Science XI

RWTH Aachen University52074 Aachen, Germany

(surname)@informatik.rwth-aachen.de

Abstract— This paper studies typical hybrid engine models andtheir variability. We sketch the main problems when dealingwith variability in hybrid engine models. In addition, we presentgeneral concepts for the modelling of variability in hybridautomata or switched ODE systems. Finally, these concepts areevaluated.

Index Terms— Variability, Hybrid engine models and Automo-tive control

I. INTRODUCTION

Modern control software development has to cope witha growing number of ECU software variants. The reasonsfor this are manifold: different car models, different engines,different national exhaust regulations etc. To solve this prob-lem one could design each variant individually or maintainone general version from which all variants are derived byparameterization. But these methods are not economical, e.g.they bear overhead for small items. Variability managementprovides an opportunity to solve this problem. The systematicplanning of re-use enables a cost saving development of ECUcontrol software.

The paper is organized as follows. In the next section, wesketch the current situation in literature with respect to theapplication of variability in hybrid engine models. Afterwards,the focus is set to the charateristics of variability in hybridengine models. Then the typical variation points in hybridengine models are presented and their influence on hybridengine models is illustrated. Section III points to general con-cepts to integrate variability in hybrid automata and differentialequations. Different extensions of this two model types areproposed and elaborated on. Finally, the new concepts areevaluated.

II. VARIABILITY OF HYBRID ENGINE MODELS INLITERATURE

Nowadays, approaches to use hybrid models in controllerdevelopment are largely available and differ strongly in theirapplication areas. They vary in the emphasis on or the com-plexity of the continuous and discrete dynamics and they focus

different levels of controller development, i.e. analysis, synthe-sis or simulation. The areas analysis, synthesis and simulationoften require switched differential equations. Conversely, finiteautomata are typically used for verification [2].

It is pretty much the same situation in the area of hybridengine models. The models found vary in the approximationlevel, the abstraction level and their notations. The mixtureof model types is a serious problem in order to understandvariability in hybrid engine models, e. g. different abstractionlevels, different notations or different semantics. Hence, it isessential to analyse the influence of the different modelingapproaches to the variability.

The use of approximation is a reasonable approach to reducecomplexity in hybrid engine models. But in the analysis ofhybrid engine models this class of approximated hybrid enginemodels is not meaningful for understanding variability. Thereason for this is the transformation of structural informationinto a numerical one and the loss of relevant information, e.g. summarization of different values.

The fact that hybrid engine models focus different levels ofabstraction is not surprising. A developer only considers thoseaspects of the system which he is interested in. For studyingthe similarities in hybrid engine models this type of modelis not useful because the differences between the individualhybrid engine models are too large.

Lastly, there is no standardised notation for describinghybrid behavior. So, in order to draw a comparison with theconsidered models it is necessary to build a common notation.But the problem is here to find a notation for the combinationof different notations of hybrid engine models. Because of thelarge difference between the hybrid engine models a generalnotation is necessary so that the individual characteristics ofthe model notation can be omitted.

In order to deal with these problems only the abstractionlevels and notations used in [3], [4], [5], [6], [7], [8], [9], [10]and [12] are focused.

After understanding the main problems in comparing vari-ability in hybrid engine models, the following chapter consid-ers the nature of variability in hybrid engine models.

Fig. 1. Restriction of variability

A. Characterisation of variability in hybrid engine models

In software engineering, variation management copes withsoftware variation over time and space. The variation in timeconsiders how the product varies over time, while variation inspace considers the differences amoung the individual productsat any fixed point in time. [15] This paper focuses on thevariation in space in order to establish understanding of thevariability in the different hybrid engine models.

Variations in hybrid engine models can be distinguishedwithout consideration of the execution time, by the timewhen they have to be implemented within a developmentprocess (e. g. presented in [8] or in [13]). In the developmenttime the static behavior of the modeling system is focusedand in the execution time the dynamic behavior. In figure1 the dependency of variability on the development time isillustrated. The aim is to maintain open decisions as long aspossible to keep the model flexible. But same decisions havesuch a strong influence on the model structure that they haveto occur in early phases of the development. To classify thiswe propose to split the development time into two types ofmodel variabilities. Firstly, the typical variants of hybrid en-gine models are generated by parameterization of continuous-time behavior. And secondly, the variants of hybrid enginemodels are generated by decisions of the discrete behavior.The mapping between a variation found in a hybrid enginemodel and the classification mentioned above is complicated.This is illustrated in the next paragraph.

B. Typical variations of hybrid engine models in literature

In the considered literature the most used variability is theparameterization , e.g.:

1) Crankshaft momentum of inertia (clutch open) [10]2) Crankshaft momentum of inertia (clutch closed) [10]3) viscous friction coefficient [10]4) Cylinders’ torque gain/offset [10]5) Torque gain/offset [10]6) throttle valve delay [1]7) air-fuel ratio [11]8) torque production delay for air [14]9) · · ·Obviously, this type of variability is easy to deal with.

But what is about the combination of discrete and continuous

Fig. 2. Local influence of variability of parameters

behavior? The first item and the second item seem to be usefulto illustrate this combination. The switching of the continuousbehavior could be seen as a substitution of the continuousbehavior depending on an external event. One special systembehavior could be considered without taking care of othervariants of the system. In figure 2 this idea is depicted.

Now we change the perspective to the discrete behavior ofthe system. The hybrid engine models as stated in [1],[3],[5]and [14] differ in the following points:

1) Number of cylinders2) Position of the clutch3) Position of the gear stick4) Spark timing5) Number of distinguished phases of the cylindersWith respect to the variations found of continuous behavior

it is obvious that there is a smaller number of variations ofdiscrete behavior. But the variations of discrete behavior havea large influence on hybrid engine models.

C. Influence of variations in hybrid engine models

The focus is now set to the main effects of the variationsin discrete behavior. To illustrate changes in hybrid enginemodels like adding of a clutch or spark timing we revert toan example mentioned in [10] depicted in figure 3. There aretwo hybrid engine models shown. The first model gets theinformation of receiving the death center dc and the insertedtorque T . This model is extended by adding the positive sparkadvance Φ. In both models the power-train’s linear continuous-time dynamics are given by:

n(t) = ann(t) + bn(Tg(t)− Tl(t))Θ(t) = kcn(t)

where an = −BJ0

, bn = 1J0

and Tl ∈ [0, Tmaxl ] ([10]). In order

to illustrate the modification of model A in figure 3 they aremarked with rectangles. Due to the fact that discrete behavioris contained in the transition space of the hybrid automata it isnecessary to adapt all transitions in the hybrid automata. Thisis a further problem in order to get common implementationcomponents.

Fig. 3. Insertion of the spark advance in a basic automaton

Fig. 4. General model to integrate Variability in Hybrid Engine Models

D. Approach to integrate Variability in Hybrid Engine Models

One approach to integrate variability of discrete behavior isalready done in [5] and shown in figure 4. The authors designa general model for an arbitrary number of cylinders. Theyuse for this their own notation and do not consider the aspectof variability further.

E. Conclusion

We can conclude that the considered hybrid engine modelsin literature do not include an explicit consideration of vari-ability and are mainly focused on only one special applicationarea. Furthermore, there does not exist a notation to describevariability in hybrid engine models. In the next section wediscuss approaches to deal with variability in hybrid automataand switched differential equations and propose a notation forvariability integration.

III. GENERAL CONCEPTS TO MODEL VARIABILITY INHYBRID SYSTEMS

It exists many approaches for modeling hybrid systems.They vary in the emphasis on or the complexity of thecontinuous and discrete dynamics [2]. This broad spectrumis focused only in two directions. On the one hand there areapproches to describe hybrid systems by hybrid automata, e.g. [8]. This approach is base on finite automata and uses agraphical representation. On the other hand a textual notationis used. The switched systems are characterized by ordinarydifferential equations with added jumps to represent discretedynamics.

In the following sections both notations are extended by newitems to integrate variability. For this, two basic modificationsto the models are proposed. Firstly, modifications are realisedby parameters, i. e. the variability is expressed by a number.Secondly, the modification consists of unsettled capacity in theconsidered node, i. e. another complex system is representedin the node.

A. Hybrid automata1) Fixed structure, variants by parameterization: Figure

5 illustrates several options to add variability in an hybridautomaton. There are three obvious possibilities:

• Generalization by extension of index• Generalization by extension of the function arguments• Generalization by extension of the guards Φ =φ1, · · · , φka) Extension of the indices: This is a typical and natural

way to insert variability in a broad spectrum of model types.In figure 6 an illustration of this kind of variability is shown.For this purpose a graphical notation is suggested. It is ableto caption this type of variability in hybrid automata.

Fig. 5. Integration of parameterization in hybrid automata

Fig. 6. State multiplication in a hybrid automaton

The effect of the introduction of index extension is obvi-ously a state multiplication. Whereas, the considered state isexchanged in dependance on the current parameter value.

b) Extension of the function arguments: This has onlyan impact on the system, represented by the automaton state.Hence, the structure of the automaton assumedly has noinfluence on it and vice versa.

c) Extension of the guards: Due to the fact that theguards are boolean expressions, broad possibilities to insertvariability are possible. Two approaches are illustred in figure7:

• Adding a picking parameter in each guard• Using a picking parameter and an additional parameteri-

zation of the guard.

Both approaches look like the same but there are not. Thefirst one allows to choose a certain state of the automaton. Incontrast, the second one enables the selection of one transition.

The guards shown in figure 8 combine the previouslyintroduced approaches. They build two equivalence classesof models. Firstly, the models with odd numbers (1,3) andsecondly, the models with even numbers (2,4). This approachdemonstrates a generalization of the new model and could findan application in the verification of the proposed models.

2) Variable structure, variants of a state: The structuresin this section have in common that changes happen in the

Fig. 8. Creation of model classes

state space of the considered hybrid automaton, i. e. the set oftransitions is fixed but states can include other systems.

In figure 9 the notation with extension by a general stateis illustrated. The start point of this general state is markedby black points. Hence, the system included has only oneentry. Two basic cases are examined in the following. It isassumed that the included system and the upper automatonrun independently. Firstly, the included system has one outputto the upper automaton and secondly, the system included hastwo outputs. In the first case it is not a problem to describe thebehavior of the automaton. The model in figure 9 uses a secondblack point to mark the output. Alternativly, the notation in the

Fig. 7. Parameterization of guards

Fig. 9. Variable structure: variants of a state

second case provides two outputs to the upper automaton andworks without a seperate marking.

B. Switched differential equations

1) Fixed structure, variants by parameterization: Typi-cal variants of differential equations by parameterization areillustrated and considered to insert variability in switcheddifferential equations:

x = fa1,···,an(x, b1, · · · , bn) if φ1 = true

· · ·x = fa2

1,···,a2n(x, b2

1, · · · , b2n) if φk = true

There are three obvious possibilities:• Generalization by extension of index

• Generalization by extension of the function arguments• Generalization by extension of the guards Φ =φ1, · · · , φka) Extension of indices: The application of the index

extension is shown in the following equations.

x =

f1,1(x) if x ≥ a ∧Mod = 1 ∧ x < bf1,2(x) if x ≥ a ∧Mod = 2 ∧ x < bf1,3(x) if x ≥ a ∧Mod = 3 ∧ x < b

Here it is important to make sure that the system is well-defined. This is done by an additional extension of the guards.

b) Extension of function arguments: A distinction be-tween two cases is suggested. Firstly, only the parameters

influence the differential equation. This case is illustrated inthe following example:

f1(x, a, b) = a · x(t) + b · x(t)

This function represents a high number of variants:

x =

f1(x, 1, 2) = 1 · x(t) + 2 · x(t) orf1(x, 2, 3) = 2 · x(t) + 3 · x(t) or

· · ·

Secondly, the parameters replace terms in the guards andeffect on the discrete dynamics of the system.

x = f1(x, a) if x ≥ a

Thus, the switching point is now variable and enables therepresentation of switching point variants, e. g. for represent-ing different exhaust regulations.

c) Extension of the guards: Due to the fact that theguards are boolean expressions, broad possibilities to insertvariability are thinkable:

1) adding a picking parameter in each guard,2) combining a picking guard and a pameterization of the

indices,3) combining a picking guard and a pameterization of the

function arguments,4) using of OR operators in the guard and5) combining all approaches.

The first three approaches have already been presented. Theforth one builds equivalence classes of equations in order touse a considered dynamical behaviour for different cases ofthe discret behaviour, e. g.

x =

f1 if x < a ∨ x < b

2) Variable structure, variants of an equation: In the pre-ceding paragraphs parameters were used to integrate variabilityin the equations. Other approaches are proposed to express thevariants of an equation.

a) Table: One idea is to introduce a symbol that repre-sents a table. The table contains all possibilities for a specialequation, e. g.

x =

f1 if x < a ∧Mod = 1 ∧ x < bf2Γ

f3 if x ≥ a ∧Mod = 1 ∧ x ≥ b

The identifyer Γ is a reference to the following table I:

Model Equation Guard1 f2 = a · x(t) + b · x(t) + c · x(t) x < a ∧ x < b2 f2 = a · x(t) + b · x(t) x > a ∧ x < b3 f2 = a · x(t) x > a ∧ x > b

TABLE ISEVERAL EQUATIONS

b) Brackets: Another suggestion is to use brackets formarking different variants. One notation is shown in thefollowing example.

x =

f1 if x < a ∧ x b ∧ x > cf2 if x ≥ a ∧ x c

f3 if x ≥ a ∧ x ≥ b ∧ x < c

Obviously, the function f2 has in the case of x > c twopossible guards.

c) Abbreviations: The last presented notation is to ab-breviate the common part of a guard and to write only thedifference. This is illustrated in the following example:

x =

f1 if x < a ∧ x b ∧ x > cf2 if x ≥ a ∧ x cf3 if x ≥ a ∧ x ≥ b ∧ x < c

This requires tree steps:

1) Identifying all common items of the guards2) Sorting all common terms in the front of the guards3) Using a new symbol for the common part

Step 1: Identifying

x =

f1 if x < a ∧ x b ∧ (x > c)f2 if (x ≥ a) ∧ x c)f3 if x ≥ a ∧ x ≥ b ∧ x < c

Step 2: Sorting

x =

f1 if x < a ∧ x c) ∧ x > bf2 if (x ≥ a ∧ x > c) ∧ x < bf3 if x ≥ a ∧ x ≥ b ∧ x < c

Step 3: Chosing an abbreviation

x =

f1 if x < a ∧ x bf2 if Γ ∧ x < bf3 if x ≥ a ∧ x ≥ b ∧ x < c

Γ = (x ≥ a ∧ x > c)

Fig. 10. Scenario 1: variants by parameterization of transitions

IV. EVALUATION OF VARIABILITY MODELING CONCEPTS

Hybrid automata and switched differential equations havedifferent areas of application. But is there a model type thathas more potential in order to integrate variability in hybridmodels? To decide this it is necessary to define the meaningof potential in our context. Our measure is modifiability, i.e. the degree to which the systems facilitate changes of themodel structure [16]. In order to compare both models we usescenarios developed in section III.

A. Scenario 1: Fixed structure, variants by parameterizationof transitions

The modeling of a simple variation by parameterization isconsidered in this paragraph. Using our approach to extendthe models by parameterization leads to the hybrid automatondepicted in figure 10 and to the differential equation 1.Obviously, there is an increase of differential equations. Thereason for this are the boolean expressions. In order to assurethat the system is well-defined and only one equation is true atthe same time it is necessary to insert for each possible case anew equation. This blows up the differential equation system.

x =

f1 if x < a ∧ x a ∧ x ≥ b ∧ x < c ∧Mod = 1f1 if x > a ∧ x ≥ b ∧ x < c ∧Mod = 3f1 if · · ·f2 if x ≥ a ∧ x < b ∧ x < c ∧Mod = 1f2 if x < a ∧ x ≥ b ∧ x < c ∧Mod = 2f2 if x < a ∧ x < b ∧ x ≥ c ∧Mod = 3

(1)

B. Scenario 2: Fixed structure, variants by parameterizationof states

The usage of our approach to integrate variants of a state byparameterization is illustrated in figure 11 and equation 2. Inorder to avoid undefined equations it is required to have in ourcase three equations for each state in the hybrid automaton.But here it would be possible to extend the differential

Fig. 11. Scenario 2: Variants by parameterization of states

equations with abbreviations, as described in section III, tosolve this problem.

x =

f1 if x < a ∧Mod = 1 ∧ x < bf1 if x < a ∧Mod = 2 ∧ x < bf1 if x < a ∧Mod = 3 ∧ x < bf1,1 if x ≥ a ∧Mod = 1 ∧ x < bf1,2 if x ≥ a ∧Mod = 2 ∧ x < bf1,3 if x ≥ a ∧Mod = 3 ∧ x < bf2 if x ≥ a ∧Mod = 1 ∧ x ≥ bf2 if x ≥ a ∧Mod = 2 ∧ x ≥ bf2 if x ≥ a ∧Mod = 3 ∧ x ≥ b

(2)

C. Scenario 3: Variable structure, variants of a state

In this section we use the example depicted in figure 9.Furthermore, we model this system in the equations 3.

x =

f1 if x < a ∧ x < b ∧ x < l ∧ case = 1f1 if x < a ∧ x < b ∧ x < l ∧ case = 2f2 if x ≥ a ∧ x < b ∧ x < l ∧ case = 1f3 if x ≥ a ∧ x ≥ b ∧ x < l ∧ case = 1f4 if x ≥ a ∧ x < c ∧ x < l ∧ x < e ∧ case = 2f5 if x ≥ a ∧ x ≥ c ∧ x ≥ l ∧ x < e ∧ case = 2f6 if x ≥ a ∧ x ≥ c ∧ x ≥ l ∧ x ≥ e ∧ case = 2f7 if x ≥ a ∧ x ≥ c ∧ x ≥ l ∧ x < e ∧ case = 2f7 if x ≥ a ∧ x ≥ b ∧ x ≥ l ∧ case = 1

(3)

D. Conclusion

We have to regard the effort to change the models in orderto consider the modifiability of both models. Obviously, inthe cases of changing the parameters the identifiers or thecomplete boolean terms the hybrid automata take advantageof the compact representation of the modeled systems. Twoprimary effects seem reasonable:

1) Multiple representation of one function (e. g. f1) in thedifferential equation

2) Multiple representation of boolean expressions (e. g.x ≥ a) in the differential equation

Thus it may be argued that in dependency on the equationnotation this result is changed. This is true, if it is possibleto avoid the multiplication of same equations, e. g. the use ofabbreviations or the introduction of rules.

V. CONCLUSION

We have presented an analysis of variability points for givenhybrid engine models. The local influence of variability ofparameters was illustrated and the large influence of variabilityof the discrete behavior was shown. Furthermore, it wasasserted that there is no explicit consideration of hybrid enginemodel variability in the literature.

Also, we proposed different concepts to add variability inhybrid automata and differential equations. Here, we focusedtwo main concepts for hybrid automata: firstly, the multiplestates and secondly, multiple transitions. To extend differentialequations we suggested different forms of parameterizationand substitutions of equations. In the last section the evaluationof the proposed concepts has been done. The need for repeat-ing one function for several times in the differential equationsleads to an increasing number of equations. Furthermore, theintegration of variants in differential equations extends thelength of the boolean terms enormously.

REFERENCES

[1] L. Albertoni, A. Balluchi, A. Casavola, C. Gambelli, E. Mosca, andA. L. Sangiovanni-Vincentelli. Idle speed control for gdi engines usingrobust multirate hybrid command governors. In Proc. CCA2003, 2003IEEE Conference on Control Applications, 2003.

[2] P. Antsaklis, X. Koutsoukos, and J. Zaytoon. On hybrid control ofcomplex systems: A survey. European Journal of Automation, 32:1023–1045, 1998.

[3] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, G. M. Miconi, U. Pozzi,T. Villa, H. Wong-Toi, and A. L. Sangiovanni-Vincentelli. Maximalsafe set computation for idle speed control of an automotive engine. InHybrid Systems: Computation and Control. Springer-Verlag, 2000.

[4] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, C. Pinello, and A. L.Sangiovanni-Vincentelli. Automotive engine and power-train control: acomprehensive hybrid model. In Proc. 8th Mediterranean Conferenceon Control and Automation - MED2000, 2000.

[5] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, C. Pinello, and A. L.Sangiovanni-Vincentelli. Automotive engine control and hybrid systems:Challenges and opportunities. Proceedings of the IEEE, 88(7):888–912,July 2000.

[6] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, and A. L. Sangiovanni-Vincentelli. Nonlinear and Hybrid Systems in Automotive Control, chap-ter Idle Speed Control Synthesis using an Assume–guarantee Approach,pages 229–243. Springer-Verlag, London, UK, 2002.

[7] A. Balluchi, L. Benvenuti, M. D. Di Benedetto, T. Villa, H. Wong-Toi, and A. L. Sangiovanni-Vincentelli. Hybrid controller synthesis foridle speed management of an automotive engine. In Proc. 2000 IEEEAmerican Control Conference, volume 2, pages 1181–1185, Chicago,IL, USA, June 2000. (invited paper).

[8] A. Balluchi, L. Benvenuti, and A. L. Sangiovanni-Vincentelli. Hybridsystems in automotive electronics design. In 44th Conference onDecision and Control and 8th European Control Conference, Seville,Spain, 2005.

[9] A. Balluchi, L. Benvenuti, T. Villa, and A. L. Sangiovanni-Vincentelli.A hybrid model of a 4-cylinder engine for idle speed control. Technicalreport, PARADES, 2002.

[10] A. Balluchi, L. Benvenuti, T. Villa, and A. L. Sangiovanni-Vincentelli.A nonlinear hybrid model of a 4-cylinder engine for idle speed control.Technical report, PARADES, 2002.

[11] A. Balluchi, A. Bicchi, C. Caterini, C. Rossi, and A. L. Sangiovanni-Vincentelli. Hybrid tracking control for spark–ignition engines. In Proc.39th IEEE Conference on Decision and Control, volume 4, Sydney,NSW, Australia, December 2000.

[12] A. Balluchi, F. Di Natale, A. L. Sangiovanni-Vincentelli, and J. H. vanSchuppen. Synthesis for idle speed control of an automotive engine. InR. Alur and G. J. Pappas, editors, Hybrid Systems: Computation andControl, volume 2993 of Lecture Notes in Computer Science, pages 80–94. Springer-Verlag, Berlin Heidelberg New York, 2004.

[13] A. Balluchi, A. L. Sangiovanni-Vincentelli, and L. Benvenuti. Chal-lenges and opportunities for hybrid systems in the automotive designflow. Technical report, PARADES, Rome, I, February 2005.

[14] D. Hrovat and J. Sun. Models and control methodologies for ic engineidle speed control design. Control Eng. Practice, 5:1093–1100, 1997.

[15] C. W. Krueger. Variation management for software production lines. InSPLC, 37-48, 2002.

[16] Software Engineering Institute. www.sei.cmu.edu, 2004. PLA web site,Carnegie Mellon University, last access: 7 October 2004.

A Hybrid Approach to Modeling, Control and State estimation ofMechanical Systems with Backlash

Femke van Belzen, Philipp Rostalski, Miroslav Baric and Manfred Morari

Abstract— Control of mechanical systems with backlash is atopic well studied by many control practitioners. This interesthas been motivated by the fact that backlash in mechanicalsystems can cause severe performance degradation and leadto instability of the control system. Furthermore, high impact-forces in backlash-systems can lead to a lower durability ofthecomponents and to strokes and peaks in the output. In thispaper a mechanical benchmark system is presented to providefacilities for testing identification and control of systems withbacklash. For the purpose of control, a hybrid MLD model ofthe system was derived and used in an on-line Model PredictiveControl (MPC) scheme. Observer-based state-estimation wasneeded to recover unmeasured states, particularly the backlashangle. Simulation results are presented to show the applicabilityof this hybrid control approach. However, limitations due tofriction prevents an experimental verification at this stage.Further investigations and a redesign of parts of the systemmight be necessary to overcome these obstacles.

I. I NTRODUCTION

Backlash is a common problem in mechanical systemsoccurring whenever there is a gap in the transmission link,e.g. in the gearbox of a powertrain.

This transmission gap causes problems when the systeminput changes from acceleration to braking or vice versa.During a short period of time the driving torque will not betransmitted to the load. When the backlash gap is traversedthe sudden contact will cause a large change in the torqueexercised on the load, causing undesirable bumps and pos-sible damages. Backlash therefore limits the performance indriving maneuvers and restricts the controller performance.To perform sophisticated control maneuvers without theundesirable effects of backlash, it needs to be taken intoaccount. A survey about systems with backlash and possi-ble control systems with a particular focus on automotivepowertrains can be found in [1]. A more detailed treatmentof different approaches towards control and estimation ofmechanical systems with backlash is given in [2].

A natural way to model a mechanical system with backlashis to distinguish two operating modes, namely the “backlashmode”, when two mechanical parts are not in contact and“contact mode”, when the contact between two mechanicalparts is established and the transmission of the momentumtakes place. The inherent switching between these two modesmakes this system a prime example for a hybrid system andmotivates the “hybrid approach” to modeling and control of

F. van Belzen is a student at the Technical University of Eindhoven, 5612AZ Eindhoven, The [email protected]

M. Baric, P. Rostalski and M. Morari are affiliated at the Au-tomatic Control Laboratory, ETH Zurich, 8092 Zurich, [email protected]

mechanical systems with backlash. Throughout this paperwe will consider a special, yet fairly general, class ofhybrid systems, namely discrete-time piecewise affine (PWA)systems [3]. An application of PWA modeling and controlparadigm to mechanical systems with backlash has beenrecently reported in [4].

The motivation of this project is twofold. The first goalis to evaluate recently developed identification and controlstrategies for discrete-time PWA systems using a mechanicalsystem with backlash as a convenient “hybrid” benchmarkproblem. Secondly, we aim at complete and efficient realiza-tion of the MPC controller, suitable for practical implemen-tation in mechanical systems typically affected by backlash,e.g. automotive powertrain systems [2].

II. M ECHANICAL SYSTEM WITH BACKLASH

A schematic representation of a rotating mechanical sys-tem with backlash is shown on Figure 1. The motor,M1

is the driving motor. The inertia,Jm represents the motorflywheel, the inertia,Jl represents the load. The spring-damper combination models a flexible shaft with damping.The dampersbm and bl represent viscous friction. Thesecond motor,M2, can be used to model disturbance torquee.g. road friction. An important parameter in a rotatingmechanical system with backlash is the size of the backlashgap, denoted as2α.

A laboratory physical model of this system has been setup and forms the basis of this project of controller designand state observation for rotating mechanical systems withbacklash. The main elements of the experimental system aretwo rotating masses, the backlash element, the spring, twomotors and two encoders that provide the measurements.The system is driven by a DC motor, another motor of thesame type is used on the load side. The backlash gap can bemechanically changed to four different values, either2, 4,6 or 10. The measurements ofθm and θl are realized byusing two incremental encoders with a maximal resolutionof 2000 counts per revolution, i.e. approximately 0.0031 rad.This resolution is sufficient to measure the position of thesystem in backlash mode for the smallest backlash gap used(2 ≈ 0.035 rad).

III. M ODELING AND PARAMETER IDENTIFICATION

The system described in the Section II has been modeledusing a first principle model. In order to have models withdifferent degree of complexity for controller design andsimulations, two physical models for the backlash systemare presented.

M1 M2

Tm Tlbm

bl

Jm Jl

Ts

θm θl

θuωmωl

b

cBacklash

Fig. 1. Schematic representation of a rotating mechanical system with backlash.

Recognizing the hybrid nature of the system makes theprocess of modeling, control and state estimation moreaccurate and systematic. The models presented will be givenin the form of Piecewise Affine (PWA) systems [5]

x[k + 1] = fPWA(x[k], u[k])

= Aix[k] + Biu[k] + fi

if

[x[k]u[k]

]∈ Di

Di :=

[x

u

]| [(Px)i(Pu)i]

[x

u

]≤ (P0)i

. (1)

The PWA description of a hybrid system is equivalent to theMixed logical dynamical (MLD) system, [5]

x[k + 1] = Ax[k] + B1u[k] + B2δ[k] + B3z[k]

y[k] = Cx[k] + D1u[k] + D2δ[k] + D3z[k]

E2d[k] + E3z[k] ≤ E1u[k] + E4x[k] + E5 (2)

which simplifies the implementation of the presented modelin a Matlab environment.

A. Dead Zone Model

The configuration of Figure 1 can be described approxi-mately by the following differential equations:

Jmωm = −bmωm + Tm − Ts, (3)

Jlωl = −blωl + Ts, (4)

whereTm is the motor torque andTs is the shaft torque. Byneglecting the shaft damping the shaft torque can be modeledas

Ts =

k(θd − α), θd ≥ α

0, |θd| < α

k(θd + α), θd ≤ −α

(5)

whereα is half the backlash gap size. The angles

θd = θm − θl (6)

θb = θm − θu (7)

are the total shaft displacement and the position in backlashrespectively.

The combination of equations (3), (4) and (5) gives apiece-wise affine (PWA) model with state vector

x = (ωm, ωl, θd)T .

The state-update equation has the following form

x[k + 1] =

Acox[k] + Bu[k] + fp[k], θd ≥ α

Ablx[k] + Bu[k], |θd| < α

Acox[k] + Bu[k] + fn[k], θd ≤ −α

(8)

whereAco, Abl, B, fp andfn are defined as

Aco =

− bm

Jm0 − c

Jm

0 − bl

Jl

cJl

1 −1 0

(9)

Abl =

− bm

Jm

0 0

0 − bl

Jl

0

1 −1 0

, (10)

B =

KJm

00

fp =

cαJm

− cαJl

0

fn = −fp. (11)

From (8) it can immediately be seen that the model hasthree discrete states or modes: positive contact, backlashandnegative contact. The same equations describe the dynamicsof positive and negative contact mode. In this modelθb is notmodeled, but approximated byθd. Throughout this article,the model discussed in this section will be referred to as thedead-zone model.

B. A More Precise Model

In most cases shaft damping can not be neglected withoutmaking a substantial modeling error. When shaft damping istaken into account, the shaft torque,Ts, is given by

Ts = c(θd − θb) + b(ωm − ωl) . (12)

The backlash position angle,θb, can be described by thefollowing nonlinear differential equation, [6]

θb =

max[0, θd + cb(θd − θb)] if θb = −α

θd + cb(θd − θb) if |θb| < α

min[0, θd + cb(θd − θb)] if θb = α

(13)

From this differential equation forθb conditions can bederived which define when the system is in backlash mode.The system is in backlash mode if either one of the followingthree conditions holds

|θb| ≤ α (14)

θb ≤ −α ∧ θd + cb(θd − θb) > 0 (15)

θb ≥ α ∧ θd + cb(θd − θb) < 0 (16)

The first part of the second and third conditions checks ifthe system is in (positive or negative) contact mode. Thesecond part of these conditions becomes true if the backlashelement starts to move away from the driving shaft, this isthe moment when the backlash mode is entered.

The differential equation (13) can be linearized as depictedin Eqn. (17), see [7]. This leads to a state-update equation

x[k + 1] =

Acox[k] + Bu[k] (Positive contact)Ablx[k] + Bu[k] (Backlash)Acox[k] + Bu[k] (Negative contact)

(18)

where the statex[k] and the matricesAco, Abl and B aregiven by

x =[

ωm ωl θd θb

]T, (19)

Aco =

− bm+bJm

− bJm

− cJm

cJm

bJl

− bl+bJl

cJl

− cJl

1 −1 0 00 0 0 0

, (20)

Abl =

− bm

Jm0 0 0

0 − bl

Jl

0 0

1 −1 0 01 −1 c

b− c

b

. (21)

Compared to dead-zone model, this model gives a moreaccurate description of the backlash phenomena, since thebacklash angle is modeled and not approximated. Hence itwill be referred to asprecise model.

These models were implemented with special tools along-side Matlab to deal with hybrid systems. All PWA sys-tems were described in the modeling language HYSDEL,developed at the Automatic Control Laboratory of the SwissFederal Institute of Technology, Zurich. The HYSDEL com-piler automatically generates the Mixed Logical Dynamical(MLD) representation of the system, equivalent to the desiredPWA form, which is used in Matlab for simulation purposes.More information on HYSDEL can be found in [8]. Forsimulation of PWA systems and design and simulation ofcontrollers for PWA systems the Multi-Parametric Toolbox(MPT) was used, see [9] for more information.

C. Parameter identification

In order to identify physical parameters for the systemused in the models above, the backlash element was removedin a first step and the shaft were connected. This leadsto a system with almost linear behavior governed by thefollowing differential equations:

Jmωm = −bmωm + Tm − c(θm − θl)

−b(ωm − ωl) (22)

Jlωl = −blωl + c(θm − θl) + b(ωm − ωl) ,

(23)

compare, Section III-B.The numerical values for the parameter can now be

obtained using linear subspace identification methods, e.g.with the Matlab Identification Toolbox. The parameters of the

linear system are the same as the corresponding parametersof the hybrid system, thus after identification of the param-eters with the linear system they can be used in the modelsthat include backlash. The derived linear model is verifiedby comparing the model simulations to measurements.

IV. CONTROLLER DESIGN

The aim of this part is to design a controller taking intoaccount the backlash phenomena and preventing bumps anddamages. These could also be prevented using a robust, butconservative linear controller designed for a linear modelwithout backlash. However, since robust control approachesusually lead to a weak performance and can not guaranteeconstraint satisfaction we will focus on model predictive con-trol approaches using the PWA model presented in SectionIII-B.

In [4] such a Model-Predictive Control (MPC) of automo-tive powertrains with backlash is presented. The difference inthe approach presented here that we aim for an integrated ap-proach. While in the paper[4] a linear acceleration controllerfor the system in contact mode and an MPC controller totraverse the backlash gap is used, we want to follow a morefundamental approach.

We will only design a single MPC controller for bothmodes and avoid any non-optimal switching between differ-ent controllers. Furthermore, the controller will be designedfor the precise model of Section III-B that includes theposition in backlash,θb, in contrast to the approach in [4],were a simplified model is used. This allows us to use allavailable information from the state observer presented inthenext section and will lead to better controller performance.The main aim of the control will be to keep the impact forcewithin certain bounds. In terms of optimal control this willbemodeled as constraints on the speed difference between thedrive and the load. In this section the details on the controllerdesign process are described.

A. Model Predictive Control for PWA systems

The general optimization problem stemming from finitehorizon optimal control with constraints is given as fol-lows [9]

minu[0],...,u[N ]

N∑k=0

‖Ru[k]‖l + ‖Qx[k]‖l

+ ‖PNx[N ]‖l

subject to

x[k + 1] = fdyn(x[k], u[k], w[k]),

umin ≤ u[k] ≤ umax,

∆umin ≤ u[k] − u[k − 1] ≤ ∆umax,

ymin ≤ gdyn(x[k], u[k]) ≤ ymax,

x[N ] ∈ Tset, (24)

wherel ∈ 1,∞ defines a linear norm. The matricesPN ,R and Q penalize the final state, the input and the states,respectively.

