and Nonlinear Systems - imag.fr fileand Nonlinear Systems A Defense Committee: DiReviewers, (Icommittee) Dr. Alain GAUTHIER, Dr. Jean JacqueANO, and s s 2009

A Polynomial Approach for Analysis and Optimal Control of Switched Nonlinear

Systems

Eduardo MOJICA-NAVAPh.D. Thesis Defense

Defense Committee:Dr. Peter CAINES and Dr. Didier HENRION, Reviewers,

Dr. Pierre RIEDINGER (Invited by the committee)

Dr. Alain GAUTHIER, Dr. Jean Jacques LOISEAU, Dr. Nicanor QUIJANO, and Dr. Naly RAKOTO, Advisors

École des Mines de Nantes, IRCCyN and Universidad de Los AndesSeptember 22 / 2009

2

Outline

1. Introduction

2. Optimal Control of Switched Nonlinear Systems

3. Stability Analysis of Switched Nonlinear Systems

4. Conclusions and Future Work

3

Introduction

4

BackgroundSwitchedSwitched Systems Systems –– Basic Basic DefinitionsDefinitions

),(1 uxf

),(2 uxf

),( uxfq

x! !

)(t"

"

xu

Switching signal

Input

State

5

BackgroundSwitchedSwitched Systems Systems –– Basic Basic DefinitionsDefinitions

),,()( tuxftx "#!

),(1 uxf

),(2 uxf

),( uxfq

x! !

)(t"

"

xu

Switching signal

Input

State

6

Thesis Statement

We are dealing with…! A new approach to solve the optimal control problem of

complex nonlinear switched systems and stability analysisof nonlinear switched systems

Why do we consider this new approach?! To solve problems avoiding difficulties by the discontinuities! To obtain a numerically efficient solution

7

Thesis Statement

How to solve the optimal control problem?

1. Transform the switched system into a polynomial system

2. Transform the polynomial representation into a relaxedconvex problem (i.e., using theory of moments)

3. Solve the new convex problem using several computationaltools as a semidefinite programming (SDP)

8

Thesis Statement

How to solve the stability analysis problem?

1. Transform the switched system into a polynomial system

2. Use the SOSTOOLS and dissipation inequalities

9

Optimal Control of Switched NonlinearSystems

10

INTRODUCTIONWhat is this about?

Control Design

Continuous-time Systems

Optimal Control

Hybrid Systems

Discrete-time Systems Anlysis and Optimal Control

of Hybrid Systems

Model Process

Real ProcessOutput

Ctrl

Input

11

BACKGROUND SwitchedSwitched Systems and Optimal ControlSystems and Optimal Control

! The solution requires both, the optimal switching sequences and the optimal continuous inputs

Qtxtx

tuxftxts

dttuxLtxtxJ ft

tf

$##

%# !

)()(

),,()(..

,),,())(),((

00

00

"

&

"

"

!

Measurablefunctions

Functional to beminimized

},...,1{ qQ #Finite set of switches

12

Equivalent Polynomial Representation

Theorem 1 Consider a switched system . Then, there existsan equivalent continuous state system with polynomial dependence in the control variable s

Where is a Lagrange quotient. And s is subject to the constraintpolynomial set

Mojica-Nava, E., Meziat, R., Quijano, N., Gauthier, A., and Rakoto-Ravalontsalama,N. (2008). Optimal control of switched systems: A polynomial approach. Proceedings of the 17th IFAC World Congress, 7808–7813.

13


Ex: A switched systems with two subsystems the equivalentrepresentation is

Where

And s is subject to the constraint polynomial set

14


Proposition 1 Consider a switched running cost . Then, there exists an equivalent continuous running cost with polynomial dependence in the control variable s

Ex:

15

Assumption

Let be the set of admissible state-control process (x(t),s(t)) up to time T

Assumption All functions and running cost functionsare polynomials

16

Assumption

Let be the set of admissible state-control process (x(t),s(t)) up to time T

Assumption All functions and running cost functionsare polynomials

17

Equivalent Optimal Control Problem (EOCP)

We obtain a Hamiltonian using the Equivalent Optimal Control Problem(EOCP)

18


We obtain a Hamiltonian using the Equivalent Optimal Control Problem(EOCP)

Objective: Obtain the global minima of the function H

19

Theory of Moments and the SDP Relaxation

Moment Matrix: For a given real sequence of real numbers, the moment matrix of order r associated with m, has its rows and columns indexed in the canonical basis and is defined by

A sequence m has a representing measure supported on only if these moments are restricted to be the entries on a positive semidefinite moment matrix defined as follows

Lasserre, J. (2001). Global optimization with polynomials and the problem of moments. SIAM J. Optimization, 11, 796–817.

