2012-7-6 Beijing

H. ChenJilin University

Applying MPC in Automotive Systems

WCICA 2012

2012-7-6 Beijing

Outline

Basic of MPC

State of the art in MPC

Examples of applying MPC in automotiveModeling driver behavior based on MPC

Data-driven MPC for AMT

MPC for controlling tire blowout vehicle

FPGA-based MPC implementation

Challenges and opportunities

2012-7-6 Beijing

past future

predictive output y

control u

model, prediction, optimization, time-domain constraints, moving horizons

2012-7-6 Beijing

Idea beyond MPC

open-loop or closed-loop

optimal or just feasible

A constrained control problem is solved online. It could be

Only a part of the resulting controls is applied into the plant

Why should be repeatedly solved?Solved in general in finite horizonDisturbances/model uncertainties real dynamics is different from the predicted dynamics.Trade-off between satisfying constraints and attenuating disturbances

The procedure is repeated, if new measurements are available

2012-7-6 Beijing

MPC Optimization problem

First principles model Empirical model Hybrid modelNeural network model Fuzzy model Data-based……

Using continuous-time model

Using discrete-time model

Prediction model could be

Constraints appear in their original form

The functionality not the form of models is important

2012-7-6 Beijing

Issues in MPC

if the constrained control problem is solvable?if we can find a feasible (optimal) solution at each time sampling time?

Feasibility and tractability. Have to answer

Robustness against

Stability of the MPC closed-loop systemsoptimality does not imply stability Mayne’96the MPC closed-loop system may be not stable even if the controller obtained ateach sampling time is stabilizing

model uncertaintiesexternal disturbances (disturbance attenuation)

AlgorithmsMPC algorithms (e.g. different prediction models, DMC, MAC,… )Optimization algorithms (e.g. active set, interior point, dual algorithm, particle swarm optimization, …)

2012-7-6 Beijing

State of the art in MPC

MPC with guaranteed stability

Robust MPC

2012-7-6 Beijing

MPC with guaranteed StabilityInfinite horizon MPC

Optimization problem

Properties Optimality (Feasibility) implies closed-loop stabilityInfinite dimensional optimization problem

Computational intractable

Keerthi/Gilbert’88

2012-7-6 Beijing

MPC with guaranteed StabilityMPC with terminal equality constraint

Optimization problem

Finite horizon = infinite horizon

Keerthi/Gilbert’88Mayne’90

Rawlings el at'93,94Genceli/Nikolaou'95

2012-7-6 Beijing

MPC with terminal equality constraint

Clear formulationNo off-line computation of controller parameters Closed-loop stability is achieved via terminal equality constraint

Properties:

Comments:

The system should be finite time controllableNumerically hard to satisfy the equality constraintSmall feasible setNo inherent robustness at all

2012-7-6 Beijing

MPC with guaranteed stabilityQuasi-infinite horizon MPC

Introduce terminal penalty and terminal constraint

Quasi-infinite horizon

Chen/Allgower’96,98

2012-7-6 Beijing

Terminal set is defined as

together with terminal penalty satisfyconstraints are satisfied in terminal set

terminal penalty function is CLF-like

monotonicity of the value functionstability

Quasi-infinite horizon MPC

2012-7-6 Beijing

Quasi-infinite horizon MPC

Feasibility at the initial time (t=0) implies feasibility at all times (t>0) Closed-loop stability is achieved via suitable choice of terminal inequality constraint and terminal penalty

Properties:

Merits:Inequality constraint is easier to implement than equality constraintFeasibility but not necessarily optimality is neededLarge feasible setInherent robustness for some disturbances or uncertainties

Yu/Marcus/Chen/Allgower’11

2012-7-6 Beijing

Robust MPC

Key point: closed-loop prediction is required

Big barrier: computation intractable, even for linear system

Campo/Morari'87Allwright/Papavasiliou'92

Zheng/Morari'93Chen/Scherer/Allgower’97

Magni et al.’03

Explicit description of uncertainties or disturbances requiredMin-max problem: maximization over a set of uncertainties/disturbances

