gain-scheduled controller design - stuba.skgain scheduling is one of the most common used controller...
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Ing. Adrian Ilka
Summary of doctoral dissertation
Gain-Scheduled Controller Design
A thesis submitted in fulfilment of the requirementsfor the degree of Doctor of Philosophy
at the
Institute of Robotics and Cybernetics
Study programme: CyberneticsStudy field: 9.2.7 Cybernetics
Bratislava, May 2015
Thesis was developed in a full-time doctoral study at Institute of Robotics andCybernetics, Faculty of Electrical Engineering and Information Technology,Slovak University of Technology in Bratislava.
Author: Ing. Adrian IlkaInstitute of Robotics and Cybernetics, Faculty of Elec-trical Engineering and Information Technology, Ilkovicova3, 812 19 Bratislava
Supervisor: prof. Ing. Vojtech Vesely, DrSc.Institute of Robotics and Cybernetics, Faculty of Elec-trical Engineering and Information Technology, Ilkovicova3, 812 19 Bratislava
Opponents: prof. Ing. Dusan Krokavec, CSc.Department of Cybernetics and Artificial Intelligence, Fac-ulty of Electrical Engineering and Informatics, TechnicalUniversity of Kosice, Letna 9/B, 042 00 Kosice
prof. Ing. Mikulas Alexık, PhD.Department of Technical Cybernetics, Faculty of Manage-ment Science and Informatics, University of Zilina, Uni-verzitna 1, 010 26 Zilina
Thesis summary was submitted on ..........................................
Date and location of PhD thesis defence: .......................................... at Facultyof Electrical Engineering and Information Technology, Slovak Universityof Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava.
prof. Dr. Ing. Milos OravecDean of FEI STU
Abstract
Thesis title: Gain-Scheduled Controller Design
Keywords: Gain-scheduled control; Lyapunov theory of stability; Guaran-teed cost control; Bellman-Lyapunov function; LPV system; Robust control;Input/output constraints
This thesis is devoted to controller synthesis, i.e. to finding a systematicprocedure to determine the optimal (sub-optimal) controller parameters whichguarantees the closed-loop stability and guaranteed cost for uncertain nonlinearsystems with considering input/output constraints, all this without on-lineoptimization. The controller in this thesis is given in a feedback structure, thatis the controller has information about the system and uses this informationto influence the system. In this thesis the linear parameter-varying basedgain scheduling is investigated. The nonlinear system is transformed to alinear parameter-varying system, which is used to design a controller, i.e. again-scheduled controller with consideration of the objectives on the system.The gain-scheduled controller synthesis in this thesis is based on the Lyapunovtheory of stability as well as on the Bellman-Lyapunov function. Severalforms of parameter dependent/quadratic Lyapunov functions are presentedand tested. To achieve performance quality a quadratic cost function and itsmodifications known from LQ theory are used. In this thesis one can find alsoan application of gain scheduling in switched and in model predictive controlwith consideration of input/output constraints. The main results for controllersynthesis are in the form of bilinear matrix inequalities (BMI) and/or linearmatrix inequalities (LMI). For controller synthesis one can use a free and opensource BMI solver PenLab or LMI solvers LMILab or SeDuMi. The synthesiscan be done in a computationally tractable and systematic way, therefore thelinear parameter-varying based gain scheduling approach presented in thisthesis is a worthy competitor to other controller synthesis methods for nonlinearsystems.
Anotacia dizertacnej prace
Nazov dizertacnej prace: Riadenie systemov metodou ”gain scheduling”
Kl’ucove slova: Riadenie s planovanym zosilnenım; Lyapunova teoria stability;Riadenie s garantovanou kvalitou; Bellman-Lyapunova funkcia; LPV systemy;Robustne riadenie; Vstupne/vystupne obmedzenia
Tato praca sa venuje problematike navrhu regulatora, tj. najst’ systematickypostup na navrh optimalnych (suboptimalnych) parametrov regulatora, ktoregarantuju stabilitu a kvalitu v uzavretej slucke, pri obmedzenı vstupno-vystupnych hodnot systemov pre nelinearne systemy s neurcitost’ami, a tobez on-line optimalizacie. Uvedeny regulator ma spatno-vazobnu riadiacustrukturu, co znamena, ze disponuje informaciami o danom systeme, ktorevyuzıva k jeho ovplyvneniu. Tato praca sa podrobnejsie zaobera s riadenıms planovanym zosilnenım, a to na baze parametricky zavislych linearnychsystemov. Nelinearny system je pretransformovany na parametricky zavislylinearny system, co sa nasledne vyuzıva na navrh regulatora, tj. regulatoras planovanym zosilnenım, s ohl’adom na poziadavky daneho systemu. Syntezaregulatora s planovanym zosilnenım sa uskutocnı na baze Lyapunovej teoriestability s pouzitım Bellman-Lyapunovej funkcie, v ramci coho su prezentovanea testovane rozne typy kvadratickej a parametricky zavislej Lyapunovej funkcie.Pre dosiahnutie pozadovanej kvality sa pouzıva kvadraticka ucelova funkciaznama z LQ riadenia, s roznymi modifikaciami. V tejto praci najdeme ajaplikaciu riadenia s planovanym zosilnenım v oblasti takzvaneho prepınaciehoriadenia (switched control), ako aj v ramci prediktıvneho riadenia (modelpredictive control). Hlavne vysledky pre syntezu regulatorov su v tvarebilinearnych maticovych nerovnıc (BMI) a/alebo linearnych maticovychnerovnıc (LMI). Na navrh regulatorov mozeme pouzıvat’ bezplatny a
”open
source“ BMI solver PenLab alebo LMI solvre LMILab a SeDuMi. Uvedeneskutocnosti umoznia vykonat’ syntezu jednoduchym a systematickym sposobom.Riadenie s planovanym zosilnenım na baze parametricky zavislych linearnychsystemov prezentovane v tejto praci je vhodnym konkurentom vo vzt’ahu k inymmetodam syntezy regulatorov pre nelinearne systemy.
Contents
Contents iii
1 Introduction 11.1 Goals & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminary pages 22.1 Linear parameter-varying systems . . . . . . . . . . . . . . . . . . 3
2.1.1 Introduction to LPV systems . . . . . . . . . . . . . . . . 32.1.2 Application of the LPV systems . . . . . . . . . . . . . . 4
2.2 Gain scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Introduction to gain scheduling . . . . . . . . . . . . . . . 52.2.2 History of gain scheduling . . . . . . . . . . . . . . . . . . 52.2.3 Application of gain scheduling . . . . . . . . . . . . . . . 62.2.4 Summary of gain scheduling . . . . . . . . . . . . . . . . . 6
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Robust Gain-Scheduled PID Controller Design for UncertainLPV Systems 83.1 Problem formulation and preliminaries . . . . . . . . . . . . . . . 83.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Robust Switched Controller Design for Nonlinear ContinuousSystems 144.1 Problem statement and preliminaries . . . . . . . . . . . . . . . . 14
4.1.1 Uncertain LPV plant model for switched systems . . . . . 144.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . 15
4.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Gain-Scheduled MPC Design for Nonlinear Systems with InputConstraints 205.1 Problem formulation and preliminaries . . . . . . . . . . . . . . . 20
5.1.1 Case of finite prediction horizon . . . . . . . . . . . . . . 215.1.2 Case of infinite prediction horizon . . . . . . . . . . . . . 24
5.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.1 Finite prediction horizon . . . . . . . . . . . . . . . . . . . 265.2.2 Infinite prediction horizon . . . . . . . . . . . . . . . . . . 27
6 Concluding remarks 276.1 Brief overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Research results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Closing remarks and future works . . . . . . . . . . . . . . . . . . 28
References 37
iii
1 Introduction
This thesis is devoted to controller synthesis, i.e. to finding a systematic proced-ure to determine the optimal (sub-optimal) controller parameters which guaran-tees the closed-loop stability and guaranteed cost for uncertain nonlinear systemswith considering input/output constraints. In consideration of the objectivesstated for the system such as tracking a reference signal or keeping the plantat a desired working point (operation point) and based on the knowledge of thesystem (plant), the controller takes decisions. The controller in this thesis isgiven in a feedback structure, which means that the controller has informationabout the system and uses it to influence the system. A system with a feedbackcontroller is said to be a closed-loop system.
To design a controller which satisfies the objectives we need an adequatelyaccurate model of the physical system. Nevertheless, real plants are hard todescribe exactly. Alternatively, the designed controller must handle the caseswhen the state of the real plant differs from what is observed by the model.A controller that is able to handle model uncertainties and/or disturbances issaid to be robust, and the theory dealing with these issues is said to be robustcontrol.
The robust control theory is well established for linear systems but almost allreal processes are more or less nonlinear. If the plant operating region is small,one can use the robust control approaches to design a linear robust controller,where the nonlinearities are treated as model uncertainties. However, for realnonlinear processes, where the operating region is large, the above mentionedcontroller synthesis may be inapplicable because the linear robust controller maynot be able to meet the performance specifications. For this reason the controllerdesign for nonlinear systems is nowadays a very determinative and importantfield of research.
Gain scheduling is one of the most common used controller design approachesfor nonlinear systems and has a wide range of use in industrial applications.Many of the early articles were associated with flight control and aerospace.Then, gradually, this approach has been used almost everywhere in control en-gineering, which was greatly advanced with the introduction of LPV systems.
Linear parameter-varying systems are time-varying plants whose state spacematrices are fixed functions of some vector of varying parameters θ(t). They wereintroduced first by Jeff S. Shamma in 1988 to model gain scheduling. Today theLPV paradigm has become a standard formalism in systems and controls withlots of researches and articles devoted to analysis, controller design and systemidentification of these models.
This thesis deals with linear parameter-varying based gain scheduling, whichmeans that the nonlinear system is transformed to a linear parameter-varyingsystem, which is used to design a controller, i.e. gain-scheduled controller. Theproblem formulation is close to the linear system counterpart, therefore usingLPV models to design a controller has potential computational advantages overother controller synthesis methods for nonlinear systems. Not to mention that
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the LPV based gain scheduling approaches comes with a theoretical validitybecause the closed-loop system can meet certain specifications. Nonetheless,following the literature it is ascertainable that there are still many unsolvedproblems. This thesis is devoted to some of these problems.