Meziat, R. (2003). The method of moments in global optimization. Journal of mathematics science, 116, 3303–3324.

20


Ex: Consider a two dimensional case. When n=2 and r=2, we obtain the polynomial ordered lexicographically , and the moment matrix has the form


21


Localizing Matrix: For a given polynomial , written as

We define the localizing matrix associated with rows and columns alsoindexed in the canonical basis of , by

The localizing matrix posivity is directly related to the existence of a representingmeasure for m with support in


22


Ex: Consider a moment matrix

and a polynomial , then we obtain a localizing matrix as follows,


23

Relaxation of the Polynomial EOCPWe apply the convexification of this function, obtaining the envelope of the function H

24

Relaxation of the Polynomial EOCP

Now, to be coherent with definitions of localizing matrix and representationsresults, we treat the polynomial Q(s) as two opposite inequalities

and we redefine the compact set

In a similar way, we obtain the relaxation of the state equation

25

Relaxation of the Polynomial EOCP



26

Semidefinite Programs-LPi

27

Optimality Conditions

•The optimum is a lower-bound on the global optimum of the original problem, since any feasible solution yields a feasible solution m of LPi.

• If any feasible point of the relaxation of order i is bounded then

• The LMI constraints of LPi state necessary conditions for m to be the vector of moments up to order 2i, of some probability measure with support constrained in S

28

Optimality Conditions

Theorem 2 Consider the polynomial equivalent optimal control problem (EOCP) and let . Then for every

29

Optimality ConditionsConsidering the previous result, we can set the correspondence between the minimizers of the original optimal switched problem and the minimizers of LPi

Theorem 3 If the moment sequence is the unique minimizerof the SDP LPi, then the polynomial EOCP admits a unique minimizer

30

Switched Optimization AlgorithmThese results are particularly convenient to obtain an algorithm for switching control laws.

31

Switched Optimization Algorithm

32

Numerical Example: The Artstein’s Circle

33

Numerical Example: The Artstein’s Circle

34

Numerical Example: The Arstein’s Circle

35

Extension Results to More General Nonlinear OCP

• We deal with elementary and nested elementary functions (these are related with explicit symbolic derivatives, i.e., exponential, logarithm, power, trigonometric, and hyperbolic functions, among others).

• We use a recasting process, which has been used for stability analysis.

Savageau, M. and Voit, E. (1987). Recasting nonlinear differential equations as S-systems: a canonical nonlinear form. Mathematical biosciences, 87(1), 83–115.

Papachristodoulou, A. and Prajna, S. (2005). Positive Polynomials in Control, chapter Analysis of nonpolynomial systems using the sum of squares decomposition, 23–43. Springer.

36

Numerical Example: Swing-up a Pendulum

We are dealing with a two-dimensional model of a pendulum.

Objective: Design a control law in such a way that the closed-loop energy presents a minimum at the desired position.

37

The Recasting Process• Step 0. Equivalent Representation: We consider the equivalentrepresentation for the switched system obtained in Theorem 1,

Ex: We obtain

• Step 1. Original State Equations: The original system is described by

38

The Recasting Process• Step 2. Decomposition of Non-Polynomial Functions:

Let , for . For each that is not already a power-law function, replace it with a new variable . An additional differentialequation is generated.

• Step 3. Recasting Process: If there are constraints, then we introduce an n-dimensional manifold on which the solutions to the original differentialequations lie.

Ex:

39

The Recasting Process

Ex: As a result we obtain

• Step 4. The Polynomial Form:

If the set of equations is in polynomial form, then the recasting process iscomplete. If not repeat steps 2-4.

40

The set of measuresWe define the set of measure related to:

• A polynomial constraint obtained from the equivalent representation

• Polynomial functions obtained by the recasting process

41


We have the Hamiltonian as defined before in the Equivalent Optimal Control Problem (EOCP)

Objective: Obtain the global minima of the function H defined on a subset , i.e.,

42

Relaxation of the Polynomial EOCPDefine the space of moments as

a measure supported in is a moment matrix associated to the vector of moments

are localizing matrices related to the vector of moments, and

Henrion, D., Lasserre, J., and Savorgnan, C. (2008). Nonlinear optimal control synthesis via occupation measures. Proceedings of the 47th IEEE Conference onDecision and Control, 4749–4754.