2012-7-6 Beijing

Introduce terminal set and terminal penalty to prove robust stability

Assume continuity of value function to prove ISS property

Introduce dissipation constraint to achieve H∞ performance

Assume the disturbance is norm-bounded to prove robust stability

Assume the disturbance is measurable to achieve H∞ performance

Robust MPCRender min-max problem solvable by parameterization of the control

Kothare/Morari’96,…

Chen/Scherer/Allgower’97,Rossiter et al’98,…

Mayne/Rakovic/Fineisen/Allgwoer’05

Goulart/Kerrigan/Maciejowski’06

Show robustness of the closed-loop system

Raimondo/Limon/Lazar/Magni/Camacho’09

Chen/Scherer/Allgower’97

Goulart/Kerrigan/Maciejowski’06

Chen/Scherer’06,Chen/Gao/Wang’07

Mayne/Rakovic/Fineisen/Allgwoer’05

2012-7-6 Beijing

Examples of applying MPC in automotive

MPC-based driver modeling

Data-driven MPC for AMT

MPC for controlling of blow-out tire vehicle

FPGA-based MPC implementation

2012-7-6 Beijing

To test and evaluate vehicles in driver-in-the-looprepeat exactly the same testsdeliver objective evaluationdo dangerous testsreduce test costs…

1. MPC-based driver modelingWhy do we need a driver model

Unmanned drive/ Autonomous vehicle

Good?+

To test control systems and ECUs in driver-in-the-loop

2012-7-6 Beijing

Predict the vehicle’s motion

Plan the trajectory by comparing

Make a decision

Steer

Brake

Accelerate

Repeat

The driver’s behavior is fit in the basic of MPC well

Road path Why MPC is suitable for modeling drivers

Driver collects/usesRoad informationTraffic informationVehicle state Driving experience

MPC-based driver modeling

2012-7-6 Beijing

Driver behavior model

DelayNeuralPhysiologic

Perception module

Knowledge and experience

Decision module

Execution module

The structure of driver behavior model

MPC-based driver modeling

2012-7-6 Beijing

:Prediction horizonPath preview (single-point preview)

Internal vehicle dynamics (bicycle model)

Optimization

Delay

:Longitudinal velocity

(minimize the lateral path error and fuel consumption)

A simple example

MPC-based driver modeling

2012-7-6 Beijing

s.t.

Parameterize

Plan trajectory based on differential flatness

Flatness outputs

Trajectory planning is simplified to plan flatness outputs.

States inputs and outputs can be expressed by and their derivatives.

MPC-based driver modeling

2012-7-6 Beijing

Simulation results

φ,θ,ψ

x y zv ,v ,v

SuspensionModels

( , )s wg z zΔ Δ

Drivingtorque

Car Body Model

( , , , , , )f x y z φ θ ψBrakingtorque

Steerangle

φ,θ,ψ

Input fromground

Vertical forces

Tire Models

Longitudinal forcesLateral force

Vx, V

y

Vertical foces of tires

( , )wq zω Δ

14 degrees of freedom vehicle dynamic model

Sinefunction

Double lane function

MPC-based driver modeling

2012-7-6 Beijing

Issues need to addressHow to describe the human decision criteria in the objective function

What for a model form is suitable to describe the driver experienceFirst principle model

Data-based model

Learning-based model

Mixed model

…

MPC-based driver modeling

2012-7-6 Beijing

Lu/Chen/Wang’10Vehicle start-up

Frequent stop-startStart-up too fast/slow

Fuel economyDriving comfort

SafetyLaunch on slope road...

...