1.1 Goals & Objectives
As already mentioned, there are many unsolved problems. Therefore, it is ne-cessary to find new and novel controller design approaches. If one wants tosummarize the main goal of this thesis in one sentence, then you would read:The main goal is to find a controller design approach for uncertain nonlinearsystems, which guarantees the closed-loop stability and guaranteed cost withconsidering input/output constraints, all this without on-line optimization andneed of high-performance industrial computers. That is why we set the followinggoals:
• Suggest a gain-scheduled PID controller design approach with guaranteedcost in continuous and discrete time state space using BMI
• Suggest a robust gain-scheduled PID controller design approach with guar-anteed cost and parameter dependent quadratic stability in state spaceusing BMI
• Suggest a variable weighting gain-scheduled approach
• Convert some BMI controller design approaches to LMI
• Suggest a switched and model predictive gain-scheduled method
• Suggest a gain-scheduled controller design approach with input/outputconstraints
• Apply methods to relevant processes
1.2 Outline
The sequel of this summary of dissertation thesis is organized as follows. InSection 2 one can find a preliminary chapter, where with review of the literaturea brief overview of linear parameter-varying systems and gain scheduling arepresented. In Section 3-5 one can find an overview of selected research resultswhich covers the main research results obtained within the last 2.5 years . Finally,in Section 6, following the selected results from selected papers, some concludingremarks and suggestions for future research are given.
2 Preliminary pages
In this section preliminaries of linear parameter-varying systems as well as gainscheduling are introduced. This section is intended to highlight the propertiesand give a short background to the tools used in the appended papers.
2
2.1 Linear parameter-varying systems
Linear parameter-varying systems are time-varying plants whose state spacematrices are fixed functions of some vector of varying parameters θ(t). It wasintroduced first by Jeff S. Shamma in 1988 [1] to model gain scheduling. ”TodayLPV paradigm has become a standard formalism in systems and controls withlot of researches and articles devoted to analysis, controller design and systemidentification of these models”, as Shamma wrote in [2]. This section deals withLPV models and presents analytical approaches for LPV systems.
2.1.1 Introduction to LPV systems
Linear parameter-varying systems are time-varying plants whose state spacematrices are fixed functions of some vector of varying parameters θ(t). Linearparameter-varying (LPV) systems have the following interpretations:
– they can be viewed as linear time invariant (LTI) plants subject to time-varying known parameters θ(t) ∈ 〈θ, θ〉,
– they can be models of linear time-varying plants,
– they can be LTI plant models resulting from linearization of the nonlin-ear plants along trajectories of the parameter θ(t) ∈ 〈θ, θ〉 which can bemeasured.
For the first and third class of systems, parameter θ can be exploited for thecontrol strategy to increase the performance of closed-loop systems. Hence, inthis thesis the following LPV system will be used:
x = A(θ(t))x+B(θ(t))u
y = Cx(1)
where for the affine case
A(θ(t)) = A0 +A1θ1(t) + . . .+Apθp(t)
B(θ(t)) = B0 +B1θ1(t) + . . .+Bpθp(t)
and x ∈ Rn is the state, u ∈ Rm is a control input, y ∈ Rl is the measure-ment output vector, A0, B0, Ai, Bi, i = 1, 2 . . . , p, C are constant matrices of
appropriate dimension, θ(t) ∈ 〈θ, θ〉 ∈ Ω and θ(t) ∈ 〈θ, θ〉 ∈ Ωt are vectors oftime-varying plant parameters which belong to the known boundaries.
The LPV paradigm was introduced by Jeff. S. Shamma in his Ph.D. thesis [1]for the analysis of gain-scheduled controller design. The authors in early works(see [1, 3, 4, 5, 6, 7, 8] and surveys [9, 10]) in gain scheduling the LPV systemframework called as the golden mean between linear and nonlinear dynamics,because ”the LPV system is an indexed collection of linear systems, in whichthe indexing parameter is exogenous, i.e., independent of the state.”(wrote J. S.
3
Shamma in his Ph.D. thesis [1]). In gain scheduling, this parameter is often afunction of the state, and hence endogenous
x = A(z)x+B(z)u
y = C(z)x
z = h(x)
(2)
2.1.2 Application of the LPV systems
Since the first publication devoted to LPV systems, the LPV paradigm has beenused in several fields in control engineering including the modeling and con-trol design. Traditionally the gain scheduling was the primary design approachfor flight control and consequently many of the first articles and papers whichapplied and improved the LPV framework were associated with flight control.Afterwards continuously many papers and articles have appeared which are us-ing LPV paradigm in several application areas such as flight control and missileautopilots [11, 12, 13, 14, 15, 16, 17], aeroelasticity [18, 19, 20, 21], magneticbearings [22, 23, 24, 25], automotive bearings [26, 27, 28], energy and powersystems [29, 30, 31, 32, 33, 34], turbofan engines [35, 36, 37, 38], microgravity[39, 40, 41], diabetes control [42, 43, 44], anesthesia delivery [45], IC manufac-turing [46].
Due to the success of LPV paradigm in 2012 for the twentieth anniversaryof the invention of LPV paradigm a gift edition book was published by JavadMohammadpour and Carsten W. Scherer Editors at Springer [2] which is fullydevoted to LPV systems.
2.2 Gain scheduling
The robust control theory is well established for linear systems but almost allreal processes are more or less nonlinear. If the plant operating region is small,one can use the robust control approaches to design a linear robust controllerwhere the nonlinearities are treated as model uncertainties. However, for realnonlinear processes, where the operating region is large, the above mentionedcontroller synthesis may be inapplicable. For this reason the controller designfor nonlinear systems is nowadays a very determinative and important field ofresearch.
Gain scheduling is one of the most common used controller design approachesfor nonlinear systems and has a wide range of use in industrial applications. Inthis section the main principles, several classical approaches and finally the linearparameter-varying based version of gain scheduling are presented and investig-ated.
4
2.2.1 Introduction to gain scheduling
In literature a lot of term are meant under gain scheduling (GS). For exampleswitching or blending of gain values of controllers or models, switching or blend-ing of complete controllers or models or adapt (schedule) controller parametersor model parameters according to different operating conditions. A commonfeature is the sense of decomposing nonlinear design problems into linear ornonlinear sub-problems. The main difference lies in the realization.
Consequently gain scheduling may be classified in different way
• According to decomposition
1. GS methods decomposing nonlinear design problems into linear sub-problems
2. GS methods decomposing nonlinear design problems into nonlinear(affine) sub-problems
• According to signal processing
1. Continuous gain scheduling methods
2. Discrete gain scheduling methods
3. hybrid or switched gain scheduling methods
• According to main approaches
1. Classical (linearization based) gain scheduling
2. LFT based GS synthesis
3. LPV based GS synthesis
4. Fuzzy GS techniques
5. Other modern GS techniques
2.2.2 History of gain scheduling
The ferret in the history of gain scheduling appears in the 1960s but a similarsimpler technique was used in World War II toat control the rockets V2 (switch-ing controllers based on measured data). It is not surprising therefore that gainscheduling as a concept or notion firstly appear in flight control and later inaerospace. Leith and Leithead in their survey [9] and likewise also Rugh andShamma in their survey paper [10] considered the first appearance of GS fromthe 1960s. Rugh stated in his survey that ”Gain control” does appear in the 25thAnniversary Index (1956–1981) published in 1981 but only one of the five listedpapers is relevant to gain scheduling. Also Automatica lists gain scheduling asa subject in its 1963–1995 cumulative index published in 1995. Of the four cita-tions given, only one dated earlier than 1990 [1]. It can be stated that increasedattention to gain scheduling appeared after introducing the LPV paradigm by
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Jeff. S. Shamma (1988). This is partly understandable because LPV paradigmallowed to describe nonlinear system as a family of linear systems and henceinvestigate the stability of these systems. Figure 1 shows the major dates withremarks in a time-line of gain scheduling.
PresentPast
1960 1969 1981 1988
1991
1943
Fist gain-scheduling
like controllers;
II. World War
First appear of notion gain-
scheduling; application in
flight and aerospace
Quiet years;
only few publications
devoted to gain-scheduling
"Gain control"
does appear in
the 25th
Anniversary
Index
Jeff. S. Shamma
introduced LPV
systmes
Rugh and Shamma
and also Leith and
Leithead survay
papers on gain-
scheduling;
Increased interest in
gain-scheduling
Gain-scheduling is one
of the most popular
approaches to nonlinear
control design
2010
Figure 1: The time-line of gain scheduling
2.2.3 Application of gain scheduling
As already noted, traditionally the gain scheduling was the primary design ap-proach to flight control and, consequently, many of the first articles and paperswere associated with flight control [47, 48, 49, 50, 51, 52, 53, 54] and aerospace[55, 56, 57]. Then gradually GS has been used almost everywhere in controlengineering which was greatly advanced with the introduction of LPV systems.
The second big bang in the history of gain scheduling was the advent of fuzzygain scheduling. Today, every second paper that appears under gain schedulingis devoted to fuzzy gain scheduling. Due to this wide range of gain-scheduleapproaches, gain scheduling is now used in several fields in practice. For examplein power systems the gain scheduling enjoyed exceptional success in control ofwind turbines [58, 59, 60, 61, 62, 63, 64]. But beside all this, some papers aredevoted to hydro turbines [65, 66], gas turbines [67], power system stabilizers [68]and generators [69]. Many papers in gain scheduling are devoted to magneticbearings [70, 71, 72, 73, 74, 75] but we can find some papers devoted to also tomicrogravity [76], turbofan engine [77] and diabetes control [78].
2.2.4 Summary of gain scheduling
The main advantage of classical gain scheduling is that it inherits the benefitsof linear controller design methods, including intuitive classical design tools andtime as well as frequency domain performance specifications. PID control isthe most used control strategy in industrial applications due to its relatively
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simple and intuitive design, hence this is a major advantage with respect to othernonlinear controller design syntheses. The approach thus enables the design oflow computational effort controllers. Conceptually, gain scheduling involves anintuitive simplification of the problem into parallel decompositions of the totalsystem.
LPV and LFT synthesis require a true LPV model as a basis. In generalhowever, gain scheduling may be employed in the absence of an analytical model,e.g. on the basis of a collection of plant linearizations. Consequently, controllerdesign based on a whitebox as well as a blackbox and even data-based ’modeling’is possible. If the possibility of fast parameter variations is not addressed in thedesign process, guaranteed properties of the overall gain-scheduled design cannotbe established. The main advantage of LPV and LFT control synthesis is thatthey do account for parameter variations in the controller design, which resultsin a priori guarantees regarding stability and performance specifications. Themain drawback of LPV and LFT control synthesis involves conservativeness,which has to be introduced to enable solving the resulting LMIs. As a resultof that, current LPV and LFT syntheses comprise specific extensions of robustcontrol techniques rather than true generalizations. However, current and futureresearch still provides and will provide less conservative solutions.