θb ≈

0, θd + cb(θd − θb) < 0 ∧ θb = −α (contact)

θd + cb(θd − θb), θd + c

b(θd − θb) ≥ 0 ∧ θb = −α (backlash)

θd + cb(θd − θb), ∧ |θb| < α (backlash)

0, θd + cb(θd − θb) > 0 ∧ θb = α (contact)

θd + cb(θd − θb), θd + c

b(θd − θb) ≤ 0 ∧ θb = α (backlash)

(17)

Alternatively, the optimal control problem can be formu-lated with a quadratic performance index

minu[0],...,u[N ]

N−1∑k=0

u[k]T Ru[k] + x[k]T Qx[k]

+ x[N ]T PNx[N ]

subject to

x[k + 1] = fdyn(x[k], u[k], w[k]),

umin ≤ u[k] ≤ umax,

∆umin ≤ u[k] − u[k − 1] ≤ ∆umax,

ymin ≤ gdyn(x[k], u[k]) ≤ ymax,

x[N ] ∈ Tset . (25)

The classical implementation of model-predictive controluses online optimization with a receding horizon policy, tocompute the optimal input-sequence for the system, [10].However, if the process under control varies rapidly, thecomputation times may become too large and the real time-optimization becomes intractable.

A different approach is to solve the optimization problem(24), resp. (25), parametrically and store the piecewise affinecontrol law e.g. in look-up table, [11]. The advantage of thisapproach is that the access to the optimal input sequence canbe speed-up significantly. Unfortunately, these calculationscan lead to large explicit control laws, depending on theproblem. Practical implementation may then be difficultbecause of large look-up times and required memory space.

The backlash system varies rapidly, so online control isnot possible for the final implementation on the experimentalsystem. Nevertheless, online controllers can be useful duringtuning and simulation steps. For online simulation of thecontroller the MLD representation of the model is used,whereas the PWA representation is used to generate theexplicit solution at a later stage of this project.

The objective of the controller is to prevent large bumpswhen the system goes from contact to backlash mode. Bumpsoccur when there is a large difference betweenωm andωl when contact is established. This can be prevented byconstraining the speed difference between drive and load

|ωm − ωl| ≤ ∆ω. (26)

The boundary value∆ω is a system-dependent design pa-rameter. If ∆ω is small, the controller is restricted toomuch and it will take a long time to bring the load angularvelocity to zero. The idea is to allow a certain level of speeddifference, that corresponds to an acceptable level of bumps,or is just small enough to prevent damage.

The controller is designed for the precise model, describedin Section III-B. A quadratic cost function is used as in (25),with no weight on the final state and the following Q and Rmatrices

Q =

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1000

, (27)

R = 0.1, (28)

with the state vector

x =[

ωm ωl θd θb s]T

. (29)

The augmented state

s =

ωm − ωl if |ωm − ωl| ≥ ∆ω

0 otherwise(30)

is used as a slack variable for implementing the soft con-straint. A prediction horizon ofN = 5 was found to be thesmallest suitable horizon.

B. State Estimation

The backlash angleθb is needed for the desired MPCcontroller. Since this state cannot be measured, a stateestimation is necessary. The size of the backlash gap isknown for the experimental setup and used as a parameterwhen estimatingθb. In practice, the size of the backlash gap,2α, may not be known. In these cases this parameter caneither be identified in the model identification process orestimated during the system operation. In [7] it is argued thatin most cases online estimation is the only feasible option.

In [7] an observer configuration is proposed to estimateθb

and the backlash gap size,α, simultaneously. This structureuses two observers: one to estimate the system parameterα

and another to estimate the position in backlash,θb. Since weare only interested in estimatingθb only the latter observerhas been implemented, [7].

V. SIMULATION RESULTS

This section describes the results obtained with the onlineoptimal controller described in Section IV and state esti-mation. The desired maneuver is to bring the load angularvelocity to zero as quickly as possible, starting from avalue of 8 rad/s. The motor velocity is also 8 rad/s and allangles are zero at the begin of the maneuver. The maximumdifference for load and drive speed, i.e. the impact velocity,was set toδω = 2 rad

s .The controller is designed for the precise model introduced

in Section III-B that includesθd. The measured variables on

the real system areθm and θl. All constraints are modeledas soft constraints, to avoid infeasibility.

In Figure 2(b) the control signal is shown. The actuator hasa range of[−10, 10]. The controller clearly prevents actuatorsaturation, still it uses the available range of inputs. Thespeed difference stays within the bounds, see Figure 2(a).

The control aim, stabilization ofωl without |ωm − ωl|exceeding∆ω, is achieved within 5 seconds. An LQRcontroller with similar performance violates the|ωm − ωl|constraint.

0 1 2 3 4 5 6 7 8 9 10

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

wm

−w

l [ra

d/s]

time [s]

(a) Speed Difference:ωm − ωl

0 1 2 3 4 5 6 7 8 9 10−10

−8

−6

−4

−2

0

2

4

6

8

10

time [s]

i [A

]

(b) Control Input, Motor Currenti

Fig. 2. Control Input and Speed Difference

VI. CONCLUSION AND OUTLOOK

A mechanical system with backlash has been specifiedand built, providing a benchmark system for hybrid controland identification strategies. Modern control strategies forhybrid systems can be verified and its realizability on a realsystem can be investigated. A first step towards a “hybrid”benchmark problem is obtained.

The full design cycle containing modeling, estimation andcontrol design for this rotational system with backlash hasbeen performed. An observer, based on [7] was implementedand tested in simulations and works satisfactory. The aim ofcontrol was to design an MPC controller that minimizes thebumps or the damages that occur when the system is operated

and backlash is not taken into account. This is realized byconstraining the difference in speed between the drive andthe load, i.e.ωm − ωl.

So far only online optimal control has been considered.Because the model used to calculate the controllers is notaccurate enough, the controllers were not yet implementedon the real system. Therefore, no explicit solutions have beencomputed and only the behavior of online controllers hasbeen examined. First attempts towards explicit solution ofthe MPC scheme presented in this paper have shown theinherent complexity of these approaches. Explicit solutionbased on the minimum time control concept [12] seems tobe more suitable for a real time implementation. Analyzingthese questions is part of the ongoing work to find an efficientrealization of the MPC control scheme for the experimentalsystem.

The derived model appeared to be inadequate for theexperimental verification on the real system, the main lim-itation being the presence of unmodeled friction. Furtherinvestigations and a redesign of parts of the system seemto be necessary to overcome the current limitations.

ACKNOWLEDGMENTS

This work has been supported by the HYCON Networkof Excellence, HYCON WP 4c

REFERENCES

[1] A. Lagerberg, “A literature survey on control of automotive power-trains with backlash,” Department of Signals and Systems, ChalmersUniversity of Technology, Goteborg, Sweden, Tech. Rep. R013/2001,December 2001.

[2] ——, “Control and estimation of automotive powertrains withbacklash,” Ph.D. dissertation, Department of Signals and Systems,Chalmers University of Technology, Goteborg, Sweden, 2004.

[3] F. Borrelli, “Discrete Time Constrained Optimal Control,” Ph.D.dissertation, Automatic Control Laboratory, ETH Zurich, 2002.

[4] A. Lagerberg and B. Egardt, “Model predictive control ofautomotivepowertrains with backlash,” in16th IFAC World Congress, Prague,Prague, Czech Republic, July 2005.

[5] M. Morari, M. Baotic, and F. Borrelli, “Hybrid Systems Modeling andControl,” European Journal of Control, vol. 9, no. 2-3, pp. 177–189,2003.

[6] M. Nordin andet. al., “New models for backlash and gear play,”In-ternational journal of adaptive control and signal processing, vol. 11,pp. 49–63, 1997.

[7] A. Lagerberg and B. Egardt, “Estimation of backlash withapplicationto automotive powertrains,” inProceedings of the 42nd IEEE Confer-ence on Decision and Control, Maui, Hawaii USA, 2003.

[8] F. Torrisi, A. Bemporad, G. Bertini, P. Hertach, D. Jost,andD. Mignone, “HYSDEL - User Manual,” Tech. Rep., August 2002.

[9] M. Kvasnica, P. Grieder, M. Baotic, and M. Morari, “Multi-ParametricToolbox (MPT),” inHybrid Systems: Computation and Control, March2004, pp. 448–462.

[10] M. J.M., Predictive Control with Constraints. Prentice-Hall, 2001.[11] F. Borrelli, M. Baotic, A. Bemporad, and M. Morari, “Dynamic

programming for constrained optimal control of discrete-time linearhybrid systems,”Automatica, vol. 41, pp. 1709–1721, October 2005.

[12] P. Grieder and M. Morari, “Complexity Reduction of Receding Hori-zon Control,” in IEEE Conference on Decision and Control, Maui,Hawaii, December 2003, pp. 3179–3184.

HYCON-CEmACS Workshop, Lund 2006 ©Reserved for authors

Hybrid Optimal Control of Dry Clutch Engagement

Authors: Arno van der Heijden*, Alex Serrarens#, Kanat Camlibel*, Henk Nijmeijer*

Affiliation: *Technische Universiteit Eindhoven, #Drivetrain Innovations BV, The Netherlands

AbstractLately, with the increasing use of AMTs (Automated Manual Transmission) the engagement control ofthe dry clutch becomes more important. The engagement control plays a crucial role, since different,conflicting objectives have to be satisfied: preservation of driver comfort, fast engagement and smallfriction losses. In this work a number of optimal control strategies for clutch engagement, based onhybrid (PWA) control principles, are compared. For developing a useful clutch control scheme, the driveline is modelled as a piecewise affine (PWA) system. The first control strategy is widely known asexplicit MPC. However, it seems that it is not suitable (yet) for this type of problems. The secondstrategy is a piecewise LQ controller, based on piecewise quadratic Lyapunov functions. Simulation results using both strategies are presented and discussed.

Nonlinear observer for vehicle lateral

velocity

Lars Imsland∗, Havard Fjær Grip†, Christoph Bohm†,‡,Tor A. Johansen∗,†, Thor I. Fossen∗,†,

Jens C. Kalkkuhl§, Avshalom Suissa§

March 15, 2006

This paper describes a nonlinear observer approach to vehicle lateral ve-locity estimation, with stability guarantees. Furthermore, the observer is ex-tended with adaptation of the maximum friction coefficient, and to providegood results on banked roads, the observer is also extended with estimationof road bank angle.

1 Introduction

Information about lateral vehicle velocity or vehicle body side-slip angle are importantfor vehicle dynamics control systems such as anti-skid systems (for instance ESP fromBosch [20]). This information is hard and expensive to measure reliably, and the onlyviable option is therefore to infer it from other measurements and a dynamic systemmodel, using an observer.

There exist several approaches to observers for lateral velocity, and most approachesare based on linear or quasi-linear techniques (see e.g. [21, 5, 19]). Due to nonlineardynamics and especially highly nonlinear friction models, there have been considerableinterest in using nonlinear techniques. The Extended Kalman Filter have been mostlyused [17, 15, 16, 2], but there has also been interest in other nonlinear techniques [7, 11,4, 8, 9].

This paper reviews the approach in [8], the only approach which gives explicit stabilityguarantees in a fully nonlinear setting, based on general conditions on a nonlinear road-tire friction model. The computational efficiency of this observer will be superior to anExtended Kalman Filter, since online solution of the Riccati equation is avoided.

∗SINTEF ICT, Applied Cybernetics, N-7465 Trondheim, Norway†NTNU, Department of Engineering Cybernetics, N-7491 Trondheim, Norway‡University of Stuttgart, Department for Systems Theory in Engineering, 70550 Stuttgart, Germany§DaimlerChrysler Research and Technology, 70456 Stuttgart, Germany

1

Furthermore, based on the results in [6], the observer is extended with adaptationof the maximum friction coefficient, an important parameter in many road-tire frictionmodels. The adaptation is based on a linearized parameterization of the friction model.Lastly, the observer is extended to banked roads by estimating the road bank angle.

2 Vehicle modeling

2.1 Rigid body dynamics

The vehicle velocity is defined in a body-fixed coordinate system (see Figure 1) with theorigin at the center of gravity (CG), which location is assumed constant. The x-axispoints forward and the y-axis points to the left. There are also wheel-fixed coordinatesystems, aligned with each wheel and with origins at the wheel centers. The distancefrom CG to each wheel center is denoted hi, with i representing wheel index. Togetherwith the angles γi, this defines the horizontal vehicle geometry.

γ2

xCG

yCG

r

CG

γ4

β

b3 b2

Fi,x

v

h4h3

h1 h2

l4

l2l1

l3

b1 b2

δi

γ1

γ3

Fi,y

43

1 2

Figure 1: Top view of vehicle: Horizontal axis systems, geometric definitions, wheelforces, speed, slip angle and yaw rate.

Vertically, roll and road bank angle will influence estimation of lateral velocity, seeFigure 2. We will neglect the roll angle (or it can be compensated for by assuming rollangle proportional to measured lateral acceleration).

The lateral vehicle velocity vy (in the body-fixed coordinate system) can be writtenvy = −vxr+ar

y, where vx is the longitudinal velocity and r is the yaw rate, and ary is the

lateral acceleration. In case of non-zero roll and road bank angle, the lateral accelerationmeasurement ay will be affected gravity, such that ar

y= ay + g sin(φ+ φR). Ignoring φ,

2

φ

yin

zCG

yUC

φR

yCG

zin

UC

CG

RC

zUC

Figure 2: Definition of road bank angle, φR (and roll angle, φ).

this leads to the following dynamic equation for lateral velocity:

vy = −vxr + ay + g sinφR (1)

In addition to vy, we will use the yaw rate r as a dynamic state for better exploiting theyaw rate measurement:

r =1

Jz

4∑i=1

gT

ifi (2)

where Jz is the moment of inertia, fi is the vector (in body-fixed coordinates) of horizontalfriction forces at wheel i, gi := (−hi sinψi, hi cosψi)

T, and the angles ψi are introducedto get a uniform representation, ψ1 = −γ1, ψ2 = γ2, ψ3 = π + γ3 and ψ4 = π − γ4.

2.2 Friction models

In most friction models, the friction forces (in wheel-fixed coordinates) are functions oftire slips, Fi = Fi(λi,x, λi,y), where the slips λi,x and λi,y are measures of the relativedifference in vehicle and tire longitudinal and lateral velocity for wheel i, see definitionsbelow. In addition, the friction will vary with

The definition of tire slips we will use herein, are

λi,x =ωiRdyn − Vi,x

Vi,x

, λi,y = sinαi,

where ωi are wheel speeds (angular velocities) and Rdyn is the dynamic wheel radius,the tire slip angles are calculated as

αi = δi − arctanvi,y

vi,x

,

and Vi,x are the velocities in x-direction of the wheel coordinate systems,

Vi,x =√v2i,x

+ v2i,y

cosαi.

The longitudinal and lateral velocities of the wheel center in the body-fixed coordinatesystem are vi,x = vx ± rbi and vi,y = vy ± rli. For the tire slip angles to be well-defined,we assume that there is no reverse motion.

3

Assumption 1 We have vi,x>0 and |αi|<π

2 , i=1, . . . , 4.

The tire slips depend on the vehicle states and the time-varying, measured steeringangles δ = (δ1, . . . , δ4)

T and wheel speeds ω = (ω1, . . . , ω4)T. We will therefore use the

notations Fi = Fi(λi,x, λi,y) = Fi(vy, r, vx, δi, ωi) interchangeably, depending on context.Of significant importance in friction modeling is the degree of adhesion between the

tire and the surface. We will assume that in the friction model, the difference in road-tireadhesion between particular surfaces is represented by the maximum road-tire frictioncoefficient, or a closely related value, denoted µH . For simplicity, the value µH willbe referred to as the friction parameter. When µH is assumed known (as in Section 3and 5), for simplicity the notation will not reflect dependence on µH .

The relation between friction forces in body-fixed and wheel-fixed coordinates, are

fi = R(δi)Fi,

where R(δi) is the usual rotation matrix. Using Newton’s law, the lateral accelerationcan be written in terms of the friction forces as

ay =1

mfy =

1

m

4∑i=1

[0 1]R(δi)Fi =1

m

4∑i=1

(Fi,y cos δi + Fi,x sin δi) (3)

We make the following assumption regarding the friction model:

Assumption 2 There exist a positive constant c1 and sets ∆, Ω such that for all δ ∈ ∆,ω ∈ Ω, vx > 0, and for all x = (vy, r)

T, the friction model is continuously differentiable

with respect to x,∥∥∥∂Fi(x,vx,δi,ωi)

∂x

∥∥∥ are (uniformly) bounded and

∂fy

∂vy

=

4∑i=1

(∂Fi,y(x, vx, δi, ωi)

∂vy

cos δi +∂Fi,x(x, vx, δi, ωi)

∂vy

sin δi

)< −c1. (4)

Remark 1 This is the assumption that allows the structure we will propose in Section 3for the lateral velocity observer. We assume here for simplicity that it holds for all vy

and all steering angles, while there for some tires might exist combinations of lateralwheel slips such that it does not hold always. This is discussed in [8], but we note herethat the condition always hold for small lateral slip values. Large lateral slip values, forwhich the assumption in some cases might not hold, can only be sustained for short timeintervals. 2

The following result can be proved (see [8]) by using the mean value theorem togetherwith Assumption 2:

Lemma 1 There exist positive constants ci, i = 1, . . . , 4 such that for all x, x, δ ∈ ∆,ω ∈ Ω, the following holds;

vy

∑i

[0 1]R(δi) (Fi(x, vx, δi, ωi) −Fi(x, vx, δi, ωi)) ≤ −c1v2y

+ c2|r||vy|, (5a)

1

Jz

∑i

gT

i R(δi) (Fi(x, vx, δi, ωi) −Fi(x, vx, δi, ωi)) ≤ c3|vy| + c4|r|, (5b)

where vy = vy − vy and r = r − r. 2

4

2.3 Measurements

The measurements used are summarized below:

Symbol Measurement

ay Lateral acceleration

vx Longitudinal velocity

r Yaw rate

ωi Wheel speed (angular velocity) for wheel i

δi Steering angle for wheel i

We will assume that vx is calculated using the wheel speed sensors (and possibly otherinformation), see [8, 12] for some approaches. The rest of the measurements can beconsidered standard in modern cars with yaw stabilization (anti-skid) systems, such asthe Bosch ESP system [20]. The acceleration and yaw rate sensor are assumed located inthe CG (or corrected if this is not the case), and to have bias and slow drift componentsremoved.

3 Observer for lateral velocity: Constant friction parameter

and flat road

For brevity we lump the time-varying parameters in θ = (vx, δT,ωT)T ∈ Θ, and use Fi

to denote Fi(x, δi, ωi) and Fi to denote Fi(x, δi, ωi). From Section 2, the model usedwhen there is no road bank (or roll) angles, is

vy = −vxr + ay, (6a)

r =1

Jz

4∑i=1

gT

iR(δi)Fi. (6b)

Based on this model, we propose the following observer (slightly different from the onein [8]):

˙vy =−vxr+ay−Kvy

(may−

4∑i=1

[0 1]R(δi)Fi

), (7a)

˙r=1

Jz

4∑i=1

gT

i R(δi)Fi +Kr (r − r) . (7b)

Define the error variables vy := vy − vy and r := r − r, and x := (vy, r)T. We will make

use of the following fact:

Fact 1 By completing squares,

−aξ21 + b|ξ1||ξ2| =b2

4aξ22 −

(√a|ξ1| −

b

2√a|ξ2|

)2

.

5

The observer error dynamics is

˙vy = Kvy

4∑i=1

[0 1]R(δi)(Fi − Fi), (8a)

˙r =1

Jz

4∑i=1

gT

i R(δi)(Fi − Fi) −Kr r. (8b)

Theorem 1 Assume that Θ is such that ∀θ ∈ Θ and ∀t ≥ 0, x(t) exists. For somekr > 0, let the observer gains be chosen such that

Kvy> 0, (9)

Kr > kr + c4 +(Kvy

c2 + c3)2

2Kvyc1

. (10)

Then the origin of the observer error dynamics is uniformly globally exponentially sta-ble. 2

Proof The proof is very similar to the proof in [8]. Define the Lyapunov functioncandidate V (x) := 1

2(v2y + r2). The time derivative along the trajectories of (8) is

V = vy

(Kvy

4∑i=1

[0 1]R(δi)(Fi − Fi)

)+ r

(1

Jz

4∑i=1

gT

iR(δi)(Fi − Fi) −Kr r

).

Using Lemma 1, V can be upper bounded:

V ≤ −Kvyc1v

2y +Kvy

c2|r||vy| + c3r|vy| + c4r|r| −Kr r2,

≤ (Kvyc2 + c3)|r||vy| −Kvy

c1v2y + c4r|r| −Kr r

2.

By Fact 1 and (9),

V ≤(Kvy

c2 + c3)2

2Kvyc1

r2 −Kvy

c1

2v2y + c4r|r| −Kr r

2 −

(√Kvy

c1

2|vy| −

(Kvyc2 + c3)√

2Kvyc1

|r|

)2

.

We see that, due to (10),

V ≤ −Kvy

c1

2v2y − kr r

2 (11)

is uniformly negative definite, and the theorem follows from standard Lyapunov theory(e.g. [10]).

Remark 2 The observer for vy (7a) could use the estimate r instead of the measuredr in the Coriolis term (replacing vxr with vxr). The bound (10) would then depend onan upper bound on vx [8]. 2

6

4 Lateral velocity estimation with friction adaptation

In this section, we will not include r as a state to be estimated. Instead, we will takethe friction parameter µH as a state, thus in this section x = (vy, µH)T. Furthermore,we will not let notation reflect dependence of the lateral force fy on the time varying,measured parameters r, vx, δ, and ω, other than letting fy be written fy(t,x).

Lemma 2 Assumption 2 implies the existence of a positive constant c5 such that ∀(t,x)

∂fy(t,x)

∂vy

≥ −c5.

The proof is straightforward. Two additional assumptions on the friction model isneeded. The first one is not restrictive, while the second one is based on an approx-imation:

Assumption 3 There exist a positive constant c6 such that ∀(t,x)∣∣∣∣∂fy(t,x)

∂µH

∣∣∣∣ ≤ c6

Assumption 4 The lateral force fy(t,x) can be written as a truncated Taylor seriesexpansion, as follows:

fy(t,x) = fy(t, x) +∂fy(t, x)

∂vy

vy +∂fy(t, x)

∂µH

µH . (12)

Obviously, this will only hold if the friction model is linear. However, if vy and µH aresmall, then the error made will be small. Lastly, the following standard assumption ismade:

Assumption 5 The friction parameter µH is constant, such that µH = 0.

For ease of notation, we define

ξvy(t, x) =

1

m

∂fy(t, x)

∂vy

, ξµH(t, x) =

1

m

∂fy(t, x)

∂µH

.

Subtracting fy(t, x) from both sides of (12) and dividing by m, we get the followingexpression for ay:

ay = ξvy(t, x)vy + ξµH

(t, x)µH . (13)

Noting that x = x − x, we also define ξvy(t, x) = ξvy

(t, x) and ξµH(t, x) = ξµH

(t, x) byconsidering x a time-varying signal.

We propose the following adaptive observer:

˙vy = ay − rvx −mKvy(ay − ay) , (14a)

˙µH = −ΓξµH(t, x)ξ−1

vy(t, x)(ay − ay), (14b)

7

with tuning gains Kvy> 0 and Γ > 0. Note that (14b) is well-defined, because, according

to Assumption 2, ξvy(t, x) ≤ − c1

m< 0. The observer error dynamics is:

˙vy = mKvyay, (15a)

˙µH = ΓξµH(t, x)ξ−1

vy(t, x)ay. (15b)

Note that Assumption 2 implies that the right-hand side of (15) is continuous in t andlocally Lipschitz continuous in x, uniformly in t.

Lemma 3 If Assumptions 2–4 hold, then the origin of (15) is uniformly globally stable(UGS). 2

Proof Define a Lyapunov function candidate V : R2 → R as V (x) = 1

2

(v2y

+mKvyΓ−1µH

2).

Its time derivative along the trajectories of (15) is

V (t, x) = mKvyay vy +mKvy

ξµH(t, x)ξ−1

vy(t, x)ayµH .

Using (13) to substitute for vy, we get that

V (t, x) = mKvyξ−1vy

(t, x)a2y≤ −

mKvy

c5a2

y≤ 0. (16)

Because V is radially unbounded and V is negative semidefinite, it follows that the originof (15) is UGS (see [14, Def. 1]).

The time derivative of the Lyapunov function is not negative definite, so we cannotdirectly conclude with asymptotic stability. If we assume that ay is uniformly continuous,it is possible to invoke Barbalat’s Lemma (see [1, 10]) to conclude that limt→∞

ay = 0.Inspecting (13), one might intuitively think that if ξvy

and ξµHvary independently with

time, this cannot happen unless vy and µH also go to zero. In the following theorem, aformal condition is given for uniform global asymptotic stability (UGAS) of the originof the error dynamics.

Theorem 2 Suppose that for each χ ∈ R2\0, there exist ε > 0 and T > 0 such that

∀t ∈ R,

∫t+T

t

ξ2vy

(τ,χ) dτ

∫t+T

t

ξ2µH

(τ,χ) dτ ≥

(∫t+T

t

ξvy(τ,χ)ξµH

(τ,χ) dτ

)2

+ ε. (17)

If Assumptions 2–4 hold, the origin of (15) is UGAS. 2

Proof (Outline) Let the error dynamics (15) be written as ˙x = Ψ(t, x). The proofis based on [14, Cor. 3], and largely follows from these facts:

• The observer error dynamics is UGS (Lemma 3).

• There exist an auxiliary negative semi-definite function Y (x,Ψ) such that theLyapunov function in Lemma 3 fulfills V ≤ Y (x,Ψ(t, x)) and Y (x,Ψ(t, x)) = 0implies that Ψ(t, x) = 0.

8

• The condition (17) implies that Ψ(t, x) is uniformly δ-persistently exciting (Uδ-PE) with respect to x, as defined in [14].

For further details, consult [6].

From the Cauchy-Schwartz Inequality, (17) is a requirement of linear independencebetween ξvy

(t, x) and ξµH(t, x) for fixed, nonzero x. This leads to an intuitive interpre-

tation of the condition for stability stated by Theorem 2. The functions ξvyand ξµH

arenonlinear functions which depend on several time-varying signals. Because ξvy

and ξµH

are substantially different from each other, they will behave differently when excited bythese time-varying signals. Therefore, if there is some variation in these signals in theinterval between t and t + T , ξvy

(t,χ) and ξµH(t,χ) will not be linearly dependent for

fixed χ over this interval. If, on the other hand, all the time-varying signals are keptconstant, the functions ξvy

(t,χ) and ξµH(t,χ) will themselves be constant and, therefore,

linearly dependent.In practical terms, this leads to a requirement of a somewhat varied driving pattern.

Rather than driving in a straight line or a circle at constant speed, UGAS requires acertain amount of turning, acceleration or braking. It is not required that this happen allthe time, but there must exist a T , arbitrarily large, such that within any time intervalof length T , there is sufficient variation to excite the system.

In [6], it is shown that if the car is driving in a straight line (for which (17) does nothold), then still vy → vy = 0. Furthermore, it is shown how the adaption law can beaugmented with a projection algorithm to ensure that the friction parameter estimateis within physical bounds.

5 Lateral velocity estimation on banked roads

There are reported approaches to lateral velocity estimation on banked roads usingthe EKF and by assuming linear friction models, using unknown input observes [13].Here, we will use a method that has similarities to [18] to construct a road bank anglemeasurement based on the available measurements, and integrate this into the observerin Section 3.

The method consists of constructing different signals that provide information aboutthe road bank angle in different maneuvers, and weight these signals to provide an overallestimate of the road bank angle.

The first signal is a “steady-state” estimate, given by setting vy = 0 in (1):

φv

R=vxr − ay

g. (18)

Here and in the rest of this section we assume (mainly for notational simplicity) sinφR ≈φR. The two next signals comes from a linearized one-track (bicycle) model of the vehicle:

ay = Gδ→ay(s)δ +GφR→ay

(s)φR ⇒ φa

R= G−1

φR→ay(s)[ay −Gδ→ay

(s)δ], (19)

r = Gδ→r(s)δ +GφR→r(s)φR ⇒ φr

R = G−1φR→r

(s) [r −Gδ→r(s)δ] . (20)

9

Some standard techniques must be used to ensure that the transfer-functions producingthese signals are proper. Practical experience shows that the signal φr

R(19) has a low

signal-to-noise ratio and we have therefore largely disregarded this signal. Thus, theweighting consist of using φv

R(18) in steady-state or slow maneuvers (in the sense that

vy is small), and putting more weight on the signal φa

R(19) for more dynamic maneuvers.

However, for large lateral tire-slip values, the nonlinear friction model implies that thelinearization in (19) results in large errors. We therefore suggest a correction factoray = kcorr(t)a

liny

, where aliny

is the lateral acceleration as calculated by the linearizedbicycle model. This results in

φa

R =1

kcorr(t)G−1

φR→ay(s)(ay − kcorr(t)Gδ→ay

(s)δ), (21)

where kcorr(t) can be calculated based on a nonlinear friction model or comparison of anacceleration calculated based on a linear model with the real (presumptively nonlinear)acceleration, or both.

We denote the calculated measurement based on φv

R, φa

Rand (possibly) φr

Ras φR, and

let the weighting process be described as

φR = f(t, φv

R, φa

R, φr

R), (22)

referring to [3] for further details.We propose the following nonlinear observer for lateral velocity on banked roads:

˙vy = −vxr + ay + gφR −Kvy

(may −

4∑i=1

[0 1]R(δi)Fi

), (23a)

˙r =1

Jz

4∑i=1

gT

iR(δi)Fi +Kr (r − r) , (23b)

˙φR = kφ

(φR − φR

). (23c)

Denote the error in the φR-measurement as ∆φR, ∆φR = φR − φR. First we show thatwith ∆φR = 0, the error dynamics (based on assuming φR = 0)

˙vy = gφR +Kvy

4∑i=1

[0 1]R(δi)(Fi − Fi), (24a)

˙r =1

Jz

4∑i=1

gT

iR(δi)(Fi − Fi) −Kr r, (24b)

˙φR = −kφφR, (24c)

is exponentially stable:

10

Theorem 3 Let the observer gains be chosen such that

Kvy> 0, (25)

Kr > c4 +

(Kvy

c2 + c3)2

4Kvyc1

, (26)

kφ >g2 (Kr − c4)

4Kvyc1 (Kr − c4) −

(Kvy

c2 + c3)2 . (27)

Then the origin of the observer error dynamics (24) is uniformly globally exponentiallystable. 2

Proof Define the Lyapunov function candidate V (x) := 12(v2

y+r2+φ2

R), x = (vy, r, φR)T.

The time derivative along the trajectories of (24) is

V = vy

(gφR +Kvy

4∑i=1

[0 1]R(δi)(Fi − Fi)

)+ r

(1

Jz

4∑i=1

gT

iR(δi)(Fi − Fi) −Kr r

)− kφφ

2R.

Using Lemma 1, V can be upper bounded:

V ≤ −Kvyc1v

2y +Kvy

c2|r||vy| + gvyφR + c3r|vy| + c4r|r| −Kr r2 − kφφ

2R

≤ −Kvyc1v

2y − (Kr − c4)r

2 − kφφ2R + (Kvy

c2 + c3)|r||vy| + g|vy||φR|

= −|x|TQ|x|,

where |x| = (|vy|, |r|, |φR|)T, and

Q =

Kvyc1 −

Kvy c2+c3

2 − g

2

−Kvy c2+c3

2 Kr − c4 0− g

2 0 kφ

.

This matrix’ principal minors should be positive,

Kvyc1 > 0,

Kvyc1 (Kr − c4) −

(Kvy

c2 + c3

2

)2

> 0,

kφ

(Kvy

c1 (Kr − c4) −

(Kvy

c2 + c3

2

)2)

−g2

4(Kr − c4) > 0.

It is seen that Kvy> 0 implies that the first principal minor is positive, which also means

that the second can be made positive by choosing Kr large enough. Due to this, it ispossible to choose kφ large enough such that the third principal minor (the determinant)also is possible. The exact conditions on the gains are given in (25)–(27).

When Q’s principal minors are positive, then Q is positive definite implying that V isnegative definite. This shows uniform global exponential stability.

11

Typically, ∆φR 6= 0, which means that (24c) must be written

˙φR = −kφφR + kφ∆φR.

Using the same Lyapunov function as above, we get

V ≤ −|x|TQ|x| + kφφR∆φR ≤ −λmin(Q)‖x‖ + kφ‖x‖|∆φR|

which shows that

V ≤ −λmin(Q)

2‖x‖2, ∀ ‖x‖ ≥

2kφ

λmin(Q)|∆φR|

implying that

‖x(t)‖ ≤ ‖x(t0)‖e−

λmin(Q)

2(t−t0) +

2kφ

λmin(Q)

(sup

to≤τ≤t

|∆φR(τ)|

). (28)

This corresponds exactly to the definition [10] of global input-to-state stability:

Theorem 4 The observer error dynamics for the observer (23) is globally input-to-statestable with respect to the input ∆φR. 2

This means that bounded errors in the computation of φR (due to model and measure-ment error) may lead to a bias on the estimate, but not loss of stability.

6 Experimental results

In this section we apply the observer to three challenging data sets:

• Data set 1: Slalom maneuver, dry asphalt.

• Data set 2: Driving in circle on ice.

• Data set 3: Making a turn on a hilly road.

For the two first sets, we both apply the non-adaptive observer (Section 3) and theadaptive (Section 4) but assume flat road. For the last set, we apply the non-adaptiveobserver (Section 3) and the observer with estimation of road bank angle (Section 5).We plot lateral velocity vy and side-slip angle β = arctan vy/vx (where vx is obtainedfrom the observer for longitudinal velocity described in [8]. For the approaches withadaptation, we also plot µH or φR.

In the observer from Section 3, we have chosen Kvy= 1 and Kr = 40. This is also

used in Section 4, with Γ = 5. In Section 5, we used kφ = 0.9. Initial conditions are

vy = r = 0, µH = 0.5 and φR = 0.For the first data set, we first apply the observer from Section 3 (without adaption

of µH and φR). We see in Figure 3 that it works good. The observer with friction

12

0 5 10 15 20 25 30−12

−10

−8

−6

−4

−2

0

2

4

6

8

Time[s]

v y [m/s

]

(a) Lateral velocity

0 5 10 15 20 25 30−40

−30

−20

−10

0

10

20

30

Time[s]

β [°

]

(b) Body side slip angle

Figure 3: Non-adaptive observer: Estimate (solid) and measurement (dashed) of lateralvelocity and side slip angle, first data set, constant µH = 1.

0 5 10 15 20 25 30−12

−10

−8

−6

−4

−2

0

2

4

6

8

Time[s]

v y [m/s

]


0 5 10 15 20 25 30−40

−30

−20

−10

0

10

20

30

Time[s]

β [°

]


0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ H

Time[s]

(c) Adapted maximum friction co-efficient

Figure 4: Observer with friction adaption: Estimate (solid) and measurement (dashed)of lateral velocity and side slip angle, and adapted µH , for first data set.

13

adaptation is applied on the same data set in Figure 4, and even though the adaptedµH is noisy, it does not affect the quality of the velocity/side slip estimate.

For the second data set, we also first apply the observer from Section 3 (with constantµH = 0.3 and φR = 0). The performance is not very good in the period around t = 15s,as can be seen in Figure 5. This can be attributed mainly to an actual change in roadsurface conditions, as the vehicle slides off the test track after the driver loses control ofit, meaning the assumption of constant µH is violated. The observer with adaptation ofµH (Figure 6) does better, and we can see the change in µH in Figure 6c.