43

Semidefinite Programs - GLPi

44

Optimality ConditionsCorrespondence between the minimizers of the original optimal switched problem and the minimizers of the GLPi

Theorem If the moment sequence is the unique minimizerof the SDP GLPi, then the polynomial EOCP admits a unique minimizer

where we can state, using the equivalent representation, that the autonomous optimal switching is given by

and the optimal trajectory is given by the first n-terms of the moments sequence

45

Simulations: Swing-up a Pendulum

46

Swing-up a pendulum

47

Stability Analysis of Switched NonlinearSystems

CDC08, SCL09

48

Unstable system!

)(xfx "#!

Stable system Stable system

The The StabilityStability ProblemProblem

49


Theorem 1 Consider a switched system . Then, there existsan equivalent continuous state system with polynomial dependence in the control variable s

Where is a Lagrange quotient. And s is subject to the constraintpolynomial set

Mojica-Nava, E., Meziat, R., Quijano, N., Gauthier, A., and Rakoto-Ravalontsalama,N. (2008). Optimal control of switched systems: A polynomial approach. Proceedings of the 17th IFAC World Congress, 7808–7813.

50

Results in Stability Analysis

Dissipation Inequalities for Polynomial constrained systems

),,( sxxaxV

!'((

a is the energy rate supplied into system

)(,, sQV )

),()(),( 2 sxsQsxpxV )'((

is satisfied for some neighborhood of the origin.

Stability Theorem The equilibrium point x*=0 of the equivalent polynomial representation is stable if there exists such that,

Ebenbauer, C. and AllgÄower, F., "Analysis and design of polynomial control systems using dissipation inequalities and sum of squares," Computersand Chemical Engineering, vol. 30, pp. 1590-1602, 2006.

Ebenbauer, C. and AllgÄower, F., ‘’Stability analysis of constrained control systems: An alternative approach," Systems & Control Letters, vol. 56, pp. 93-98, 2007.

51


Ebenbauer, C. and AllgÄower, F., "Analysis and design of polynomial control systems using dissipation inequalities and sum of squares," Computersand Chemical Engineering, vol. 30, pp. 1590-1602, 2006.

Ebenbauer, C. and AllgÄower, F., ‘’Stability analysis of constrained control systems: An alternative approach," Systems & Control Letters, vol. 56, pp. 93-98, 2007.

It is difficult to search for a Lyapunov function V

Methods based on semidefinite programmingand the sum of squares decomposition

To verify Lyapunov inequalities

We assume that functionsare all polynomials

)(,, sQV )

52


* *# ##

n

i

d

jj

iji xx1 1

2,)( +&

SOS Proposition Given a polynomial V(x) of degree 2d,

Let such that,

With . Then the condition

Guarantees the positive definiteness of V(x)

SOSisxxV )()( &,

,,...,11 , nim

j ji #-.* #/+

jiji ,00 , -0. +/

Papachristodoulou, A., Prajna, S., and Parrilo, P., Introducing SOSTOOLS: A general purpose sum of squares programming solver," Proceedingsof the IEEE Conference on Decision and Control, pp. 741-746, 2002.

53

Results in Stability Analysis (3)

SOSisxxV )()( &,

),(),( sxxV )

SOSissxsQsxpxV ),()(),( 2)%((

,

The polynomials and the positive denite function are

computing using SOSTOOLS

)(x&

),(),( sxxV )

Proposition Dissipation Inequalities and SOS for Polynomial Switched systems. Suppose that for the polynomial representationsystem there exist polynomial functions , a positive definite function such that

Papachristodoulou, A., Prajna, S., and Parrilo, P., Introducing SOSTOOLS: A general purpose sum of squares programming solver," Proceedingsof the IEEE Conference on Decision and Control, pp. 741-746, 2002.

)(x&

54

Numerical Example (1)

Consider the set of systems decribed by the drift vector field

1 20)1()(:)1()( 10

#,##3$%,#

sssQsssfsftx!

)]()([)( 10 xfxfxF #with

45

678

9

%%,,%%,,

#45

678

9

,,,,%%%

# 322

21

221

31

322

21

221

31

1322

21

221

31

322

21

221

31

0 635.0242

)(,632422

)(xxxxxxxxxxxx

xfxxxxxxxxxxxx

xf

We use the polynomial equivalent representation

55

Numerical Example

Polynomial Representation

SOSisxxV )()( &,

: ; SOSissxssssfsfxV ),()2()1( 234

10 )%,%%,((

,

Using the MATLAB toolbox SOSTOOLS, a Lyapunov function of degree six is found, so that the equilibrium point is stable.