The main drawback of fuzzy gain scheduling involves the lack of a relationbetween the dynamic characteristics of the original nonlinear model and thefuzzy model. Even locally, the dynamics of the fuzzy model can not be relatedto the original nonlinear model. Fuzzy gain scheduling techniques may involveclassical gain scheduling alike as well as LPV techniques.
The analysis and theorems stated herein are presented in an informal manner.Technical details may (and should) be found in the associated references.
2.3 Discussion
This thesis is devoted to gain scheduling within this to LPV based gain schedul-ing because in our opinion the biggest potential between gain scheduling ap-proaches is in the LPV based gain scheduling. Despite this, we described allmain historical approaches to gain scheduling as classical gain scheduling, LFTbased gain scheduling and novel fuzzy gain scheduling.
As we mentioned LPV based gain scheduling appear in 1988 when Jeff. S.Shamma introduced the LPV paradigm in his Ph.D. thesis [1]. Today LPVparadigm has become a standard formalism in systems and controls with lots ofresearches and articles devoted to analysis, controller design and system identific-ation of these models. Due to this nowadays the LPV gain scheduling belongs tothe most popular approaches to nonlinear control design. But, as we mentionedin Introduction, there are still a lot of unsolved problems. Browsing throughliterature we cannot find any general LPV based gain-scheduled approach whichwill involve guaranteed cost and affine quadratic stability. In addition there arevery rudimentary approaches in switched and predictive control not to mentionthe robust and discrete design approaches.
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Currently, nowhere it is solved how to affect the performance quality sep-arately in each working point when direct LPV controller approach is used.Furthermore, there are only few papers devoted to output feedbacks and theyalso not use fixed order output feedbacks like PID/PSD controllers.
Most of the papers devoted to LPV based gain scheduling convert the stabilityconditions into LMI problems. But currently we cannot find a general LMIapproach with guaranteed stability and guaranteed cost. Furthermore, nowhereit is solved how to consider input/output constraints without need of on-lineoptimization.
Among other things, to find a solution for some of these unsolved problems(deficiencies) were the main goals of the research which is summarized in thisthesis.
3 Robust Gain-Scheduled PID Controller Design for Un-certain LPV Systems
A novel methodology is proposed for robust gain-scheduled PID controller designfor uncertain LPV systems. The proposed design procedure is based on theparameter-dependent quadratic stability approach. A new uncertain LPV sys-tem model has been introduced in this paper. To access the performance quality,the approach of a parameter-varying guaranteed cost is used which allowed toreach the desired performance for different working points. Several forms of para-meter dependent quadratic stability are presented which withstand arbitrarilyfast model parameter variation or/and arbitrarily fast gain-scheduled parametervariation.
3.1 Problem formulation and preliminaries
Consider a continuous-time linear parameter-varying (LPV) uncertain system inthe form
x = A(ξ, θ)x+B(ξ, θ)u
y = Cx
yd = Cdx
(3)
where linear parameter-varying matrices
A(ξ, θ) = A0(ξ) +
s∑i=1
Ai(ξ)θi ∈ Rn×n
B(ξ, θ) = B0(ξ) +
s∑i=1
Bi(ξ)θi ∈ Rn×m(4)
x ∈ Rn, u ∈ Rm, y ∈ Rl denote the state, control input and controlled output,respectively. Matrices Ai(ξ), Bi(ξ), i = 0, 1, 2, . . . , s belong to the convex set:
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a polytope with N vertices that can be formally defined as
Ω =
Ai(ξ), Bi(ξ) =
N∑j=1
(Aij , Bij) ξj
,
i = 0, 1, 2, . . . , s,
N∑j=1
ξj = 1, ξj ≥ 0
(5)
where s is the number of scheduled parameters; ξj , j = 1, 2, . . . , N are constantor possibly time-varying but unknown parameters; matrices Aij , Bij , C, Cd areconstant matrices of corresponding dimensions, where Cd is the output matrixfor D part of the controller. θ ∈ Rs is a vector of known (measurable) constant orpossibly time-varying scheduled parameters. Assume that both lower and upperbounds are available. Specifically
1. Each parameter θi, i = 1, 2, . . . , s ranges between known extremal values
θ ∈ Ωs =θ ∈ Rs : θi ∈ 〈θi, θi〉, i = 1, 2, . . . , s
(6)
2. The rate of variation θi is well defined at all times and satisfies
θ ∈ Ωt =θi ∈ Rs : θi ∈ 〈θi, θi〉, i = 1, 2, . . . , s
(7)
Note that system (3), (4), (5) consists of two type of vertices. The first one isdue to the gain-scheduled parameters θ with T = 2s vertices – θ vertices, and thesecond set of vertices are due to uncertainties of the system – N , ξ vertices. Forrobust gain-scheduled ”I” part controller design the states of system (3) need tobe extended in such a way that a static output feedback control algorithm canprovide proportional (P) and integral (I) parts of the designed controller. Formore details see [79]. Assume that system (3) allows PI controller design with astatic output feedback.
To access the system performance, we consider an original scheduling quad-ratic cost function
J =
∫ ∞0
J(t)dt =
∫ ∞0
(xTQ(θ)x+ uTRu+ xTS(θ)x
)dt (8)
where
Q(θ) = Q0 +
s∑i=1
Qiθi, S(θ) = S0 +
s∑i=1
Siθi
The feedback control law is considered in the form
u = F (θ)y + Fd(θ)yd (9)
where
F (θ) = F0 +
s∑i=1
Fiθi, Fd(θ) = Fd0 +
s∑i=1
Fdiθi
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Matrices Fi, Fdi, i = 0, 1, 2, . . . , s are the static output PI part and the out-put derivative feedback gain-scheduled controller. The structure of the abovematrices can be prescribed.
The respective closed-loop system is then
Md(ξ, θ)x = Ac(ξ, θ)x (10)
where
Md(ξ, θ) = I −B(ξ, θ)Fd(θ)Cd
Ac(ξ, θ) = A(ξ, θ) +B(ξ, θ)F (θ)C
Let as recall some results about an optimal control of time-varying systems [80].
Lemma 1. Let there exists a scalar positive definite function V (x, t) such thatlimt→∞ V (x, t) = 0 which satisfies
minu∈Ωu
δV
δxAc(θ) +
δV
δt+ J(t)
= 0 (11)
From (11) obtained control algorithm u = u∗(x, t) ensure the closed-loop stabilityand on the solution of (3) optimal value of cost function as J∗ = J(x0, t0) =V (x(0), t0).
Eq. (11) is known as Bellman-Lyapunov equation and function V (x, t) whichsatisfies to (11) is Lyapunov function. For a given concrete structure of Lyapunovfunction the optimal control algorithm may reduces from ”if and only if ” to ”if ”and for switched systems, robust control, gain-scheduled control and so on toguaranteed cost.
Definition 1. Consider a stable closed-loop system (10). If there exists acontrol law u (9) which satisfies (13) and a positive scalar J∗ such that thevalue of closed-loop cost function (8) J satisfies J < J∗ for all θ ∈ Ωs and ξj ,j = 1, 2 . . . , N satisfying (5), then J∗ is said to be a guaranteed cost and u issaid to be a guaranteed cost control law for system (10).
Let us recall some parameter dependent stability results which provide basicfurther developments.
Definition 2. Closed-loop system (10) is parameter dependent quadraticallystable in the convex domain Ω given by (5) for all θ ∈ Ωs and θ ∈ Ωt if andonly if there exists a positive definite parameter dependent Lyapunov functionV (ξ, θ) such that the time derivative of Lyapunov function with respect to (10)is
dV (ξ, θ, t)
dt< 0 (12)
Lemma 2. Consider the closed-loop system (10). Control algorithm (9) is theguaranteed cost control law if and only if there exists a parameter dependentLyapunov function V (ξ, θ) such that the following condition holds [80]
Be(ξ, θ) = minu
(dV (ξ, θ, t)
dt+ J(t)
)≤ 0 (13)
10
Uncertain closed-loop system (10) conforming to Lemma 2 is called robustparameter dependent quadratically stable with guaranteed cost.
We proceed with the notion of multi-convexity of a scalar quadratic function[81].
Lemma 3. Consider a scalar quadratic function of θ ∈ Rs
f(θ) = α0 +
s∑i=1
αiθi +
s∑i=1
s∑j>i
βijθiθj +
s∑i=1
γiθ2i (14)
and assume that if f(θ) is multiconvex that is
∂2f
∂θ2i
= 2γi ≥ 0, i = 1, 2, . . . , s
Then f(θ) is negative in the hyper rectangle (6) if and only if it takes negativevalues at the vertices of (6), that is if and only if f(θ) < 0 for all vertices ofthe set given by (6). For decrease the conservatism of Lemma 3 the approachproposed in [81] can be used.
In this paragraph for uncertain gain scheduling system (3) we have proposedto use a model uncertainty in the form of a convex set with N vertices definedby (5). Furthermore, we consider the new type of performance (8) to obtain theclosed-loop system guaranteed cost.
3.2 Main Results
This section formulates the theoretical approach to robust PID gain-scheduledcontroller design for polytopic system (3), (4), (5) which ensures closed-loopsystem parameter dependent quadratic stability and a guaranteed cost for allgain scheduling parameters θ ∈ Ωs, and θ ∈ Ωt. The main result on robuststability for the gain-scheduled control system is given in the next theorem.