2. Data-driven MPC for AMT

2012-7-6 Beijing

Control requirements

The range of engine speed is limited

The maximum friction clutch torque is restricted

Frequency response of the clutch actuator is limited

minimize clutch lockup timeensure smooth accelerationminimize friction losses

Avoid stalling

the engine

Vehicle start-up

Data-driven MPC for AMT

Hard constraints

FastSmooth

Friction loss

Clutch engagement

2012-7-6 Beijing

Engine torque map Engine friction torque map

Clutch spring characteristics mapslip rate and tire/road friction map

Powertrain of a rear driven truck

Data-driven MPC for AMT

Complex dynamics due to combustionvibrationfrictiontire-ground mechanics

Long-term aging of rotating partsBacklash in gearsSwitching due to gear shift

2012-7-6 Beijing

Subspace Identification

Data-driven predictive control (Huang’03)

I/O Data

Hankel matrix

MPC

Orthogonal/oblique projection

State sequence,Extended

Observability matrix

Least square

System matricesA,B,C,D

mea

sure

men

tOptimization

Control input

System matrices A,B,C,D

Prediction equation

Data-driven Predictive Control

I/O Data

Hankel matrix

Least square

Optimization

Control input

Predictionequation

mea

sure

men

t

Data-driven MPC for AMT

2012-7-6 Beijing

Data design:excite the vibration of clutch, drive shaft, tyre…

Input

0 50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

140

160

180

Tc (N

m)

0 50 100 150 200 250 300 350 400 450 500-50

0

100

200

300

Δ ω

(rad

/s)

Excite dynamics relevant to the control goal

fast “engaged”fast “disengaged”

AMESim ModelMedium Truck6.2L Diesel EngineDry Clutch

Output

Data-driven MPC for AMT

2012-7-6 Beijing

Validation

Predictive Output vs Actual Output

Prediction Equation

Clutch Speed

Data-driven MPC for AMTPrediction Equation

validation data

off-line data

2012-7-6 Beijing

Data-driven predict control

Optimization

anti-jerk (smooth)fast engagementTime-domain constraints

limitation on clutch friction torque

limitation of clutch actuator

limitation of engine speed

Control InputSolve the optimization problem at each sampling time

Data-driven MPC for AMT

Predicted output

2012-7-6 Beijing

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

Tc (

Nm

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

55

100

150

200

ωe, ω

c (ra

d/s)

ωe

ωc

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

Δ ω

(rad

/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

time (s)

a (ra

d/s2

),da

(rad/

s3)

ada

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

Tc (

Nm

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

55

80

100ω

e, ω

c (ra

d/s)

ωe

ωc

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

Δ ω

(rad

/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

time (s)

a (ra

d/s2

),da

(rad/

s3)

ada

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

Tc (

Nm

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

55

100

150

ωe, ω

c, (ra

d/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

Δ ω

, (ra

d/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-10

0

10

20

time (s)

a (ra

d/s2

),da

(rad/

s3)

ωeωc

ada

Simulation results in different conditions

Data-driven MPC for AMT

2012-7-6 Beijing

Issues need to addressHow to use on-line data effectivelyHow to predict nonlinear dynamics How to guarantee stability…

Data-driven MPC for AMT

2012-7-6 Beijing

MPC

3. MPC for controlling tire blowout vehicle

stay safe, avoid roll-overkeep the vehicle in the lane

Control requirements

What is “tire blowout”The sudden deflation of a vehicle tire is called as tire blowout.

path following safety constraint

2012-7-6 Beijing

rolling resistance coefficient cornering stiffness

MPC for controlling tire blowout vehicleModeling (bicycle model) without tire blowout

with tire blowoutDescribe the effect of tire blowout

Lateral force is changedIntroduce an additional yawing moment

2012-7-6 Beijing

MPC for controlling tire blowout vehicleOptimization (minimize the lateral path error)

Path preview (single-point preview) : Prediction horizon

Model predictive control problem description

Objective function

Safety constraint

Control block diagram

2012-7-6 Beijing

on a straight roadfront left tire blowout

velocity: 120 km/htire-road friction coefficient: 0.8

Lateral displacement of vehicle on the straight road Sideslip angle of vehicle on the straight road

Yaw rate of vehicle on the straight road Tire slip angle of vehicle on the straight road

Simulation result

MPC for controlling tire blowout vehicle

2012-7-6 Beijing

velocity: 120 km/htire-road friction coefficient: 0.8

Lateral displacement of vehicle on the crooked road Sideslip angle of vehicle on the crooked road