0 5 10 15 20 25−14

−12

−10

−8

−6

−4

−2

0

2

Time[s]

v y [m/s

]


0 5 10 15 20 25−40

−35

−30

−25

−20

−15

−10

−5

0

5

Time[s]β

[°]


Figure 5: Non-adaptive observer: Estimate (solid) and measurement (dashed) of lateralvelocity and side slip angle, second data set, constant µH = 0.3.

0 5 10 15 20 25−14

−12

−10

−8

−6

−4

−2

0

2

Time[s]

v y [m/s

]


0 5 10 15 20 25−40

−35

−30

−25

−20

−15

−10

−5

0

5

Time[s]

β [°

]


0 5 10 15 20 250.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ H

Time[s]

(c) Adapted maximum friction co-efficient

Figure 6: Observer with friction adaption: Estimate (solid) and measurement (dashed)of lateral velocity and side slip angle, and adapted µH , second data set.

For the third data set, we again apply the observer without adaptation, see Figure 7.In the period 6–9s the observer is inaccurate, even giving the wrong sign of the lateralvelocity and side-slip angle. We expect this is due to a road bank angle affecting theacceleration measurement, and this is confirmed when we apply the observer from Sec-tion 5, see Figure 8. By estimating the road bank angle, we are able to get good qualitylateral velocity and side slip angle.

14

It is notable that since the road bank angle does not affect the injection term in (7a),we can get better performance in the case of Figure 7 (without estimation of φR) byincreasing the gain Kvy

. However, this comes at a cost of worse overall performanceof the observer since it increases the sensitivity to measurement noise and errors in themeasurement equation (the friction model).

0 2 4 6 8 10 12−0.5

0

0.5

1

Time[s]

v y [m/s

]


0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

3

4

5

6

Time[s]

β [°

]


Figure 7: Observer without estimation of road bank angle: Estimate (solid) and mea-surement (dashed) of lateral velocity and side slip angle, third data set, constant φR = 0.

0 2 4 6 8 10 12−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time[s]

v y [m/s

]


0 2 4 6 8 10 12−2

−1

0

1

2

3

4

5

6

Time[s]

β [°

]


0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

φ R [°

]

Time[s]

(c) Estimated road bank angle

Figure 8: Observer with estimation of road bank angle: Estimate (solid) and measure-ment (dashed) of lateral velocity and side slip angle, and estimated φR, third data set.

7 Concluding remarks

In this paper, we first reviewed a nonlinear observer for lateral velocity. The observerrelies on a certain property of road-tire friction models ( 2). Herein, we assumed it tohold globally. For many tires, there will be combinations of large lateral tire slip valuesfor which the condition does not hold. However, such a situation cannot be sustainedover more than a few seconds, and it is possible to prove [9] that in this period, theobserver will not diverge.

This observer works well for constant and known friction characteristics, and on flat

15

road, as the experimental results showed. However, the experimental results also in-dicates worse performance for changing/unknown friction characteristics, and bankedroads. Therefore, the observer was first extended with adaptation of the maximumfriction coefficient, and thereafter extended with estimation of the road bank angle.

The experimental results shows promise for both these extensions, separately. Furtherwork will consist of integrating both in a common observer, adapting tire-road frictionand estimating road bank angle at the same time.

Acknowledgments

This research is supported by the European Commission STREP project CEmACS,contract 004175.

References

[1] I. Barbalat. Systemes d’equations differentielles d’oscillations non lineaires. Rev.Math. Pures Appl., 4(2):267–270, 1959.

[2] M. C. Best, T. J. Gordon, and P. J. Dixon. An extended adaptive Kalman filter forreal-time state estimation of vehicle handling dynamics. Vehicle System Dynamics,34:57–75, 2000.

[3] Christoph Bohm. Lateral velocity estimation on banked roads xxx. Master’s thesis,University of Stuttgart, 2006. Draft version.

[4] Jim Farrelly and Peter Wellstead. Estimation of vehicle lateral velocity. In Proc.35th IEEE Conf. Decision Contr., pages 552–557, 1996.

[5] Yoshiki Fukada. Slip-angle estimation for stability control. Vehicle Systems Dy-namics, 32:375–388, 1999.

[6] Havard F. Grip, Lars Imsland, Tor A. Johansen, Thor I. Fossen, Jens C. Kalkkuhl,and Avshalom Suissa. Nonlinear vehicle velocity observer with road-tire frictionadaptation. Submitted to CDC 2006.

[7] Marcus Hiemer, Anne von Vietinghoff, Uwe Kiencke, and Takanori Matsunaga.Determination of vehicle body slip angle with non-linear observer strategies. InProceedings of the SAE World Congress, 2005. Paper no. 2005-01-0400.

[8] L. Imsland, T. A. Johansen, T. I. Fossen, J. Kalkkuhl, and A. Suissa. Vehiclevelocity estimation using modular nonlinear observers. In Proc. 44th IEEE Conf.Decision Contr., 2005.

[9] L. Imsland, T. A. Johansen, T. I. Fossen, J. Kalkkuhl, and A. Suissa. Nonlinearobserver for vehicle velocity estimation. In Proc. SAE World Congress, 2006.

16

[10] Hassan K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 3rdedition, 2002.

[11] Uwe Kiencke and Armin Daiss. Observation of lateral vehicle dynamics. ControlEngineering Practice, 5(8):1145–1150, 1997.

[12] Uwe Kiencke and Lars Nielsen. Automotive Control Systems. Springer, 2000.

[13] Chia-Shang Liu and Huei Peng. Inverse-dynamics based state and disturbanceobservers for linear time-invariant systems. Journal of Dynamic Systems, Measure-ment, and Control, 124:375, 2002.

[14] Antonio Lorıa, Elena Panteley, Dobrivoje Popovic, and Andrew R. Teel. A nestedMatrosov theorem and persistency of excitation for uniform convergence in stablenonautonomous systems. IEEE Trans. Automat. Contr., 50(2):183–198, 2005.

[15] Laura R. Ray. Nonlinear state and tire force estimation for advanced vehicle control.IEEE Transactions on Control Systems Technology, 3(1):117–124, 1995.

[16] Laura R. Ray. Nonlinear tire force estimation and road friction identification: sim-ulation and experiments. Automatica, 33(10):1819–1833, 1997.

[17] A. Suissa, Z. Zomotor, and F. Bottiger. Method for determining variables charac-terizing vehicle handling. Patent US 5557520, 1996.

[18] H.E. Tseng. Dynamic estimation of the road bank angle. Vehicle Systems Dynamics,36(4–5):307–328, 2001.

[19] Ali Y. Ungoren and Huei Peng. A study on lateral speed estimation methods. Int.J. Vehicle Autonomous Systems, 2(1/2):126–144, 2004.

[20] Anton T. van Zanten. Bosch ESP system: 5 years of experience. In In Proceedingsof the Automotive Dynamics & Stability Conference (P-354), 2000. Paper no. 2000-01-1633.

[21] Paul J. Th. Venhovens and Karl Naab. Vehicle dynamics estimation using Kalmanfilters. Vehicle System Dynamics, 32:171–184, 1999.

17

Improved road grade estimation using sensor fusion

Henrik Jansson, Ermin Kozica, Per Sahlholm and Karl Henrik Johansson

Abstract— A method for estimation of the road grade basedon standard mounted sensors in a heavy duty vehicle ispresented. The method combines information from a barometerand a GPS with velocity and torque measurements of conven-tional type. The sensor information is adaptively integratedusing extended Kalman filtering. This provides a systematicmethod for dealing with varying uncertainty in the sensors.The method can handle periods of missing or unreliable datafrom one or several sensors, e.g., occasions when the satellitecoverage is low or when the brakes are applied.

I. INTRODUCTION

Embedded control systems have become standard in to-day’s cars and heavy duty vehicles (HDV). Several of theimplemented algorithms in these embedded systems arebased on state information of the vehicle. As a consequence,the need for reliable estimation of state parameters thatinfluence the performance of the vehicle has increased. Themass and the road grade are two parameters that largelyinfluence a vehicle’s performance. This is particularly truefor heavy duty vehicles where the loadings due the massand the grade can be significant. Many of today’s systems forengine control, transmission control and brake managementthus include algorithms to estimate the mass and the drivingresistance. Beside traditional applications, new advanceddriver assistance systems are under development where thefuture dynamics of the vehicle is predicted. These predictionscombine the current state of the vehicle with informationabout the road ahead, e.g., the topology, the curvature or thetraffic situation. There are several contributions in this arearelated to advanced cruise control systems where knowledgeabout the road profile ahead of the vehicle is used to varythe reference speed in order to reduce the fuel consumption.Some examples are [1], [2] and [3]. The desire for good pre-dictions will increase the quality demands on the parameterestimates further. Reliable sensors for the solely purpose ofestimating one parameter are often too costly, which furtherincrease the demands on developing methods for parameterestimation.

In this contribution we will focus on estimation of theroad grade based on standard mounted sensors in a HDV. Aprincipal sketch of the longitudinal forces acting on a HDVis shown in Figure 1. Fengine is the pull force produced bythe engine, Fbrake is the applied brake force, Fdrag is theair drag, Froll is the rolling resistance and Fgravity is the

H. Jansson and K.H. Johansson are with the Automatic Control Lab,School of Electrical Engineering, Royal Institute of Technology (KTH),SE-100 44 Stockholm, Sweden. [email protected]

P. Sahlholm and E. Kozica are with Scania CV, SE-151 87 Södertalje,Sweden

gravity induced force given by

Fgravity = mg sinα

where m is the mass and α is the road grade. Using Newton’slaw of motion the force balance for the HDV in Figure 1 isgiven by

mv = Fengine − Fbrake − Fdrag − Froll − mg sin α (1)

where v is the velocity. This equation illustrates the couplingof the mass and the grade. Besides the coupling of theseparameters, their time-variance make the estimation morecomplicated, e.g., the mass of a HDV can vary more than400% depending on the load it carries. Before we proceedto describe our contribution to road grade estimation we willbriefly describe some related work.

A. Related work

To deal with the coupling of the mass and the grade ithas been suggested to directly or indirectly measure thegrade. In [4] the grade is determined using a GPS receiverby calculating the ratio of the vehicle’s vertical velocity toits horizontal velocity. A GPS receiver need good satellitecoverage to obtain decent estimates. This is, however, aconstraint that is difficult to sustain in areas with buildings,trees, tunnels or other large objects. There are also othertypes of disturbances such as multi-pathing that may corruptthe GPS readings leading to deteriorated estimates of thegrade.

In [5] a barometer is used instead of a GPS to obtainelevation readings which together with the vehicle’s velocityyields an estimate of the grade. A barometer or a pressuresensor reacts on changes in the surrounding air pressure.For altitude measurements fast changes in the pressure areassumed to correspond to changes in the vertical positionof the sensor. However, the sensor is exposed to all kind ofchanges in air pressure, not only those induced by verticalmovements. This will lead to disturbances in the elevationreadings. Furthermore, a quite high resolution is required toobtain good estimates of the grade. Most of the barometersthat are mounted in vehicles today have a low resolution.It is difficult to motivate replacing them by more expensivesensors with better resolution and accuracy.

An alternative way of determining the grade is by using anaccelerometer. Such a solution is presented in [6] where thelongitudinal acceleration is measured using an accelerometer.The gravity is then determined by subtracting the drivingforce from the measured value by the accelerometer. Withknown mass it is then possible to compute an estimate ofthe grade. In [6] the driving resistance due to air drag and

Froll

FgravityFdrag

Fbrake Fengineα

Fig. 1. Longitudinal forces acting on the vehicle.

rolling friction has been neglected. It is, however, easy toinclude these terms to increase the accuracy of the gradeestimate. With an accelerometer it is possible to estimate thegrade when the vehicle is standing still. Most other methodsrequire the vehicle to move. A major drawback is that thissolution requires an extra sensor. The sensor is also sensitiveto vibrations and other disturbances induced in the chassis.

Two methods that do simultaneous estimation of mass andgrade are presented in [7] and [8]. In [7] the road gradeis estimated using a Kalman filter based on measured orestimated propulsion force, estimated retardation forces andmeasured velocity. In extension, the mass is simultaneouslyestimated with the grade using an extended Kalman filter.In [8] the grade and mass is simultaneously estimated usingRecursive Least Squares (RLS) based on a simple motionmodel. The big advantage with these methods is that noextra sensors are required. Furthermore, the estimation doesnot require to be performed during certain occasions suchas gearshifts. There are however certain occasions whenthese two methods fail, or have big difficulties, to deliverreliable estimates. This is the case when the friction brakesare applied or when gearshifts are performed. The brakeforce is difficult to measure or estimate and oscillations areintroduced in the driveline during gearshifts that are notcaptured by the motion models.

B. Contribution

We will in this paper present a method that estimates theroad grade. A major contribution is that the method combinesinformation from several standard mounted sensors. Anoverview of the filter architecture is shown in Figure 2.All measured signals are assumed to be available on thevehicle’e data bus. The information from the measurementsis integrated adaptively using extended Kalman filtering withtime-varying covariances. This provides a systematic methodfor dealing with varying uncertainty in the sensors. Themethod can handle periods of missing or unreliable datafrom one or several sensors, e.g., occasions when the satellitecoverage is low or when the brakes are applied. Thus, thecombination of sensors leads to redundancy.

For the grade estimation we primarily measure barometer-pressure, GPS-velocity, GPS-altitude, vehicle-velocity andvehicle-torque. The vehicle-velocity can be obtained fromthe tachograph or the ABS-system. The vehicle-torque thatwill give a certain propulsion force, can be estimated basedon throttle position together with the engine speed. A barom-

eter is standard in many vehicles with injection engines.The barometer measures the ambient air pressure and theinformation is used to adjust the engine inlet. These barom-eters have typically a resolution that is too low for gradeestimation. But the barometer is useful together with othersensors as will be illustrated. GPS-receivers are becomingmore and more common. They are primarily utilized fornavigation or tracking applications. In the future they willbe used for other applications as well, e.g., applications thatutilizes information from maps for vehicle control, see [1].

The algorithm also use other signals on the data bus.They are primarily used for detection of certain occassionssuch as loss of satellite connection, applied friction brakesor gearshifts. This information is used to weight the sensorinformation depending on their reliability.

C. Outline

The method for road grade estimation using sensor fusionis described in Section II. The sensor fusion is based on ex-tended Kalman filtering that is a model-based method. Usedmodels are presented in Section II-A. The measurements andtheir relation to the estimated states are given in Section II-B. Extended Kalman filtering is reviewed in Section II-C.Section II-D shows results where the road grade estimationalgorithm has been applied on data collected during a driveon a Swedish highway.

In Section III it is further shown how the algorithm can beextended to include estimation and tuning of certain vehicleparameters such as the mass, the absolute velocity and therolling resistance. The paper is concluded in Section IV.

II. ESTIMATION OF ROAD GRADE

In this section we introduce the method for road grade esti-mation using sensor fusion based on measurements availableon the vehicle’s data bus.

A. Model

Some basic relations that describes the connection betweenthe estimated states are necessary for the model-based sensorfusion. The model consists of three parts. The first partidescribes the longitudinal dynamics of the vehicle. Thesecond part describes the topology of the road by relatingthe altitude, with the grade and the speed of the vehicle.The third part describes the relation between the ambient airpressure and differences in altitude.

The dynamics in the longitudinal direction can be de-scribed by

mv = D(M, rw) + R(v, rw) + G(α, rw) (2)

where rw corresponds to a general wheel radius. D is thepropulsion force, which is a function of the torque Mproduced by the engine. R denotes the resistive forces suchas air drag and rolling resistance and G is the road load dueto gravity. This is a more general description of the drivelinethan (1). For specific examples of D, R and G we refer to[9]. A requirement is that D, R and G are differentiable withrespect to chosen states.

Extended Kalman Filter

sats (GPS)

EMS GMS GPSEmbedded systems

BMSABS TCO EMS GMS GPSEmbedded systems

BMSABS TCO

shifting (GMS)

v (ABS,TCO)M (EMS)v (GPS)z (GPS)p (EMS)

R

noisecovariances

QK

gain

Kgain

braking (BMS)

data

bus

gear (GMS)

state update

α

xk = Fkxk−1 + Kk(yk − h(xk−1))

Fig. 2. Overview of filter architecture for road grade estimation. Informa-tion from the vehicle’s data bus is integrated using an extended Kalmanfilter. Information is delivered to the data bus from onboard embeddedsystems such as the ABS, the tachograph (TCO), engine-, gear- and brakemanagement systems (EMS,GMS,BMS) and a GPS-receiver. Consideredsignals are velocity (v), altitude (z), engine torque (M), ambient air pressure(p), number of available satellites for the GPS (sats), current gear (gear) andboolean variables indicating shift and brake in process (shifting,braking).The grade estimate is denoted α.

The torque M represents the engine produced torqueminus the internal friction in the driveline. It is possible toconsider the torque M as a state. This increase the possibili-ties to handle uncertainties in the generated propulsion force.The alternative is to consider M as a known variable. Whenthe torque M is modelled we will use the relation

M = 0. (3)

To describe the topology of the road we will use two states,the altitude z and the grade α. The dynamics for these twostates are

z = v sin α

α = 0.(4)

An alternative is to consider the dynamics of the grade as afirst-order process as in [7].

Finally, two states are introduced to describe changes inair pressure. The first state pz represents the part of theair pressure that changes due to elevation changes and thesecond state pw represents the bias that is primarily due tothe weather conditions. Their dynamics are described by

pz = A(z)pw = 0

(5)

where A is a function that describes changes in air pressurebased on an incremental change in altitude.

By combining (2)-(5) together with a first-order Euler ap-proximation a discrete-time state-space model with samplingtime Ts is obtained as⎡⎢⎢⎢⎢⎢⎢⎣

vk

Mk

αk

zk

pz,k

pw,k

⎤⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸xk

=

⎡⎢⎢⎢⎢⎢⎢⎣

vk−1 + Ts

m ∆vk

Mk−1

αk−1

zk−1 + Tsvk−1 sin αk−1

pz,k−1 + TsAd(zk−1, vk−1, αk−1)pw,k−1

⎤⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸f(xk−1)

+

⎡⎢⎢⎢⎢⎢⎢⎣

wv,k

wM,k

wα,k

wz,k

wpz,k

wpw,k

⎤⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸wk

(6)

where to the deterministic part also process noise wk hasbeen added. The process noise is used to describe the uncer-tainty in the model. In (6) Ad(·) is the discrete counterpartof A(·). Furthermore, ∆vk is given by

∆vk = D(Mk−1, rw) + R(vk−1, rw) + G(αk−1, rw). (7)

Subscript k denotes the discrete time instant.

B. Measurements

The relation between the measurements and the states isdescribed by

yk =

⎡⎢⎢⎢⎢⎣

vveh,k

vGPS,k

Meng,k

zGPS,k

pk

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

vk

vk

Mk

zk

pz,k + pw,k

⎤⎥⎥⎥⎥⎦

︸︷︷︸h(xk)

+

⎡⎢⎢⎢⎢⎣

evveh,k

evGPS,k

eM,k

ezGPS,k

ep,k

⎤⎥⎥⎥⎥⎦

︸︷︷︸ek

(8)

where ek represents stochastic measurement noise. Here vveh

is the measured vehicle-velocity from either the tachographor the wheel speed sensors. The velocity and the altitudefrom the GPS are denoted vGPS and zGPS , respectively.The propulsion force is given by Meng and the barometer-pressure by p.

C. Estimation

The measurements and the state update are thus describedby the state-space system

xk = f(xk−1) + wk

yk = h(xk) + ek.(9)

Extended Kalman filtering can be applied to estimate thestate vector xk based on the measurements yk when the noisesources wk and ek are considered to be zero-mean Gaussiannoise. In extended Kalman filtering the non-linear system islinearized and the standard recursions for Kalman filteringare applied on the linearized system. These recursions aredescribed by two update steps: a time update and a mea-surement update. In the first step, the time update, the stateestimate xk−1 and the error covariance Pk−1 are updatedaccording to

Fk =∂f

∂x(xk−1)

xk = Fkxk−1

Pk = FkPk−1FTk + Qk

(10)

Fig. 3. Scania R420 test vehicle.

where Qk is the covariance of the process noise wk. Thesecond step is a measurement update where the estimate iscorrected based on the measurements according to

Hk =∂h

∂x(xk)

Kk = PkHTk

(HkPkHT

k + Rk

)−1

xk = xk + Kk (yk − h(xk))Pk = Pk + KkHkPk

(11)

where Rk is the covariance of the process noise ek.Hence extended Kalman filtering provides a systematic

method for sensor fusion that includes disturbance attenua-tion. The covariances Qk and Rk are tuning variables thatcan be adjusted to describe the reliability of different partsof the system or the measurement equations. Furthermorethe reliability changes typically over time and it is thereforevery useful to use time-varying covariances. A high relativevariance of some states or measured signals will lead to adecreased influence of them on the state estimate.

D. Results

In this section we will present results where extendedKalman filtering as described in Section II-C has beenapplied on data collected from the CAN bus of a Scaniatractor R420, see Figure 3. A batch of measured data duringa drive on a typical Swedish highway is shown in Figure 4.The objective is to estimate the road profile.

The system equations in (6) have been used with (7)replaced by

∆vk = c1(c2Mk−1 − c3v2k−1 − c4 − c5 sin(αk−1)),

where c1, . . . , c5 are vehicle parameters. The altitude-causedpressure change is given by

A(∆zk) = µ(pb,k + pz,k)vk sin αk.

This represents a linearization of the standard altitude pres-sure function.

The implementation is now straightforward, with the ex-ception of choosing noise covariances. As described earlier,the covariance matrices Qk and Rk are chosen to give desiredfilter behaviour. To simplify the design we have assumedthat the covariance matrices are diagonal. To handle certain

2000 4000 6000 8000 10000 12000 14000

30

40

50

60

70

Alti

tude

[m]

2000 4000 6000 8000 10000 12000 140000

20

40

60

80

Vel

ocity

[km

/h]

Tor

que

[%]

2000 4000 6000 8000 10000 12000 140004

6

8

10

12

G

ear

S

atel

lites

Position [m]

Fig. 4. Batch of measurements from the CAN bus during a test driveon the Swedish highway E20 between Strängnäs and Kjula. The top plotshows the GPS-altitude (solid) and the barometer-pressure (dashed) scaledto a corresponding altitude. The middle plot shows the GPS-velocity (solid)and the vehicle-torque (dashed). The bottom plot shows the current gear(thick solid) and the number of available satellites.

events, e.g., braking (which change the system characteristicsin (6) and (8)), the covariance matrices are made dependentof such events. For example, the quality of the measurementsfrom the GPS-receiver, especially the altitude, is highlydependent on the number of available satellites. Thus thesize of the variance of ezGPS,k can be varied according tothe current number of available satellites, e.g., being inverseproportional to this number. In this way, the state estimatewill not rely on corrupt GPS measurements during periodswith low satellite coverage. The state estimate will insteadbe based on the system equations blended with informationfrom the other sensors. A similar reasoning can be appliedfor other situations. One example is when the friction brakesare applied. It is difficult to estimate the brake force thatacts on the vehicle. As a consequence, the equation forthe longitudinal dynamics (2) becomes uncertain. A way tohandle this is to increase the process noise wv,k, wheneverthe friction brakes are applied. The same holds for gearshifts.During gearshifts, the produced torque in the driveline aredifficult to model. Oscillations can then be introduced thatare not included in (2).

We will now show estimation results for two drivingscenarios, steady driving and gear shifting.

1) Steady driving: When driving at a steady pace with allsensors active, Qk and Rk are chosen so that road altitudeand grade vary smoothly. Estimation on the Swedish roadE20 from Strängnäs to Kjula gives the results depicted inFigure 5 with all sensors available.

At least four satellites are needed for the GPS-altitude tobe reliable. When there are less than four satellites available,the noise variance of teh measured GPS-altitude is increasedto a large value which removes the dependence of the GPS-altitude in the estimation. Figure 6 depicts the result without

2000 4000 6000 8000 10000 12000 14000

30

40

50

60

70

Alti

tude

[m]

2000 4000 6000 8000 10000 12000 14000−4

−3

−2

−1

0

1

2

3

Gra

de [%

]

Fig. 5. Estimation of the road profile of the Swedish highway E20between Strängnäs and Kjula using the sensor fusion algorithm describedin Sections II-A–II-C with all sensors enabled. Thin solid line correspondsto measured altitude by the GPS. Thick solid lines are estimated altitudeand grade.

2000 4000 6000 8000 10000 12000 14000

30

40

50

60

70

Alti

tude

[m]

2000 4000 6000 8000 10000 12000 14000−4

−3

−2

−1

0

1

2

3

Gra

de [%

]

Position [m]

Fig. 6. Dashed lines are estimated altitude and grade with no GPS dataavailable, simulating loss of satellite connection. Barometer data is relied onto a greater extent giving a more quantized estimation of road altitude. Theresult is compared with the estimates obtained with all sensors are enabled.

GPS data.2) Gear shifting and braking: No torque is delivered to

the wheels when shifting gears. This event is handled bysimply setting a high variance on wvveh. Furthermore, thevariance of the road grade is reduced to assure that it doesnot change too much while shifting. When a gear is in placeagain, process variance is restored to its value before theshift. Braking adds another torque to the velocity evolutionequation, and is hard to estimate correctly. That is why braketorque has not been modelled and why the equation for thelongitudinal dynamics is not valid when braking. Hence, allbraking is handled exactly the same way as shifting gears.Figure 7 compares the result when all sensors are enabledcompared to the case with both the barometer and the GPSdisabled. The grade is in this case underestimated during the

1.65 1.7 1.75 1.8 1.85

x 104

35

40

45

50

55

60

Alti

tude

[m]

1.65 1.7 1.75 1.8 1.85

x 104

−2

0

2

Gra

de [%

]

1.65 1.7 1.75 1.8 1.85

x 104

10

10.5

11

11.5

12

12.5

Gea

r

Fig. 7. Grade estimation during gearshift is illustrated. Thick solid linesin the two upper plots corresponds to altitude and grade estimates withall sensors enabled. This is compared with the grade estimate when thebarometer and the GPS are disabled (dotted line). Thin solid line in theupper plot is GPS-altitude and dashel line in the lower plot indicates thegear.

gear shift when the barometer and the GPS are disabled.

III. TUNING OF VEHICLE PARAMETERS

There are several parameters in the system descriptionthat are uncertain and time-varying. Here we will discussthree such parameters, the rolling resistance, the mass andthe wheel radius.

A. Tuning of velocity

Another extension of the framework is to include a cor-rection of the wheel radius rw. This parameter is used in theequation for the longitudinal dynamics (2) to translate theproduced torque into a longitudinal force. The longitudinaldynamics is relatively sensitive to changes in the wheelradius. There is typically an offset in rw compared to itsnominal value due to load and wear combined with possiblechanges in tire pressure. In [7] and [8] the wheel radius isassumed to be constant. One reason for this is that theirchoice of sensors restricts their possibilities to include acorrection. In our system, velocity measurements from aGPS-receiver are occasionally available. Together, with thevelocity measurements from the vehicle it is possible to applya recursive algorithm to estimate the offset in the wheelradius.

For estimation of the offset we assume that the wheelradius rw consists of a constant nominal radius rnom anda time-varying offset δ according to

rw = rnom + δ.

The dynamics of the offset can in discrete time with processnoise included be described as

δk = δk−1 + wδ,k−1. (12)

2000 4000 6000 8000 10000 12000 14000

−40

−20

0

20

40

60

Alti

tude

[m]

2000 4000 6000 8000 10000 12000 14000−4

−3

−2

−1

0

1

2

3

Gra

de [%

]

Position [m]

Fig. 8. Dotted lines are estimated altitude and grade with both the GPSand the barometer disabled. There are significant errors in the vehicle modelleading to deteriorated estimates. The result is compared with the estimatesobtained with all sensors are enabled.

The measurement equation for the vehicle velocity can thenafter linearization aorund v = v and δ = 0 be rewritten as

vveh,k = vk + vδk. (13)

The offset can then be estimated using extended Kalmanfiltering by including (12) and (13) in (6) and (8). An alter-native is to estimate this offset separately using a recursivemethod such as Kalman filtering or Recursive Least Squaresbased on velocity measurements from the vehicle and theGPS. Correcting the offset in the wheel radius will improveaccuracy of the equation for the longitudinal dynamicsespecially during periods when velocity information fromthe GPS is not available.

B. Tuning of rolling resistance

One can argue that the onboard barometer does not pro-vide much information about road grade since it outputs aquantized signal with a very low resolution. The resolutionfor the used barometer is 4.2 meters/bit. Figure 8 shows theestimation result when both GPS and barometer data are ne-glected. Observe the trend in the estimated altitude comparedto the case where barometer but not GPS data is used, seeFigure 6. One explanation for this is that the coefficient forthe rolling resistance is too low. The coefficient depends onthe vehicle configuration. The rolling resistance is differentbetween a tractor alone or if there is a tractor and trailercombination. As can be seen in Figure 8, the estimationalgorithm is vulnerable with respect to errors in the vehiclemodel when both GPS and barometer data are neglected.This also shows the redundancy that the sensor fusion gives,where errors in the vehicle model are compensated by theinformation retrieved from the GPS and the barometer, cf.,the results in Figure 5 and Figure 6. This illustrates one of themajor strengths with the algorithm. The result in Figure 8 isnot satisfactory. The algorithm should produce more accurateresults during periods of unreliable GPS and barometer data.To overcome the problem with inaccuracies in the vehicle

2000 4000 6000 8000 10000 12000 14000 1600020

30

40

50

60

70

Alti

tude

[m]

2000 4000 6000 8000 10000 12000 14000−4

−3

−2

−1

0

1

2

3

Slo

pe [%

]

Position [m]

Fig. 9. Dotted lines are estimated altitude and grade with both the GPSand the barometer disabled. Here the rolling resistance has been tuned. Theresult is compared with the estimates obtained with all sensors are enabled.

model, periods with all sensors available should be utilized totune certain important parameters. Figure 9 shows the resultwhen the rolling resistance has been adjusted based on acomparison between the case with all sensors enabled andthe case where the barometer and the GPS are disabled.

C. Simultaneous estimation of mass and grade

In this paper we have assumed that the mass is known. Itis straightforward to include estimation of the mass in theframework presented in Section II. The only difference isthat the state-space model (6) needs to be enlarged with oneadditional state m corresponding to the vehicle mass. Withthe assumption that m = 0 we obtain the time discrete modelfor the mass as

mk = mk−1 + wm,k (14)

Once the covariance Qk has been modified to include thevariance of wm,k, it is easy to apply the recursions (10)–(11) based on the system description (14) together with themeasurements given by (8) to obtain an estimate of the roadgrade as well as the mass. Some care has to be taken sincegood excitation is necessary to obtain good simultaneousestimates.

IV. CONCLUSIONS

We have demonstrated a method for estimation of theroad grade based on sensor fusion using extended Kalmanfiltering. The method combines information from a barom-eter and a GPS with velocity and torque measurements ofconventional type. The sensor fusion provides redundancy,e.g., the method is able to deliver reliable estimates when thefriction brakes are applied as long as the satellite coverageis good. Furthermore, information about the vehicle velocityand the propulsion force can be used to compensate for errorsin the GPS or alternatively in the barometer readings. Thecorrectness of the estimated grades are hard to establish sincethe true road grade is not known. For tuning of the algorithm

and the covariances a reference road with known profilecould be very useful. This is an issue for future research.

In the end of the paper it is briefly illustrated how thealgorithm can be extended to include estimation and tuningof certain vehicle parameters such as the mass, the absolutevelocity and the rolling resistance. All these parameters areused in the model-based filtering and plays therefore animportant for the quality of the grade estimates. To exploitthese ideas further is a topic for future work.

V. ACKNOWLEDGMENTS

The work was partially supported by Scania CV, theSwedish Program Council for Vehicle Research and byEuropean Commission through the Network of ExcellenceHYCON.

REFERENCES

[1] F. Lattemann, K. Neiss, S. Terwen, and T. Connolly, “The predictivecruise controlUa system to reduce fuel consumption of heavy dutytrucks,” ser. SAE Technical Paper Series, 2004.

[2] E. Hellström, A. Fröberg, and L. Nielsen, “A real-time fuel-optimalcruise controller for heavy trucks using road topography information,”ser. SAE World Congress 2006, no. 2006-01-0008, 2006.

[3] A. Fröberg, E. Hellström, and L. Nielsen, “Explicit fuel optimal speedprofiles for heavy trucks on a set of topographic road profiles,” ser.SAE World Congress 2006, no. 2006-01-1071, 2006.

[4] H. S. Bae, J. Ruy, and J. Gerdes, “Road grade and vehicle parameterestimation for longitudinal control using GPS.” in Proceedings ofthe 4th IEEE Conference on Intelligent Transportation Systems, SanFrancisco, CA, 2001.

[5] M. Panzer, “Verfahren und verrichtung zur ermittlung der fahrzeug-masse.” European Patent Application EP1387153A1, 2003.

[6] H. Ohnishi, J. Ishii, M. Kayano., and H. Katayama, “A study on roadslope estimation for automatic transmission control,” JSAE Review,vol. 21, pp. 322–327, 2000.

[7] P. Lingman and B. Schmidtbauer, “Road slope and vehicle massestimation using Kalman filtering.” in Proceedings of the 19th IAVSDSymposium, Copenhagen, Denmark, 2001.

[8] A. Vahidi, A. Stefanopolou, and H. Peng, “Recursive least squares withforgetting for online estimation of vehicle mass and road grade: Theoryand experiments,” Journal of Vehicle System Dynamics, vol. 43, pp. 31–57, 2005.

[9] U. Kiencke and L. Nielsen, Automotive Control Systems. SpringerVerlag, Berlin, 2003.

Development of a controller to perform anautomatic lateral emergency collision

avoidance manoeuvre for a passenger car

Geraint Bevan, Henrik Gollee and John O’ReillyCentre for Systems and Control

University of Glasgow, GLASGOW. G12 8QQ Scotland.

+44 (0)141 330 T: 4723 F: 4343 [email protected]

March 21, 2006

AbstractAn automatic controller is being developed to cause a passenger car to performa lateral emergency collision avoidance manoeuvre: a single lane change at highspeed, while operating at the vehicle’s physical limits.

The car is stabilised about a predetermined velocity profile by a feedforwardsteering action based on the Ackermann steering angle, combined with a feedbackcontrol loop which uses the anti-lock braking system to apply differential torquesto each of the wheels. The forces to be applied to each wheel are calculated usingthe pseudo-inverse of a velocity-based linearisation of a system model.

This inverse controller acts upon a signal containing the lateral and yaw veloc-ity error, fed back through a scheduled gain matrix; the matrix, obtained by poleplacement and scheduled according to the vehicle state, causes the car to exhibituniform dynamic behaviour as its speed increases.

An additional control loop augments the steering angle of the front wheels,using actuators which form part of a steer-by-wire system, to correct for errorsin the lateral position and heading angle for which the force/velocity control loopdoes not account.

The control system is evaluated in simulation experiments which show thatthe performance requirements are met over a wide range of velocities.

1

IntroductionAs roads become busier and technology improves, there is a growing potentialfor driver assistance systems to improve the safety of road users. It is becomingincreasingly common for luxury cars to be fitted with longitudinal collision avoid-ance systems, where cruise control functions are integrated with forward lookingobstacle detection sensors to automatically slow the car when necessary.