Theorem 1. The closed-loop system (10) is robust parameter dependent quad-ratically stable with a guaranteed cost if there exist positive definite matrixP (ξ, θ) ∈ Rn×n, matrices N1, N2 ∈ Rn×n positive definite (semidefinite)matrices Q(θ), R, S(θ) and gain-scheduled controller (9) such that
a)
L(ξ, θ) = W0(ξ) +s∑i=1
Wi(ξ)θi+
+
s∑i=1
s∑j>i
Wij(ξ)θiθj +
s∑i=1
Wiiθ2i < 0
(15)
b)Wii(ξ) ≥ 0, θ ∈ Ωs, i = 1, 2, . . . , s (16)
11
where we consider parameter dependent Lyapunov matrix
P (ξ, θ) = P0(ξ) +
s∑i=1
Pi(ξ)θi > 0 (17)
the above matrices (15) and (16) are given as follows:
W0(ξ) =
[W110(ξ) W120(ξ)∗ W220(ξ)
]W110(ξ) = S0 + CTd Fd
T0 RFd0Cd
+NT1 (I −B0(ξ)Fd0Cd)
+ (I −B0(ξ)Fd0Cd)T N1
W120(ξ) = −NT1 (A0(ξ) +B0(ξ)F0C)
+ (I −B0(ξ)Fd0Cd)T N2 + P0(ξ)
+CTd FdT0 RF0C
W220(ξ) = −NT2 (A0(ξ) +B0(ξ)F0C)
− (A0(ξ) +B0(ξ)F0C)T N2 +Q0
+CTFT0 RF0C +∑sj=1 Pj(ξ)θi
Wi(ξ) =
[W11i(ξ) W12i(ξ)∗ W22i(ξ)
]W11i(ξ) = Si + CTd
(Fd
T0 RFdi + Fd
Ti RFd0
)Cd
−NT1 (B0(ξ)Fdi +Bi(ξ)Fd0)Cd
− [(B0(ξ)Fdi +Bi(ξ)Fd0)Cd]T N1
W12i(ξ) = −NT1 (Ai(ξ) +B0(ξ)Fi +Bi(ξ)F0)C
− (Bi(ξ)Fd0Cd)T N2 + Pi(ξ)
+CTd(Fd
Ti RF0 + Fd
T0 RFi
)C
W22i(ξ) = −NT2 (Ai(ξ) + (B0(ξ)Fi +Bi(ξ)F0)C)
− [Ai(ξ) + (B0(ξ)Fi +Bi(ξ)F0)C]T N2
+Qi + CT(FT0 RFi + FTi RF0
)C
Wij(ξ) =
[W11ij(ξ) W12ij(ξ)∗ W22ij(ξ)
]W11ij(ξ) = CTd
(Fd
Ti RFdj + Fd
Tj RFdi
)Cd
−NT1 (Bi(ξ)Fdj +Bj(ξ)Fdi)Cd
−CTd (Bi(ξ)Fdj +Bj(ξ)Fdi)T N1
W12ij(ξ) = −NT1 (Bi(ξ)Fj +Bj(ξ)Fi)C
−CTd (Bi(ξ)Fdj +Bj(ξ)Fdi)T N2
+CTd(Fd
Ti RFj + Fd
Tj RFi
)C
W22ij(ξ) = −NT2 (Bi(ξ)Fj +Bj(ξ)Fi)C
−CT (Bi(ξ)Fj +Bj(ξ)Fi)T N2
+CT(FTi RFj + FTj RFi
)C
Wii(ξ) =
[W11ii(ξ) W12ii(ξ)∗ W22ii(ξ)
]W11ii(ξ) = CTd Fd
Ti RFdiCd −NT
1 Bi(ξ)FdiCd−CTd FdTi Bi(ξ)TN1
12
W12ii(ξ) = −NT1 Bi(ξ)FiC − CTd FdTi BTi (ξ)N2
+CTd FdTi RFiC
W22ii(ξ) = −NT2 Bi(ξ)FiC − CTFTi BTi (ξ)N2
+CTFTi RFiC
The proof is based on Lemma 2 and 3. The time derivative of the Lyapunovfunction V (ξ, θ) = xTP (ξ, θ)x is
dV (ξ, θ)
dt=[xT xT
] [ 0 P (ξ, θ)
P (ξ, θ) P (ξ, θ)
] [xx
](18)
where
P (ξ, θ) =
s∑i=1
Pi(ξ)θ
To isolate two matrices (system and Lyapunov) introducing matrices N1, N2 inthe following way
[2N1x+ 2N2x]T [Md(ξθ)x−Ac(ξ, θ)] = 0 (19)
and substituting (19), (18), J(t) (8) and control law (9) to (13), after somemanipulation one obtains
Be(ξ, θ) =[xT xT
] [ W11(ξ) W12(ξ)W12
T (ξ) W22(ξ)
] [xx
](20)
whereW11 = S(θ) + CTd F
Td (θ)RFd(θ)Cd +NT
1 Md(ξ, θ)+MT
d (ξ, θ)N1
W12 = −NT1 Ac(ξ, θ) +MT
d (ξ, θ)N2 + P (ξ, θ)+CTd F
Td (θ)RF (θ)C
W22 = −NT2 Ac(ξ, θ)−ATc (ξ, θ)N2 +Q(θ)
+CTFT (θ)RF (θ)C + P (ξ, θ)
Eq. (20) immediately implies (15), which proves the sufficient conditions ofTheorem 1.
Eq.’s (15) and (16) are linear with respect to uncertain parameter ξj , j =1, 2, . . . , N , therefore (15) and (16) have to hold for all j = 1, 2, . . . , N . For theknown gain-scheduled controller parameters, inequalities (15) and (16) reduceto LMI, for gain-scheduled controller synthesis problem (15) (16) are BMI.
Remark 1. Theorem 1 can be used for a quadratic stability test, whereLyapuunov function matrices (matrix) are either independent of parameter ξj ,j = 1, 2, . . . , N or parameter θi, i = 1, 2, . . . , s or both as listed below.
1. Quadratic stability with respect to model parameter variation. For thiscase one has P (θ) = P0 +
∑si=1 Piθi. This Lyapunov function should
withstand arbitrarily fast model parameter variation in the convex set (5)
13
2. Quadratic stability with respect to gain-scheduled parameters θ. For thiscase Pi → 0, i = 1, 2 . . . , s and Lyapunov matrix is P (ξ, θ) = P0(ξ).This Lyapunov function should withstand arbitrarily fast θ parameter vari-ations.
3. Quadratic stability with respect to both ξ and θ parameters. Lyapunovmatrix is P (ξ, θ) = P0 and it should withstands arbitrarily fast model andgain-scheduled parameter variation.
4 Robust Switched Controller Design for Nonlinear Con-tinuous Systems
A novel approach is presented to robust switched controller design for nonlin-ear continuous-time systems under an arbitrary switching signal using the gainscheduling approach. The proposed design procedure is based on the robustmulti parameter dependent quadratic stability condition. The obtained switchedcontroller design procedure for nonlinear continuous-time systems is in the bi-linear matrix inequality form (BMI). In the paper several forms of parameterdependent/quadratic Lyapunov functions are proposed.
4.1 Problem statement and preliminaries
4.1.1 Uncertain LPV plant model for switched systems
Consider family of nonlinear switched systems
z = fσ(z, v, w) σ ∈ S = 1, 2, . . . , Ny = h(z)
(21)
where z ∈ Rn is the state, the input v ∈ Rm, the output y ∈ Rl, exogenous inputw ∈ Rk which captures parametric dependence of the plant (21) on exogenousinput. The arbitrary switching algorithm σ ∈ S is a piecewise constant, rightcontinuous function which specifies at each time the index of the active system,[82]. Assume that f(.) is locally Lipschitz for every σ ∈ S. Consider thatthe number of equilibrium points for each switching modes is equal to p, thatis for each mode σ ∈ S the nonlinear system can be replaced by a family ofp linearized plant. For more details how to obtain the gain-scheduled plantmodel see excellent surveys [9], [10]. To receive the model uncertainty of thegain-scheduled plant it is necessary to obtain other family of linearized plantmodels around the p equilibrium points. Finally, one obtains the gain-scheduleduncertain plant model in the form
x = Aσ(ξ, θ)x+Bσ(ξ, θ)u σ ∈ Sy = Cx
(22)
where x = z − ze, u = v − ve, y = y − ye, (ze, ve, ye) define the equilibriumfamily for plant (21). Assume, that for i − th equilibrium point one obtain the
14
sets x ∈ Xi, u ∈ Ui, y ∈ Yi, i = 1, 2, . . . , p. Summarizing above sets we getx ∈ X =
⋃pi=1 Xi, u ∈ U =
⋃pi=1 Ui, y ∈ Y =
⋃pi=1 Yi.
Aσ(ξ, θ) = Aσ0(ξ) +
p∑j=1
Aσj(ξ)θj ∈ Rn×n
Bσ(ξ, θ) = Bσ0(ξ) +
p∑j=1
Bσj(ξ)θj ∈ Rn×m(23)
Matrices Aσj(ξ), Bσj(ξ), j = 0, 1, 2, . . . , p belong to the convex set a polytopewith K vertices that can formally defined as
Ωσ =
Aσj(ξ), Bσj(ξ) =
K∑i=1
(Aσij , Bσij)ξi
j = 0, 1, 2, 3, . . . , p,
K∑i=1
ξi = 1, ξi ≥ 0, ξi ∈ Ωξ
(24)
where ξi, i = 1, 2, . . . ,K are constant or possible time-varying but unknownparameters, Aσij , Bσij , C are constant matrices of corresponding dimensions,θ ∈ Rp is a vector of known constant or time-varying gain-scheduled parameter.Assume that both lower and upper bounds are available, that is
θ ∈ Ωs = θ ∈ Rp : θj ∈ 〈θj , θj〉
θ ∈ Ωt = θ ∈ Rp : θj ∈ 〈θj , θj〉(25)
4.1.2 Problem formulation
For each plant mode consider the uncertain gain-scheduled LPV plant model inthe form (22),(23) and (24)
x =
(Aσ0(ξ) +
p∑j=1
Aσj(ξ)θj
)x+
(Bσ0(ξ) +
p∑j=1
Bσj(ξ)θj
)u
y = Cx
(26)
For a robust gain-scheduled I part controller design, the states x of (26) needto be extended in such a way that a static output feedback control algorithmcan provide proportional (P) and integral (I) parts of the designed controller, formore detail see [79]. Assume that system (26) allows PI controller design witha static output feedback. The feedback control law is considered in the form
u = Fσ(θ)y =
(Fσ0 +
p∑j=1
Fσjθj
)Cx (27)
15
where Fσ(θ) is the static output feedback gain-scheduled controller for mode σ.The closed loop system is
x = Aσc(ξ, θ, α)x (28)
whereAσc(ξ, θ, α) =
N∑σ=1
(Aσ(ξ, θ) +Bσ(ξ, θ)Fσ(θ)C
)ασ =
N∑σ=1
Aσ(ξ, θ)ασ
αT = [α1, α2, ...αN ],
N∑σ=1
ασ = 1,
N∑σ=1
ασ = 0
αj = 1 when σj is active plant mode, else αj = 0. Assume α ∈ Ωα, α ∈ Ωd.To access the system performance, we consider an original weighted scheduledquadratic cost function
J =
∫ ∞t=0
J(t)dt (29)
where J(t) = xTQ(θ)x+ uTRu, and
Q(θ) = Q0 +
p∑j=1
Qjθj , Qj ≥ 0, R > 0
.