Yaw rate of vehicle on the crooked road Tire slip angle of vehicle on the crooked road

on a left crooked roadrear right tire blowout

Simulation result

MPC for controlling tire blowout vehicle

2012-7-6 Beijing

Issues need to addressHow to create more accurate models for a tire blowout vehicleHow to validate the modelHow to test the controllerHow to reduce the computation cost and the code size…

MPC for controlling tire blowout vehicle

2012-7-6 Beijing

MPC needs to solve an optimization problem on line, but

fast dynamics dominated in automotive systemslow-cost computation and memory (controller on a chip)low-cost development

Solution efficient MPC algorithmsfast optimization algorithmshardware architectures for parallel computations

4. FPGA-based MPC implementation

2012-7-6 Beijing

Embedded implementation of MPC on an FPGA

Hardware accelerationcustom instructions: fine-tune the system hardwarecustom peripherals: coprocessor, parallel computing

Hard processor cores: higher performancelow power consumption

Soft cores (Nios II): more flexibleeasy to use

FPGA-based MPC implementation

SoPC technique

2012-7-6 Beijing

Design flow on FPGA/SoPCAnalysis of system requirements

MPC algorithm analysis: floating point matrix operations, the sizes of matrices, …system analysis:

Nios II fast core, standard IP cores, …

Hardware designbuild Nios II systemdefine custom instructions (floating point operations, …) design custom peripherals (matrix operations, …)

Software design (Nios II IDE)program C/C++ codes of the MPC algorithmapply the macros of custom instructions

Software

Analyze system requirements

Define and create the SoPC system

Configurable soft core processor and IP-cores

Add custom instructions and peripherals

Compilation andsynthesis

Download the file to FPGA device

Apply the macros of custom instructions

Compile and generate executable files

ISS run and debug

Write MPC code in C/C++ language

Hardware

Real-Time simulation on target board

FPGA-based MPC implementation

2012-7-6 Beijing

Prototyping environment

FPGA board: hardware implementation of the MPC controllerCyclone II, Stratix III, …

Real-time simulation systemrun plant model and monitor resultsdSPACE, xPC-Target, …

PC1 and PC2design the SoPC system of MPC controllerQuartus II, SoPC builder, Nios II IDE

UART

MPC controller

UART

Model

dSPACE orxPC-Target

Monitor

ComputerFPGA RS 232PCI

PCI

Cyclone II FPGA and dSPACE

Stratix III FPGA and xPC-Target

FPGA-based MPC implementation

2012-7-6 Beijing

Results of controlling an electronic throttle Chen/Xu/Xi’12

Time taken to solve one QP by using custom instructions

solving one QP

completely in software

solving one QP by using custom

instructions

Time taken to solve one QP by using different optimization methods

NPSOL: commercial

software packages

FPGA-based MPC implementation

2012-7-6 Beijing

Challenges and opportunities

Various models due tocomplex dynamics: fuel/air mixing, compression, combustion, aftertreatment, vibration of rotating parts, friction….tires - ground mechanics …heavy coupled

Model-based predictionFunctionality is importantDo not care the model form

Automotive controlProperties of MPC

Trade-off various requirementsdrivabilitycomfortfuel economy …

Hard constraintssafety constraintsemission regulationactuator saturation…

On-line optimization

Explicit handling of constraints

MPC is a suitable solution for automotive control

But applying MPC in automotive is not trivial !

2012-7-6 Beijing

Challenges and opportunities Efficient algorithms ( MPC and optimization)

formulate various requirements in the optimization problemdrivabilityfuel economycomfortemission regulation…

predict nonlinear dynamics based on on-line dataattack numerical difficulties due to using various prediction models

data-based modelphysically/data mixed model …

do fast prediction/computation do fast optimization….

develop satisfying control system on a chip

2012-7-6 Beijing

Challenges and opportunities

Robustness: an open problem!!!

Stability: existing stability results are invalid, due to the use of various prediction models

data-based modelphysically/map mixed modelswitching model…

objective function being not positive definite

Data-driven MPCPhysically/data mixed MPCHybrid MPC Economic MPC…..

2012-7-6 Beijing

Thank You !

WCICA 2012