Such devices can be a valuable aid if an impending rear-end collision betweencars travelling in the same lane is due to driver inattention and the vehicles areamply separated in space and time for the aft vehicle to brake. They are, however,of limited benefit for preventing head-on collisions or avoiding obstacles whichappear suddenly in front of a moving vehicle. In these circumstances, aggressivelateral manoeuvres are more appropriate; as well as altering the path of the vehicleto move it out of danger, the manoeuvre can be completed in a shorter distancethan that required to bring the vehicle to a stop.

The availability of steer-by-wire augmentation and electronic stability pro-grammes can enable such manoeuvres to be initiated and performed under theguidance of a vehicle management computer. One of the aims of the CEMACSproject is to produce such a controller which will cause a car to change laneswhile operating at its physical limits. There are clearly many technologies thatmust come together to bring such a system to fruition on production vehicles -particularly with regard to situational awareness - as well as legal issues that mustbe considered, but these issues are beyond the scope of the CEMACS project.

Vehicle design modelThe development of an appropriate linear model, or family of models, is an im-portant part of conventional controller design. During an aggressive lane changemaneouvre the vehicle is expected to operate far from equilibrium conditions andwith potentially large and fast changing control inputs.

Non-linear modelA two track non-linear model of a vehicle was developed, with three vehicle bodystates and five control inputs. This non-linear model is of the form x = F(x,u)where the state and input vectors are defined as x = (X,Y, Ψ)T and u = (fX , δ)T .The state vector x comprises of the vehicle longitudinal position, X; lateral po-sition, Y ; and yaw angle, Ψ. The input vector u comprises of the longitudinalforces on each of the four wheels, fX ; and the steering angle of the front wheels,δ.

2

Velocity based linearisationThe velocity based method of Leith and Leithead [1998] enables linearisations ofmodels to be obtained under non-equilibrium conditions and is not constrained bythe restriction to small inputs that can arises from conventional small perturba-tion linearisation. The suitability of the method has been demonstrated for othervehicle dynamics problems which demand high performance [Leith et al., 2001].

Differentiating the non-linear model with respect to time t results in a familyof linear models defined by a scheduling vector ρ = (x,u)T :

x = A(ρ)x + B(ρ)u

where A(ρ) = ∂F∂x

and B(ρ) = ∂F∂u

.For this particular model, the longitudinal wheel forces do not appear in the

state and input matrices of the linearised model so the scheduling parameter canbe reduced to ρ = (x, δ)T .

ImplementationThe non-linear model was implemented in C++ and wrappers were written toenable it to be dynamically loaded and called from the Matlab and GNU Octavematrix algebra tools. Functions were created to calculate the velocity-linearisedstate and input matrices for any operating condition. The A and B matrices cantherefore be used for linear controller design, while the non-linear model can beused to perform a basic evaluation of performance by simulation.

Controller architectureThere are three distinct elements to the controller architecture shown in figure 1.

The simplest is a feedforward controller D(xref ) which assigns a nominalsteering angle δ0 on the basis of a predetermined profile, scheduled according tothe longitudinal position of the vehicle relative to its reference trajectory. Thisreference steering profile is derived from the Ackermann steering angle, i.e. thegeometrically-derived angle which would cause the vehicle to follow the pathin the absence of slip. However, there is nothing particularly special about thisprofile and any suitable alternative could be substituted without impacting thecontroller architecture.

Two parallel loops provide stabilisation about the reference trajectory. A ve-locity loop, shown in blue, controls the vehicle lateral and yaw velocity (Y and Ψ)by altering the longitudinal tyre forces, using the ABS slip controller. A positionloop, shown in red, controls the lateral position of the vehicle by augmenting thesteering angle of the front wheels with an additional steering input ∆.

3

Force/Velocity loopThe force/velocity loop consists of a scheduled gain matrix L(ρ) and a pseudo-inverse B

−1

f of that part of the linearised input matrix that relates to vehicle forces.L is designed to give the vehicle uniform dynamic behaviour as its operating

condition changes. The open-loop poles of the plant were identified from themodel when the car was driven at low speed. Pole placement was then used toforce the poles to remain at these locations as the velocity changes. The behaviourof the closed loop is approximately

θ

θref

≈1

s (s + 0.015) (s + 0.03)

where θ = (Y , Ψ)T .It is not necessary to explicitly control the longitudinal velocity of the vehicle

and attempting to do so would reduce the available traction which can be used forproviding lateral acceleration. The reference longitudinal velocity is therefore setto be equal to the measured velocity, effectively setting the error in this signal tozero.

Steering/Position loopThe feedforward Cf and feedback Cb controllers of the position loop eliminatethe steady state error that would result if the velocity loop existed in isolation.They were designed by direct synthesis, using an approximation to the vehicle’sdynamic behaviour. The approximation was obtained by averaging the frequencyresponse of the car over a range of conditions. The closed loop specification waschosen to be of the form

Y

Yref

=K ω2

τ(

s + 1

τ

)

(s2 + 2ζωs + ω2)

Sensitivity and robustnessThe controller L does not attempt to control the forward velocity X so only thesensitivity and robustness of the channels with lateral and yaw velocity outputsneed be considered.

4

Figure 1: Controller architecture: A feedforward controller (D) provides a pre-determined nominal steering angle to coerce the car to follow its target trajectory.A lateral position control loop augments the nominal steering angle using feed-forward and feedback controllers Cf and Cb. In parallel, a force/velocity loopconsisting of a scheduled gain matrix L and pseudo-inverse B−1

f of the linearisedinput matrix stabilises the car about its reference velocity profile and providesuniform dynamics as the velocity changes.

Disturbance rejectionSensitivity functions S(iω) for the channels with lateral and yaw velocity out-puts are shown in figure 2. It can be seen that both of these outputs are entirelyinsensitive to disturbances in the vehicle longitudinal velocity at all frequencies.

It should be noted here that the nominal plant changes with the speed of thevehicle and that this does not therefore provide any guarantee that the model willperform well at velocities far removed from 80 kph but does provide some confi-dence that the system will cope reasonably well with minor speed changes duringthe course of the manoeuvre.

The lateral velocity ceases to reject disturbances to itself at fairly low fre-quencies. This channel is seen to have a bandwidth varying from approximately5 × 10−3 radians per second with the wheels pointing straight ahead, increasing

5

10−5 10−4 10−3 10−2 10−1 100−450

−440

−430

−420

−410

−400

−390

−380

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(a) Y

wX

10−5 10−4 10−3 10−2 10−1 100−35

−30

−25

−20

−15

−10

−5

0

5

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(b) Y

wY

10−5 10−4 10−3 10−2 10−1 100−120

−100

−80

−60

−40

−20

0

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(c) Y

wΨ

10−4 10−3 10−2 10−1 100−450

−440

−430

−420

−410

−400

−390

−380

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(d) Ψw

X

10−4 10−3 10−2 10−1 100−90

−80

−70

−60

−50

−40

−30

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(e) Ψw

Y

10−5 10−4 10−3 10−2 10−1 100−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(f) ΨwΨ

Figure 2: Sensitivity functions, showing the response of the lateral and yaw ve-locities of the nominal system to disturbances in the states at a forward speed of80 kph for four steering angles δ.

by an order of magnitude to 5 × 10−2 radians per second as the front wheels areturned to a steering angle of π

4radians.

Disturbances in the yaw velocity are rejected by the lateral velocity at all fre-quencies, with very low sensitivity at frequencies higher than approximately 10−3

radians per second for all steering angles.The yaw velocity rejects disturbances on all input channels, with little sensi-

tivity to longitudinal and lateral velocity disturbances and very low sensitivity toyaw rate disturbances above 10−2 radians per second

Sensitivity to model errorsAssuming that the uncertainty in the vehicle model can be characterised as G =G (I + ∆G) where G is the nominal model and ∆G the model uncertainty at theoutput, the system will be robust to model uncertainties if the complementarysensitivity T and uncertainty satisfy |T (iω)∆G(iω)| < 1 ∀ω.

Complementary sensitivity functions are shown for the in figure 3 for the chan-

6

10−6 10−5 10−4 10−3 10−2 10−1 100−450

−440

−430

−420

−410

−400

−390

−380

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(a) Y

nX

10−6 10−5 10−4 10−3 10−2 10−1 100−180

−160

−140

−120

−100

−80

−60

−40

−20

0

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(b) Y

nY

10−6 10−5 10−4 10−3 10−2 10−1 100−120

−100

−80

−60

−40

−20

0

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(c) Y

nΨ

10−6 10−5 10−4 10−3 10−2 10−1 100−450

−440

−430

−420

−410

−400

−390

−380

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(d) Ψn

X

10−6 10−5 10−4 10−3 10−2 10−1 100−90

−80

−70

−60

−50

−40

−30

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(e) Ψn

Y

10−6 10−5 10−4 10−3 10−2 10−1 100−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de (d

B)

Frequency (rad/sec)

δ = 0δ = π/32δ = π/16δ = π/4

(f) ΨnΨ

Figure 3: Complementary sensitivity functions, showing the response of the lat-eral and yaw velocities of the nominal system to noise in the output signals at aforward speed of 80 kph for four steering angles δ.

nels with lateral and yaw velocity outputs.The maximum sensitivities to uncertainty on any of the six channels that out-

put lateral or yaw velocity are of magnitude one, occurring on the channels linkinglateral velocity noise to lateral velocity output, and yaw rate noise to yaw rate out-put. In both of these cases, the sensitivity to noise rolls off significantly as itsfrequency increases.

SpecificationThe lane change manoeuvre to be performed was adapted from the ISO standardfor severe lane change manoeuvres ISO [2002].

The lane change manoeuvre is to be conducted as quickly as possible, takingthe vehicle to its physical limits.

Five control inputs are available to manouevre the vehicle: front wheel steer-ing and control of the longitudinal forces on each wheel, through the anti-lockbraking system (ABS). Actuator constraints were provided by the vehicle manu-

7

u uu u u

u u u u u

u u

u u u u u

uuu

u

1

32

cones

Figure 4: Test track layout for demonstration of severe lane change derived fromISO 3888-2:2002 Passenger cars – Test track for a severe lane-change manoeuvre– Part 2: Obstacle avoidance

Section Length (metres) Width (metres)1 12.0 1.1 × vehicle width + 0.252 13.5 2.1 × vehicle width + 1.253 11.0 1.0 × vehicle width + 1.00

Table 1: Test track dimensions

facturer.For the purposes of this controller, it is assumed that the vehicle position, ve-

locity and acceleration are well known and available from the vehicle managementcomputer.

Taking into account the width of the car, a step change of 2.26 metres wouldallow a margin of 0.55 metres either side of the centre line as the car movesinto the final section (3 on the diagram), corresponding to a maximum over-shoot/undershoot of approximately 24%.

A suitable reference trajectory to manouevre the car through the test sectionscan be defined by the function:

Y0(X) =

0 ( 0.0 ≤ X < 12.0)

1.13 ×(

1 + sin(

πX−12.01.5

− π2

))

(12.0 ≤ X < 25.5)

2.26 (25.5 ≤ X)

which provides a margin of approximately four metres in the X direction betweenthe reference trajectory and the first cone at the start of the final section. Conse-quently, the vehicle must not lag the reference trajectory by a time greater than 4

X

seconds.

8

Simulation resultsThe results of simulations using the non-linear vehicle model are shown in fig-ures 5 to 8 for four speeds: 80, 60, 40 and 20 kph. In each case, the vehicle startedwith the given longitudinal speed and zero lateral and yaw velocity.

Subfigure (a) in each figure shows the feedforward steering angle (δ0, shownin blue) and the combined steering angle (δ = δ0 +∆, shown in green) that resultsfrom the action of the steering/position loop. The steering controller was designedto operate at speeds of about 80 kph so there is some poor performance at verylow speeds, as is evident in the oscillatory behaviour seen in figure 8.

Subfigure (b) shows the reference (blue) and actual (green) yaw angle as thecar performs the manoeuvre.

Subfigure (c) shows the longitudinal forces that act on each wheel. Theseforces are the output of the inverse model D, acting upon the error signals thatresult from the gain matrix L. It can be seen that in each case the wheels onopposite sides of the vehicle act in opposite directions as the controller stabilisesthe yaw rate. As expected, the output of the force controller decreases as thevehicle velocity decreases.

Subfigure (d) shows the trajectory taken by the vehicle. The coloured dashedlines depict the boundary of the manouevre area while the dotted black lines showthe approximate limits for the vehicle centre of gravity if the side of the car is notto exceed the coned boundary. The blue solid line shows the reference trajectorywhile the green line shows the path taken by the vehicle. In all cases, the vehiclestays with the limits and completes the manoeuvre successfully.

ReferencesISO-3888-2:2002 passenger cars - test track for a severe lane-change manoeuvre

– part 2: Obstacle avoidance, 2002.

D. J. Leith and W. E. Leithead. Gain-scheduled and nonlinear systems: dynamicanalysis by velocity-based linearization families. International Journal of Con-trol, 70(2):289–317, May 1998.

D.J. Leith, A. Tsourdos, B.A. White, and W.E. Leithead. Application of velocity-based gain-scheduling to lateral auto-pilot design for an agile missile. Con-trol Engineering Practice, 9(10):1079 – 1093, 2001. ISSN 0967-0661. URLhttp://dx.doi.org/10.1016/S0967-0661(01)00077-6. Autopilot design;.

9

http://dx.doi.org/10.1016/S0967-0661(01)00077-6

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

X (m)

δ (d

eg)

δ0 (Ackermann)

δ = δ0 + ∆

(a) Steering angle

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X (m)

Ψ (r

ad)

referenceΨ

(b) Yaw angle

0 10 20 30 40 50 60−1500

−1000

−500

0

500

1000

1500

X (m)

f x (N)

front leftfront rightrear leftrear right

(c) Tyre forces

0 10 20 30 40 50 60−2

−1

0

1

2

3

4

X (m)

Y (m

)

referenceoutput

(d) Trajectory

Figure 5: Simulation results at 80 kph

10

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

X (m)

δ (d

eg)

δ0 (Ackermann)

δ = δ0 + ∆

(a) Steering angle

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X (m)

Ψ (r

ad)

referenceΨ

(b) Yaw angle

0 10 20 30 40 50 60−1500

−1000

−500

0

500

1000

1500

X (m)

f x (N)


(c) Tyre forces

0 10 20 30 40 50 60−2

−1

0

1

2

3

4

X (m)

Y (m

)

referenceoutput

(d) Trajectory


11

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

X (m)

δ (d

eg)

δ0 (Ackermann)

δ = δ0 + ∆

(a) Steering angle

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

X (m)

Ψ (r

ad)

referenceΨ

(b) Yaw angle

0 10 20 30 40 50 60−800

−600

−400

−200

0

200

400

600

800

X (m)

f x (N)


(c) Tyre forces

0 10 20 30 40 50 60−2

−1

0

1

2

3

4

X (m)

Y (m

)

referenceoutput

(d) Trajectory


12

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

X (m)

δ (d

eg)

δ0 (Ackermann)

δ = δ0 + ∆

(a) Steering angle

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

X (m)

Ψ (r

ad)

referenceΨ

(b) Yaw angle

0 10 20 30 40 50 60−400

−300

−200

−100

0

100

200

300

400

X (m)

f x (N)


(c) Tyre forces

0 10 20 30 40 50 60−2

−1

0

1

2

3

4

X (m)

Y (m

)

referenceoutput

(d) Trajectory


13

DESIGN AND PROBABILISTIC VALIDATIONOF A SYSTEM FOR COOPERATIVE DRIVING

O. Gietelink ∗,∗∗,1 B. De Schutter ∗∗ M. Verhaegen ∗∗

∗ TNO Science& Industry, P.O. Box 756, 5700 ATHelmond, The Netherlands, email: [email protected]

∗∗ DCSC, Delft Univ. of Technology, Delft, TheNetherlands, email: b.deschutter,[email protected]

Abstract:This paper presents a cooperative longitudinal control system for a cluster ofvehicles using environment sensors and vehicle-to-vehicle communication. Thehybrid control system consists of discrete state automatons, each with its owncontrol laws. These controllers compute a desired acceleration that is realizedby a lower-level controller to obtain a smooth and safe traffic flow in a stringof vehicles. The system is evaluated using a new methodological approach,based on randomized algorithms. This new methodology is more efficient thanconventional validation by simulations and field tests, especially with increasingsystem complexity.

Keywords: vehicle-to-vehicle communication, hardware-in-the-loop simulation,longitudinal vehicle control, adaptive cruise control, randomized algorithms

1. INTRODUCTION

With the increasing demand for safer passengervehicles, the development of advanced driver assis-tance systems (ADASs) is a major research topicin the automotive industry. An ADAS is a vehiclecontrol system that uses environment sensors (e.g.radar, laser, vision) to improve driving comfortand traffic safety by assisting the driver in rec-ognizing and reacting to potentially dangeroustraffic situations. Examples of ADASs that havealready been introduced to the market are driverwarning systems that actively warn the driver ofa potential danger, e.g. lane departure warning,and forward collision warning systems.

1 Research supported by TNO, TRAIL Research School,the Transport Research Centre Delft program “TowardsReliable Mobility”, and the European 6th Framework Net-work of Excellence “HYbrid CONtrol: Taming Hetero-geneity and Complexity of Networked Embedded Systems(HYCON)” under contract number FP6-IST-511368.

Another widely available system since the 1990’sis adaptive cruise control (ACC) (Winner et al.,1996), which provides a more active support tothe driver. ACC is a comfort system that main-tains a set cruise control velocity, unless an en-vironment sensor detects a slower vehicle ahead.The ACC then controls the vehicle to follow theslower vehicle at a safe distance xd, see Figure 1.Since the available literature on ACC systems isvast, the interested reader is referred to (Vahidiand Eskandarian, 2003) for further details. Somedrawbacks of ACC are mentioned though:

• ACC systems have a maximum range ofabout 200m, which is insufficient for warningabout traffic jams or other potential dangerfurther ahead.

• False and missed alarms can occur whendriving in curves or when other vehicles orroad infrastructure are blocking the line-of-sight of the sensor.

xr

[x2 v2 a2][x1 v1 a1]

xdex

driving direction1: leading vehicle 2: following vehicle

Fig. 1. Schematic representation of an ACC sys-tem.

• In addition, the sensor signals can be unre-liable, due to multi-path reflections, weatherconditions, and sensor noise.

Therefore, ACC could be greatly enhanced whenthe field-of-view of the sensorial platform is ex-tended to include information from other preced-ing vehicles. This can be achieved by implement-ing a communication system between vehicles, so-called vehicle-to-vehicle communication (VVC).

Current research is focussed on extending ACCsystems to cooperative adaptive cruise control(CACC) systems (Lu et al., 2002), where theinter-vehicle distance is accurately estimated us-ing VVC and environment sensors. The advan-tage of CACC is that it has an increased controlbandwidth and reliability with respect to ACC.However, CACC is only operational with regard tothe directly preceding vehicles that are sensed bythe environment sensors. The ACC will thereforenot directly respond to other preceding vehiclesfurther ahead.

Another possibility of VVC is a cooperative colli-sion warning and avoidance systems, which hasbeen the topic of research within several EUprojects, such as CARTALK (Morsink and Gi-etelink, 2002) and WILLWARN (Schulze et al.,2005). An advantage of these systems is thatthey can receive collision warnings and trafficinformation from vehicles further ahead, insteadof only the directly preceding vehicle. The newcooperative driving system to be presented in thispaper, therefore combines the ACC function witha cooperative system that looks multiple vehiclesahead.

The demand for safety naturally increases withincreasing automation of the driving task, sincethe driver must fully rely on a flawless operation ofthe ADAS. Therefore, the ADAS should be testedfor the wide variety of complex traffic situationsthat the system should be able to recognize andhandle (Kiefer et al., 1999). To address this is-sue, efficient methods are required for the designof ADAS controllers and the validation of theirsafety and performance.

The objective of this paper is to present a hybridcontrol system for cooperative driving with severalpreceding vehicles based on GPS-based naviga-tion, environment sensing, and VVC. In orderto validate the system’s safety, a methodological

approach based on randomized algorithms (RAs)is used.

The organization of this paper is as follows. Sec-tion 2 discusses a database with stochastic in-formation on traffic scenarios. Section 3 thenpresents the demonstrator vehicles with their sen-sors, actuators, and sensor post-processing algo-rithms. The signals from the post-processing mod-ule are then used as an input for the hybrid controlarchitecture, presented in Section 4. Subsequently,Section 5 presents the methodological approachfor validation of this system. Finally, Section 6concludes the paper.

2. RELEVANT TRAFFIC SCENARIOS FORCOOPERATIVE DRIVING

An ADAS should be tested for a representative setof traffic scenarios. For this purpose a databasehas been developed with stochastic informationon typical scenarios that are of interest for longi-tudinal vehicle control. Since a longitudinal con-trol system only considers target vehicles that arein (or entering into) the host vehicle’s lane, thefocus is on single-lane scenarios. The single-lanescenario can be split up into four sub-scenarios:

• Free-flow A vehicle is considered in free-flow when it has no vehicle directly in front,i.e. the time headway th is larger than 8 s.

• Car-following A car-following scenario ap-plies when th is less than 8 s.

• Slow-down A slow-down scenario occurswhen a slower target vehicle is detected infront of the host vehicle (e.g. after a cut-in or approaching manoeuvre), and the hostvehicle makes a transient manoeuvre to alower velocity.

• Speed-up Contrary to slow-down, speed-upcan occur when the host vehicle is precededby a faster (or no) vehicle, or when the legalspeed limit increases.

Figure 2 illustrates the presented subdivision,based on the relation between distance xr andrelative velocity vr. The actual traffic scenarios arecharacterised by particular scenario parameters,as illustrated in Figure 1. This figure defines theparameters for car-following: the position x(t), ve-locity v(t), and acceleration a(t) of both vehicles,the relative velocity vr(t) = v1(t)−v2(t), the head-way xr(t) = x1(t) − x2(t) (neglecting the lengthof the vehicles), all in longitudinal direction. Thetime headway th is defined as xr/vr. Furthermore,a driver reaction time tr is defined.

For these four sub-scenarios, a representative setof scenarios can be defined by a stochastic de-scription. For example, the velocity and acceler-ation profiles during free-flow can be described by

vr

xr

0

Danger Danger

free-flowfree-flow

car-following

slow

-dow

nsp

eed-u

p

Fig. 2. Division of the single-lane scenario intoseveral sub-scenarios.

distribution functions (e.g. normal or lognormal).However, dynamic manoeuvres like slow-down canonly be described by a dynamic model, where themodel parameters are sampled from a stochasticdistribution. An example is the well-known Gazis-Herman-Rothery model (Brackstone and McDon-ald, 1999), given by:

a(t) = cvm(t)vr(t− tr)

xlr(t− tr)

, (1)

where m, l, and c are stochastic parameters.

Obviously, there is an infinite number of combi-nations of scenario parameters possible, so theADAS cannot be tested exhaustively. In Section 5we will therefore present a probabilistic approachfor testing the ADAS for a representative set ofscenarios. First we will present the demonstratorvehicles and their hybrid control architecture.

3. DEMONSTRATOR VEHICLES

This section presents the hardware and sensorfusion algorithms of the demonstrator vehiclesthat are used in this research.

3.1 Hardware implementation

The demonstrator vehicles are Smart vehicles,small 2-seat vehicles, depicted in Figure 3. TheSmarts are instrumented with electronically con-trolled actuators: throttle, brake, gearbox andsteering. In addition, they are equipped with sev-eral sensors: differential GPS (DPGS), accelerom-eters, wheel speed sensors and a gyroscope. BothSmarts are also equipped with an environmentsensor (radar or lidar). Table 1 provides anoverview of the sensor characteristics. In addition,the Smarts are equipped with wireless local areanetwork (WLAN) modules such that they can

Fig. 3. The two Smart vehicles.

Table 1. Smart instrumentation.

Sensor Signal Noise variance

DGPS xgps 10m2

ygps 10m2

ψgps 0.4rad2

Accelerometer along 0.02(m/s2)2

alat 0.03(m/s2)2

Gyroscope ψ 1.2·10−6 (rad/s)2

Wheel encoders vij 3·10−4(m/s)2

Steer angle encoder δs 1·10−4(m/s)2

Radar xr,radar 0.03 m2

vr,radar 0.01(m/s)2

ψr,radar 0.004rad2

receive and transmit information to other vehi-cles within a range of several hundreds of meters,depending on the environmental conditions. Thesignal processing and control algorithms, whichare described below, are implemented on a real-time Linux based, PC/104 computer system.

3.2 On-board filtering

Cooperative vehicle control requires accurate andreliable knowledge of the vehicle state of the hostvehicle and target vehicles. However, the infor-mation from only one sensor is usually not reli-able and accurate enough for longitudinal vehiclecontrol. Therefore, the information from multiplesensors must be fused to obtain more completeand more accurate vehicle states of the vehiclesinvolved in the cooperative setup. For example,DGPS alone is not sufficient for real-time positioninformation, because of the 1 Hz update rate andthe inaccuracy. Therefore, additional sensors arefused in an Extended Kalman Filter (EKF). ThisEKF is based on a non-linear dynamic model ofthe vehicle motion, taking into account that thesensor measurements are perturbed by Gaussianwhite noise, as indicated in Table 1. In this wayall desired vehicle states can be obtained withsufficient frequency and accuracy. In (Hallouzi etal., 2004) we have elaborated on the integration ofthe different signals for the vehicle state. In thispaper, focus is put on the estimation of the inter-vehicle states.

Navigation frame

θ

ψ1

ψ2

x-position

y-p

osi

tion

vx,1

vx,2

vy,1

vy,2

x1

y1

x2

y2

Vehicle 1

Vehicle 2

L1

L2

xrψr

Fig. 4. Model of the relative motion between twovehicles.

3.3 Estimation of the relative motion

For the development of a cooperative driving sys-tem, information on the relative motion betweenvehicles is required, characterised by the distancexr, relative speed vr, and relative angle ψr. In thedemonstrator vehicles two methods for obtaininginformation on the relative motion between ve-hicles are available: using direct sensor measure-ments or with VVC. For obtaining relative motioninformation via VVC a kinematic model is used,as shown in Figure 4. In this figure, xi, yi, andψi represent the vehicle position and orientation,where the subscript indicates the i-th vehicle; vx,i

and vy,i are the longitudinal and lateral velocity atthe vehicle center of gravity (COG), respectively;L1 is the distance between the COG and the rearend of vehicle 1; L2 is the distance between theCOG and the front end of vehicle 2. The kinematicequations for the relative motion are given by

xr =√

(x1 − x2)2 + (y1 − y2)2 − (L1 + L2) (2)

ψr = ψ2 − θ (3)

vr = vx,1 − vx,2 cos(|ψr |) − vy,2 sin(|ψr|), (4)

where θ can be computed with

θ = arctany1 − y2

x1 − x2. (5)

Using the kinematic equations from Eq. (2)-Eq.(5), the local states computed by the on-boardEKF, which are available to vehicles in the vicinityvia VVC, can be used to compute the relative mo-tion states xr,comm, vr,comm, and ψr,comm. In addi-tion, the same states are also available directlyfrom radar or lidar measurements, denoted byxr,radar, vr,radar, and ψr,radar. In order to providean accurate and reliable estimation, the relative

Radar

KFvehicle 1 Relative

motionmodel

KFvehicle 2

Faultmanagement

x1, y1, ψ1,vx,1, vy,1

x2, y2, ψ2,vx,2, vy,2

xr,radar, vr,radar, ψr,radar

xr,comm,vr,comm,ψr,comm

xr,est, vr,est, ψr,est

Fig. 5. Fault management system for estimationof the relative motion.

0 50 100 150 200 2500

10

20

30

40

50

60

70Estimated distance between two Smarts

time [s]

dis

tan

ce [

m]

Est. KF 1Est. KF 2Non−fault Est.

Fig. 6. Combination of distance information.

motion information from both sources is com-bined. Figure 5 shows this scheme to obtain the es-timated states xr,est, vr,est, and ψr,est. In the faultmanagement block discrepancies between the twosignals are detected and the corrected signal isused as the fault-free estimate.

Figures 6 and 7 the distance and relative velocityduring a test run with the 2 Smarts is depicted.In each of these two figures, the values obtainedfrom radar measurements, the values computedwith the relative motion model using VVC andthe fault-free values are depicted. In these figuresit can be seen that the fault-free signal is muchmore reliable than the two separate sensor signals.

4. LONGITUDINAL CONTROL FORCOOPERATIVE DRIVING

In this section, an algorithm for ACC is describedand extended to cooperative driving in a clusterof vehicles. In the control architecture two hier-archical levels can be distinguished. The outerloop (high level controller) consists of the co-operative longitudinal controller that computesa reference acceleration ad based on the sensorinformation. The inner loop (low level controller)consists of an acceleration controller that tracks

0 50 100 150 200 250−10

−8

−6

−4

−2

0

2

4

6

Estimated relative velocity between two Smarts

time [s]

velo

city

[m

/s]

Est. KF1Est. KF2Non−fault Est.

Fig. 7. Combination of relative speed information.

CACCAcceleration

control

Vehicle

Inner loop

Outer loop

Fault-freestates

Radar

VVC

vehicle states

ad

a2

Throttle

Brakes

Fig. 8. Block diagram of the cooperative longitu-dinal vehicle controller.

an acceleration command ad from the outer loopas good as possible. The advantage of this con-figuration is that these two loops can be designedseparately, reducing the overall complexity of thedesign. Furthermore, this structure can also bewell motivated from the relation between driverand vehicle. The inner loop corresponds to thevehicle dynamics and the outer loop correspondsto the driving behaviour. A schematic overviewof the control structure is given in Figure 8. Inthis figure, the block ‘fault-free states’ denotes thesystem described in Section 3.3 and a2 denotes theacceleration of the host vehicle.

4.1 Longitudinal Control Algorithms

The ACC longitudinal control problem consistsof two vehicles, as shown in Figure 1. In velocitycontrol mode, the ACC operates as a conventionalcruise control, where the desired acceleration ad isgiven by a simple proportional controller

ad = kcc(vcc − v2), kcc > 0. (6)

where vcc is the set-point for the velocity.

For distance control, the desired acceleration ad

for vehicle 2 is usually given by feedback controlof the distance separation error ex = xr − xd andits derivative ev = ex = vr − vd

ad = k2ev + k1ex, k1, k2 > 0, (7)

3 2 1xr,2 xr,1

lvxd,2 xd,1

VVC

driving direction

Fig. 9. Overview of a cluster of three vehicles,using vehicle-to-vehicle communication.

to obtain a desired acceleration ad that controlsboth ex and vr to zero. In order to achieve anatural following behaviour, the desired clearanceis chosen as xd = max(v2th, s0) and the feedbackgains are calculated by nonlinear functions k1 =f1(v2, xr, th, s0) and k2 = f2(v2, th), where s0 is adistance safety margin and th is the driver-selectedtime gap.

4.2 Algorithms for Cooperative Driving

A control law for cooperative driving can be simi-lar to Eq. (7). However, the main advantage of co-operative driving is that there is more informationavailable, such as the acceleration of the precedingvehicle. Using VVC, the acceleration of the leadvehicle (which is difficult to estimate with onlyan environment sensor) can be communicated tothe following vehicle. With information on theacceleration a1, as well as more reliable estimatesfor the range and range rate, the ACC control lawEq. (7) can be modified to

ad = k3a1 + k2ev + k1ex, k1, k2, k3 > 0, (8)

where k1, k2, and k3 are nonlinear feedback gains.The availability of an acceleration signal in thefeedback control law provides an opportunity toreact faster to emergency braking of a precedingvehicle.

Because the reference acceleration from Eq. (8)only considers a single vehicle, a method hasbeen developed to consider more vehicles in front,which is the aim of cooperative driving. A con-ceptual overview of cooperative driving with theSmart vehicles using VVC is shown in Figure 9.

In order to use Eq. (8) with respect to multiplepreceding vehicles, we propose the following al-gorithm. The idea is that vehicle 3 in Figure 9should not only keep a headway of xd,2 to vehicle2, but it should also keep a headway of xd,1 +lv + xd,2 to vehicle 1. Based on this idea, thealgorithm computes a desired acceleration ad,i foreach preceding vehicle i, according to Eq. (8). Thedesired acceleration ad to be sent to the vehicle’s

CCEq. (6) ACC2CC

CC2ACCACC

Eq. (7)CACCEq. (8)

CDEq. (10)

nocontrol

Fig. 10. State automatons for cooperative driving.

lower-level controller is then calculated by takingthe minimum of all ad,i for all n preceding vehicles

ad = min(ad,n−1 , . . . , ad,1). (9)

Equation Eq. (9) implies that the host vehicle willrespond to the preceding vehicle that produces thelowest ad,i. However this may result in a very con-servative controller. Therefore, the accelerationsad,i for vehicles further ahead (so for i ≥ 2) haveto cross a threshold value amin or amax, beforethey are used in Eq. (9), according to

ad,i =

0, if amin < ad,i < amax

ad,i , else, i ≥ 2.

(10)

4.3 Design of the inner control loop

Contrary to the outer loop the design of theinner control loop is specific for each vehicle.The acceleration (inner loop) controller has beenrealized by using a feedforward control based onthe vehicle model that includes engine, brakeand gearbox dynamics. In addition, feedback ofthe acceleration signal and a simple PI controllerrealizes ad accurately.

4.4 Hybrid control system

In order to combine the various controllers pre-sented in this section, we implement them as hy-brid automatons in the structure, shown in Fig-ure 10. This allows to design and fine-tune eachcontroller for a specific mode of operation. Thisstate machine logic is included in a supervisor,which also includes a fault management systemto detect and isolate faults. In this way, gracefuldegradation of control functions in the presenceof sensor faults can be distinguished. For exam-ple, when the communication fails, the systemcan degrade to a conventional ACC. Vice-versa,when the environment sensor fails, the system canuse the communication to obtain a pseudo-rangemeasurement.

Obviously, the controller should not apply thethrottle and brake at the same time. Furthermore,the driver input at the throttle or brake shouldalways be followed. Therefore, the supervisor alsoincludes state automatons for the lower-level con-troller, as shown in Figure 11.

on

brakeoverrule

throttleoverrule

off

Fig. 11. State automatons for the lower levelcontrol, depending on the driver input.

The integration of several controllers and theswitching between various operating modes (off,standby, CC, ACC, driver overrule) creates a verycomplex system. These interactions may intro-duce unforseen failure modes and complicate thedesign and validation of ADASs. The control sys-tem therefore needs to be validated for not onlythe separate states, but especially for the switch-ing between those states.

5. PROBABILISTIC APPROACH FORCONTROLLER VALIDATION

5.1 Motivation for a probabilistic approach

An iterative process of simulations and test drivesis often used for validation. Test drives give real-istic results, but can never cover the entire setof operating conditions. Results are also diffi-cult to analyze and not reproducible (Fancher etal., 1998). On the other hand, simulations havetheir limitations as well. For a realistic nonlinearmodel and multiple traffic disturbances, the vali-dation problem will become difficult to solve, andeventually become intractable (Vidyasagar, 1998).

An alternative approach for solving this problemexactly, is to solve it approximately by using a ran-domized algorithm (RA). An RA is an algorithmthat makes random choices during its execution(Motwani, 1995). The use of an RA can turn anintractable problem into a tractable one, but atthe cost that the algorithm may fail to give acorrect solution. The probability δ that the RAfails can be made arbitrarily close to zero, butnever exactly equal to zero. This probability δ

mainly depends on the sample complexity N , i.e.the number of simulations performed, but also onthe specification of the problem to be solved.