Definition 3. Consider a stable closed loop switched system (28) with N modes.If there is a control algorithm (27) and a positive scalar J∗ such that the closedloop cost function (29) satisfies J ≤ J∗ for all θ ∈ Ωs, α ∈ Ωα, then J∗ is said tobe a guaranteed cost and ”u” is said to be a guaranteed cost control algorithmfor arbitrary switching algorithm σ ∈ S.
Theorem 2. [80] Control algorithm (27) is the guaranteed cost control law forthe switched closed loop system (28) if and only if there is Lyapunov functionV (x, ξ, θ, α) > 0, matrices Q(θ), R and gain matrices Fσk; k = 0, 1, . . . , p suchthat for σ ∈ S the following inequality holds
Be =dV (x, ξ, θ, α)
dt+ J(t) ≤ −εxTx, ε→ 0 (30)
4.2 Main results
This section formulates the theoretical approach to the robust switched gain-scheduled controller design with control law (27) which ensure closed loop multiparameter dependent quadratic stability and guaranteed cost for an arbitraryswitching algorithm σ ∈ S. Assume that in Theorem 2 the Lyapunov functionis in the form
V (x, ξ, θ, α) = xTP (ξ, θ, α)x (31)
16
where the Lyapunov multi parameter dependent matrix is
P (ξ, θ, α) =
K∑i=1
(P0i +
N∑σ=1
(Pσ0i +
p∑j=1
Pσijθj)ασ
)ξi (32)
Time derivative of the Lyapunov function(31) is
V (.) = [xT xT ]
[0 P (ξ, θ, α)
P (ξ, θ, α) P (ξ, θ, α)
] [xx
](33)
where
P (.) =
K∑i=1
N∑σ=1
DPσiασξi (34)
DPσi =
N∑σ=1
Pσ0iασ +
p∑j=1
Pσij θj +
p∑j=1
N∑σ=1
Pσijασθj
Using equality
(2N1x+ 2N2x)T(x−
N∑σ=1
Aσ(ξ, θ)ασx
)= 0 (35)
equation (33) can be rewritten as
dV (.)
dt=
N∑σ=1
[xT xT
]Lσ(ξ, θ)
[xx
](36)
Lσ(ξ, θ) = lσ(i, j)2×2
lσ(1, 1) = NT1 +N1
lσ(1, 2) = −NT1 Aσ(ξ, θ) +N2 +
K∑i=1
(P0i + Pσ0i +
p∑j=1
Pσijθj
)ξi
lσ(2, 2) = −NT2 Aσ(ξ, θ)−ATσ (ξ, θ)N2 +
K∑i=1
DPσiξi
where N1, N2 ∈ Rn×n are auxiliary matrices.On substituting (27) to (29) one obtains
J(t) = xTS(θ)x (37)
where
S(θ) = S0 +
p∑j=1
Sjθj +
p∑j=1
p∑k>j
Skjθkθj +
p∑k=1
Skkθ2k
S0 = Q0 + CTFTσ0RFσoC
17
Sj = Qj + CT (FTσ0RFσj + FTσjRFσ0)C
Sjk = CT (FTσjRFσk + FTσkRFσj)C
Skk = CTFTσkRFσkC
The switched model plant (28) can be rewritten to the form
Aσ(ξ, θ) = M0(ξ) +
p∑j=1
Mj(ξ)θj +
p∑j=1
p∑k>j
Mjk(ξ)θjθk+ (38)
p∑k
Mkk(ξ)θ2
whereM0(ξ) = Aσ0(ξ) +Bσ0(ξ)Fσ0C
Mj(ξ) = Aσj + (Bσ0(ξ)Fσj +Bσj(ξ)Fσ0)C
Mjk(ξ) = (Bσj(ξ)Fσk +Bσk(ξ)Fσj)C
Mkk(ξ) = Bσk(ξ)FσkC
Due to Theorem 2 the closed loop switched gain-scheduled system is multiparameter dependent quadratically stable with guaranteed cost for σ ∈ S, ξi,i = 1, 2, . . . ,K if the following inequalities hold
Be = [xT xT ]W (ξ, σ, θ)[xT xT ]T ≤ 0 (39)
where W (ξ, σ, θ) = wij(σ, ξ)2×2
w11(σ, ξ) = NT1 +N1
w12(σ, ξ) =∑Ki=1
(P0i + Pσ0i +
∑pj=1 Pσijθj
)ξi
− NT1 Aσ(ξ, θ) +N2
w22(σ, ξ) = −NT2 Aσ(ξ, θ)−Aσ(ξ, θ)TN2
+∑Ki=1 DPσiξi + S(θ)
Inequality (39) implies :- for all σ ∈ S the inequality is linear with respect to uncertain parameter ξi,i = 1, 2, . . . ,K,- for all σ ∈ S the inequality is a quadratic function with respect to the gain-scheduled parameters θi, i = 1, 2, . . . , p.For the next development the following theorem is useful.
Theorem 3. [81] Consider a scalar quadratic function of θ ∈ Rp
f(θ) = a0 +
p∑j=1
ajθj +
p∑j=1
p∑k>j
ajkθjθk +
p∑k
akkθ2k (40)
18
and assume that if f(θ) is multiconvex, that is
δ2f(θ)
δθ2k
= 2akk ≥ 0, k = 1, 2, . . . , p
then f(θ) is negative definite in the hyper rectangle (25) if and only if it takesnegative values at the vertices of (25), that is if and only if f(θ) < 0 for allvertices of the set given by (25).
Due to (34), (37) and (38) the robust stability conditions of switched systemcan be rewritten as
W (ξ, σ, θ) =
N∑σ=1
L(θ, ξ)ασ =
N∑σ=1
(Wσ0(ξ)+
+
p∑j=1
Wσj(ξ)θj +
p∑j=1
p∑k>j
Wσjk(ξ)θjθk+
+
p∑k=1
Wσkkθ2k)ασ ≤ 0
(41)
where Wσ0(ξ) = wσ0ij2×2, Wσj(ξ) = wσjik2×2
wσ011 = NT1 +N1
wσ012 = −NT1 M0(ξ) +N2 +
∑Ki=1(P0i + Pσ0i)ξi
wσ022 = −NT2 M0(ξ)−MT
0 (ξ)N2 + S0+
+∑Ki=1
(Pσ0iασ +
∑pj=1 Pσij θj
)ξi
wσj11 = 0; wσj12 = −NT1 Mj(ξ) +
∑Ki=1 Pσijξi
wσj22 = −NT2 Mj(ξ)−Mj(ξ)
TN2 + Sj+
+∑Ki=1
(∑Nσ=1 Pσijασ
)ξi
Wσjk(ξ) =
[0 −NT
1 Mjk(ξ)” ∗ ” −NT
2 Mjk(ξ)−Mjk(ξ)TN2 + Sjk
]Wσkk(ξ) =
[0 −NT
1 Mkk(ξ)” ∗ ” −NT
2 Mkk(ξ)−Mkk(ξ)TN2 + Skk
]The main results on the robust stability condition for the switched gain-scheduledcontrol system is given in the next theorem.
Theorem 4. Closed loop switched system (28) is robust multi parameter depend-ent quadratically stable with guaranteed cost if there is a positive definite matrixP (ξ, θ, α) ∈ Rn×n (32), matrices N1, N2 ∈ Rn×n, positive definite (semidefinite)matrices Q(θ), R and gain-scheduled controller matrix Fσ(θ), such that for σ ∈ S
1.Lσ(ξ, θ) < 0 (42)
19
2.
Wσkk ≥ 0, σ ∈ S, θ ∈ Ωs, k = 1, 2, . . . , p
The proof of theorem sufficient conditions immediately follows from eqs. (32)-(41).Notes.
• Lσ(θ, ξ) is linear with respect to uncertain parameter ξi, i = 1, 2, . . . ,K,it holds Lσ(θ, ξ) =
∑Ki=1 Lσi(θ)ξi, therefore inequality (42) for each σ ∈ S
split to K inequalities of type Lσi(θ) < 0 and Wσikk ≥ 0.
• Eq. (32) implies that in dependence on the chosen structure of the Lya-punov matrix P (ξ, θ, α) one should obtained different types of stabilityconditions from quadratic to multi parameter dependent quadratic stabil-ities. Different types of stability conditions determine the conservatism ofthe design procedure and the rate of change of corresponding variables.
5 Gain-Scheduled MPC Design for Nonlinear Systemswith Input Constraints
A novel methodology is proposed for discrete model predictive gain-scheduledcontroller design for nonlinear systems with input(hard)/output(soft) constraintsfor finite and infinite prediction horizons. The proposed design procedureis based on the linear parameter-varying (LPV) paradigm, affine parameter-dependent quadratic stability and on the notion of the parameter-varying guar-anteed cost. The design procedure is in the form of BMI (we can use a free andopen source BMI solver).
5.1 Problem formulation and preliminaries
Consider a nonlinear plant x(k + 1) = F (x(k), u(k), θ(k)) which is identified inseveral working points. The identified family of linear systems in discrete-timespace is given as follows
x(k + 1) = Ai x(k) +Bi u(k)
y(k) = Ci x(k) i = 1, 2, . . . , N(43)
where x(k) ∈ Rn is the state vector, u(k) ∈ Rm is the controller output, y(k) ∈ Rlis the measured plant output vector at step k ∈ R+, matrices Ai, Bi, Ci,i = 1, 2, . . . , N are system matrices with appropriate dimension and N is thenumber of identified plants model. Assume that a known vector θ(k) ∈ Ω existswhich captures the parametric dependence of the linearized model (43) on theequilibrium (working) points of the original nonlinear system.
20
5.1.1 Case of finite prediction horizon
The identified family of linear systems (43) for a given prediction and controlhorizon Nk can be transformed to the following form [83]
z(k + 1) = Af i z(k) +Bf i v(k)
yf (k) = Cf i z(k) i = 1, 2, . . . , N(44)
wherezT (k) =
[xT (k|k) xT (k + 1|k) · · · xT (k +Nk − 1|k)
]vT (k) =
[uT (k|k) uT (k + 1|k) · · · uT (k +Nk − 1|k)
]yTf (k) =
[yT (k|k) yT (k + 1|k) · · · yT (k +Nk − 1|k)
]Af i =
Ai 0 · · · 0A2i 0 · · · 0
......
. . . 0Ai
Nk 0 · · · 0
, Cf i =
Ci 0 · · · 00 Ci · · · 0...
.... . .
...0 0 · · · Ci
Bf i =
Bi 0 · · · 0AiBi Bi · · · 0A2iBi AiBi · · · 0...
.... . .
...ANk−1i Bi A
Nk−2i Bi · · · Bi
and x(k + j|k) is the step ahead prediction of the state, calculated in sampletime k. From the family of linear systems (44) one obtains [84] a gain-scheduledplant model in the form
z(k + 1) = Af a(θ(k)) z(k) +Bf a(θ(k)) v(k)
yf (k) = Cf a z(k)(45)
where Cf a = Cf 1 = Cf 2 = . . . = CfN and
Af a(θ(k)) = Af a0+
p∑i=1
Af aiθi(k)
Bf a(θ(k)) = Bf a0+
p∑i=1
Bf aiθi(k)
and Af ai, Bf ai, i = 0, 1, . . . , p− 1 are system matrices with appropriate dimen-
sion, Af ap = 0, Bf ap = 0, θ(k)T = [θ1(k), θ2(k), . . . , θp−1(k)] ∈ Ω is the vector of
p−1 known independent scheduling parameters at step k and θp ∈ 〈0, Hm〉 is thescheduled parameter which is used to ensure I/O constraints, where Hm ∈ (0, 1).The control law for the model predictive gain-scheduled controller design for agiven prediction and control horizon Nk is considered in the form
v(k) = F (θ(k)) yf (k) = F (θ(k))Cf a zf (k) (46)
where F (θ(k)) = F0 +∑p−1j=1 Fjθj(k)− F0θp(k).
21
Note 1. We can extend system (45) to PS or PSD control, for more informationsee [85].
The procedure to ensure the input/output constraints is very simple. Ifthe system input or output approach the maximal or minimal value, using thescheduling parameter θp one can affect the controller output. There are severalsolutions how to generate the scheduling parameter θp, it is depending on thesystem. We will deal with this issue in the examples. If we substitute controllaw (46) to system (45), a closed-loop system is obtained
z(k + 1) = Ac(θ(k))z(k) (47)
where Ac(θ(k)) = Af a(θ(k)) +Bf a(θ(k))F (θ(k))Cf a.To assess the performance quality with possibility to obtain different per-
formance quality in each working point a quadratic cost function described inpaper [85] will be used
Jdf (θ(k)) =
∞∑k=0
zf (k)TQ(θ(k))zf (k) + v(k)TRv(k)
+ ∆zf (k)TS(θ(k))∆zf (k) =
∞∑k=0
Jd(θ(k))
(48)
where ∆zf (k) = zf (k + 1) − zf (k), Q(θ(k)) = Q0 +∑pi=1 Qiθi(k), S(θ(k)) =
S0 +∑pi=1 Siθi(k), Qi = QTi ≥ 0, Si = STi ≥ 0, R > 0 and Qp = Sp = 0.
Note 2. Using the cost function (48) we can affect the performance qualityseparately in each working point with defining different weighting matrices foreach working point which then are transformed to affine form and depend onthe scheduled parameters as system matrices. [85]
Definition 4. Consider system (45) with control algorithm (46). If a controllaw v∗ and a positive scalar J∗d exist such that the closed-loop system (47) isstable and the value of closed-loop cost function (48) satisfies Jd ≤ J∗d , then J∗dis said to be a guaranteed cost and v∗ is said to be guaranteed cost control lawfor system (45).
Substituting the control law (46) to the quadratic cost function (48) one canobtain
Jd(θ(k)) = zT[Jd11(θ(k)) Jd12(θ(k))JdT12(θ(k)) Jd22(θ(k))
]z (49)
where zT =[zT (k + 1) zT (k)
]and
Jd11(θ(k)) = S(θ(k)), Jd12(θ(k)) = −S(θ(k)),
Jd22(θ(k)) = Q(θ(k)) + CfTa F (θ(k))TRF (θ(k))Cf a
+ S(θ(k))
22
To ensure the Affine Quadratic Stability (AQS) [81] the following Lyapunovfunction has been chosen
V (θ(k)) = zTf (k)P (θ(k))zf (k) (50)
The first difference of Lyapunov function (50) is given as follows
∆V (θ(k)) = zTf (k + 1)P (θ(k + 1)) zf (k + 1)−
− zTf (k)P (θ(k)) zf (k)(51)
where
P (θ(k)) = P0 +
p∑i=1
Piθi(k) (52)
On substituting θ(k+1) = θ(k)+∆θ(k) to P (θ(k+1)) one obtains the followingresult
P (θ(k + 1)) = P0 +
p∑i=1
Piθi(k) +
p∑i=1
Pi∆θi(k) (53)
where if assuming that Pi > 0, ∆θi ∈ 〈∆θi,∆θi〉 ∈ Ωt, i = 0, 1, . . . , p andmax |∆θi| < ρi, one can write
P (θ(k + 1)) ≤ P0 +
p∑i=1
Piθi(k) + Pρ = Pρ(θ(k)) (54)
where Pρ =∑pi=1 Piρi. The first difference of the Lyapunov function (51) using
the free matrix weighting approach [84] is in the form
∆V (θ(k)) = zT[V11(θ(k)) V12(θ(k))V T12(θ(k)) V22(θ(k))
]z (55)
whereV11(θ(k)) = Pρ(θ(k)) +N1 +NT
1
V12(θ(k)) =NT2 −N1Ac(θ(k))
V22(θ(k)) =−P (θ(k))−N2Ac(θ(k))−ATc (θ(k))NT2
where N1, N2 ∈ Rn×n are auxiliary matrices.
Definition 5. [81] The linear closed-loop system (47) for θ(k) ∈ Ω and∆θ(k) ∈ Ωt is affinely quadratically stable if and only if p+1 symmetric matricesP0, P1, . . . , Pp exist such that P (θ(k)) (52), Pρ(θ(k)) (54) are positive definedand for the first difference of the Lyapunov function (55) along the trajectory ofclosed-loop system (47) it holds
∆V (θ(k)) < 0 (56)
From LQ theory we can introduce the well known results:
23
Lemma 4. Consider the closed-loop system (47). Closed-loop system (47) isaffinely quadratically stable with guaranteed cost if and only if the following in-equality holds
Be(θ(k)) = minu∆V (θ(k)) + Jd(θ(k)) ≤ 0 (57)
for all θ(k) ∈ Ω. For proof see [80].
5.1.2 Case of infinite prediction horizon
The system described by (45) for the case of Nk = 0 can be transformed to thegain-scheduled plant model
x(k + 1) = A(θ(k))x(k) +B(θ(k))u(k)
y(k) = C x(k)(58)
where A(θ(k)) = A0 +∑pi=1 Aiθi(k), B(θ(k)) = B0 +
∑pi=1 Biθi(k), Ap = 0 and
Bp = 0. For the case Nk →∞ and S = 0 the cost function (48) can be rewrittenas
J =
∞∑k=0
J(k) =
∞∑k=0
(∞∑j=0
xT (k + j)qjx(k + j)
+uT (k + j)rju(k + j)
)=
∞∑k=0
∞∑j=0
J(k)
(59)
where qj ∈ Rn×n, rj ∈ Rm×m are positive definite matrices. The control law forthe model predictive gain-scheduled controller design for the infinite predictionhorizon is considered in the form
u(k) = F (θ(k)) y(k) = F (θ(k))C x(k) (60)
where F (θ(k)) = F0 +∑p−1j=1 Fjθj(k) − F0θp(k). To guarantee the stability and
performance of the closed-loop gain-scheduled system, due to Lemma 4 it issufficient to ensure
Be(θ(k)) = ∆V (x(k + j), θ(k)) + J(k) ≤ 0 (61)
where ∆V (x(k+ j), θ(k)) = V (x(k+ j + 1), θ(k) + ∆θ(k))− V (x(k), θ(k)) is thefirst difference of the Lyapunov function for j horizon prediction. Summing (61)from j = 0 to j →∞, the upper bound on J(k) is obtained
J(k) ≤ V (x(k), θ(k)) (62)
On the basis of (62) the following gain-scheduled MPC design procedure is given
minF (θ(k))
V (x(k), θ(k)) (63)
24
with constraints to system model (58), stability model and performance (61)and other constraints. Assume that the Lyapunov function is in the formV (x(k), θ(k)) = xT (k)P (θ(k))x(k), where P (θ) = P0 +
∑pi=1 Piθi(k). Due to
(63), the predicted control design procedure can be modified as
minF (θ(k))
xT (k)P (θ(k))x(k) ≤ xT (k)x(k)γ (64)
which leads to the inequality
P (θ(k)) ≤ minF (θ(k))
γI (65)
In the paper [86] inequality (64) is in the form
min xT (k)P (θ(k))x(k) ≤ γ (66)
which needs to know the state vector x(k) and on-line optimization of (66) atevery sample time.
The stability of the closed-loop system is guaranteed if
Be = ∆V (x(k), θ(k)) + αV (x(k), θ(k)) ≤ 0 (67)
where α ∈ 〈0, 1) is a coefficient with an influence on the closed-loop systemperformance. If we substitute the Lyapunov function and its first difference to(67), we can obtain
Be(θ(k)) = xTW (θ(k)) x ≤ 0 (68)
where xT =[xT (k + 1) xT (k)
], β = 1− α and
W (θ(k)) =
[W11(θ(k)) W12(θ(k))W12(θ(k))T W22(θ(k))
]W11(θ(k)) =NT
1 +N1 + Pρ(θ(k))W12(θ(k)) =−NT
1 Ac(θ(k)) +N2
W22(θ(k)) =−N2Ac(θ(k))−ATc (θ(k))N2 − P (θ(k))(β)
5.2 Main results
In this section the discrete predictive gain-scheduled controller design procedureis presented which guarantees the affine quadratic stability and guaranteed costfor θ(k) ∈ Ω with pre-defined maximal rate of change of the scheduled parametersρ. The main result of this section – the discrete model predictive gain-scheduledcontroller design procedure – relies on the concept of multi-convexity, that isconvexity along each direction θi(k), i = 1, 2, . . . , p of the parameter space. Theimplications of multiconvexity for scalar quadratic functions are given in thenext lemma [81].
Lemma 5. Consider a scalar quadratic function of α ∈ Rp.
f (α) = a0 +
p∑i=1
aiαi +
p∑i=1
p∑j>i
bijαiαj +
p∑i=1
ciα2i (69)
25
and assume that f (α1, . . . , αp) is multi-convex, that is ∂2f(α)
∂α2i
= 2ci ≥ 0 for
i = 1, 2, . . . , p. Then f(α) is negative for all α ∈ Ω and α ∈ Ωt if and only if ittakes negative values at the corners of α.
5.2.1 Finite prediction horizon
Using Lemmas 4 and 5 the following theorem is obtained for discrete modelpredictive gain-scheduled controller design for finite horizon.