A popular example of an RA is a Monte Carlostrategy, where the system is simulated for arepresentative, though very large, set of oper-ating conditions, based on the probability thatthese conditions occur (Stengel and Ray, 1991).In (Gietelink et al., 2005) we have introducedthe use of importance sampling (IS) to makeMonte Carlo simulation more efficient, i.e. usinga much smaller number of tests. However, the

γ

ρQ

qiqmin qρsim qmax

ρQ,max

ρQ,sim

ρQ,min

Fig. 12. Illustration of the dependency betweenthe performance characteristic ρQ and a pa-rameter q ∈ Q.

performance of Monte Carlo importance samplingheavily depends on the reliability of the proba-bility density functions (pdf’s) that are used torandomly generate and simulate traffic scenarios.

In this section we propose a generic methodolog-ical approach for validation of ADASs, consistingof the following steps: (1) problem specification;(2) simulation; and (3) model validation.

5.2 Problem specification

In this paper the controller validation is restrictedto the measure of safety, expressed as the prob-ability p that a collision will occur for a wholerange of traffic situations. The safety measure fora single experiment is ρs ∈ 0, 1, where ρs = 0means that the control system manages to followthe preceding vehicle at a safe distance, and ρs = 1means that the traffic scenario would require abrake intervention by the driver to prevent a col-lision 2 .

The value of ρs for a particular traffic scenariodepends on the perturbations imposed by thatscenario. The disturbance to the system is formedby the motion of other vehicles that are detectedby the environment sensors, as described in Sec-tion 2. These scenario parameters, together withmeasurement noise, unmodelled dynamics, statemachine transitions, and various types of faultsconstruct an n-dimensional perturbation set Q.It is then of interest to determine the probabil-ity of performance satisfaction pN : that is checkwhether ρ is below threshold γ with a certainprobability level p for the whole perturbation setQ, based on N samples. The first step in themethod is thus to identify Q and its pdf fQ, asshown in Section 2.

2 Please note the difference between the performance levelρ for a particular experiment and its probability p for awhole range of experiments.

5.3 Simulation

After Q has been defined, the control system issimulated for a wide range of traffic scenarios,where the scenario parameters are sampled fromQ in a Monte Carlo simulation. In order to reducethe sample complexity N , it makes sense to givemore attention to operating conditions that aremore likely to cause a collision than others.

One possibility for using a priori knowledge oninteresting samples is importance sampling, whichis a technique to increase the number of occur-rences of the event of which the probability p

should be estimated. Using importance sampling,the simulation parameters are sampled, such thatthey reflect the importance of the events. Theresulting estimation for p is then afterwards cor-rected by dividing it by the increased probabilityof the occurrence of the event. A rare but dan-gerous event can thus be equally important as amore frequent but less dangerous event. Figure 13shows the results of a benchmark example usingconventional Monte Carlo sampling. The resultsof importance sampling are shown in Figure 14,which illustrates that a more reliable estimate ofp is obtained with a lower number of samples N .The reader is referred to (Gietelink et al., 2005)for a more detailed description of this simulationmethod.

5.4 Model validation

The most critical scenarios Qi (that were identi-fied with the simulation phase), are then chosento be reproduced in a laboratory environment andin test drives with the real vehicles. This is alsodone in a randomised approach to efficiently coverQ. These particular Qi are selected in the interval[Qmin, Qmax].

Using this approach, a test program is conductedthat minimises the number of hardware tests tobe done, in order to obtain a cost- and time-efficient evaluation phase. In addition, the testresults can also be used for model validation. Theestimate pN may indicate necessary improvementsin the system design regarding fine-tuning of thecontroller parameters.

In an iterative process the simulation results instep 2 and thus the estimate pN can be improved.Subsequently, the vehicle test program in step 3can be better optimised by choosing a smallerinterval [Qmin, Qmax]. From the combination ofsimulations and laboratory tests, the performancepN of the system can then be estimated with ahigh level of reliability, and the controller designcan be improved.

0.9 0.92 0.94 0.96 0.98 1 0

20

40

60

80

100

120

140

Histogram of 500 simulation sets, with N = 100 each Gaussian distribution of acceleration profile

Num

ber

of s

imul

atio

n se

ts

pNj

Fig. 13. Histogram of 500 estimates pNj, with

N = 100 each, where the acceleration profileis sampled from a Gaussian pdf N (0, 1.5).

0.9 0.92 0.94 0.96 0.98 1 0

50

100

150

200

250

Histogram of 500 simulation sets, with N = 100 each Gaussian distribution of acceleration profile

Empirical mean with Importance Sampling

Num

ber

of s

imul

atio

n se

ts

pNj

Fig. 14. Histogram of 500 estimates pNj, with

N = 100 each, where the acceleration pro-file is sampled from an importance samplingdistribution ϕ = −0.005a1 + 0.05.

6. CONCLUSIONS

A cooperative vehicle control system has beenpresented that uses state estimation of individualvehicles combined with vehicle-to-vehicle commu-nication and measurements from an environmentsensor. This provides the host vehicle with rel-evant information of more than one precedingvehicle. In addition, this redundancy allows afault-tolerant longitudinal control in case of sensorfaults or communication outage.

REFERENCES

Brackstone, M. and M. McDonald (1999). Car-following : a historical review. TransportationResearch Part F: Traffic Psychology and Be-haviour 2(4), 181–196.

Fancher, P., R. Ervin, J. Sayer, M. Hagan, S. Bog-ard, Z. Bareket, M. Mefford and J. Haugen(1998). Intelligent cruise control field opera-tional test. Final report DOT HS 808 849.

National Highway Traffic Safety Administra-tion. Washington, DC, USA.

Gietelink, O., B. De Schutter and M. Verhaegen(2005). Probabilistic validation of advanceddriver assistance systems. In: Proc. of the16th IFAC World Congress. Prague, CzechRepublic. paper nr. 4254.

Hallouzi, R., V. Verdult, H. Hellendoorn, P.L.J.Morsink and J. Ploeg (2004). Communicationbased longitudinal vehicle control using anextended Kalman filter. In: Proceedings of theIFAC Symposium on Advances in AutomotiveControl. Salerno, Italy.

Kiefer, R.J., D.J. LeBlanc, M.D. Palmer,J. Salinger, R.K. Deering and M.A. Shulman(1999). Development and validation of func-tional definitions and evaluation proceduresfor collision warning/avoidance systems. Fi-nal Report DOT HS 808 964. National High-way Traffic Safety Administration. Washing-ton, DC, USA.

Lu, X.Y., J.K. Hedrick and M. Drew (2002).ACC/CACC - control design, stability androbust performance. In: Proc. of the Amer-ican Control Conference. Anchorage, Alaska,US. pp. 4327–4332.

Morsink, P.L.J. and O.J. Gietelink (2002). Pre-liminary design of an application for CBLC inthe CarTALK2000 project: Safe, comfortableand efficient driving based upon inter-vehiclecommunication. In: Proc. of the e-Safety Con-ference. Lyon, France.

Motwani, R. (1995). Randomized Algorithms.Cambridge University Press. New York.

Schulze, M., G. Noecker and K. Boehm (2005).PReVENT: A European program to improveactive safety. In: Proc. of the 5th Interna-tional Conference on Intelligent Transporta-tion Systems Telecommunications. Brest,France.

Stengel, R.F. and L.R. Ray (1991). Stochas-tic robustness of linear time-invariant con-trol systems. IEEE Trans. Automatic Control36(1), 82–87.

Vahidi, A. and A. Eskandarian (2003). Researchadvances in intelligent collision avoidance andadaptive cruise control. IEEE Trans. on Intel-ligent Transportation Systems 4(3), 143–153.

Vidyasagar, M. (1998). Statistical learning theoryand randomized algorithms for control. IEEEControl Systems Magazine 18(6), 69–85.

Winner, H., S. Witte, W. Uhler and B. Lichten-berg (1996). Adaptive cruise control systemaspects and development trends. SAE Tech-nical Paper Series.

A methodology for the design of robust rollover prevention controllersfor automotive vehicles: Part 2-Active steering

Selim Solmaz, Martin Corless and Robert Shorten

Abstract— In this paper we apply recent results from robustcontrol to the problem of rollover prevention in automotivevehicles. Specifically, we exploit the results of Pancake, Corlessand Brockman, which provide controllers to robustly guaranteethat the peak magnitudes of the performance outputs of anuncertain system do not exceed certain values. We use thedynamic Load Transfer Ratio LT Rd as a performance output forrollover prevention, and design active-steering based rollovercontrollers to keep the magnitude of this quantity below acertain level, while we use control input u as an additionalperformance output to limit the maximum amount of controleffort. We present numerical simulations to demonstrate theefficacy of our controllers.

I. INTRODUCTION

It is well known that vehicles with a high center ofgravity such as vans, trucks, and the highly popular SUVs(Sport Utility Vehicles) are more prone to rollover accidents.According to the 2004 data [8], light trucks (pickups, vans,SUV’s) were involved in nearly 70% of all the rolloveraccidents in the USA, with SUV’s alone responsible foralmost 35% of this total. The fact that the composition of thecurrent automotive fleet in the U.S. consists of nearly 36%pickups, vans and SUV’s [9], along with the recent increasein the popularity of SUV’s worldwide, makes rollover animportant safety problem.

There are two distinct types of vehicle rollover: trippedand un-tripped rollover. Tripped rollover is usually causedby impact of the vehicle with something else resulting inthe rollover incident. Driver induced un-tripped rollover canoccur during typical driving situations and poses a realthreat for top-heavy vehicles. Examples are excessive speedduring cornering, obstacle avoidance and severe lane changemaneuvers, where rollover occurs as a direct result of thewheel forces induced during these maneuvers. It is however,possible to prevent such a rollover incident by monitoringthe car dynamics and applying proper control effort aheadof time. Therefore there is a need to develop driver assistancetechnologies which would be transparent to the driver duringnormal driving conditions, but which act when needed torecover handling of the vehicle during extreme maneuvers[9].

We present in this paper a robust rollover preventioncontroller design methodology based on active steering. Theproposed control design is an application of recent results on

S. Solmaz is with the Hamilton Institute, National University of Ireland-Maynooth [email protected]

M. Corless is with the School of Aeronautics & Astronautics, PurdueUniv. West Lafayette, IN [email protected]

Prof. R. Shorten is with the Hamilton Institute, National University ofIreland-Maynooth [email protected]

the design of control systems which guarantee that the peakvalue of the performance output of a plant does not exceedcertain thresholds. [2]. The selected performance output forthe rollover problem is the dynamic Load Transfer RatioLT Rd . This measure of performance is related to tire lift-offand it can be considered as an early indicator of impendingvehicle rollover. The aim of our control strategy is to limitthe peak value of this performance output. The additionalperformance output on u(t) minimizes the maximum amountattenuation with the controller while achieving the objectiveperformance on LT Rd . We indicate how our design canbe extended to account for other sources of uncertaintysuch as unknown vehicle center of gravity, and tire stiffnessparameters.

II. RELATED WORK

Rollover prevention is a topical area of research in theautomotive industry and several studies have recently beenpublished. Relevant publications include that of Palkovics etal. [11], where they proposed the ROP (Roll-Over Preven-tion) system for use in commercial trucks making use of thewheel slip difference on the two sides of the axles to estimatethe tire lift-off prior to rollover. Wielenga [10] suggestedthe ARB (Anti Roll Braking) system utilizing braking ofthe individual front wheel outside the turn or the full frontaxle instead of the full braking action. The suggested controlsystem is based on lateral acceleration thresholds and/ortire lift-off sensors in the form of simple contact switches.Chen et al. [6] suggested using an estimated TTR (Time ToRollover) metric as an early indicator for the rollover threat.When TTR is less than a certain preset threshold value forthe particular vehicle under interest, they utilized differentialbreaking to prevent rollover. Ackermann et al. and Odenthalet al. [4], [5] proposed a robust active steering controller,as well as a combination of active steering and emergencybraking controllers. They utilized a continuous-time activesteering controller based on roll rate measurement. They alsosuggested the use of a static Load Transfer Ratio (LT Rs)which is based on lateral acceleration measurement; this wasutilized as a criterion to activate the emergency steering andbraking controllers.

III. VEHICLE MODELLING AND LT Rd

We use a linearized vehicle model for control design.Specifically, we consider the well known single-track (bi-cycle) model with a roll degree of freedom. In this modelthe steering angle δ , the roll angle φ , and the vehicle sideslipangle β are all assumed to be small. We further assume that

all the vehicle mass is sprung, which implies insignificantwheel and suspension weights. The lateral forces on the

Fig. 1. Linear bicycle model with roll degree of freedom.

front and rear tires, denoted by Sv and Sh, respectively, arerepresented as linear functions of the tire slip angles αv andαh, that is, Sv = Cvαv and Sh = Chαh, where Cv and Ch arethe front and rear tire stiffness parameters respectively. Inorder to simplify the model description, we further definethe following auxiliary variables

σ , Cv +Ch ,

ρ , Chlh−Cvlv , (1)κ , Cvl2

v +Chl2h ,

where lv and lh are defined in Figure 1. For simplicity, it isassumed that, relative to the ground, the sprung mass rollsabout a horizontal roll axis which is along the centerline ofthe body and at ground level. Using the parallel axis theoremof mechanics, Jxeq , the moment of inertia of the vehicle aboutthe assumed roll axis, is given by

Jxeq = Jxx +mh2 (2)

where h is the distance between the center of gravity (CG)and the assumed roll axis and Jxx is the moment of inertia ofthe vehicle about the roll axis through the CG. We introducethe state vector ξ =

[vy ψ φ φ

]T , where descriptions areas follows• vy : lateral velocity of the CG,• ψ : yaw rate of the undercarriage,• φ : roll rate of the sprung mass,• φ : roll angle of the sprung mass.

The equations of motion corresponding to this model aredescribed as follows

ξ = Aξ +Bδ , (3)

A =

− σmvx

JxeqJxx

ρmvx

JxeqJxx− vx − hc

Jxx

h(mgh−k)Jxxρ

Jzzvx− κ

Jzzvx0 0

− hσJxxvx

hρvxJxx

− cJxx

mgh−kJxx

0 0 1 0

,

B =[

Cvm

JxeqJxx

CvlvJzz

hCvJxx

0]T

.

TABLE IMODEL PARAMETERS AND DEFINITIONS

parameter descriptionm vehicle mass, [kg]vx vehicle longitudinal speed, [m/s]δ steering angle,[rad]

Jxx,Jzz roll and yaw moment of inertia of the sprungmass measured at the CG, respectively, [kg ·m2]

lv, lh longitudinal CG position measured w.r.t the frontand the rear axles, respectively, [m]

h CG height measured over the ground, [m]c suspension damping coefficient, [kg ·m2/s]k suspension spring stiffness, [kg ·m2/s2]

Cv,Ch linear tire stiffness coefficients for the frontand the rear tires, respectively, [N/rad]

Further definitions of the parameters appearing in (3) aregiven in Table I. Also see [13] for a detailed description andderivation of this vehicle model.

A. The Load Transfer Ratio, LT Rd

In [1] we define the a dynamic version of LTR. To aidexposition we repeat the derivation here. Traditionally, asdiscussed in the related work section, some estimate of thevehicle load transfer ratio (LTR) has been used as a basisfor the design of rollover prevention systems. The LTR [5],[7] can be simply defined as the load (i.e., vertical force)difference between the left and right wheels of the vehicle,normalized by the total load (i.e., the weight of the car). Inother words

LT R =Load on Right Tires-Load on Left Tires

Total Weight. (4)

It is apparent that LT R varies within [−1,1], and for aperfectly symmetric car that is driving straight, it is 0. Theextremum are reached in the case of a wheel lift-off of oneside of the vehicle, in which case LT R becomes 1 or −1depending on the side that lifts off. If roll dynamics areignored, it is easily shown [5] that the corresponding staticLTR (which we denote by LT Rs) is approximated by

LT Rs ≈ 2ay

ghT

. (5)

where ay is the lateral acceleration of the CG.Note that rollover estimation based upon (5) is not suf-

ficient to detect the transient phase of rollover (due to thefact that it is derived ignoring roll dynamics.) Consequently,we obtain an expression for LTR which does not ignore rolldynamics. We denote this by LT Rd . In order to derive LT Rdwe write a torque balance equation. Recall that we assumedthe unsprung mass weight to be insignificant and the mainbody of the vehicle rolls about an axis along the centerline ofthe body at the ground level. We can write a torque balancefor the unsprung mass about the assumed roll axis in termsof the suspension torques and the vertical wheel forces asfollows:

−FRT2

+FLT2− kφ − cφ = 0 . (6)

Now substituting the definition of LT R from (4) and rear-ranging yields the following expression for LT Rd :

LT Rd = − 2mgT

(cφ + kφ

). (7)

In terms of the state vector, LT Rd can be represented by thefollowing linear matrix equation

LT Rd = Cξ , where (8)

C =[

0 0 − 2cmgT − 2k

mgT

].

B. Actuators, Sensors and Parameters

We are interested in robust control design based onactive steering actuators. There are two types of activesteering methods: full steer-by-wire and mechatronic-angle-superposition types. Steer-by-wire actuators do not contain aphysical steering column between the steering wheel and thetires, which enable them to be flexible and suitable for var-ious vehicle dynamics control applications. However, strin-gent safety requirements on such systems prevent them fromentering today’s series-production vehicles. Mechatronic-angle-superposition type active steering actuators howeverhave been recently introduced to the market. They containa physical steering column and act cooperatively with thedriver, while they permit various functions such as speeddependent steering ratio modification, and active response tomild environmental disturbances. It is plausible that activesteering actuators will become an industry standard in thenear future, due to their capability of directly and mostefficiently affecting the lateral dynamics of the car. Activesteering based lateral control methods can be perfectlytransparent to the driver and they are likely to cause theleast interference with the driver intent unlike the control ap-proaches based on differential braking and active suspension.Moreover use of active steering actuators do not result in asignificant velocity loss, therefore they are likely to enter themarket initially for the high performance vehicle segment. Inthis paper we assume mechatronic-angle-superposition typesteering actuators; however results can easily be extended tothe use of steer-by-wire actuators.

We also assume full state feedback information for thedesign of the reference robust controllers and that all themodel parameters m,Jxx,Jzz, lv, lh,Cv,Ch,k,h,c are known.This is an unrealistic assumption: yet our control design iseasily extended to account for uncertainty in these parame-ters. As a side note, although we assumed all the vehiclemodel parameters to be known, it is possible to estimatesome of these that are fixed (but unknown) using the sensorinformation available for the control design suggested here;this however is outside the scope of this work [12].

IV. STATE FEEDBACK CONTROLLERS FOR ROBUSTDISTURBANCE ATTENUATION

We are interested in designing a controller to preventrollover that is robust with respect to parameter uncertainty.Our starting point is in results obtained by Pancake, Corless

and Brockman [2], [3] for uncertain systems of the form

x = A(θ)x+B(θ)ω +Bu(θ)u (9)z j = C j(θ)x+D j(θ)ω +Du j(θ)u , (10)

where θ is some parameter that captures the plant nonlinear-ity/uncertainty, x∈Rn is the state at time t ∈R and ω ∈R is abounded disturbance input while z j ∈R are the performanceoutputs for j = 1, . . . ,r. We wish to synthesize a stabilizingcontroller which prevents the peak value of the performanceoutputs exceeding a certain value. In other words, we wantto design a feedback controller, which guarantees a boundedperformance output given a bounded uncertain disturbance,that is, ||ω|| ≤ ωmax. In order to keep the problem simple,we consider linear state feedback controllers of the form

u = Kx , (11)

where K is a constant matrix. We can now define closed loopsystem matrices Acl and Ccl j as follows

Acl(θ) = A(θ)+Bu(θ)K, Ccl j(θ) = C j(θ)+Du j(θ)K, (12)

for all j = 1, . . . ,r. Applying (11) to system (9)-(10) andusing the closed loop matrix definitions (12) we obtain thefollowing closed loop system:

x = Acl(θ)x+B(θ)ω (13)z j = Ccl j(θ)x+D j(θ)ω, j = 1, . . . ,r. (14)

Assumption 1: For each θ and j = 1, . . . ,r, the matrixsextuple

(A(θ),B(θ),Bu(θ),C j(θ),D j(θ),Du j(θ))

can be written as a convex combination of a finite numberof matrix sextuples

(A1,B1,Bu1,C j,1,D j,1,Du j,1), . . . ,(AN ,BN ,BuN ,C j,N ,D j,N ,Du j,N )

for each j. That is for each θ there exists non-negative scalarsξ1, . . . ,ξN such that ∑N

i=1 ξi = 1, and

A(θ) =N

∑i=1

ξiAi , C j(θ) =N

∑i=1

ξiC j,i,

B(θ) =N

∑i=1

ξiBi , Bu(θ) =N

∑i=1

ξiBui, (15)

D j(θ) =N

∑i=1

ξiD j,i, Du j(θ) =N

∑i=1

ξiDu j,i .

We have now the following result which is useful forcontrol design.

Theorem 1: Consider a nonlinear/uncertain system de-scribed by (9)-(10) and satisfying Assumption 1. Supposethat there exists a matrix S = ST > 0, a matrix L andpositive scalars β1, . . .βN and µ j,0,µ j,1,µ j,2 such that foreach j = 1, . . . ,r the following matrix inequalities hold[

βi(SATi +AiS +LT BT

ui +BuiL)+S βiBiβiBT

i −µ j,0I

]≤ 0, (16)

−µ j,1S 0 SCTj,i +LT DT

u j,i

0 −µ j,2I DTj,i

C j,iS +Du j,i L D j,i −I

≤ 0, (17)

for all i = 1, . . . ,N. Then the controller

u = Kx , where K = LS−1 (18)

results in a closed loop nonlinear/uncertain system (13)-(14)which is L∞ stable with L∞ gains less than or equal to

γ j =√

µ j,0µ j,1 + µ j,2. (19)

The above means that for a bounded disturbance input,that is, ‖ω(t)‖ ≤ ωmax for all t, and zero initial state, theperformance outputs z1, . . . ,zr of the closed loop systemare bounded and satisfy ‖z j(t)‖ ≤ γ jωmax for all t. Thescalars γ1, . . . .γr are called levels of performance and canbe regarded as measures of the ability of the closed loopsystem to attenuate the effect of the disturbance input on theperformance outputs; a smaller γ j means better performancein the sense of increased attenuation. For a proof of thetheorem, see [3].

V. ROLLOVER PREVENTION CONTROLLERS

Here we use the results of the previous section to obtainrobust rollover prevention controllers using active steering asthe sole control input.

For the implementation of an active steering state feedbackcontroller, we used the reference model (3) along with anadditional control input term that is superimposed on thedriver steering input (i.e., disturbance input); this is describedby

ξ = Aξ +Bω +Bu, (20)

where ξ (t) ∈ R4 is the state at time t ∈ R, and matricesA and B are fixed and are described as in (3). u(t) ∈ R isthe control input and ω(t) ∈ R is disturbance input. In thispaper we designate the driver commanded input δd to be adisturbance input and active steering input δc as the controlinput. i.e.,

ω = δd (21)u = δc, (22)

where the total steering angle is the sum of these two inputssuch that δ = δc +δd . Note that this is where we make use ofthe mechatronic-angle-superposition type steering actuators.For this problem we considered proportional-integral (PI)type state feedback controller of the form

u = KPξ +KIξI , (23)

where the integrator state ξI is the integral of the yaw ratetracking error with a zero initial condition, that is,

ξI = ψ− ψre f , ξI(0) = 0 . (24)

The reference yaw rate ψre f is the steady yaw rate whichresults from a constant driver input δd and zero control input;thus

ψre f = αδd , (25)

for a constant gain α . The above control structure is schemat-ically depicted on Figure 2 below.

Fig. 2. Flow diagram of the PI active steering controller.

We can describe the system resulting from (20), (24) and(25) by

ξ = Aξ +Bδd +Bu (26)ξI = ψ−αδd . (27)

We introduce the performance outputs z1,z2 which are theLT Rd given by (7) that helps in detecting the rollover like-lihood, and the control effort u that enables us to bound themaximum control effort. We are interested in synthesizing aL∞ stabilizing controller with closed loop performance mea-sures γ1 and γ2 for z1 and z2, respectively. These performanceoutputs can be expressed as follows:

z1 = Cξ (28)z2 = u, (29)

where C is given as in (7). We can now define a newaugmented state x = [ξ T ξI ]T and express (26)-(29) as

x = Ax+ Bδd + Buu

z1 = C1x (30)z2 = u,

with

A =[

A 0h 0

], B =

[B−α

], Bu =

[B0

](31)

C1 =[

C 0], (32)

where h = [ 0 1 0 0 ]. Also, the proposed controllerstructure (23) can be described by u = Kx where

K =[

KP KI]. (33)

We used Theorem 1 to design an L∞ controller withperformance levels γ j where j = 1,2. The model parametersfor (20) were tuned against a highly accurate commercialvehicle dynamics simulator to provide an exact steady statematch. The tuning was performed at vx = 40m/s with aδd = 30 step steering input, and the steering ratio wasassumed to be 1:17.5. The tuned parameters given in TableII are for a compact car. State responses to a step steeringinput and zero control input are given in Figure 3.

In order to find a controller gain matrices KP and KI sothe resulting closed loop system has desirable performance,we used an iterative solution algorithm based on the onedescribed in [2], [3] to obtain solutions to the matrix inequal-ities of Theorem 1. We attempted to minimize the level of

TABLE IIFIXED MODEL PARAMETERS

parameter descriptionm 1224.1 [kg]

Jxx,Jzz 362, 1279 [kg ·m2], respectivelylv, lh 1.102, 1.254 [m], respectively

T 1.51 [m]h 0.375 [m]c 4000 [kg ·m2/s]k 36075 [kg ·m2/s2]

Cv,Ch 90240, 180000 [N/rad], respectively

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Sta

tes

time [sec]

vy [m/s]

dψ/dt [rad/s]

dφ/dt [rad/s]

φ [rad]

Fig. 3. State responses to a step steering input.

performance γ1 for a specified level of performance γ2. In thenumerical simulations we simulated an obstacle avoidancemaneuver that is known as the elk-test, which takes placeat a speed of vx = 40m/s and the peak steering magnitudeof 100. The results are presented in Figures 4-6, whichdemonstrates the effectiveness of the controller.

Comment : Our design is easily extended to incorpo-rate compensation for parameter uncertainties such as theunknown vehicle parameters, velocity variations, unknownmass and center of gravity height.

VI. CONCLUSIONS

We have presented a methodology for the design of ve-hicle rollover prevention systems using differential braking.By introducing the load transfer ratio LT Rd , we obtain asystem performance output whose value provides an accuratemeasure for determining the onset of rollover. Our rolloverprevention system is based upon recent results from Pancake,Corless and Brockman, which provide controllers to robustlyguarantee that the peak value of the performance output of anuncertain system does not exceed a certain value. Simulationresults are presented to illustrate the benefits of the proposedapproach. Future work will proceed in several directions. Weshall extend the methodology to include differential braking,active suspension and combinations thereof to refine ourrollover prevention strategy. We shall also examine the effi-cacy of our controllers in the presence of conditions which

0 2 4 6 8 10 12 14 16 18 20−1.5

−1

−0.5

0

0.5

1

1.5

LTR

time [sec]

LTRd−uncontrolledLTRd−controlled

Fig. 4. Comparison of LT Rd for the controlled and uncontrolled systems.

0 2 4 6 8 10 12 14 16 18 20−20

−15

−10

−5

0

5

10

15

20

φ [d

eg]

time [sec]

φ−uncontrolledφ−controlled

Fig. 5. Comparison of φ for the controlled and uncontrolled systems.

can result in a tripped rollover. A second strand of workwill investigate refinement of the synthesis procedure. Inparticular, we shall investigate whether feasibility conditionscan be developed to determine the existence of control gainsto achieve certain pre-specified performance parameters γ j.Finally, we hope to implement and evaluate our controlsystem in real production vehicles in collaboration with ourindustrial partners.

REFERENCES

[1] Solmaz S., Corless M., and Shorten R., “A methodology for thedesign of robust rollover prevention controllers for automotivevehicles: Part 1-Differential Braking”, In Review for CDC 2006,San Diego, 2006.

[2] Pancake T., Corless M., and Brockman M., “Analysis and Con-trol of Polytopic Uncertain/Nonlinear Systems in the Presenceof Bounded Disturbance Inputs”, Proceedings of the AmericanControl Conference, Chicago, IL, June 2000.

[3] Pancake T., Corless M., and Brockman M., “Analysis and Controlof a Class of Uncertain/Nonlinear Systems in the Presence ofBounded Disturbance Inputs”, In Preperation.

[4] Ackermann J. and Odenthal D.,“Robust steering control foractive rollover avoidance of vehicles with elevated center ofgravity”,Proceedings of International Conference on Advances inVehicle Control and Safety,Amiens, France, July 1998.

0 2 4 6 8 10 12 14 16 18 20

−100

−80

−60

−40

−20

0

20

40

60

80

100δ

[deg

]

time [sec]

δ

d, driver input

δc, control input

δ , total input

Fig. 6. Comparison of the steering commands and the resulting steeringangle.

[5] Odenthal D., Bunte T. and Ackermann J.,“Nonlinear steering andbreaking control for vehicle rollover avoidance”,Proceedings ofEuropean Control Conference, Karlsruhe, Germany, 1999.

[6] Chen B. and Peng H.,“Differential-Breaking-Based Rollover Pre-vention for Sport Utility Vehicles with Human-in-the-loop Eval-uations”,Vehicle System Dynamics,36(4-5):359-389,2001.

[7] Kamnik R., Bottiger F., Hunt K., “Roll Dynamics and LateralLoad Transfer Estimation in Articulated HeavyFreight Vehicles:A Simulation Study”, Proceedings of the Institution of MechnaicalEngineers, Part D, 2003.

[8] National Highway Trafic Safety Administration (NHTSA), TraficSafety Facts 2004: A Compilation of Motor Vehicle Crash Datafrom the Fatality Analysis Reporting System and the GeneralEstimates System,Technical Report, 2006.

[9] Carlson C.R. and Gerdes J.C., “Optimal Rollover Prevention withSteer by Wire and Differential Braking”,Proceedings of ASMEInternational Mechanical Engineering Congress and Exposition,IMECE’03,Washington, D.C.,November 16-21,2003.

[10] Wielenga T.J., “A Method for Reducing On-Road Rollovers: Anti-Rollover Braking”, SAE Paper No. 1999-01-0123, 1999.

[11] Palkovics L., Semsey A. and Gerum E., “Roll-Over PreventionSystem for Commercial Vehicles-Additional Sensorless Functionof the Electronic Brake System”, Vehicle System Dynamics, 1999,Vol.4, pp.285-297.

[12] Akar M., Solmaz S. and Shorten R., “Method for Determining theCenter of Gravity for an Automotive Vehicle”, 2006, Irish Patent.

[13] Kiencke U. and Nielsen L., Automotive Control Systems for En-gine, Driveline and Vehicle, Springer-Verlag & SAE Int., Berlin,2000.

Hybrid Modelling and Control of the Common

Rail Injection System

Andrea Balluchi1,2, Antonio Bicchi2, Emanuele Mazzi1,2,Alberto L. Sangiovanni Vincentelli1,3, and Gabriele Serra4

1 PARADES, Via S. Pantaleo, 66, 00186 Roma, Italyballuchi, emazzi, [email protected]

2 Centro Interdipartimentale di Ricerca “Enrico Piaggio”,Universita di Pisa, 56100 Pisa, Italy

[email protected] Dept. of EECS., University of California at Berkeley, CA 94720, USA

[email protected] Magneti Marelli Powertrain, Via del Timavo 33, 40134 Bologna, Italy

[email protected]

Abstract. We present an industrial case study in automotive controlof significant complexity: the new common rail fuel injection system forDiesel engines, currently under production by Magneti Marelli Power-train. In this system, a flow–rate valve, introduced before the High Pres-sure (HP) pump, regulates the fuel flow that supplies the common railaccording to the engine operating point. The standard approach followedin automotive control is to use a mean–value model for the plant and todevelop a controller based on this model. In this particular case, this ap-proach does not provide a satisfactory solution as the discrete–continuousinteractions in the fuel injection system, due to the slow time–varying fre-quency of the HP pump cycles and the fast sampling frequency of sensingand actuation, play a fundamental role. We present a design approachbased on a hybrid model of the Magneti Marelli Powertrain common–railfuel–injection system for four-cylinder multi–jet engines and a hybrid ap-proach to the design of a rail pressure controller. The hybrid controller iscompared with a classical mean–value based approach to automotive con-trol design whereby the quality of the hybrid solution is demonstrated.

1 Introduction

Common–rail fuel–injection is the dominant system in diesel engine control. Incommon–rail fuel–injection systems (see Figure 1), a low-pressure pump locatedin the tank supplies an HP pump with a fuel flow at the pressure of 4–6 bars.The HP pump delivers the fuel at high pressure (from 150 to 1600 bars) to thecommon rail, which supplies all the injectors. The fuel pressure in the commonrail depends on the balance between the inlet fuel flow from the HP pump andthe outlet fuel flow to the injectors. The common–rail pressure is controlledto achieve tracking of a reference signal that is generated on–line (it dependson the engine operating point) to optimize fuel injection and to obtain propercombustion with low emissions and noise.

Joao Hespanha and A. Tiwari (Eds.): HSCC 2006, LNCS 3927, pp. 79–92, 2006.c© Springer-Verlag Berlin Heidelberg 2006

80 A. Balluchi et al.

Fig. 1. Common rail fuel injection system developed by Magneti Marelli Powertrain

In the novel fuel–injection system developed by Magneti Marelli Powertrain,a flow–rate valve located before the HP pump allows for effective control ofthe amount of fuel that is compressed to high pressure and delivered to therail. The HP pump and, hence, the rail are supplied with the precise amountof fuel flow that is necessary for fuel injection, achieving high efficiency of theinjection system. The previous fuel injection system, which was not equippedwith the flow–rate valve, was characterized by a high power consumption by theHP pump, which always supplied with the maximum fuel flow for the currentoperating condition (rail pressure control was achieved by a regulation valvelocated on the rail).

To control the rail pressure efficiently, we need to model accurately the in-teraction between discrete and continuous behaviours of the injection systemcomponents, exhibiting the pulsating evolution of the rail pressure due the dis-continuous inlet fuel flow from the HP pump and outlet fuel flows to the injectors.To do so, we present in this paper a hybrid model of the Magneti Marelli Pow-ertrain common–rail fuel–injection system for four-cylinder multi–jet engines.Motivated by the success in solving other automotive control problems using hy-brid system methodologies, e.g. cut-off control [1], intake throttle valve control[2], actual engaged gear identification [3], and adaptive cruise control [4], we de-veloped a hybrid rail pressure controller that exhibits excellent performance. Tocompare our solution with the standard design methodology adopted in the auto-motive industry based on mean–value models of the plant, we present a classicalSmith Predictor discrete–time controller. Simulations of the closed–loop systemshow that the mean–value model design approach does not achieve the samequality of design as the hybrid approach.

Hybrid Modelling and Control of the Common Rail Injection System 81

We believe this paper udnerlines the important role played by hybrid systemsin solving complex industrial control problems in a domain as economically rel-evant as the automotive sector.