Theorem 5. Closed-loop system (47) is affinely quadratically stable if p + 1symmetric matrices P0, P1, . . . , Pp exist such that P (θ(k)) (52), Pρ(θ(k)) (54)are positive definite for all θ(k) ∈ Ω, with pre-defined ρi, matrices N1, N2, Qi,R, Si, i = 1, 2, . . . , p and gain-scheduled matrices F (θ(k)) satisfying
M(θ(k)) < 0; θ(k) ∈ Ω
Mii ≥ 0; i = 1, 2, . . . , p(70)
where (at sample time k)
M(θ) =M0 +∑pi=1 Miθi +
∑pi=1
∑p−1j>i Mijθiθj +
∑pi=1 Miiθ
2i
M0 =[M110 M120
M12T0 M220
], Mi =
[M11i M12i
M12Ti M22i
]Mij =
[M11ijM12ij
M12TijM22ij
],M110 = P0 +N1 +NT
1 + S0 + Pρ
M11i =Pi, M11ij = 0, M11ii = 0M120 =N2 −NT
1 (Af a0+Bf a0
F0Cf a)− S0
M12i =−NT1 (Af ai +Bf aiF0Cf a +Bf a0
FiCf a)− SiM12ij =−NT
1 (Bf aiFj +Bf ajFi)Cf aM12ii =−NT
1 Bf aiFiCf aM220 =Q0 + S0 − P0 −NT
2 (Af a0+Bf a0
F0Cf a)−(Af a0
+Bf a0F0Cf a)TN2 + Cf
Ta F
T0 RF0Cf a
M22i =−Pi −NT2 (Af ai +Bf aiF0Cf a +Bf a0
FiCf a)−(Af ai +Bf aiF0Cf a +Bf a0
FiCf a)TN2
+CfTa (FT0 RFi + FTi RF0)Cf a +Qi + Si
M22ij =−NT2 (Bf aiFj +Bf ajFi)Cf a − Cf
Ta (Bf aiFj
+Bf ajFi)TN2 + Cf
Ta (FTi RFj + FTj RFi)Cf a
M22ii =−NT2 Bf aiFiCf a − (Bf aiFiCf a)TN2
+CfTa F
Ti RFiCf a
Proof. The proof of the Theorem 5 is clear from the previous derivations. Here,the proof is repeated only in basic steps. The proof is based on the Lemmas 4 and5. When substituting the first difference of the Lyapunov function (55) and thequadratic cost function (49) to the Bellman-Lyapunov function (57), after somemanipulation, using Lemma 5 we obtain (70) which proofs the Theorem 5.
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5.2.2 Infinite prediction horizon
Using inequalities (65), (68) and Lemma 5 the following theorem is obtained fordiscrete model predictive gain-scheduled controller design for infinite horizon.
Theorem 6. Closed-loop system is affinely quadratically stable if there existp + 1 symmetric matrices P0, P1, . . . , Pp such that P (θ(k)) (52), Pρ(θ(k)) (54)are positive definite for all θ(k) ∈ Ω, with pre-defined ρi, matrices N1, N2, andgain-scheduled matrices F (θ(k)) satisfying
W (θ(k)) < 0; θ(k) ∈ Ω
Wii ≥ 0; i = 1, 2, . . . , p
P (θ(k)) ≤ minF (θ(k))
γ
(71)
where (at sample time k)
W (θ) =W0 +∑pi=1 Wiθi +
∑pi=1
∑p−1j>i Wijθiθj +
∑pi=1 Wiiθ
2i
W0 =[W110 W120
W12T0 W220
], Wi =
[W11i W12i
W12Ti W22i
]Wij =
[W11ij W12ij
W12Tij W22ij
], W110 = P0 +N1 +NT
1 + Pρ
W11i = Pi, W11ij = 0, W11ii = 0W120 =N2 −NT
1 (A0 +B0F0C)W12i =−NT
1 (Ai +BiF0C +B0FiC),W12ij =−NT
1 (BiFj +BjFi)CW220 =−NT
2 (A0 +B0F0C)− (A0 +B0F0C)TN2 − P0βW22i =−NT
2 (Ai +BiF0C +B0FiC)−(Ai +BiF0C +B0FiC)TN2 − Piβ
W22ij =−NT2 (BiFj +BjFi)C − CT (BiFj +BjFi)
TN2
W12ii =−NT1 BiFiC, W22ii = −NT
2 BiFiC − (BiFiC)TN2
Proof. The proof of the Theorem 6 regarding to space limitations is sketched onlyin basic steps. The proof is based on the Lemmas 4 and 5. If we substitute theLyapunov function and its first difference to (67), we can obtain (68), after somemanipulation, using Lemma 5 we obtain (71) which proofs the Theorem 6.
6 Concluding remarks
6.1 Brief overview
This thesis deals with controller design for nonlinear systems. The controller isgiven in a feedback structure, that is the controller has informations about thesystem and use it to influence the system. The nonlinear system is transformed tolinear parameter-varying system, which is used to design a controller, i.e. gain-scheduled controller. The gain-scheduled controller synthesis in this thesis isbased on the Lyapunov theory of stability as well as on the Bellman-Lyapunovfunction. To achieve a performance quality a quadratic cost function and itsmodifications known from LQ theory are used. The obtained gain-scheduled
27
controller guarantees the closed-loop stability and the guaranteed cost. Themain results for controller synthesis are in the form of bilinear matrix inequalitiesand/or linear matrix inequalities. For controller synthesis one can use a free andopen source BMI solver PenLab or LMI solvers LMILab or SeDuMi.
6.2 Research results
In the initial stage of our research I with my supervisor Prof. Ing. VojtechVesely, DrSc. developed a gain-scheduled controller design which guaranteesthe closed-loop system stability and a guaranteed cost for continuous-time lin-ear parameter-varying (LPV) systems for all scheduled parameter changes withpre-defined rate of scheduled parameter changes. These results were publishedin Journal of Process Control and presented at several conferences (ICCC’13,ICPC’13, ELITECH’13, IN-TECH’13). After that we expanded this theory torobust controller design for continuous and discrete-time uncertain LPV sys-tems with possibility for variable weighting in cost function. Some of theseresults were published in Journal of Electrical Engineering, in Journal of Elec-trical Systems and Information Technology and they were presented at severalconferences such as the European Control Conference 2014 (ICCC’14, CPS’14,SSKI’14, ELITECH’14, ICPC’15).
In the middle stage of our research we modified our approaches from BMI(bilinear matrix inequality) to LMI (linear matrix inequality) problem. Thiscaused that our controller synthesis works for high-order systems (50-60th orderwas tested). We successfully ported our approaches to switched and model pre-dictive controller design. Some of these results have been published in Journalof the Franklin Institute, in Journal of Electrical Engineering, in InternationalReview of Automatic Control, in Asian Journal of Control and will be presentedat several IFAC symposiums and conferences as MICNON’15 or ROCOND’15(ICPC’15, ELITECH’15). Also some of these results are under review processin journals International Journal of Control, Automation and Systems and inArchives of Control Sciences.
In the final stage we successfully developed a new stability condition where wecould bypass the multi-convexity that significantly reduced the conservativenessof the controller synthesis. In addition we added to the controller synthesis theinput (hard) / output (soft) constraints as well as input rate (soft) / outputrate (soft) constraints where we do not need online optimization. Publicationsof these results are only in the preliminary stage but hopefully they will bepublished in high impact factor journals, too.
6.3 Closing remarks and future works
The main goal for this thesis (and also to our research in last 2,5 – 3 years) wasto find a systematic controller design approach for uncertain nonlinear systems,which guarantees the closed-loop stability and guaranteed cost with considering
28
input/output constraints, all this without on-line optimization and need of high-performance industrial computers.
This summary presents the selected results from 3 published papers. Onecan find the main results for robust gain-scheduled controller design, for robustswitched controller design and for discrete gain-scheduled MPC design with in-put constraints. Dissertation thesis consists from 9 papers, which covers themain research results obtained within the last 2.5 years. We tried to selectthose publications, which best reflect the achieved results. The first includedpaper (Chapter 4 ) presents a simple gain-scheduled controller design for nonlin-ear systems, which guarantees the closed-loop stability and guaranteed cost. Onecan include the maximal value of the rate of gain-scheduled parameter changes,which allows to decrease conservativeness and obtain the controller with a givenperformance. In the next chapter one can find a simple modification of these res-ults, where a new quadratic cost function is used, where weighting matrices aretime-varying and depends on scheduled parameter. Using these original variableweighting matrices we can affect performance quality separately in each workingpoints and we can tune the system to the desired condition through all parameterchanges. Chapters 6, 7 presents the robust versions of the obtained results fromChapters 4, 5, where in Chapter 6 the design procedure is transformed from thebilinear matrix inequality form to linear matrix inequality, which caused that ourcontroller synthesis works for high order systems. In Chapter 8 a simplified ver-sion of the robust controller design in discrete time domain is presented, wherea new LPV description of T1DM Bergman’s minimal model with two additionalsubsystems (absorption of digested carbohydrates and subcutaneous insulin ab-sorption) is created. The controller synthesis in this paper is also transformed toLMI problem. In Chapters 9 and 10 a gain-scheduled controller designs adop-ted to switched control are presented in continuous time. In the proposed designprocedures there is no need to use the notion of the ”dwell-time” for arbitraryswitching, which significantly simplifies the switched controller design comparedto approaches in the literatures. In Chapter 11 a novel gain scheduling basedmodel predictive controller design procedure for nonlinear systems is presentedfor finite and infinite prediction horizons with considering input/output con-straints. Finally a novel unified robust gain-scheduled and switched controllerdesign approach is presented in Chapter 12 where the conservativeness frommulti-convexity is eliminated.
The stated objectives was reached successfully, but there are many unsolvedproblems yet. For example, in this thesis it is hypothesized, that the scheduledparameters can be measured and the measurement is accurate. It is true, thatif one use the robust version, then the measurement inaccuracy can be coveredas model uncertainty, but this should be studied in more detail. Furthermore,it would be good to study, how can be used the informations from disturbancesto improve the performance quality under disturbances. Moreover it would bean interesting study how to reduce the time required to controller synthesis,because it is well known, that the required time for controller design using LMIand especially BMI solvers, rapidly increases for higher order systems. It follows,
29
that this thesis opens new possibilities for further studies and research in thisspecific area.
List of publications
Original research papers in international scientific journals (peer re-viewed)
[1] VESELY, Vojtech – ILKA, Adrian. Gain-Scheduled PID Controller Design. Journalof Process Control, 2013, vol. 23, p. 1141-1148.