2 Hybrid Model of the Common Rail Injection System

The proposed hybrid model of the injection system, shown in Figure 2, consistsof: the flow–rate valve, the HP pump, the injectors and the common rail [5]. Theproposed hybrid model describes accurately the interacting discrete and contin-uous behaviours of the injection system components, reproducing the pulsatingevolution of the rail pressure due the discontinuous inlet fuel flow from the HPpump and outlet fuel flows to the injectors. The rail pressure p [bar] is the con-trolled output. The flow–rate valve duty cycle u ∈ [0, 1] is the control input. Theinjectors fuel flow qINJ [mm3/sec] , which depends on the injectors opening timesET [sec], is considered as a disturbance to be compensated. The models of thecomponents of the system are described in the next sections using the hybridautomaton formalism [6].

Fig. 2. Hybrid model of the fuel injection system

2.1 The Flow–Rate Valve

The hybrid model of the flow–rate valve is depicted in Figure 3 and includes:the valve PWM1 electrical driver; the dynamics of the coil current I [A]; andthe relation between the coil current and the fuel flow–rate qM [mm3/sec] acrossthe valve.

The PWM electrical driver model is a hybrid model with as output a squarewave voltage vPWM(t) ∈ 0, Vbat given by pulse–width modulation of the batteryvoltage Vbat with duty cycle defined by the control input signal u(t) ∈ [0, 1]. Itsimplementation is based on a triangular wave generator with period T0 and

1 Pulse Width Modulation.


Fig. 3. Flow–rate valve hybrid model

output α(t), modelled as a hybrid system. The dynamics of the coil current Idepends on the coil resistance R and inductance L. The relation between thecoil current I and the fuel flow rate qM is given by a nonlinear function

qM = fM (I) (1)

represented as a piecewise affine expression (see [7]).

2.2 The HP Pump

The HP pump consists of three identical hydraulic rams mounted on the sameshaft with a relative phase of 120o (see Figure 4). Since the pump is powered by

Fig. 4. HP pump hybrid model


η(p, n)

n [rpm]

Fig. 5. HP pump efficiency

the camshaft, its revolution speed depends on the engine speed n [rpm]. Pumpefficiency reduces the fuel flow qI [mm3/sec] to the rams, i.e.

qI = η(p, n)qM (2)

where the efficieny η(p, n) depends on the rail pressure and the engine speed asdepicted in Figure 5. The HP pump fuel flow to the rail qP [mm3/sec] is obtainedby adding the contributions qP

i of the three rams: qP = qP1 + qP

2 + qP3 .

The partial closure of the flow–rate regulation valve produces the cavitationphenomenon in the pump, which affects both the intake and compression phases.For small effective area of the flow–rate valve, the pressure reduction in theram during the intake phase causes fuel vaporization [8]. As a consequence, theamount of fuel charge in volume is lower than the geometric displacement of thecylinder. The partial fuel charge depends on the amount of fuel vapor in thecylinder. In a first part of the compression phase, the ram does not deliver anyfuel to the rail. In fact, at the beginning of the compression phase, the increase ofpressure inside the cylinder causes fuel condensation only. The outlet flow to therail starts when the fuel is completely in the liquid state, i.e. when the geometricvolume of the cylinder (which decreases during compression) equals the fuelcharge in volume. From this time on, pressure increase in the ram produces theopening of outlet valve and the exit of the compressed fuel to the rail.

The hybrid model of the i-th ram of the HP pump is depicted in Figure 6. Itsevolution is determined by the ram angle φi [o]. Since the camshaft revolutionspeed is half the engine speed n, then the ram angle dynamics is φi = 360

2n60 =

3n, where n is the engine speed in rpm.The hybrid model contains two macro discrete states corresponding to the

intake and compress phases, which have durations of half camshaft cycle. Thepumping cycle starts with the beginning of the intake phase, which is triggered bythe guard φi = 180o. The camshaft sensor detects the beginning of the pumpingcycle by emitting the output event triggeri at transition time.

Since the intake duration is 180o and the three rams are mounted with arelative phase of 120o, then the intake phases of the rams partially overlap.Intake overlapping results in different supplying fuel flow to the rams. Rams


Fig. 6. Hybrid model of the i-th ram of the HP pump

overlapping is modelled in the i-th ram hybrid model by including three discretestates I1, I2 and I3 inside the intake state. In each state the model dwells fora duration of 60o of the ram angle φi. Concurrent intake with one of the otherrams occurs in the first and the last part of the intake, i.e. in I1 and I3. Assumingthat, in case of concurrent intake, both rams receive half of the flow qI givenby (2), then the amount of fuel vi [mm3] inside the i-th ram is subject to thedynamics: vi = qI/2 in I1 and I3; and vi = qI in I2.

The compression state consists of two different states: C1, modeling fuelcondensation, and C2, modeling fuel delivery to the rail. On entering the com-pression state, the ram angle φi is reset. During fuel condensation in state C1,the fuel charge in the ram remains constant (vi = 0) and the fuel flow–rate tothe rail qP

i is zero. The system remains in state C1 while the geometric volumeof the ram V (cos(φi) − 1) is greater than the fuel charge vi. When all fuel isat the liquid state (i.e. vi = V (cos(φi) − 1)), the model switches to state C2

where: the outlet valve is open, the compressed fuel flows towards the rail withflow–rate qP

i = V sin(φi), and the ram fuel charge decreases as vi = −V sin(φi).The compression state is left when the ram angle φi reaches 180o.

2.3 Injectors

The common rail supplies four injectors, one for each cylinder of the engine.In multi–jet engines, each injection phase is composed by a sequence of 3 to 5


distinct injections. However, in most of the engine operating conditions onlythree injections are used. For the sake of simplicity, we consider this case. Thethree injections are: a pilot injection (applied to reduce combustion time by in-creasing cylinder temperature and pressure), a pre-injection (used to reduce pro-duction of emissions by optimizing combustion conditions) and a main injection(which produces the desired engine torque). Having the engine four cylinders,the frequency of injection sequences is twice the engine speed. The engine torquecontroller implemented in the engine control unit defines the amount of fuel to beinjected and, consequently, the durations ET = (τPIL, τPRE , τMAIN ) [sec] andphases (θPIL, θPRE , θMAIN ) (expressed in crank angle) of each fuel injection,depending on the engine operating condition.

The amount of fuel that flows from the common rail to each injector is the sumof three different terms: the flow that enters the combustion chamber Qinj , a flownecessary to keep the injector open Qserv, and a leakage flow Qleak. The lattertwo are collected into the tank. While the leakage flow–rate Qleak is a continuoussignal, the flow–rate Qinj and Qserv are not zero only when the injector isopen. Since the common rail model is zero-dimensional and in each engine strokeonly an injector is operated, then there is no loss of generality in referring thequantities Qinj , Qserv, Qleak to the overall contribution of the four injectors tothe common rail balance, with injection frequency twice the engine speed.

The fuel flow–rate qINJ [mm3/sec] out of the common rail is represented bythe hybrid model reported in Figure 7, where qL denotes the leakage flow Qleak

and qJ stands for the sum of the Qinj and Qserv flows.

Fig. 7. Hybrid model of the injectors


The three states on the top of the model represent the synchronization phasesfor the opening of the injectors, which are defined in terms of guards on thecrankshaft angle θ [o] that evolves from 0 to 180o with dynamics θ = 6n. Pa-rameters θPIL, θPRE , θMAIN denote the corresponding start of injection an-gles. In these states, the fuel flow to the injectors is due to leakage only, i.e.qJ = 0.

As soon as the guard conditions θ = θPIL, θ = θPRE , θ = θMAIN becometrue, a transition to the corresponding state on the bottom takes place, and thetimer τ is initialized to the current injection duration time τPIL, τPRE , τMAIN .The three states on the bottom model the system with one injector open. Theflow to the open injector depends on the engine speed and the rail pressure:qJ = fJ(n, p) = Qinj(p, n) + Qserv(p, n). The system remains in the injectionstates until the injection time elapses, i.e. τ = 0.

2.4 Common Rail

The dynamics of the rail pressure is obtained by considering the balance betweenthe HP pump inlet flow and injectors outlet flows. Under the assumption of notdeformable rail, the fuel volume is constant, while the capacity depends on thepressure and temperature of the fuel in the rail according the Bulk module,which takes into account fuel compressibility. The evolution of the rail pressureis given by:

p(t) =KBulk

Vrail

(qP (t) − qINJ(t)

), (3)

where the HP pump fuel flow qP is given by the hybrid model in Figure 4 andthe injector fuel flow qINJ is given by the hybrid model in Figure 7.

p[bar]

qP

[mm3/sec]

qINJ

[mm3/sec]

t [sec]

Fig. 8. Rail pressure pulsating profile and HP pump and injectors fuel flows


Simulation results obtained with the proposed common rail hybrid modelshow that it nicely represents the pulsating behaviour of the common rail pres-sure due to the HP pump and injectors discontinuous evolutions. Figure 8 reportsa typical evolution of the common rail pressure, along with the pulsating fuelflows of the HP pump and the injectors. When the pump delivers the fuel, thepressure increases while when the injectors open, the pressure decreases.

3 Control Design

The objective is to design a feedback controller for the rail pressure that achievestracking of a reference pressure signal. The latter is generated on-line by anouter loop control algorithm so to optimize fuel injection and obtain proper fuelcombustion, with low emissions and noise, for the current engine operating point.The specifications for the rail pressure controller are:

– steady state rail pressure error lower than 30 bar;– settling time lower than 150 mseconds;– undershoot/overshot lower than 50 bar, for a ramp of rail pressure reference

with rate 800 bar/sec, at 1000 rpm, with 15 mm3/stroke fuel injection.

The most important aspect to be taken into account in the design of the controlalgorithm is the varying time delay between the flow–rate valve control commandu and the pulsating fuel flow from the HP pump to the rail. This delay is dueto HP pump cycles and is roughly in inverse proportion to engine speed. As aconsequence, the control task is particularly critical during cranking and at lowengine speed.

3.1 Controller Based on the Smith Predictor

In this section, we develop a “standard” controller based on a mean–value modelof the plant. To cope with the large and time–varying loop delay, the controlleris based on the Smith Predictor. The rail pressure Smith Predictor controller(see e.g. [9, 10]) is obtained following the standard approach to controller designadopted in the automotive industry that is based on mean–value modelling ofthe plant. The following continuous time model is considered:

I(t) = −R

LI(t) +

vPWM(t)L

(4)

p(t) =Kbulk(p)

Vrail

[(qP (t − Td) − qINJ(t)

](5)

where Td = 120/n is an estimate of the loop delay. The controller includes amodel of the high pressure circuit and a PID with anti–windup and feedforwardterms. The control algorithm is implemented in discrete time, with a samplingtime of 5 mseconds. Satisfactory rail pressure tracking is achieved provided thatthe rate of variation of the reference pressure is not too large. Figure 9 reportsa typical rail pressure evolution.


p[bar]

p[bar]

t [sec] t [sec]

Fig. 9. Closed–loop hybrid system simulation results with the Smith Predictor: forslow (left) and fast (right) pressure references

However, the tracking performance significantly degrades and large overshootsare produced for fast rail pressure reference signals, as described in Figure 9. Onthe other hand, the simulation of the Smith Predictor controller against themean–value model exhibits the expected behaviour showing that the controlleris able to compensate properly the time delay. Hence, the poor tracking perfor-mances shown in the simulations with the common rail hybrid model demon-strate that mean–value modelling is not accurate enough to design high qualitycontrol. In fact, major difficulties in the calibration of mean–value model–basedcontrollers for fast reference pressure signals were observed by Magneti MarelliPowertrain. From the closed-loop hybrid model simulation shown in Figure 9,to be able to efficiently track fast pressure references, the controller should bedesigned taking into account each single fuel delivery of the HP pump. In fact,in the reported simulation, only three compression phases of the HP pump drivethe pressure close to the target value. From a physical point of view, the HPpump combines a sequence of control actions to determine the fuel charge foreach single cycle. However, this behaviour is not taken into account by the pres-sure controller designed on the basis of the mean–value model of the system,which then exhibits large overshoot.

This analysis motivates the search for a better solution that can be offeredby designing a hybrid controller that is based on the accurate hybrid modelpresented above.

3.2 Hybrid Multi–rate Controller

During the intake phases, the HP pump combines a sequence of control actions todetermine the fuel charge for each single cycle. Hence, the HP pump introducesan under–sampling of the control actions. The slow frequency of intake anddelivery of the HP pump is time varying since it depends on the engine speed.A hybrid system approach to controller design allows us to effectively handlethe under–sampling produced by the HP pump cycles and properly handle thedrift between the fast frequency of sensing and actuation (at 5 mseconds) andfrequency of the HP pump [11].


Fig. 10. Hybrid multi–rate controller

The proposed hybrid multi–rate controller, showed in Figure 10, consists ontwo regulators:

– The CM pressure controller is event–based and is synchronous with theHP pump fuel intake phases (it receives the HP pump trigger event fromthe camshaft sensor). This controller defines the desired fuel mass QHP (k)[mm3/stroke] needed to control the rail pressure error perr(k) to zero. A PIcontrol with anti-windup and feedforward terms is used for this purpose.

– The flow–rate valve controller runs at 5 mseconds. Its task is to feed the highpressure circuit with the amount of fuel QHP (l) requested by the outer loopcontroller. Due to the lack of a fuel flow–rate sensor downstream the valve,the flow–rate valve controller has to be open-loop. The duty cycle control uis obtained by abstracting away the coil current dynamics and inverting theflow–rate valve characteristic (1) and the PWM model, i.e.

u =23

R

VbattfM

−1(QHP (l)). (6)

The factor 23 is introduced to take into account the partial overlapping of

the intakes phases of the rams in HP pump.

Smooth and effective coupling between the different time domains of pressuresensing, CM pressure control and flow–rate valve control is achieved by using adecimator and an interpolator [12].

– The decimator converts the high frequency pressure error perr(l) = p(l) −pref (l), having sampling time 5 mseconds, to the time–varying HP pumpfrequency. An IIR low–pass filter is employed (see Figure 11).


pref (l)pref (k)

[bar]

QHP (k)

QHP (l)[mm3/sec]

t [sec]

Fig. 11. Signal conversions provided by the decimator and the interpolator

p[bar]

t [sec]

Fig. 12. Comparison between the proposed hybrid multi-rate controller and a controllerbased on the Smith Predictor developed using a mean–value model of the plant

– The interpolator converts the fuel mass signal QHP (k) in [mm3/stroke],synchronous with the time–varying HP pump frequency to the 5 mseconddiscrete–time domain, QHP (l) in [mm3/sec] used by the flow–rate valve con-troller. An IIR low–pass filter is employed in the interpolator. The inter-polator produces a smooth and uniform input signal to the flow–rate valvecontroller as illustrated in Figure 11.


Both the decimator and the interpolator implement a gain scheduling of thecut–off frequency based on engine speed to compensate the variation of the HPpump frequency.

The simulation results presented in Figure 12 show the improvement obtainedby the proposed hybrid multi–rate controller with respect to a controller basedon the Smith Predictor presented in the previous section. Both controllers havebeen tuned to meet the specification on bounded overshoot. The settling timeof the hybrid multi–rate controller is significantly shorter than the one of theSmith Predictor controller. Moreover, the hybrid multi–rate regulator, whichimplements a PI algorithm and two low–pass filters, is significantly simpler thanthe Smith Predictor that includes an internal model of the plant. Finally, whilethe Smith Predictor is affected by a time delay estimation error, in the multi-ratecontroller the loop delay is simply represented by a one step delay. Simulationresults show that the hybrid multi–rate controller is robust to phase errors be-tween the CM pressure controller execution and the beginning of intake phasesof the rams.

4 Conclusions

We presented a relevant problem in diesel engine control that has been solvedwith a hybrid system approach. We first developed a hybrid model that takesinto account the interactions between the discrete dynamics of the componentsof the common rail system.

Then we demonstrated the superiority of a hybrid multi–rate control al-gorithm versus the standard mean-value model approach to controller designadopted in the automotive industry. To do so, we designed a Smith Predic-tor controller to compensate the loop delay. Simulation results show that suchcontroller achieves satisfactory tracking only for slow rail pressure reference sig-nals. Figure 12 illustrates the improvement achieved by using the multi–ratecontroller.

In summary, we demonstrated how the use of hybrid models and control al-gorithms can produce superior results versus standard control approaches basedon mean–value models for a relevant and complex industrial problem.

References

1. Balluchi, A., Benedetto, M.D.D., Pinello, C., Rossi, C., Sangiovanni-Vincentelli,A.L.: Hybrid control in automotive applications: the cut-off control. Automatica:a Journal of IFAC 35 (1999) 519–535

2. Baotic, M., Vasak, M., Morari, M., Peric, N.: Hybrid theory based optimal controlof electronic throttle. In: In Proc. of the IEEE American Control Conference,Denver, Colorado, USA, ACC (2003) 5209–5214

3. Balluchi, A., Benvenuti, L., Lemma, C., Sangiovanni-Vincentelli, A.L., Serra, G.:Actual engaged gear identification: a hybrid observer approach. In: 16th IFACWorld Congress, Prague, CZ, IFAC (2005)


4. Mobus, R., Baotic, M., Morari, M.: Multi-object adaptive cruise control. HybridSystems: Computation and Control 2623 (2003) 359–374

5. Millo, F.: Il sistema common rail. Technical report, Dipartimento di Energetica,Politecnico di Torino (2002)

6. Henzingerz, T.A.: The theory of hybrid automata. Technical report, ElectricalEngineering and Computer Sciences University of California, (Berkeley)

7. Bosch: Injection Systems for Diesel Engines. Technical Customer Documents.(2003)

8. Knapp, R.: Cavitation. McGraw-Hill (1970)9. Rath, G.: Smith’s method for dead time control. Technical report (2000)

10. Mirkin, L.: Control of dead-time systems, K.U.Leuven - Belgium, MathematicalTheory of Networks and Systems (2004)

11. Glasson, D.: Development and applications of multirate digital control. IEEEControl Systems Magazine 3 (1983) 2–8

12. Vaidyanathan, P.: Multirate digital filters, filter banks, polyphase networks andapplications: a tutorial. Proceedings of the IEEE 78 (1990) 56–92

Submitted to the HYCON & CEMaCS Workshop on Automotive Systems and Control Control

Lund University, Lund Sweden, June 1–2, 2006

Error Feedback Nonlinear Control of

Electromagnetic Valves for Camless Engines

S. Di Gennaro, B. Castillo–Toledo, and M. D. Di Benedetto∗

March 13, 2006

Abstract

Conventional internal combustion engines use mechanical camshafts to command the openingand closing phases of the intake and exhaust valves. The lift valve profile is designed in orderto reach a good compromise among various requirements of the engine operating conditions. Inprinciple, optimality in every engine condition can be attained by camless valvetrains. In this context,electromagnetic valves appear to be promising, although there are some relevant open problems. Infact, in order to eliminate acoustic noises and avoid damages of the mechanical components, thecontrol specifications require sufficiently low impact velocities between the valve and the constraints(typically the valve seat), so that “soft–landing” is obtained. In this paper, the soft–landing problem istranslated into a regulation problem for the lift valve profile, by imposing that the valve position tracksa desired reference, while the modeled disturbances are rejected. Both reference and disturbance aregenerated by an autonomous system, usually called exosystem. The submanifold characterized bythe zeroing of the tracking error and the rejection of the disturbance, is determined. Finally, thestabilization problem of the system trajectory on such a manifold is solved, both in the case of stateand output measurement.

1 Introduction

Conventional internal combustion engines use mechanical camshafts to command the opening and closingphases of the intake and exhaust valves. The lift valve profile, connected with the crankshaft angle andobtained with a proper cam profile, is designed in order to find a compromise among various require-ments, such as the engine efficiency, pollution emissions, fuel economy, valvetrain noise and vibration,maximization of the output torque and power. In fact, the different engine operating conditions wouldneed different lift valve profiles and valve timings, which can not be dynamically changed in mechanicallydriven camshaft. To overcome these limitations and optimize the aforementioned requirements, a solutionbased on variable valve timing can be pursued.

In this context, camless valvetrains are devices recently considered to decouple the camshaft and thevalve lift dynamics ([4], [17], [15], [1]). Their main advantages are fuel saving, increase of maximum andlow speed torque, flatting of the toque characteristic and improve of driveability, pollution and energyconsumption reductions, increase of the burn rate, possible variability of the compression rate, improvingof the combustion stability at low speed [10]. On other hand, the electrohydraulic or electromechanicalactuators of camless engines present considerable problems, which still remain open. In fact, theseactuator dynamics are highly nonlinear and unstable near the valve’s terminal positions, while the controlspecifications require that impact velocities between the valve and the constraints (typically the valve seat)be sufficiently low in order to eliminate acoustic noises and avoid damages of the mechanical components.These problems are complicated by the short time (typically 3 – 5 s) available at high engine speed tomake a transition between the two valve’s terminal positions, and the constraint in terms of actuatorcost and space limitations. These last aspects imply that one typical request is the absence of the valveposition sensor.

∗This work was partially supported by European Commission under Project IST NoE HyCON contract n. 511368. S. DiGennaro and M. D. Di Benedetto are with Department of Electrical Engineering and Center of Excellence DEWS, Universityof L’Aquila, Poggio di Roio, 67040 L’Aquila, Italy. E-mail: dibenede, [email protected]. B. Castillo–Toledois with the Centro de Investigacion y de Estudios Avanzados – CINVESTAV del IPN Unidad Guadalajara, A.P. 31–438,Plaza La Luna, Guadalajara, Jalisco 44550, Mexico. E. mail: [email protected]

In [5] a precise control of the voltage applied to the coils has been proposed. A controller with aconstant preset voltage, augmented by a voltage command based on a linear feedback, is determined andthen modified by an Iterative Learning Controller (ILC). In [18] a control–oriented linear model for anEMCV has been considered, based on a gray-box approach which combines mathematical modeling andsystem identification. In [19] a physics–based model for an EMCV is derived. Moreover, a sensitivitystudy has been conducted to characterize the ability of the control signal to affect the reduction on contactvelocities. This model has been enriched by [12] by introducing the impact dynamics, and a self–tuningnonlinear controller has been design. An observer based output feedback controller has been proposed in[13], while in [14] linear, non linear and cycle–to–cycle self–tuning controllers have been considered.

Our work use the model of an electromechanical actuator presented in [11], [16]. The soft–landingproblem is translated into a regulation problem for the lift valve profile, by imposing that the valve positiontracks a desired reference, while the modeled disturbances are rejected. Both reference and disturbanceare generated by an autonomous system, usually called exosystem. The submanifold characterized by thezeroing of the tracking error and the rejection of the disturbance, is determined. Finally, the stabilizationproblem of the system trajectory on such a manifold is solved. This stabilization problem is solved firstin the case of measurement of the whole state. Indeed this is an unrealistic situation. In fact, as alreadynoted, a technological requirement is the elimination of the valve position and velocity sensors. Theselimitations are solved by an output stabilizing controller, which makes use of output measurements only.The limitation of this last controller is due to the fact that it needs the knowledge of the disturbanceacting on the valve. Nevertheless, this controller represents a first step in the development of a dynamiccontroller using only output measurements.

2 Mathematical Model of the Electromagnetic Valves and Prob-lem Formulation

In this section we briefly recall the mathematical model of an electromagnetic valve [3], representedin Figure 1. This valve is composed of an anchor moving between two electromagnets. Thanks to theattractive forces applied by the electromagnets on the anchor rigidly connected with the stem of the valve,this latter can be opened or closed. In what follows we first present the dynamics of the electromagnets,then the mechanical dynamics.

Valve spring

Engine Head

AnchorCoils

Torsion bar

Electromagnet 2

Electromagnet 1

Coils

xv

Vs1

V1 ½

¡½

0

Fm2,ma

Fm1,ma Fm1

Fm2

Fg

Valve

Valve Stem

Fel,v

V2

Vs2

Ffr,v

d

d1

d2

Figure 1: Scheme of a Electromagnetic Valve System

The equation describing the dynamics of the electric windings of the electromagnetic valve are [11], [16]

φmj =1N

(Vmj −RjImj), j = 1, 2 (1)

where φmj are the magnetic fluxes, Rj is the electrical resistances, and N is the number of turns of thewindings, which is the same for both windings for each magnet. Moreover, Vmj , Imj are the voltages andcurrents of the electric circuits.

The currents Imj can be given as a function of φmj by expressing the magnetomotive force Mmj asthe sum of two terms [16]

Mmj(xa, φmj) = Hj(φmj) +Rj(xa)φmj (2)

j = 1, 2, with Hj(φmj) a nonlinear function of the flux φmj , describing the nonlinear effects in the iron,while the magnetic reluctance Rj(xa) takes into account the air–gap portion, and is a nonlinear functionof the anchor tip position xa ∈ [−ρ, ρ]. Here 2ρ is the anchor tip displacement (coinciding with the valvevertical stroke), and xa is positive in the same direction of xv, see Figure 1. Experimentally, one obtainsthe following expressions

Rj(xa) = −aj

bje−bjxa + cjxa + dj , j = 1, 2

where aj , bj , cj , dj are parameters obtained by identification.Denoting by Ipj the eddy currents in parasitic circuits with resistance Rpj and inductance Lpj coupled

with the magnetic circuits, having dynamics [11]

−RpjIpj = φmj + Lpj Ipj (3)

j = 1, 2, one obtains the expression of the magnetomotive forces

NImj + Ipj = Mmj = Hj(φmj) +Rj(xa)φmj (4)

j = 1, 2, where (2) has been used. Therefore, by substituting (4) into (1), one gets the flux dynamics

φmj =1N

Vmj − Rj

N2

(Mmj(xa, φmj)− Ipj

)(5)

j = 1, 2. Moreover, using (3), (5), one works out the eddy current dynamics

Ipj = − 1Lpj

(Rpj +

Rj

N2

)Ipj +

Rj

LpjN2Mmj(xa, φmj)

− 1LpjN

Vmj , j = 1, 2.

(6)

The mechanical dynamics of the actuator can be obtained by writing the balance of the forces actingon the anchor, of mass ma, and on the valve, of mass mv, supposed always connected. Moreover, thecontribution due to the gravity is supposed to be negligible, and all the mechanical parts of the valve areconsidered rigid bodies.

The (attractive) forces developed by the electromagnets on the valve are

Fmj = −12R′j(xa)φ2

mj (7)

with

R′j(xa) =dRj(xa)

dxa= aje

−bjxa + cj

j = 1, 2. Clearly R′1(xa) < 0 and R′2(xa) > 0, so that Fm1 > 0 (closing the valve) and Fm2 < 0 (openingthe valve), accordingly to the convention on the force signs. It is clear that both electromagnets maybe active at the same time in order to impose some specific behaviors at the anchor. For the sake ofthe simplicity and without loss of generality, we suppose that only the electromagnet 1 (see Figure 1) isactive during the closing phase of the valve, while during the opening phase that only the electromagnet2 is active.

A torsional spring keeps the ancor in an intermediate position. This force can be rendered equivalent(by equating the momenta) to a force Fel,a = −ka(xa − xa) acting on the tip of the anchor, with xathe intermediate position, and ka the elastic coefficient for the torsion bar. Hence, considering the

(equivalent) spring elastic force Fel,a, the viscous friction force Ffr,a = −baxa, the net magnetic forceFm = Fm1 + Fm2, and the constraint force Fc,a due to the electromagnet surface, each considered withits sign, the anchor motion equation is

maxa = Fel,a + Ffr,a + Fm + Fc,a

= −ka(xa − xa)− baxa + Fm + Fc,a

(8)

where ba is the viscous friction for anchor. Note that Fc,a is always zero except for xa = ±ρ.The valve has two springs, one closing it and another opening it, which can be modeled equivalently by

a single linear spring, preloaded to keep the valve in the center of its stroke xv when the two electromagnetsare not supplied, so that the elastic force Fel,v, due to the (equivalent) valve spring is Fel,v = −kv(xv−xv),with kv the elastic coefficient of the (equivalent) linear spring. Considering the other forces acting onthe valve of mass mv, namely the viscous friction Ffr,v = −bvxv, the force Fg due to the exhaust gasesexiting the cylinder, and the force Fc,v due to constraint given by the valve seat, the valve dynamics are

mvxv = Fel,v + Ffr,v + Fg + Fc,v

= −kv(xv − xv)− bvxv + Fg + Fc,v

(9)

with bv the viscous friction for the valve. The force Fg is a disturbance acting on the valve. It dependson the crankshaft angle ϑ and parameters, such as the valve equivalent area, the load, the gas turbulence,etc. A typical behavior of Fg is given in Figure 2 as function of ϑ, along with a valve lift reference. Themodel of Fg in a specific case will be given in the following sections. Note that Fc,v is always zero exceptfor xv = ρ.

Combining (8), (9) one gets the mechanical dynamics. Since the anchor is connected with the valveduring its stroke, xa = xv. Hence, adding (8) and (9) and considering (7), we work out the mechanicaldynamics

Mxv = −kxv − bxv + Fm + Fg + Fc

Fm = −12

(δm1(t)R′1(xv)φ2

m1 + δm2(t)R′2(xv)φ2m2

) (10)

where M = ma +mv, b = ba +bv, k = ka +kv, Fc = Fc,a +Fc,v, and δmj(t) is a function of time describingwhen the electromagnet j = 1, 2 is on. Here we have set the center of the valve stroke xv as the originfor xv. Equation (10) holds for xv ∈ [−ρ, ρ], where ρ is the maximal displacement with respect to xv. Itis worth noting that Fc is always zero except for x = ±ρ, in which

Fc(ρ) = kρ− Fm(ρ)− Fg

Fc(−ρ) = −kρ− Fm(−ρ)− Fg.

The mathematical model of the system is given by equations (5), (6), (10). The dynamics for thecrankshaft angle ϑ and velocity ω have not been considered here for the sake of simplicity, even if theapproach presented in the following sections can be applied also for the complete engine dynamics.Therefore, in what follows we consider that the engine has reached a steady–state with ω = ω0 known.

The control problem is to determine a controller such that the valve is opened and closed following adesired valve lift reference trajectory (see Figure 2), while the disturbance due to Fg is rejected.

A central aspect connected with the proper valve motion is the so–called soft landing of the valve.This problem comes into play either when the valve is closing or is reaching its maximum aperture,namely when the anchor is approaching one of the electromagnet surfaces (we have supposed the ancorand the valve rigidly connected). Clearly, this ensures also a soft landing for the valve in its seat duringthe closing phase, and avoids the valve chatter in the opening phase. Typical values of the valve velocityapproaching the mechanical constraints (seating velocity) is 0.05 – 0.1 m/s.

One can easily translate the control problem with soft landing into a regulation problem for the valvestem, imposing that the valve position xv tracks a desired reference xr, while the disturbance due toFg is rejected. As it will be clearer in the next sections, in the following we will suppose that both thereference xr and the disturbance Fg will be functions xr(w), Fg(w) of the state w of a system modelingperturbations and references.

Although various policy can be followed for the supply of the electromagnets (for instance, someauthors propose also that the closed–loop control is unnecessary when the anchor is in the intermediateposition, since high currents would be necessary to impose an effectual force [10]), in order to reducethe control effort and for the sake of simplicity we will suppose that only one electromagnet is activewhen attracting the anchor. Therefore, when the magnetomotive force is positive (negative) only theelectromagnet 1 (electromagnet 2) is active, and δm1(t) = 1 (δm2(t) = 1) in (10). This yields a description

xrj(#c)

j=1

j=2

j=3

j=4

#c0 #c1 #c2 #c3

#c

#c0+2¼

½

¡½

h=1

h=2

#c

Fg,h(#c)

Fg0

#c4 #c4+¢d

#c0+2¼#c0

#c4+¢d¡2¼

Figure 2: Reference trajectory xr and disturbance Fg

of the system by four models, each corresponding to a state q ∈ Q = q1, q2, q3, q4 of a finite stateautomaton (see Figure 3): one for the closing phase, one when the valve is completely close, one for theopening phase, and finally one when the valve is completely open. The transitions among these modelsdepend on the value of the system state x =

(xv vv φm Ip

)T so that the resulting system is hybrid andof the form

x = fi(x,w, u) (11)e = hi(x,w) (12)

for i ∈ I = 1, 2, 3, 4, with hi = xv − xr,

fi =

A

(xv

vv

)+ B

(Fm + Fg + Fc

)

RN2

(Ip −Mm(xv, φm) + N

R u)

− 1Lp

RpIp − 1Lp

φm

where

A =(

0 1

− kM − b

M

), B =

(01M

)(13)

Fm = − 12R′(xv)φ2, and where for the various functions, parameters and invariant conditions see Table 1,

with Fc2 = kρ + 12R′1(ρ)φ2

m1 − Fg, Fc4 = −kρ + 12R′2(−ρ)φ2

m2 − Fg.

q1 q2 q3 q4

R = R1 R = R1 R = R2 R = R2

H = H1 H = H1 H = H2 H = H2

R = R1 R = R1 R = R2 R = R2

Rp = Rp1 Rp = Rp1 Rp = Rp2 Rp = Rp2

Lp = Lp1 Lp = Lp1 Lp = Lp2 Lp = Lp2

I = I1 I = I2 I = I3 I = I4

Fc = 0 Fc = Fc2 Fc = 0 Fc = Fc4

Table 1

whereI1 =

xv ∈ (−ρ, ρ), Fm ≥ 0, φm = φm1, Ip = Ip1, u = Vm1

I2 =xv ≡ ρ, Fm ≥ 0, φm = φm1, Ip = Ip1, u = Vm1

I3 =xv ∈ (−ρ, ρ), Fm < 0, φm = φm2, Ip = Ip2, u = Vm2

I4 =xv ≡ −ρ, Fm ≤ 0, φm = φm2, Ip = Ip2, u = Vm2

.

Note in particular that when the system is in the states q2, q4, the first two equations in (11) are identicallyverified, since the constraint force Fc takes the values shown in Table 1.

The transitions among these four models are forced when the invariant conditions x ∈ Ii, i = 1, · · · , 4,are violated [9]. Moreover, the transitions leaving a state qi ∈ Q are regulated by the so–called guardtransitions, namely rules stating when a certain transition can take place. Referring to Figure 3,Gj,k =

x ∈ Ik

for j, k = 1, 2, 3, 4. Finally, after a transition x can possibly undergo a reset of

(some of) its components. In the case under study the reset functions are the identities.

q1

.x = f1(x,w,u) x 2 I1

q2

.x = f2(x,w,u) x 2 I2

R12G12

G23

G41

R41

R23

q4

.x = f4(x,w,u) x 2 I4

G34R34

q3

.x = f3(x,w,u) x 2 I3

R13

G13

G31

R31

Figure 3: Hybrid system modelling the EMVS

3 The Regulation Problem for EMVS

In this section we will solve the regulation problem for a hybrid systemH, characterized by the continuousnonlinear dynamics (11), (12) along with the autonomous nonlinear dynamics

w = si(w) (14)

with i ∈ I = 1, 2, · · · , N, I a finite index set, x(t) ∈ IRn the state of the ith subsystem (11),w(t) ∈ Wi ⊂ IRr the state of a Poisson stable external signal generator described by (14), calledexosystem [6], which provides the reference and perturbation signals, and u(t) ∈IRm the input signal.

3.1 The Reference Trajectory and Disturbances for an EMVS

The exosystem models the reference trajectory and the disturbances acting on the system. For the sakeof simplicity and without loss of generality, some simplifications will be considered in the following inorder to better illustrate the proposed approach. For instance, we consider that all the moving parts ofthe valve are rigid bodies, and we consider as disturbances only the pressures of the intake/exaust gases.Flexibility in the valve mechanism, usually modeled as disturbance, and/or other perturbations could beeasily taken into account at the expense of a more complicated presentation.

As far as the reference trajectory is concerned, we will consider a trajectory ensuring the soft landingof the valve in its seat and of the anchor on the surfaces of the electromagnets. Such a trajectory is, forinstance, that imposed by a mechanical cam. In our case the EMVS allows for more general references,satisfying some condition of obvious physical interpretation. In fact, it is possible to impose a trajectorycomposed of four parts (see Figure 2), depending on the cam’s angle ϑc = ϑ/2 ∈ [0, 2π), where ϑ ∈ [0, 4π)is the crankshaft’s angle, solution of ϑ = ω0 (since ω0 is here considered constant, this dynamic equationhas not been considered in the system model). We will require that the reference be at least a C4 functionof ϑc (and hence of time). Later on it will be clearer that this condition will imply that the input ensuringthe tracking will be a continuous function.

The first part of the reference (corresponding to j = 1 in Figure 2) is parameterized as follows

xr1(ϑc) =7∑

i=0

a1iϑi

c

i!

with the coefficients in a1i, i = 0, 1, · · · , 7, determined so that at ϑc = ϑc0 = 0 the valve (completelyopen) starts to close with velocity, acceleration and jerk (namely up to the third derivative) equal to zero,and is completely closed (with zero velocity, acceleration and jerk) at ϑc = ϑc1 (8 conditions)

xr1(ϑc0) = −ρ =7∑

i=0

a1iϑi

c0

i!,

dkxr1

dϑkc

∣∣∣ϑc0

= 0 =7∑

i=k

a1iϑi−k

c0

(i− k)!,

xr1(ϑc1) = ρ =7∑

i=0

a1iϑi

c1

i!

dkxr1

dϑkc

∣∣∣ϑc1

= 0 =7∑

i=k

a1iϑi−k

c1

(i− k)!, k = 1, 2, 3.

In this way one obtains the expressions of the coefficients a1i, i = 1, · · · , 7 (see Appendix). In a similarway, it is easy to check that the second part of the reference (corresponding to j = 2 in Figure 2), is simplyxr2(ϑc) = ρ, for ϑc ∈ [ϑc1, ϑc2), the third part (j = 3) is xr3(ϑc) = −r1(ϑc − ϑc2) for ϑc ∈ [ϑc2, ϑc3),while the fourth part (j = 4) is simply xr4(ϑc) = −ρ, for ϑc ∈ [ϑc3, ϑc0 + 2π).

These four references can be generated by (recall that ϑc = ω02 t)

w0 = 0, wi =ω0

2wi−1, i = 1, · · · , 7 (15)

with w0(0) = 1, wi(0) = 0, and with appropriate reset of the initial conditions and switching among thefollowing different outputs (see Figure 4)

xrj(w) =7∑

i=0

ajiwi, j = 1, · · · , 4. (16)

Here a20 = ρ, a21 = · · · = a25 = 0, a40 = −ρ, a41 = · · · = a45 = 0, while a3i = −a1i, i = 1, · · · , 7.As far as the disturbance Fg is concerned, it can be generated by

w8 = 0, w9 =ω0

2w8, w8(0) = 1, w9(0) = 0 (17)

with appropriate reset of the initial conditions and switching among the following disturbance signals(see Figure 4)

Fg,h(ϑc) =

Fg,1(ϑc) = Fg0w8 ϑc ∈ Iϑ1

Fg,2(ϑc) ϑc ∈ Iϑ2(18)

Fg,2(ϑc) = Fg0w8 + paw81+(w9−pc)2/pb

, Iϑ1 = [ϑc4 + ∆d − 2π, ϑc4), Iϑ2 = [ϑc4, ϑc4 + ∆d), where we have usedthe so–called Agnesi’s versiera (or Agnesi’s witch) to approximate the gas pressure profile during thecombustion phase in the cylinder.

The exosystem is hence given by systems (15), (16) and (17), (18), each of which constitutes a hybridsystem, defined by state transitions, resets, invariant and guard conditions, as described in Figure 4.

3.2 Solution of the Regulator Equations for EMVS

The regulator equations for obtained setting x = π(w) =(πxv πvv πφm πIp

)T and u = cm(w) into (11),(12), and setting to zero the tracking error (12). Considering system (11) in the states q1, q3, the regulatorequations are

Lsπxv = πvv

Lsπvv=

1M

(− kπxv

− bπvv+ πFm

+ Fg

)

Lsπφm=

R

N2

(πIp

−Mm(πxv, πφm

) +N

Rcm

)

LsπIp= −Rp

LpπIp

− 1LpLsπφm

0 = πxv− xr

(19)

with Mm(πxv, πφm

) = H(πφm)+R(πxv

)πφm, and where L stands for the classical Lie derivative [6], s(w)

is given by the right–hand functions of (15), (17), w =(w1 · · · w9

)T , πFm= − 1

2R′(πxv)π2

φm. From (19)

it is easy to calculate πxv= xr(w), πvv

= Lsπxvand

πφm= (−1)j sign(α)

√|α|

α =Fg − kπxv

− bLsπxv−ML2

sπxv

R′(πxv ), j = 1, 2

(20)

where we recall that R′(πxv) 6= 0. Here the sign function is defined as usual.

The fourth of (19) admits solution πIp since the corresponding differential equations (3) have eigen-values −Rpj

Lpjwith negative real parts. These solutions could possibly be calculated in an approximated

way. Alternatively, one can consider a numeric resolution; the problem is that the initial conditions areunknown. Therefore, one can consider the following observers

LsπIp = −Rp

LpπIp −

1LpLsπφm (21)

where πIp(0) is known. It is easy to show that the error dynamics of eIp = πIp − πIp are exponentiallystable, i.e. πIp tend exponentially to πIp .

Finally, from the third equation of (19) one works out the steady–state input

cm = NLsπφm − R

NπIp +

R

NMm(πxv

, πφm). (22)

Analogously, the regulator equations when system (11) is in the states q2 (x = ρ) or q4 (x = −ρ)are similar to (19), with the only difference due to the fact that the first two equations are Lsπxv = 0and Lsπvv = 0, while in the remaining one has to set πxv = ±ρ. Clearly, πvv = 0, while πφm

remainsundetermined. It is hence possible to choose πφm as in (20), with

α = 2Fg − k(±ρ)R′(±ρ)

.

Moreover, for πIp one can repeat the same discussion, and hence it is possibile to consider an estimateπIp given by (21), while the steady–state control is given by (22) with πxv = ±ρ.

4 Full Information Output Regulation

In this section we will determine a control law ensuring that the reference is followed, rejecting thedisturbance. This control will need the knowledge of the whole state. This assumption will be removedin the next section. Let us first consider the following Lyapunov function candidate

Vs,1 =12‖zm‖2Ps

, zm =(

xv − πxv

vv − πvv

), Ps = PT

s > 0

where ‖zm‖2Ps= zT

mPszm. Considering the mechanical subsystem (first two equations of the EMVSmodel), and the equations (Lsπxv

Lsπvv

)= A

(πxv

πvv

)+ B(πFm + Fg).

the derivative of Vs,1 along the trajectories of this subsystem is

Vs,1 = −‖zm‖2Qs+ BT Pszm(Fm − πFm)

with Ps solution of (PsA + AT Ps)/2 = −Qs, Qs = QTs > 0. Since R′(xv) 6= 0, ∀ xv, we can write

Fm − πFm = −12R′(xv)

(φ2

m − κ2sπ

2φm

)

with κs(xv, w) =√R′(πxv )/R′(xv). Therefore, considering the subsystem given by the mechanical

plus the flux dynamics (first three equations of the EMVS model) and the following Lyapunov functioncandidate

Vs = Vs,1 +N2

2Rz2φ, zφ = φm − κs πφm

if the control law is

u = κs cm +R

N

[12R′(xv)

(φm + κs πφm

)BT Pszm − k3zφ

− zI +M(xv, φm)− κs M(xv, πφm) + κs πφm

]

zI = Ip − κs πIp, k3 > 0, where κs = − bj

2 κs

(vv − πvv

), j = 1, 2, one finally obtains

Vs = −‖zm‖2Qs− k3z

2φ ≤ −λs

∥∥∥∥zm

zφ

∥∥∥∥2

. (23)

This shows that xv, vv, φm exponentially tend to πxv, πvv , πφm , since κs tend to 1. Finally, it is easy to

check that Ip exponentially tends to πIp. In fact, from equations (3) one gets

Ip − LsπIp = −Rp

Lp

(Ip − πIp

)− 1

Lp

(φm − Lsπφm

)

which are exponentially stable dynamics forced by a term going to zero.The stability of the whole switching system can be stated on the basis of simple considerations about

the so called dwell time [8]. In fact, the ith system remains in the state qi for a known time τd,i, namelyfor the time interval Ti = [ti, t′i = ti + τd,i). Let τd = min

i∈Iτd,i. Moreover, for the ith ∈ I system we have

determined a Lyapunov function Vs,i such that

k1,i

∥∥∥∥zm

zφ

∥∥∥∥2

≤ Vs,i ≤ k2,i

∥∥∥∥zm

zφ

∥∥∥∥2

, Vs,i ≤ −λs,i

∥∥∥∥zm

zφ

∥∥∥∥2

for appropriate values k1,i, k2,i. Note that λs,i can be fixed arbitrarily large. Therefore, Vs,i ≤ −λs,i

k2,iVs,i,

so that

Vs,i(ti+1) ≤ Vs,i(ti + τd) ≤ e−λs,i

k2,iτdVs,i(ti)

ti+1 = t′i. When in the state qi+1

Vs,i+1(ti+1) ≤ k2,i+1

k1,iVs,i(ti+1) ≤ kie

−λiτdVs,i(ti).

ki = k2,i+1k1,i

, λi = λs,i

k2,i. Since i is generic, it is easy to check that

Vs,i+4(ti+4) ≤ kiki+1ki+2ki+3e−

3∑h=0

λi+hτd

Vs,i(ti).

It is now sufficient to show that for every pair of time intervals Ti, Ti+4, in which the switching systemis in qi, one has

Vs,i(ti+4)− Vs,i(ti) ≤ −Wi(ti)

for a (family) of positive definite continuous functions Wi, i ∈ I. For instance, if one cosiders Wi(ti) =

εki,1

∥∥∥∥zm

zφ

∥∥∥∥2

, for an arbitrarily small ε > 0, the stability of the switching system is ensured taking

λs,i = · · · = λs,i+3 = λs and

λs >1

τd∆iln

(kiki+1ki+2ki+3

), ∆i =

3∑

h=0

1k2,i+h

.

5 Simulation Results

The control low has been implemented on a digital computer to test the performance. A seating velocityof 0.05 m/s, a transition time of 3 ms, maximal current of 30 A, and a maximal dissipated power of 1kW have been considered. In the following we give the values of the system’s parameters.

k= 117000 N/m b = 6 Ns/mM= 0.1054 Kg N = 50

Lp1= 4.8000× 10−6 Lp2 = 8.0745× 10−6 HRpm1= 0.0451 Ω Rpm2 = 0.0234 Ω

R1= 0.2040 Ω R2 = 0.2440 Ω

rlm1= 0.57813 rlm2 = 0.452Hj0= 3000 cj1 = 0.2, cj2 = 3

aj= 1.95× 104 cj3 = 3000

φm10= 1.05× 10−3 φm20 = 1.22× 10−3

ρ= 0.004 m

j = 1, 2, and (in Nm/Wb2)p11= 1.5167× 106 p21 = 1.5330× 106

p12= 1.1473× 103 p22 = 996.5755

p13= 3.8869× 108 p23 = 1.8226× 108.

The following figures summarize the simulation results. The tracking error is of the order 10−6 m.Moreover, the maximal seating velocity results to be 0.0018 m/s. This implies the respect of the controlrequirements.

6 Conclusions

In this paper a controller has been designed for a camless engine, in which the main control problemis represented by the so–called soft landing. The approach follows the regulation theory. We havedetermined two control laws. The first needs the whole state measurement, which usually is not realisticin practice. In fact, a technological requirement is the elimination of the valve position and velocitysensors. The second controller is based only on the measurements of the output, namely the fluxes andthe parasitic currents. This removes the limitation of the first controller. Indeed, this second law is basedon the knowledge of the disturbance acting on the valve. Work is in progress to eliminate this hypothesison the disturbance.

References

[1] M. S. Ashhab, A. G. Stefanopoulou, J. A. Cook, and M. Levin, Camless Engine Control for RobustUnthrottled Operation, SAE, New York, SAE Paper 960 581, 1996.

[2] L. Benvenuti, M. D. Di Benedetto, S. Di Gennaro and A. Sangiovanni–Vincentelli, Individual Cylin-der Characteristic Estimation for a Spark Injection Engine, Automatica, No. 39, pp. 1157–1169,2003.

[3] B. Castillo–Toledo, M. D. Di Benedetto and S. Di Gennaro, Nonlinear Control of ElectromagneticValves for Camless Engines, 2nd IFAC Conference on Analysis and Design of Hybrid Systems –ADHS’06, Alghero, Sardinia, Italy, June 7-9, 2006, Submitted.

[4] C. Gray, A Review of Variable Engine Valve Timing, SAE, New York, SAE Paper 880 386, 1988.

[5] W. Hoffmann, A. G. Stefanopoulou, Iterative Learning Control of Electromechanical Camless ValveActuator, Proceedings of the American Control Conference Arlington, VA June 25–27, 2001.

[6] Nonlinear Control Systems,Third Edition,Springer–Verlag, London.

[7] C.I. Byrnes, F. Delli Priscoli and A.Isidori(1997). Output Regulation of Uncertain Nonlinear Systems,Birkauser, Boston.

[8] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, USA, 2003.

[9] J. Lygeros, C. Tomlin, S. Sastry, Controllers for reachability specifications for hybrid systems, Au-tomatica, Special Issue on Hybrid Systems, vol. 35, 1999.

[10] M. Monatnari, F. Ronchi, C. Rossi and A. Tonielli, Control of a Camless Engine Electromechani-cal Actuator: Position Reconstruction and Dynamic Performance Analysis, IEEE Transactions onIndustrial Electronics, Vol. 51, No. 2, pp. 299–311, April 2004.

[11] M. Marchi, A. Palazzi and M. Panciroli, Innovative Valve Control (IVC) Model, Variable Ventils-teuerung: Ein Verfahren zur Reduzierung von Kraftstoffverbrauch und Emissionen, Stefan PischingerEd., Expert Verlag, Renningen, pp. 114–129, Expert Verlag, Renningen, 2002.

[12] K. Peterson, A. Stefanopoulou, Y. Wang, M. Haghgooie, Nonlinear Sel–Tuning Control For SoftLanding of an Electromechanical Valve Actuator, 2001.

[13] K.Peterson, A.Stefanopoulou, T. Megli, M. Haghgooie, Output Observer Based Feedback for Soft-Landing of Electromechanical Camless Valvetrain Actuator, Proceedings of the American ControlConference 2002, Vol. 2, pp. 1413–1418.

[14] K. Peterson, A. Stefanopoulou, Y. Wang, Control of Electromechanical Actuators: Valves Tappingin Rhythm, 2003.

[15] M. Pischinger, W. Salber, F. van der Staay, H. Baumgarten, and H. Kemper, Benefits of Electro-mechanical Valve Train in Vehicle Operation, SAE, New York, SAE Paper 2000-01-1223, 2000.

[16] F. Ronchi, C. Rossi and A. Tilli, Sensing Devices for Camless Engine Electromagnetic Actuators,IEEE Conference of the Industrial Electronics Society, Vol. 2, pp. 1669–1674, 2002.

[17] M. Schechter, M. B. Levin, Camless Engine, SAE, New York, SAE Paper 860 581, 1996.

[18] C.Tai, A. Stubbs, T.-C. Tsao,Modeling and Controller Design of an Electromagnetic Engine Valve,Proceedings of the American Control Conference, Arlington, VA June 25–27, 2001.

[19] Y. Wang, A, Stefanopoulou, M. Haghgooie, I. Kolmanovsky, and M. Hammoud, Modeling of anElectromechanical Valve Actuator for a Camless Engine, Proceedings of the American Control Con-ference, Arlington, VA June 25–27, 2001.

[20] Y. Wang, T. Megli, M. Haghgoogie, K. S. Peterson and A. G. Stenopoulou, Modeling and Controlof Electromechanical Valve Actuator, Society of Automotive Engineers, 2002.

Appendix

a10 = ρ

(−1 + 70

ϑ40

∆4+ 168

ϑ50

∆5+ 140

ϑ60

∆6+ 40

ϑ70

∆7

)

a11 = −280∆

ρϑ3

0

∆3

(1 + 3

ϑ0

∆+ 3

ϑ20

∆2+

ϑ30

∆3

)

a12 =840∆2

ρϑ2

0

∆2

(1 + 4

ϑ0

∆+ 5

ϑ20

∆2+ 2

ϑ30

∆3

)

a13 = −1 680∆3

ρϑ0

∆

(1 + 6

ϑ0

∆+ 10

ϑ20

∆2+ 5

ϑ30

∆3

)

a14 =1680∆4

ρ

(1 + 12

ϑ0

∆+ 30

ϑ20

∆2+ 20

ϑ30

∆3

)

a15 = −20 160∆5

ρ

(1 + 5

ϑ0

∆+ 5

ϑ20

∆2

)

a16 =100 800

∆6ρ

(1 + 2

ϑ0

∆

)

a17 = −201 600∆7

ρ

with ∆ = ϑc1 − ϑc0.

Reference hybrid system

Gr12 Rr12

Gr12: #ç #c1

Ir1 = f#c0, #c1g Ir2 = f#c1 , #c2g

j = 1 j = 2

Gr23

Rr23

Gr23: #ç #c2

Rr12, Rr23, Rr34, Rr41: w0(0) = 1, w1(0) = ... = w5(0) = 0

j = 4 j = 3

.w1 = w0

.w0 = 0

xr= xr1

.w2 = w1.w3 = w2.w4 = w5.w5 = w4

.w1 = w0

.w0 = 0

.w2 = w1.w3 = w2.w4 = w5.w5 = w4

.w1 = w0

.w0 = 0

.w2 = w1.w3 = w2.w4 = w5.w5 = w4

.w1 = w0

.w0 = 0

.w2 = w1.w3 = w2.w4 = w5.w5 = w4

Gr34 Rr34

Rr41

Gr41

Ir3 = f#c2, #c3g Ir2 = f#c3 , #c0+2¼g

Gr12: #ç #c3 Gr23: #ç #c0+2¼

Gd12: #ç #c4+¢d¡2¼

Id1 = f#c4+¢d¡2¼ , #c4g Id2 = f#c4 , #c4+¢dg

h=1

.w7 = w6

.w6 = 0

Fg= Fg,1h=2

.w7 = w6

.w6 = 0

Fg= Fg,2

Gd12

Gd21Rd21

Rd12

Gd21: #ç #c4+¢d

Rd12: w6(0) = 1, w7(0) = 0

Rd21: Identity

Disturbance hybrid system

xr= xr2

xr= xr3xr= xr4

#c2Ir1 #c2Ir2

#c2Ir3#c2Ir4

#c2Id1 #c2Id2

Figure 4: Reference and disturbance hybrid systems

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

Figure 5: Valve position xv and reference xr = πxv

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−500

−400

−300

−200

−100

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14−1

0

1

2

3

4

5

6

7x 10

4

Figure 6: Control inputs

Multilinear Identification and Model Predictive Control

of a Turbocharged Diesel Engine

Univ. Prof. Dr. L. del Re Institut für Design und Regelung mechatronischer Systeme

Johannes Kepler Universität Linz Altenbergerstr. 69 – A 4040 Linz – Tel: +43+70+2468-9772 Fax: +43+70+2468-9759

http://desreg.jku.at email: [email protected]

Abstract

Air path control of diesel engines is classically performed using decoupled controls for air flow and air fuel ration.Several multivariable contributions are known, the main problems being the fact that the plant is strongly nonlinear,that there are hard bounds on the actuator, that the design must work for the all production lot, but also that somecomponents – in particular exhaust gas recirculation – do provide a really switching behaviour. A possible way tocope with these conditions is to define different operating regions in the engine map and then to use an explicitmodel predictive framework to cope with the constraints, something which leads to a hybrid control scheme. Whilethe second step is quite straightforward if methods a la Bemporad/Borrelli/Morari are used, the first one is morecomplicated, because the choice of the region is critical. In the presentation this problem is addressed both usingheuristical choices and automatic clustering based approaches (à la Ferrari-Trecate), and compared with actualglobal results. Then the resulting structure is applied to a BMW production engine and yields substantialimprovements both in NOx and particulate if compared with the production ECU.

1

Model Predictive Control of

Magnetically Actuated Mass Spring Dampers

for Automotive Applications

Stefano Di Cairano†, Alberto Bemporad†, Ilya Kolmanovsky‡ and Davor Hrovat‡

Abstract

Mechatronic systems such as those arising in the automotiveapplications are characterized by

significant nonlinearities, tight performance specifications as well as by state and input constraints

which need to be enforced during the system operation. This paper takes a view that Model Predictive

Control (MPC) and hybrid models can be an attractive and systematic methodology to handle these

challenging control problems. In addition, the piecewise affine (PWA) explicit form of MPC solutions

avoids on-line optimization and can make this approach computationally viable even in situations with

rather constrained computational resources. To illustrate the MPC design procedure and the underlying

issues, we focus on a specific example of a mass spring damper system actuated by an electromagnet.

Such a system is one of the most common elements of mechatronic systems arising in the automotive

applications, with fuel injectors representing a concreteexample. We first consider the possibility of

decoupling the control problem to reduce the controller complexity, and to apply the MPC approach

only at the mechanical subsystem. In this case, we compare two different designs which are based,

respectively, on a linear MPC approach and on a hybrid MPC. The linear MPC approach can account

for all but one of the constraints in the system; this remaining constraint can be subsequently enforced

via a state-dependent saturation element. The hybrid MPC approach can handle all the constraints in

the design phase and achieves better performance, but results in an higher complexity of the controller.

Finally, we consider the possibility of designing an MPC controller for the full system, which optimizes

the whole performance but that is much more complex than the decoupled ones.

† S. Di Cairano and A. Bemporad are with Dipartimento di Ingegneria dell’Informazione, Universita di Siena, Italy

dicairano,bemporad,dii.unisi.it

‡ I. Kolmanovsky and D. Hrovat are with Ford Motor Company, Dearborn, Michigan, USA.

ikolmano,[email protected]

28th April 2006 DRAFT

2

I. I NTRODUCTION

During the last few years the major advances in automotive applications have been enabled

by “smart” electronic devices that monitor and control the mechanical components. Cars have

become complex systems in which electronic and mechanical subsystems are tightly connected

and interact to achieve optimal performance. Automotive actuators, in particular, have become

mechatronicsystems [1]–[3] in which mechanical components coexist with electronics and

computing devices. These mechatronic automotive systems are characterized are characterized

by tight operating requirements (such as high precision, low power consumption, fast transition

time), significant nonlinearities, as well as input and state constraints which need to be enforced

during the system operation. On the other hand, their dynamics may often be characterized by

relatively low-dimensional dynamical models.

Model Predictive Control (MPC) [4]–[6] is a systematic feedback control design technique

which determines the control input via receding horizon optimal control by exploiting the

knowledge of a (possibly approximated) system model, called prediction model. Its main appeal

is in being able to enforce pointwise-in-time constraints,while providing the control designer

with direct capability to shape the transient response by adjusting the weights in the objective

function being minimized. MPC controllers can handle continuous-valued and discrete-valued

control inputs, accommodate system parameter changes or subsystem faults, as long as they are

reflected in the model used for on-line optimization.

Automotive actuators can often be adequately characterized by low dimensional models, and

in this case an explicit implementation of the MPC controller becomes possible [7], [8], whereby

the solution is pre-computed off-line and its representation is stored for on-line application. The

on-line optimization is not required and the computationaleffort can be reduced to the point

where the implementation of these control algorithms becomes feasible within the stringent

memory and chronometric constraints of automotive micro-controllers.

In this paper, we discuss and illustrate the MPC-based controller design in more detail in

application to an electromagnetically actuated mass spring damper. Such a system arises very

frequently in automotive actuation mechanisms, includingfuel injectors, see [9] and references

therein. In this device the mechanical mass-spring-dampersubsystem interacts with an electro-

magnetic subsystem, which provides the force for controlling the first one. The mass position


3

has to track as fast as possible an external reference, whileminimizing the control effort and

external power consumption. Moreover, many different constraints must be enforced on both the

electromagnetic and the mechanical subsystems. In particular, it is assumed that the electromagnet

can only attract but not repel the mass, that the mass is moving within a constrained region space,

that the velocity must be bounded, and that the control inputis limited.

The force from the coil decays inversely proportionally to the square of the distance between

the mass and the coil. This force is also proportional to the square of the current, so that the

system is nonlinear and it would require a nonlinear-MPC controller. However, if an inner-loop

controller is applied to the electrical subsystem so that the resulting closed-loop dynamics are

sufficiently fast, the overall system may be viewed as a second order system (with states being

position and velocity of the moving mass) with the magnetic force obtained by the coil being

the control input.

The paper is organized as follows. In Section II the compoundsystem model of a magnetically

actuated mass-spring damper system is introduced, and the operating constraints on the system

are defined. Due to different response characteristics of the subsystems, a decoupling approach

is presented first, which results in the control system architecture described in Section III. In

this approach, a linear MPC controller acts as a reference governor for the electromagnetic

system, which is controlled by an inner-loop controller. The inner-loop controller imposes on

the electromagnetic subsystem the dynamics much faster than those of the mechanical system.

In Section IV two MPC designs for the decoupled architecturepresented. The first is based

on a linear MPC scheme and it cannot take into account the constraint on the maximum

available magnetic force, which in this case is enforced by asaturation block in cascade, so

that a modelling error is introduced. The second is a hybrid MPC controller that is capable

of taking into account also the constraint on the maximum available force. The decoupled

controller architecture implementation and the closed-loop experiments, that show effect of the

inner loop controller on the trajectory planned by the MPC, are reported in Section V. In the

decoupling approach the MPC controller optimizes only the mechanical subsystem behavior.

Thus, in Section VI a hybrid MPC controller based on the complete system model is designed,

for comparison purposes. This approach permits the optimization of the whole system’s behavior,

but the resulting controller is more complex than the previous ones. Finally, the key conclusions

are summarized in Section VII.


4

II. PHYSICAL MODEL AND CONSTRAINTS

i

dd

c

k

F

x

z

m

Fig. 1. The schematics of a magnetically actuated mass spring damper system.

The magnetically actuated mass-spring-damper system, with the schematics shown in Figure 1,

is a heterogenous system composed by a mechanical subsystemand an electromagnetic subsystem

that influence each other. A massm [kg] moves linearly into a bounded region under the effect

of a controlled magnetic forcef [N]. Such a force is generated by a coil placed at one of the

boundary of the region. Additional forces acting on the massare generated by a spring and a

damper. The equations that define the full system are

mx = F − cx − kx, (1a)

V = Ri + λ, (1b)

λ =2kai

kb + z, (1c)

F =kai

2

(z + kb)2=

λ2

4ka

, (1d)

z = d − x. (1e)

Equation (1a) represents the dynamics of the mass positionx [m] under the effect of an external

force F [N], of a spring with stiffnessk [N/m] and of a damper with coefficientc [N· s/m].

Equation (1b) represents a resistive circuit with resistanceR [Ω], in which the effects of magnetic

flux variations are considered. The relation between magnetic flux λ [V ·s] and currenti [A] is

defined by (1c), whereka, kb are constants, while Equation (1d) defines the magnetic force either


5

as a function of the current or as a function of the magnetic flux. Finally, Equation (1e) defines

the relation between position coordinates in the mechanical (x) and in the electromagnetic (z)

subsystem. The first has the origin at the neutral position ofthe spring, while the second at the

position in which the mass is in contact with the coil. Moreover, sincex takes its maximum value

at the contact position, andd is the distance between the contact position and the spring neutral

position, z ≥ 0, and the force is always bounded wheni is bounded. The physical model (1)

can be expressed as a dynamical system by takingλ, x, x as state variables

x =1

4kamλ2 −

k

mx −

c

mx, (2a)

λ = −R(kb + d)

2ka

λ +R

2ka

λx + V , (2b)

and it is clearly nonlinear.

The magnetically actuated mass spring damper is subject to several constraints related to

physical limits and performance. Even though the moving mass cannot penetrate into the coil

or into the symmetric stop on the other end, the constraint

−d ≤ x ≤ d [m], (3)

whered = 4 · 10−3 for the system considered, is imposed explicitly to preserve the validity of

model. If this is not done, undesirable moving mass bouncingcan create noise and increase wear

of the parts.

In a number of practical applications for which the problem that we consider here serves as

a prototype, it is actually desirable to control the moving mass so that it is positioned against

the coil with x = d. In this case, as the moving mass approaches the coil, its velocity needs

to be carefully controlled: this problem is calledsoft-landing. The purpose of the soft landing

is to avoid high collision velocities and, by keeping the velocity of approach relatively low, to

reduce the disturbance to the current control loop. In this paper we enforce the soft-landing by

the constraint

−ε − β(d − x) ≤ x ≤ ε + β(d − x) [m/s], (4)

whereε andβ are constants. When the mass is at the contact position (x = d), the velocity is

constrained in[−ε, ε], while the absolute value of the bounds increases linearly with coefficient

β, as the mass moves away from the coil. For the mass-spring-damper system we consider here,


6

ε andβ are chosen so that forx = 0, x ∈ [−10.2, 10.2], i.e., the constraint is essentially inactive,

while for x = d, x ∈ [−0.2, 0.2], i.e., the constraint is quite tight and difficult to meet. Note

that the soft-landing constraint is not required for position x1 = −d since the moving mass will

never be controlled to the symmetric stop. In fact, because of the specification considered, the

magnetic force is insufficient to counteract the spring force atx1 = −d.

The current in the circuit must be positive and, as a consequence of (1d), the magnetic force

is able to only attract the mass

i ≥ 0 [A], (5a)

F ≥ 0 [N]. (5b)

In addition, a constraint on the maximum voltage is considered

0 ≤ V ≤ Vmax [V], (6)

enforcing the physical limits and the safety of the electrical circuit.

III. D ECOUPLEDCONTROL SYSTEM ARCHITECTURE

The system model (2) is nonlinear so that complex control techniques must be applied to

achieve the specifications and the performance required. However, it is reasonable that the

dynamics of the electrical subsystem are much faster than the ones of the mechanical subsystem.

This suggests the possibility of decoupling the control problem. An inner-loop controller acting

only on the electrical subsystem can be introduced. If the closed-loop electrical dynamics are

much faster than the mechanical dynamics, the MPC controller can be designed based on the

reduced system model,

x = −c

mx −

k

mx +

F

m, (7)

where the position (x) and the velocity (x) of the mass are the state components and the magnetic

force F is the controlled input. The constraints enforced are (3), (4), (5b) and

F ≤ ka

i2max

(d + kb − x)2, (8)

which defines an upper bound on the available force, related to the maximum available current.

In fact, the input computed by the MPC is applied as referencefor the closed-loop electrical

subsystem, which generates the voltage profile required to obtain such a force. Constraint (8)


7

enforces the feasibility of the force, with respect to the maximum allowed current on the circuit.

The valueimax is computed from equation (1b) in static conditions, considering the maximum

voltageVmax. Note that (8) defines a nonconvex set, being the hypograph ofa convex function.

Mechanical MPC

rF → ri

inner-loop

Electromagnetic

i → F

ri

rF

x,x

F

V

i

rx

subsystem

subsystem

controller

controller

Fig. 2. Controller architecture for the decoupled MPC design

When considering the decoupling approach, the control system has the architecture reported

in Figure 2 structured as follows.

- The MPC receives measurement from the mechanical subsystem (7) and position reference

rx, and generates the force profilerF for optimally trackingrx.

- rF is converted into a current reference profile (ri), which is sent as reference to the

electromagnetic subsystem in closed-loop with the inner-loop controller.

- The inner-loop controller actuates the voltageV to make the electromagnetic subsystem

track ri.

- The closed-loop electromagnetic subsystem tracksri generating the desired currenti.

- The current generates the forceF that tracksrF , and that makes the mass trackrx.

In the block diagram depicted in Figure 2, the white blocks represent the dynamical subsys-

tems, and the dark grey blocks represent the controllers. The light gray blocks represent the

static blocks which act as interfaces between the previous ones. TherF → ri block converts


8

the force reference into the current reference by invertingEquation (1d). Thei → F block

represents the transduction of the current into the magnetic force acting on the mass computed

by Equation (1d).

A. Static Components

The static elements in Figure 2 are represented as light grayblocks. TherF → ri block

converts the force reference generated by the MPC controller into the current reference which

is fed in the inner-loop controller. Such a block is obtainedby inverting Equation (1d) in the

current domain defined by (5a)

ri =

√rF

(z + kb)2

ka

. (9)

The i → F block represents the transduction of the current, obtainedby Equation (1b) in

closed-loop with the inner-loop controller, into the magnetic force, computed by Equation (1d),

acting on the mass.

B. Inner-loop controller

The current dynamics defined by (1b) and (1c) are

di

dt=

kb + z

2ka

V −kb + z

2ka

Ri +1

kb + zidz

dt. (10)

The simplest way to control such nonlinear dynamics is to design the inner-loop controller

V = g(i, z, dz

dt, ri

)via feedback linearization. Letdi

dt= f(i, z, dz

dt, V ), we impose thatdi

dt=

f(i, z, dz

dt, g

(i, z, dz

dt, ri)

)= −βi + γri, β, γ > 0. When in closed-loop with the feedback

linearization controller, the current dynamics are first-order with a stable polepi = −β and

steady state gainγβ.

For the current dynamics (10), the feedback linearization controller is defined by the law

V =2ka

kb + z

[kb + z

2ka

Ri −1

kb + zidz

dt

]− βi + γri, (11)

where the referenceri is obtained from the force referencerF , produced by the MPC controller,

by inverting Equation (1d). The behavior of the whole systemlargely depends on the dynamics

imposed by the feedback linearization controller for the electromagnetic subsystem. In the

decoupled MPC design, the effect of the mass position and velocity on the electromagnetic


9

subsystem are treated as disturbances, thus, they must be much slower than the electromagnetic

subsystem dynamics.

A drawback of the feedback linearization controller is thatthe voltage command can take

large values and vary rapidly and that small modelling errors can cause loss of stability or

performance degradation. Since the main concerns here are the properties of the decoupled

MPC design, for simplicity we proceed with the assumption that the inner-loop controller is the

feedback linearization controller (11), while noting thatan alternative, more robust design, for

instance based on backstepping techniques, could have beenused instead.

C. Model Predictive Controller

The overall idea behind the decoupled design is to exploit a nonlinear controller to obtain a

linearization of the dynamics, and to use a model predictivecontroller to meet specifications

and performance. In fact, it is difficult to meet tight specification constraints and performance

with a nonlinear controller. In the other hand, the MPC guarantees constraint satisfaction and

performance, but if the system is nonlinear the control algorithm [10] is complex and slow. In

fact only the linear MPC and a particular class of nonlinear MPC discussed later can be applied

to automotive actuators, because of the short sampling period.

In the decoupled design the MPC controller acts as referencegovernor for the inner-loop

controller, which has the aim of actuating the force indicated by the MPC. Obviously, there

is a tracking error related to the inner-loop controller dynamics, because of the time required

to reach the desired value ofi. If the electrical dynamics imposed by the inner-loop controller

are fast with respect to the mechanical dynamics, the effectof such an error is limited and the

performance is only slightly degraded with respect to the nominal MPC behavior. On the other

hand if the electrical dynamics are too slow, the performance is largely degraded and constraint

violations and instability phenomena may even occur.