[2] ILKA, Adrian – VESELY, Vojtech. Robust Gain-Scheduled Controller Design forUncertain LPV systems: Affine Quadratic Stability Approach. Journal of ElectricalSystems and Information Technology, 2014, vol. 1, no. 1, p. 45-57.
[3] ILKA, Adrian – VESELY, Vojtech. Gain-Scheduled Controller Design: VariableWeighting Approach. Journal of Electrical Engineering, 2014, vol. 65, no. 2, p.116-120.
[4] VESELY, Vojtech – ILKA, Adrian. Robust Gain-Scheduled PID Controller Designfor Uncertain LPV Systems. Journal of Electrical Engineering, 2015, vol. 66, no. 1,p. 19-26.
[5] VESELY, Vojtech – ILKA, Adrian. Design of Robust Gain-Scheduled PI Controllers.Journal of the Franklin Institute, 2015, vol. 352, no. 1, p. 1476-1494.
[6] ILKA, Adrian – OTTINGER, Ivan – LUDWIG, Tomas – TARNIK, Marian – VESELY,
Vojtech – MIKLOVICOVA, Eva – MURGAS, Jan. Robust Controller Design forT1DM Individualized Model: Gain Scheduling Approach. International Review ofAutomatic Control (I.RE.A.CO), 2015, vol. 8, no. 2, p. 155-162.
[7] VESELY, Vojtech – ILKA, Adrian. Novel Approach to Switched Controller Design forLinear Continuous-Time Systems. Accepted in Asian Journal of Control, Acceptanceday: 22. May, 2015.
Published contributions on scientific conferences or symposiums (peerreviewed)
[1] HOLIC, Ivan – ILKA, Adrian. Robust State Feedback Controller Design for DC-MotorSystem. Oxford: Elsevier, 2013. Advances in Control Education – ACE 2013: 10thIFAC Symposium on Advances in Control Education. Scheffield, UK, August 28-30,2013, p. 120-125. ISBN 978-3-902823-43-4.
[2] VOZAK, Daniel – ILKA, Adrian. Application of Unstable System in Education ofModern Control Methods. Oxford: Elsevier, 2013. Advances in Control Education –ACE 2013: 10th IFAC Symposium on Advances in Control Education. Scheffield, UK,August 28-30, 2013, p. 114-119. ISBN 978-3-902823-43-4.
[3] VESELY, Vojtech – ROSINOVA, Danica – ILKA, Adrian. Decentralized GainScheduling Controller Design: Polytopic System Approach. Shanghai: IFAC, 2013.Large Scale Complex Systems Theory and Applications: 13th IFAC Symposium.Shanghai, China, July 7-10, 2013, p. 401-406. ISBN 978-3-902823-39-7.
[4] ILKA, Adrian – VESELY, Vojtech. Discrete Gain-Scheduled Controller Design: Guar-anteed Cost and Affine Quadratic Stability Approach. Proceedings of the Interna-tional Conference on Innovative Technologies IN-TECH 2013, Budapest, Hungary 10.-13.09.2013, p. 157-160. ISBN 978-953-6326-88-4.
[5] VESELY, Vojtech – ILKA, Adrian. Gain-Scheduled Controller design: MIMO Sys-tems. Proceedings of the 14th International Carpathian Control Conference ICCC2013. Rytro, Poland, May 26-29, 2013, p. 417-422. ISBN 978-1-4673-4489-0.
30
[6] ILKA, Adrian – VESELY, Vojtech. Discrete gain-scheduled controller design: Vari-able weighting approach. Proceedings of the 15th International Carpathian ControlConference ICCC 2014. Velke Karlovice, Czech Republic, May 28-30, 2014, p. 186-191.ISBN 978-1-4799-3527-7.
[7] VESELY, Vojtech – ILKA, Adrian. PID robust gain-scheduled controller design. Pro-ceedings of the 13th European Control Conference 2014, June 24-27, 2014, Strasbourg,France, p. 2756-2761. ISBN 978-3-9524269-2-0.
[8] VESELY, Vojtech – ILKA, Adrian. Gain-Scheduled Controller Design: GuaranteedQuality Approach. Proceedings of the 19th International Conference on Process Con-trol 2013. Strbske Pleso, Slovakia, June 18-21, 2013, p. 456-461. ISBN 978-80-227-3951-1.
[9] VESELY, Vojtech – ILKA, Adrian – KOZAKOVA, Alena. Frequency Domain GainScheduled Controller Design for SISO Systems. Proceedings of the 19th InternationalConference on Process Control 2013. Strbske Pleso, Slovakia, June 18-21, 2013, p.439-444. ISBN 978-80-227-3951-1.
[10] ILKA, Adrian – VESELY, Vojtech. Gain-Scheduled Controller Design: VariableWeighting Approach. Bratislava. 15th Conference of Doctoral Students ELITECH’13.Bratislava, Slovakia, 5 June 2013, p. 1-6. ISBN 978-80-227-3947-4.
[11] ILKA, Adrian – ERNEK, Martin. Modeling and Control System of Wind Power Plantswith Pumped Storage Hydro Power Plant. Renewable Energy Sources 2013: 4th In-ternational Scientific Conference OZE 2013. Tatranske Matliare, Slovakia, May 21-23,2013, p. 495-500. ISBN 978-80-89402-64-9.
[12] ILKA, Adrian – VESELY, Vojtech. Robust Gain-Scheduled Controller Design forUncertain LPV Systems: Quadratic Stability Approach. International Conference onCybernetics and Informatics SSKI 2014. Oscadnica, Slovakia, 5.-8. 2. 2014, ISBN978-80-227-4122-4.
[13] VESELY, Vojtech – ILKA, Adrian. Switched system controller design: Quadraticstability approach. International Conference on Cybernetics and Informatics SSKI2014; Oscadnica, Slovakia, 5.-8. 2. 2014, ISBN 978-80-227-4122-4.
[14] ILKA, Adrian – VESELY, Vojtech. Decentralized gain-scheduled PSS design on thebase of experimental dates. Proceedings of the 11th International Scientific Conferenceon Control of Power Systems 2014; Tatranske Matliare, Slovakia; 20-22 May 2014, p.15-20. ISBN 978-80-89402-72-4.
[15] ILKA, Adrian – VESELY, Vojtech. Robust discrete gain-scheduled controller design:LMI approach. 16th Conference of Doctoral Students ELITECH’14; Bratislava, Slov-akia, 4 June 2014, ISBN 978-80-227-4171-2.
[16] VESELY, Vojtech – ILKA, Adrian. Robust Switched Controller Design for NonlinearContinuous Systems. Accepted to the 1st IFAC Conference on Modeling, Identifica-tion and Control of Nonlinear Systems – MINCON 2015, Saint Petersburg, RussianFederation, June 24-26, 2015, Acceptance day: 16. March, 2015.
[17] ILKA, Adrian – VESELY, Vojtech. Gain-Scheduled MPC Design for Nonlinear Sys-tems with Input Constraints. Accepted to the 1st IFAC Conference on Modeling,Identification and Control of Nonlinear Systems – MINCON 2015, Saint Petersburg,Russian Federation, June 24-26, 2015, Acceptance day: 16. March, 2015.
[18] VESELY, Vojtech – ILKA, Adrian. Robust Controller Design with Input Constraints,Time Domain Approach. Accepted to the 8th IFAC Symposium on Robust ControlDesign – ROCOND 2015, Bratislava, Slovakia, July 15-17, 2015, Acceptance day: 20.April, 2015.
[19] ILKA, Adrian – LUDWIG, Tomas – OTTINGER, Ivan – TARNIK, Marian – MIK-
LOVICOVA, Eva – VESELY, Vojtech – MURGAS, Jan. Robust Gain-Scheduled Con-troller Design for T1DM Individualised Model. Accepted to the 8th IFAC Symposiumon Robust Control Design – ROCOND 2015, Bratislava, Slovakia, July 15-17, 2015,Acceptance day: 20. April, 2015.
31
[20] ILKA, Adrian – VESELY, Vojtech. Robust Discrete Gain-Scheduled Controller Designfor Uncertain LPV Systems. Accepted to the 20th International Conference on ProcessControl, Strbske pleso, Slovak Republic, June 9-12, 2015, Acceptance day: 20. April,2015.
[21] ILKA, Adrian – VESELY, Vojtech. Observer-Based Output Feedback Gain-ScheduledController Design. Accepted to the 20th International Conference on Process Control,Strbske pleso, Slovak Republic, June 9-12, 2015, Acceptance day: 20. April, 2015.
[22] ILKA, Adrian – VESELY, Vojtech. Gain-Scheduled Model Predictive ControllerDesign. Accepted to the 17th Conference of Doctoral Students, ELITECH’15; Bratis-lava, Slovakia, May 25, 2015, Acceptance day: 11. May, 2015.
Original research papers submitted to international scientific journals(peer reviewed)
[1] ILKA, Adrian – VESELY, Vojtech. Decentralized Power System Stabilizer Design:Gain Scheduling Approach. Submitted to the Journal of Electrical Systems and In-formation Technology. Submitted on 29. March, 2015.
[2] VESELY, Vojtech – ILKA, Adrian. Output Feedback Control of Switched NonlinearSystems: A Gain Scheduling Approach. Submitted to International Journal of Control,Automation and Systems, Submitted on 7. May, 2014.
[3] ILKA, Adrian – VESELY, Vojtech. Unified Robust Gain-Scheduled and SwitchedController Design for Linear Continuous-Time Systems. Submitted to the journalInternational Review of Automatic Control (I.RE.A.CO). Submitted on 25. May,2015.
[4] ILKA, Adrian – VESELY, Vojtech. Robust Switched Controller Design for LinearContinuous-Time Systems. Submitted to the journal Archives of Control Sciences,Submitted on 25. May, 2015.
Other publications
[1] ILKA, Adrian. Modeling and Control of Wind Turbine. BSc thesis, Slovak Universityof Technology in Bratislava 2010, 86 p., in Slovak.
[2] ILKA, Adrian. Modeling and Control System of Wind Power Plants with PumpedStorage Hydro Power Plant, MSc thesis, Slovak University of Technology in Bratislava,2012, 128 p., in Slovak.
[3] ILKA, Adrian – ERNEK, Martin. Modeling and Control of Wind Turbine using Mat-Lab/SimPowerSystems. Proceedings of the SVOC 2011, Bratislava, Slovakia, 4.5.2011,p. 355-361, ISBN 978-80-227-3508-7.
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Notes/Comments
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