IV. L INEAR AND HYBRID DECOUPLEDMODEL PREDICTIVE CONTROL

Conventional linear feedback controllers, such as LQR controllers, do not explicitly handle

pointwise-in-time constraints on system’s input, state and output. On the other hand, it is

relatively easy to embed such pointwise-in-time constraints into a finite horizon optimal control


10

problem formulated for a discrete-time model, but the resulting strategy is open-loop. Model Pre-

dictive Control [4], [5] is an optimization-based closed-loop control strategy in which pointwise-

in-time design constraints on system’s state, input and output can be explicitly embedded into

the controller and, at the same time, it is a closed-loop strategy, since at each time instant the

optimization is repeated using the most recent measurements.

We consider two designs for the model predictive controllerin the decoupled architecture. In

the first the controller is a linear model predictive controller obtained by neglecting the nonlinear

constraint (8), enforced externally through a state-dependent saturation. In the second the an

hybrid model predictive controller is designed, which consider a piecewise affine approximation

of (8).

In this Section we assume that the dynamics of the electricalsubsystem in closed-loop with

the inner-loop controller are infinitely fast. Thus, we willuseF to indicate the output of the

MPC controller sinceF = rF , because of this assumption. In Section V we analyze the effect

of finitely fast electrical dynamics.

We consider discrete-time MPC, thus the moving mass dynamics(7) are discretized in time

with sampling periodTs = 5 · 10−3 [s]

x(k + 1) = Ax(k) + Bu(k), (12)

wherex = [ x1x2 ] ∈ R

2, x1, x2 are the position and the velocity of the mass, respectively,u ∈ R is

the applied magnetic forceF . In addition, an output-equationy(k) = Cx(k) is defined, which

is used in the cost function and in the constraints.

The MPC strategy is based on the solution of the optimal control problem

minyk,uk

N−1k=0

NJ−1∑k=0

(yk − ry(t))′Qy(yk − ry(t))+

∆u′

kQ∆u∆uk

subject to ymin ≤ yk ≤ ymax, k = 1, ..., NC

umin ≤ uk ≤ umax, k = 0, ..., NU

∆umin ≤ ∆uk ≤ ∆umax, k = 0, ..., NU

∆uk = 0, k ≥ NU

xk+1 = Axk + Buk

yk = Cxk + Duk, k = 0, . . . , NJ − 1

(13)


11

where∆uk = uk−uk−1, u−1 = u(t−1) is the previous input, andry(t) is the output reference at

time stept. NJ is the prediction horizon along which performance is computed,NC is the horizon

along which the output constraints are enforced, andNU is the number of free control actions, so

thatNU ≤ NJ anduk = uNU, ∀k = Nu +1, . . . , NJ

1. The MPC strategy enforces the constraints

pointwise-in-time. However, it is possible that those are violated during the intersampling period.

In order to prevent infeasibility and a halt of the control algorithm, we consider soft output

constraints in (13). In this case, problem (13) is always feasible2.

The MPC algorithm can be summarized as follows: at each sampling instantt

1) Setx0 = x(t).

2) Solve problem (13) obtainingu∗(x(t)) = [u∗

0, . . . , u∗

N−1].

3) Apply the inputu(t) = u∗

0 and discard the remaining elements ofu∗(x(t)).

The complexity of the MPC algorithm clearly depends on the structure of the optimization

problem. In particular, if the system dynamics and the design constraints are linear and Problem

(13) involves only continuous variables, the MPC algorithmrequires, at each time stept, the

solution of a Quadratic Program (QP), for which solution algorithms of polynomial complexity

exist [11]. On the other hand if some variables in Problem (13) are integer-valued, which is

the case when the system model in (13) is a hybrid model, mixed-integer programming (MIP)

techniques are required, such as branch-and-cut, which have combinatorial complexity [12].

A. Decoupled Linear Model Predictive Control

When designing model-based control systems, there is a natural trade-off between model

complexity and computation required. In particular, the more complex (and presumably accurate)

is the model, the more complex Problem (13) may become. In view of this trade-off, in the sequel

we first design an MPC controller, disregarding constraint (8). This can be enforced externally

to the MPC block, by adding a saturation

u =

u if u ≤ ka

i2max

(kd+kb−x)2

ka

i2max

(kd+kb−x)2if u > ka

i2max

(kd+kb−x)2

, (14)

1In the MPC literature and in many MPC algorithms usuallyNC = NJ = NU = N .

2In this paper the constraintAx ≤ b is softened asAx ≤ b + ε1, where1 is the vector consisting of all1. The constraint

violation penaltyρ ·ε2, is added in the cost function where the weightρ is to be two orders of magnitude higher than the higher

weight in the objective function.


12

where u is the force (reference) obtained by the linear MPC which does not consider the

maximum force constraint (8) andu is the input applied to the system in closed loop with

controller (11). Since dynamics (12) are linear and constraints (5b), (3), and (4) are also linear,

the MPC algorithm requires the solution of QP problems only.If constraint (8) is almost never

active, the resulting MPC controller, cascaded by a state-dependent input-saturation, may be

sufficient for adequately controlling the system. Furthermore, such a controller based on linear

MPC solution can be simple and fast. On the other hand, if constraint (8) is often active, the

predicted trajectory will largely differ from the actual one, because of the unmodeled state-

dependent input saturation (14). In the latter case, the system performance will be most likely

degraded.

For the magnetically actuated mass spring damper, the linear-MPC controller was designed

using the Hybrid Toolbox [13]. To make the moving mass position track a given reference signal

while enforcing the constraints (3) and (4) as output constraints, we define the output equation

y(t) =[

1 02500 1−2500 1

]x(t) (15)

where the first component defines the measured state,x1, while the second and the third outputs

are useful for defining the soft landing constraint (4) as an output constraint. Accordingly, we

setQy =

[104 0 00 0 00 0 0

], Qu = 10−10,

ymin =[−4·10−3

−∞

−10.2

], ymax =

[4·10−3

10.2+∞

],

umin = 0, umax = 104, −∆umin = ∆umax = ∞,

NJ = 30, NC = 5, NU = 3,

and use the linear dynamic model (12), (15) as prediction model.

Figure 3 shows the behavior of closed-loop formed by the linear model and the MPC controller

when tracking a desired reference profile over a simulation time interval of0.1 seconds from the

initial statex(0) = [ 00 ]. Figure 3(a) reports the position, velocity and input profiles with respect

to time, and Figure 3(b) reports the phase plane in which satisfaction of velocity constraint (4)

is shown. This velocity constraint only becomes active nearthe contact positionx = d. Because

the controller cannot provide quick decelerations due to unidirectionality of the magnetic force,

it keeps the constraint (4) inactive in parts of the trajectory away from the contact point.


13

0 0.02 0.04 0.06 0.08 0.1−2

0

2

4

time (s)

x 1 (m

m)

0 0.02 0.04 0.06 0.08 0.1

−2000

0

2000

time (s)

x 2 (m

m/s

)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

time (s)

u (k

N)

(a) State and input trajectories

−1 0 1 2 3 4 5−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

x1 (mm)

x 2 (m

m/s

)

(b) Phase-plane trajectories

Fig. 3. Ideal linear MPC simulation

Figure 4(a) shows that the simplification introduced by removing constraint (8) may be

unsatisfactory. In Figure 4(a), the curved line representing the upper bound on the force of

Eq. (8) as a function of the moving mass positionx1 is superimposed on the input signal

generated by the MPC, showing that constraint (8) is often violated.

Figure 4(b) shows the behavior of the closed-loop system when the position-dependent satura-

tion block (14), enforcing constraint (8), is cascaded withthe MPC controller. The performance

clearly degrades, especially when the reference is decreasing. The reason for performance degra-

dation is that the linear MPC controller does not recognize that braking the mass at large distances

away from the coil is impossible because of the state-dependent input saturation. This is in fact

seen from the input plot in Figure 4(b) where the dashed line corresponds to the output of the

MPC controller, while the solid line corresponds to the output of the saturator.

To avoid wide oscillations and long settling periods, the saturation constraint (8) should be

taken into account in the MPC setup. Unfortunately, (8) is a nonconvex constraint that cannot

be handled by standard linear MPC. Next section shows how sucha constraint can be handled

by a hybrid MPC approach.

B. Decoupled Hybrid Model Predictive Control

An approach for formulating optimal control problems involving nonlinear dynamics/constraints

is to find a piecewise linear approximation of the nonlinearities and to formulate the optimization


14

−1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

x1 (mm)

u (k

N)

(a) Input value generated by the linear MPC controller vs.

position

0 0.02 0.04 0.06 0.08 0.1−2

0

2

4

time (s)

x 1 (m

m)

0 0.02 0.04 0.06 0.08 0.1

−2000

0

2000

time (s)

x 2 (m

m/s

)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

time (s)

u (k

N)

(b) Input trajectory when (8) is enforced a posteriori

Fig. 4. Effects of saturation (??) cascaded to the linear MPC

problem on the approximated model. Such an approximated model is a piecewise affine (PWA)

system [14]. PWA systems have been shown to be equivalent, under mild conditions, to mixed

logical dynamical systems (MLD) [15], that are easy to embedinto mixed-integer programs.

The result is that the nonlinear dynamics are approximated as a hybrid dynamical system (this

is referred to as “hybridization”), that can be used as a prediction model in a hybrid model

predictive controller [16], [17].

In order to design an MPC controller for a nonlinear system, the following approach is applied:

1) The nonlinear dynamics are piecewise linearized;

2) the piecewise affine system is modeled as an MLD system;

3) the MLD is used as a prediction model for the hybrid MPC algorithm.

The resulting hybrid MPC controller uses an approximated prediction model, but if the approx-

imation is close enough to the real system, the controller can enforce the constraints and obtain

a good performance.

1) Hybridization of nonlinear functions:We consider here the simple case of hybridization

of one-dimensional functions. Letg : R → R be a nonlinear function, we approximateg(·) by

a (continuous) piecewise affine functionf(χ) = riχ + qi, if χ ∈ [χi, χi+1), i = 0 . . . ℓ, where

χi < χi+1 andχiℓ−1i=1 are the function breakpoints. Next, we introduceℓ − 1 binary variables

15

δ1, . . . δℓ−1 ∈ 0, 1 defined by the logical conditions

[δi = 1] ↔ [χ ≤ χi],

i = 1, . . . , ℓ,(16)

and ℓ − 1 continuous variablesz1, . . . , zℓ−1 ∈ R defined by

zi =

(ri − ri+1)χ + (qi − qi+1) if δi = 1

0 otherwise

i = 1, . . . ℓ − 1, (17a)

zℓ =

rℓ−1χ + qℓ−1 if δℓ = 1

rℓχ + qℓ otherwise(17b)

Then, the piecewise affine approximation ofg(χ) is

f(χ) =ℓ−1∑i=1

zi, (18)

and (16), (17), (18) can be embedded into an MLD system, usingfor instance the modelling

languageHYSDEL [18], along with the logical constraints

[δi = 1] → [δi+1 = 1], ∀i = 1 . . . ℓj − 1 (19)

can be included in the optimization problem.

2) Hybrid MPC design:The force constraint (8) is nonlinear and defines a nonconvexset,

the hypograph of a convex function. Hybridization can be applied to obtain a piecewise affine

approximation of such a constraint, that ensures that (8) issatisfied without being excessively

conservative.

We have considered a piecewise affine approximation with three segments (ℓ = 3), and as a

consequence, twoδ and twoz auxiliary variables have been introduced. The force constraint (8)

is defined as

u ≤ z1 + z2, (20)

wherez1 andz2 are defined by (16) and (17) withχ = x1 andℓ = 3. Clearlyf(x1) = z1 + z2 is

the function that approximates the right-hand side of (8). Model (7) with (15), (16), (17), (20),

can be modelled in HYSDEL [18], and the equivalent Mixed Logical Dynamical (MLD) hybrid


16

model [19]

x(k + 1) = Ax(k) + B1u(k) + B2δ(k) + B3z(k), (21a)

y(k) = Cx(k) + D1u(k) + D2δ(k) + D3z(k), (21b)

E2δ(k) + E3z(k) ≤ E1u(k) + E4x(k) + E5, (21c)

corresponding to the saturated magnetic actuator is obtained, where the matricesA, Bi, i =

1 . . . 3, Ej, j = 1, . . . 5, are generated automatically in MATLAB using the Hybrid Toolbox [13].

The hybrid MPC optimization problem is formulated as

minuk

N−1k=0

(xN − rx)T QN(xN − rx)+

N−1∑k=0

(xk − rx)T Qx(xk − rx) + ukQuuk (22a)

subject to MLD dynamics (21), (22b)

ymin ≤ yk ≤ ymax, k = 1 . . . N, (22c)

umin ≤ uk ≤ umax, k = 0 . . . N − 1 , (22d)

where (22c) models (3) and (4), and (22d) models (5b). For thehybrid MPC controller of the

mass-spring-damper system,Qx = QN =[

2·106 00 0

], Qu = 10−7, N = 3. Output constraints,

whereymin and ymax are the same as for problem (13), and input constraints, where umin = 0

andumax = +∞, are enforced as soft constraints, while the approximationof (8) is enforced as

a hard constraint and embedded into the MLD model.

Because of the binary variablesδ, the hybrid MPC strategy (22) requires the solution of

mixed-integer quadratic programs. In our case only two binary variables are considered for each

prediction step, so that the resulting optimization problem is of very small size.

The resulting closed-loop trajectories of the simulation scenario proposed in Section IV-A

when the hybrid MPC control algorithm is applied are reported in Figure 5.

Note that the PWL approximation (16), (17) is a lower bound to the maximum force profile, so

that the force generated by the hybrid MPC algorithm never exceeds the saturation limits. With

respect to the simulation of the linear MPC cascaded by the saturation block (reported as dashed

line in the position trajectory plot in Figure 5) we note thatthe system reacts a little slower when

starting from the neutral positionx1(0) = 0, with null velocity x2(0) = 0. This is the effect


17

0 0.02 0.04 0.06 0.08 0.1−2

0

2

4

time (s)

x 1 (m

m)

0 0.02 0.04 0.06 0.08 0.1

−2000

0

2000

time (s)

x 2 (m

m/s

)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

time (s)

u (k

N)

(a) State and input trajectories

−1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

x1 (mm)

u (k

N)

(b) Nonlinear force constraint, its PWL approximation, and

input values generated by hybrid MPC

Fig. 5. Closed-loop system using the hybrid MPC controller (22)

MPC controller Cumulated position Input energy

error (mm2) (kN2)

Linear (ideal) 51.4679 29.1918

Linear saturated 97.8608 26.6314

Hybrid 83.1005 26.6588

TABLE I

COMPARISON OF THE THREEMPC SCENARIOS

of the conservative approximation of the force constraint.While such negative effect can be

eliminated by introducing a more refined approximation, thepositive effects of the hybrid MPC

controller are clear when the reference decreases. Both the overshoot and the settling period are

reduced, because the controller is now aware of the limited available force and it provides the

braking action in the region where a larger magnetic force isavailable.

Table I compares the cumulated squared position error,∑

k(x1(k) − r(k))2, and cumulated

squared inputs (=actuator’s energy),∑

ku(k)2, for the different MPC control scenarios. The

tracking performance clearly degrades from the linear MPC controller in the ideal case of

no force saturation, to the saturating one, while the hybridcontroller has better performance

(15%) with respect to the linear-saturated one, despite the slightly conservative approximation of


18

constraint (8). Moreover one must consider that a certain component of the tracking error is due to

the one-step delay in reacting to reference changes, due to the non-anticipative implementation

of the MPC algorithms. Such an error, that with respect to data in Table I has a value of

25.5, is independent of the controller applied, and thus should not be considered in comparing

performances. Following this reasoning, the increase of net performance of the hybrid MPC

algorithm is about20% with respect to the linear-saturated one.

V. I MPLEMENTATION AND CONTROL ARCHITECTURE TESTS

We consider now the implementation problem. First we derivea form of the controllers that

can be implemented into automotive applications hardware.Then we analyze the effects of a

finitely fast dynamics of the current dynamics in the electrical subsystem in closed-loop with

the feedback linearization controller.

A. Explicit Implementation of the Controller

The implementation of the MPC controllers described in Section IV in a typical automotive

micro-controller with the sampling timeTs = 0.5 ms can be very difficult because of the time

required for the online solution of the underlying optimization problem. With the motivation to

complete off-line a large part of the computations we developed explicit versions of the MPC

controllers.

In [20] it is shown that the solution to Problem (13) can be obtained as a function of the

parametersx0 and ry (i.e., the actual state and output reference) by using multiparametric

quadratic programming (mp-QP). Using the mp-QP solver in the Hybrid Toolbox, we obtain

an explicit feedback lawu(x, ry) in continuous piecewise affine form consisting of80 regions,

which can be evaluated on-line very quickly. The mpQP algorithm also returns the value function

V (x, ry) = J∗(x, ry), which is a piecewise quadratic function.

It must be stressed that the implicit MPC controller and the explicit one produce the same

results, but there is a difference in the amount of computation required at each sampling step.

More specifically, this difference is between the solution of an online optimization problem

versus the evaluation of a set of inequalities and the computation of an affine state feedback

term.


19

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

−10000

−8000

−6000

−4000

−2000

0

2000

4000

6000

x1 (mm)

x 2 (m

m/s

)

Polyhedral partition − 80 regions

(a) Linear explicit MPC controller

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

−10000

−8000

−6000

−4000

−2000

0

2000

4000

6000

x1 (mm)

x 2 (m

m/s

)

Polyhedral partition − 671 regions

(b) Hybrid explicit MPC controller

Fig. 6. Section of the explicit controller partitions obtained forry = 0.

Figure 6(a) shows a section of the three-dimensional polyhedral partition of the explicit linear

MPC controller (where hard input/state/output constraints are enforced) obtained forry = 0.

There is an affine state feedback controller associated to each region in the partition. Figure 6(a)

also shows the state trajectory superimposed over the polyhedral partition.

In the case of hybrid MPC, we use the algorithm of [13] to obtaina representation of the

MPC controller as a set of (possibly overlapping) piecewiseaffine controllers. During the on-

line operation, at each step for each controller the value function is evaluated [17], and only

the input obtained from the controller corresponding to minimum cost is applied to the system.

Thus the explicit hybrid MPC solution involves the additional operation of comparing online

the value functions. In addition, the number of regions increases to more than650 regions, thus

the controller requires a larger storage memory in the micro-controller and a larger number of

comparison operations to find the active region.

B. Simulation of the decoupled MPC architecture

We analyze now the effect of a finitely fast dynamics in the controlled current dynamics. We

have tested the decoupled linear/hybrid MPC approach against a squared reference and we have

compared the results with the ones obtained in the ideal case, in which the dynamics of the

electromagnetic subsystem are infinitely fast.

We have designed the inner-loop controller (11) withβ = γ = 1.5 · 105. Since the mechanical


20

subsystem is a second order system with resonance frequencyωr = 950 rad/s and3db-bandwidth

BW3 = 3 · 103 rad/s, the feedback linearization controller imposes a current dynamics (BW3 =

1.5 · 105 rad/s) which is much faster than the mechanical one.

We consider as position referencerx a square wave between the critical value4 mm and0 mm

and frequency15 Hz. The initial state isx0 = [0 0]T .

0 0.025 0.05 0.075 0.1 0.125−3

−2

−1

0

1

2

3

4

5

time (s)

posi

tion

(mm

)

(a) tracking performance

0 0.025 0.05 0.075 0.1 0.125−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

time (s)

d y(t)

(mm

)

(b) position differencedy(t) between the decoupled linear

MPC and the nominal MPC

Fig. 7. Decoupled linear MPC of the mass-spring-damper system.

Figure 7 reports the results obtained with the linear MPC controller (13) with superimposed

saturator (14), where the inner-loop controller is (11). InFigure 7(a) the mass trajectory (solid)

when tracking the reference (dashed) is shown. Figure 7(b) reports the differencedy(t) = y1(t)−

y(MPC)1 (t), wherey1 is the position obtained by the decoupled linear MPC, in whichthe current

dynamics are imposed by the feedback linearization controller, while y(MPC)1 is the nominal

linear MPC position, with superimposed saturator but assuming infinitely fast current dynamics.

The difference is very small, because of the fast response ofthe controlled current dynamics.

Note that the constraints are slightly violated: this is mainly due to the current dynamics and only

for a limited amount due to the soft constraints in the optimization problem. However, thanks to

the soft constraints, even in the cases in which a solution that respects all the constraints does

not exist, the control algorithm does not get stuck with infeasibility.

Figure 8 reports the situation in which the hybrid MPC (22) isused with (11) as an inner-

loop controller. Figure 8(a) reports the mass trajectory (solid) and the reference (dashed), and


21

0 0.025 0.05 0.075 0.1 0.125−3

−2

−1

0

1

2

3

4

5

time (s)

posi

tion

(mm

)


0 0.025 0.05 0.075 0.1 0.125−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

time (s)

d y(t)

(mm

)

(b) position differencedy(t) between the decoupled hybrid


Fig. 8. Decoupled hybrid MPC of the mass-spring-damper system.

Figure 8(b) reports the differencedy(t) between position of the decoupled hybrid MPC and of

the nominal MPC. It is evident that the performance of the hybrid controller is higher than the

linear one in both the difference, which is reduced, and the tracking performance, especially in

the transition from4 mm to 0 mm. The increase of performance is paid by a higher complexity

of the controller, since the hybrid MPC algorithm requires at each step the solution of a mixed-

integer program, that has a combinatorial complexity. The average CPU time on a Pentium-IV

2 GHz with 1 GB ram, Cplex9 and Matlab7 for simulating the decoupled linear MPC is4

seconds, while for the decoupled hybrid MPC is6.5 seconds.

In case slower dynamics of the current are imposed by the feedback linearization controller,

the performance clearly degrades. Figure 9 shows the results for the linear decoupled MPC

that tracks a square reference with frequency15 Hz, higher value3.1 mm and lower value

0 mm. In this case the feedback linearization controller parameters areβ = γ = 5 · 103, and the

closed-loop current dynamics have bandwidthBW3 = 5 ·103 rad/s, the same order of magnitude

than the one of the mechanical subsystem. The tracking performance is largely degraded and

the difference between the decoupled MPC and the nominal MPCtrajectory is increased by an

order of magnitude. When the reference has larger amplitude or the imposed current dynamics

are slower, larger constraint violations and numerical instability of the MPC algorithm also occur.

The feedback linearization controller does not enforce constraints on the voltage, and in fact,


22

0 0.025 0.05 0.075 0.1 0.125−3

−2

−1

0

1

2

3

4

5

time (s)

posi

tion

(mm

)


0 0.025 0.05 0.075 0.1 0.125−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

d y(t)

(mm

)

(b) position differencedy(t) between the decoupled linear


Fig. 9. Performance degradation caused by slow current dynamics.

0 0.025 0.05 0.075 0.1 0.125−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time (s)

d y(t)

(mm

)

(a) position differencedy(t) between the saturated and the

unsaturated decoupled linear MPC

0 0.025 0.05 0.075 0.1 0.1250

50

100

150

200

250

300

350

time (s)

volta

ge (

V)

(b) voltage dynamics with saturation

Fig. 10. Voltage dynamics of the feedback linearization controller.

values even higher than1 kV have been reached in our simulations. The voltage can be limited by

inserting a saturation block between the feedback linearization controller and the electromagnetic

subsystem at the price of reducing the tracking performanceof the current and, as a consequence,

of the mass position. Figure 10(a) shows the differencedy(t) = y(sat)1 (t) − y

(uns)1 (t) between

the decoupled linear MPC position with and without a saturation block which enforces the

constraint0 ≤ V ≤ 350 [V]. However, Figure 10(b) shows that even with the saturation block


23

the electromagnetic subsystem dynamics may be unacceptable, because of the extremely rapid

variations in the voltage.

VI. COUPLED MODEL PREDICTIVE CONTROL

The decoupled MPC has been proven to be feasible even by usinga simple feedback lineariza-

tion controller as an inner-loop controller. However, in this approach the MPC algorithm does

not take into account the current dynamics. As a consequence, it cannot optimize the behavior

of the entire system and enforce constraints on the voltage.

In order to analyze the degree of optimality of the decoupledMPC approach we can design

an MPC controller that does not require any additional element, that is a controller which takes

into account both the mechanical and the electromagnetic subsystems. The control loop becomes

a standard feedback loop, in which the MPC receives the reference and the measurements on

the mass position/velocity and on the electromagnetic subsystem state, and chooses the voltage

to be applied to the system for optimizing the performance, while satisfying all the mechanical

and electrical constraints.

A. Coupled System model

The nonlinear dynamics (2) cannot be used as a prediction model for linear/hybrid MPC, since

it cannot be embedded into an LP/QP nor in an MILP/MIQP. However, we can apply again the

approach of Section IV-B.1 to find a piecewise affine approximation of (2).

To this end consider (2b) and the following change of variable, Λ = ln λ

λ0, whereλ0 = 1 [V ·s]

is used to make the argument of the logarithm adimensional. Since Λ = λ−1λ, Equation (2b)

becomes

Λ =R

2ka

x + u −R(kb + kd)

2ka

, (23)

whereu = V

λ= V

λ0eΛ [s−1] is the input. Thus, takingx and Λ as state variables, system (1) is


24

described by

x = −c

mx −

k

mx +

F

m, (24a)

Λ =R

2ka

x + u −R(kb + kd)

2ka

, (24b)

F =λ2

0e2Λ

4ka

, (24c)

u =V

λ0eΛ≤

Vmax

λ0eΛ, (24d)

which consists of two affine dynamical equations, modellingthe mechanical and electromagnetic

subsystems, and of two nonlinear static equations that act as interfaces. In order to obtain a

piecewise affine model of such system, a piecewise affine approximation of static equations

(24c), (24d) as functions ofΛ is needed. In particular, Equation (24d) is used to enforce the

constraint0 ≤ V ≤ Vmax as a piecewise affine constraint onu.

Remark 1:From a mathematical point of view the nonlinear change of coordinateΛ = ln λ

λ0

is valid only in the intervalλ ∈ (0,∞). Constraint (5a) enforcesi ≥ 0, so that we have to discuss

only the casei = 0. Such an error can be considered as a modelling error, since model (24)

is used only for prediction by the MPC controller, and it can be arbitrarily small be leavingΛ

unbounded below. However, in order to maintain the possibility of having a force exactly null, in

the piecewise linearization we can imposeF = 0 for Λ ≤ Λ, whereΛ is a negative number. As

a consequence the modelling error occurs for0 ≤ λ ≤ λ0eΛ, while for λ = 0 the approximation

error in the force is null.

The piecewise linearization of Equations (24c) and (24d) isperformed with the approach

described in Section (IV-B.1), whereχ = Λ. An approximation with four segments for each

function is considered,

fj(Λ), j = 1, 2, (25)

wherej = 1, 2 indicates the approximation of (24c) and (24d), respectively. Hence,ℓj = 3, j =

1, 2, and in total6 discrete auxiliary variables (16) and6 continuous auxiliary variables (17) have

been introduced. However, because of the additional constraints (19), only7 discrete variables

combinations are feasible. In particular we have approximated (24c) so that[δ1 = 1] → [F = 0]

and [δ1 = 1] ↔ [Λ ≤ Λ], in order to have an exact approximation of the force wheni = 0.


25

Equations (24a), (24b) and the linearization (25) of equations (24c), (24d) are modelled as an

MLD system (21), with three states, three outputs, one inputand12 (6 + 6) auxiliary variables.

B. Coupled controller simulations

A hybrid MPC controller (22) can be designed, where (22b) is the MLD system computed

in section (VI-A) that approximates (24). Constraints (3), (4) are enforced as soft constraints,

while the input constraint

0 ≤ V ≤ Vmax, (26)

whereVmax = 350 V, is enforced as a hard constraint embedded in the MLD model,by exploiting

the piecewise affine approximation of (24d).

The prediction horizon isN = 3, the cost matrices and the input/output bounds are

Qx = QN =[

2·1010 0 00 5 00 0 1

], Qu = 10−8,

ymin =[−4·10−3

−∞

−10.2

], ymax =

[4·10−3

10.2+∞

],

umin = −∞, umax = ∞.

The maximum voltage constraints are embedded into the MLD model. With regards to the third

state component,Λ − Λ0 is weighted in the cost function. By settingΛ0 = Λ = −7 a null cost

is associated to the situation in which the force in the approximated model is null.

Figure 11 reports the nominal results obtained for the coupled MPC approach. The tracking

performance and the mechanical subsystem trajectories arereported in Figure 11(a), while

Figure 11(b) shows the trajectories of the electromagneticsubsystem and that of the input.

The performance is even higher than in the ideal case of the decoupled linear/hybrid MPC. This

is due to the fact that the voltage constraint is less conservative then the current constraint. The

peaks of the input signalu occur whenΛ reaches large negative values. This depends on the

fact thatu ≤ V

λ0eΛ and for Λ → −∞, u becomes unbounded. In Figure 11(c) the phase plane

behavior of the mechanical subsystem is illustrated. The soft landing constraints (4) are slightly

violated (they are soft constraints). Since the state dimension has been increased, there is one

additional step of delay in the effects of the input on the position. However, the violation is

small, because of the large cost associated to the constraint violation, which forces the system

to avoid as much as possible such situations.


26

0 0.02 0.04 0.06 0.08 0.1−2

0

2

4

6

time (s)

posi

tion

(mm

)

0 0.02 0.04 0.06 0.08 0.1−4

−2

0

2

4

time (s)

velo

city

(m

/s)

(a) Mechanical subsystem trajectory

0 0.02 0.04 0.06 0.08 0.1−20

−15

−10

−5

0

time (s)

Λ

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8x 104

time (s)

u=V

/λ (

s−1 )

(b) Electromagnetic subsystem trajectory

−1 0 1 2 3 4 5−15

−10

−5

0

5

10

15

position (mm)

velo

city

(m

/s)

(c) Mechanical subsystem phase plane

Fig. 11. Coupled hybrid model predictive control results.

The practical advantage of the coupled MPC approach is that the two controllers are fused

into a single one, so that the overall system dynamics are optimized, while enforcing constraints

also on the voltage. On the other hand, the optimal control problem becomes more complex, so

that in our preliminary tests the explicit version of the coupled MPC controller has more than

10000 regions, while the linear and the hybrid controller in the decoupled approach have about

80 and650 regions, respectively. Even if it is possible to reduce the controller complexity, for

instance, by absorbing the small regions (i.e., the regionsthat have a Chebyshev radius smaller

than a given value) into the neighboring ones, the coupled MPC controller will remain certainly

much more complex than the decoupled MPC controllers.

VII. C ONCLUSIONS

We have presented different model predictive control techniques for controlling a magnetically

actuated mass-spring-damper system. Such system arises frequently in automotive applications

and an experimental setup is currently under development atthe Automatic Control Laboratory


27

of Siena. We have presented two different strategies based on the decoupling of the controllers

for the mechanical and electromagnetic subsystems. These approaches produce MPC controllers

that optimize only the mechanical subsystem behavior, while the electromagnetic subsystem is

controlled by an inner-loop controller that provides a fastcurrent dynamics. The decoupled MPC

controllers have been tested in closed-loop with the nonlinear system, and with the inner-loop

controller implemented through feedback linearization. The difference between the real and the

ideal case have been presented and the problems arising whenthe current dynamics are too slow

or the command input of the inner-loop controller is saturating have been discussed. Finally, the

coupled MPC approach has been analyzed. In this case, the MPCcontroller optimizes the whole

system, and in fact the performance is higher, at the price ofan increased controller complexity.

Thus, the choice of the design strategy to be applied dependson the specific application, on the

required performance, and on budget constraints.

REFERENCES

[1] L. Guzzella and A. Sciarretta,Vehicle Propulsion Systems Introduction to Modeling and Optimization. Springer Verlag,

2005.

[2] D. Hrovat, J. Asgari, and M. Fodor, “Automotive mechatronic systems,” in Mechatronic Systems, Techniques and

Applications: Volume 2–Transportation and Vehicle Systems, C. Leondes, Ed. Gordon and Breach Science Publishers,

2000, pp. 1–98.

[3] M. Barron and W. Powers, “The role of electronic controls for future automotive mechatronic systems,”IEEE/ASME

Transactions onMechatronics, vol. 1, no. 1, pp. 80–88, June 1996.

[4] S. Qin and T. Badgwell, “A survey of industrial model predictive control technology,”Control Engineering Practice, vol.

93, no. 316, pp. 733–764, 2003.

[5] J. Maciejowski,Predictive control with constraints.Englewood Cliffs, NJ: Prentice Hall., 2002.

[6] E. Camacho and C. Bordons,Model Predictive Control. Springer-Verlag, 2004.

[7] N. Giorgetti, A. Bemporad, E. H. Tseng, and D. Hrovat, “Hybrid model predictive control application towards optimal

semi-active suspension,” inProc. IEEE Int. Symp. on Industrial Electronics, Dubrovnik, Croatia, 2005.

[8] N. Giorgetti, A. Bemporad, I. Kolmanovsky, and D. Hrovat, “Explicit hybrid optimal control of direct injection stratified

charge engines,” inProc. IEEE Int. Symp. on Industrial Electronics, Dubrovnik, Croatia, 2005.

[9] D. Dyntar and L. Guzzella, “Optimal control for bouncing suppression of cng injectors,”ASME Journal of Dynamic

Systems, Measurement and Control, vol. 126, no. 1, pp. 47–45, 2004.

[10] A. Zheng and F. Allgower, Eds.,Nonlinear model predictive control. Birkhauser, 2000.

[11] S. Boyd and L. Vandenberghe,Convex optimization. Cambridge University Press, 2004.

[12] ILOG, Inc., CPLEX 8.1 User Manual, Gentilly Cedex, France, 2003.

[13] A. Bemporad,Hybrid Toolbox – User’s Guide, Dec. 2003, http://www.dii.unisi.it/hybrid/toolbox.


28

[14] E. Sontag, “Nonlinear regulation: The piecewise linear approach,” IEEE Trans. Automatic Control, vol. 26, no. 2, pp.

346–358, April 1981.

[15] W. Heemels, B. de Schutter, and A. Bemporad, “Equivalence ofhybrid dynamical models,”Automatica, vol. 37, no. 7,

pp. 1085–1091, July 2001.

[16] A. Bemporad, F. Borrelli, and M. Morari, “Piecewise linear optimalcontrollers for hybrid systems,” inProc. American

Contr. Conf., Chicago, IL, June 2000, pp. 1190–1194.

[17] ——, “On the optimal control law for linear discrete time hybrid systems,” in Hybrid Systems: Computation and Control,

ser. Lecture Notes in Computer Science, M. Greenstreet and C. Tomlin,Eds., no. 2289. Springer-Verlag, 2002, pp.

105–119.

[18] F. Torrisi and A. Bemporad, “HYSDEL — A tool for generating computational hybrid models,”IEEE Trans. Contr. Systems

Technology, vol. 12, no. 2, pp. 235–249, Mar. 2004.

[19] A. Bemporad and M. Morari, “Control of systems integrating logic,dynamics, and constraints,”Automatica, vol. 35, no. 3,

pp. 407–427, 1999.

[20] A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,”

Automatica, vol. 38, no. 1, pp. 3–20, 2002.