+ Dissipation Inequalitiesproposition

+SOSTOOLS

56

A Generalization for More General Switched NonlinearSystems

The Recasting ProcessGeneralization

The Dissipation Inequality

State the SOS conditions from it

57

Conclusions and Future Work

58

ConclusionsChapter 2. (CDC’08, SCL’09)

1. A new method for stability analysis of switched nonlinear systems based on a polynomial approach using SOSTOOLS

2. An alternative way to search for a common Lyapunov function for switched systems

Chapter 3. (IFACWC’08)

1. A new method for solving the optimal control problem of nonlinear switched systems based on a polynomial approach

2. Necessary and sufficient conditions for the existence of minimizer by using particular features of the relaxed, convex formulation

3. An algorithm using SDP relaxation is used to solve the nonlinear optimal control problem

For both:The recasting process extend the results to a more general nonlinear switched systems

59

Future Work

Chapter 2.

Several powerful tools from the algebraic polynomial system theory can be extend to switched systems using the polynomial representation

Chapter 3.

1. Prove the computational efficiency of the proposed method

2. SDP relaxation should be done for subsystems modeled by differential-algebraic equations

60

Publications

Journal articles

! Mojica-Nava E., Quijano N., Gauthier A., Rakoto-Ravalontsalama N., “A Polynomial Approach for Stability Anlysis of Switched Systems”. Submitted to Systems and Control Letters, Feb 2009

! Mojica-Nava E., Gauthier A., Rakoto-Ravalontsalama N., “Piecewise-linear Approximation to Nonlinear Cellular Growth Control”. Submitted to Control Engineering Practice, May 2008.

! Pantoja A., Mojica-Nava E., Quijano N., “Técnica de Suma de Cuadrados y sus Aplicaciones en Control”, Submitted to Revista de Ingeniería UNAL, March 2009.

Conference Proceedings

! Mojica-Nava E., Quijano N., Gauthier A., Rakoto-Ravalontsalama N.; “A Generalization of a Polynomial Optimal Control of Switched Systems”. Proceeding of the ADHS 2009, Zaragoza, Spain

! Mojica-Nava E., Quijano N., Gauthier A., Rakoto-Ravalontsalama N.; “Stability Analysis of Switched Polynomial Systems using Dissipation Inequalities”. Proceeding of the IEEE CDC 2008, Cancún, Mexico.

! Mojica-Nava E., Meziat R., Quijano N., Gauthier A., Rakoto-Ravalontsalama N.; “Optimal Control of Switched Systems: A Polynomial Approach”. Proceeding of IFAC World Congress 2008, Seoul, Korea.

! Mojica E., Gauthier A., Rakoto-Ravalontsalama N.; “Canonical Piecewise-linear Approximation of Nonlinear Cellular Growth”. Proceeding of IEEE CDC 2007, USA.

61

Publications

! Mojica E., Gauthier A., Rakoto-Ravalontsalama N.; “Probing Control for PWL Approximation of Nonlinear Cellular Growth”. Proceeding of IEEE MSC 2007, Singapore, Singapore.

! Mojica-Nava E., Quijano N., Rakoto-Ravalontsalama N., Gauthier A., “Análisis y Control de Sistemas Híbridos”, Proceeding of thethe Congreso Colombiano de Automática, 2009.

! Mojica E., Gauthier A., Rakoto-Ravalontsalama N.; “Piecewise-linear Approximation of Nonlinear Cellular Growth”. Proceeding of IFAC SSSC 2007, Foz de iguazu, Brazil.

! Caro C., Quijano N., Mojica-Nava Eduardo., “Sistemas Híbridos y el Ciclo de Carnot: Modelo y Análisis”. Proceeding of the IV IEEE Colombian Workshop on Robotics and Automation, Cali, Colombia, 2008.

62

Thank you for your attention!Gracias, Merci

63

Literature Review of Optimal Control of Hybrid Systems

! Seminal Papers. ! Seminal Papers improved.! Recent Papers.

64

LITERATURE REVIEW (2)Seminal Papers

H. Sussman (’99)

P. Riedinger (’00)

M. Branicky (’95, ‘98)

Xu & Antsaklis (’02,’04)

Basic algorithms, generalconditions for the existence of optimal control law for HS

Max. Princ. Fixed seq of finite length

Two-stage optim.

Lincoln & Rantzer (’06)

Relaxing dynamicprogramming

Rantzer & Johansson(’00)

Bemporad & Morari (’99)

Bengea & DeCarlo (’05)

Egerstedt (’03)

Optimal control of PWL

Mixed IntegerProgramming

Real time applications Autonomous case

Convex Embedded solution

Convexification

Witsenhausen (’66)

65

LITERATURE REVIEW (3)Papers Improved

MPC – Model Predictive Control (PWA, HS)

Basic algorithms, generalconditions for the

existence of optimal control law for HSSeatzu & Corona (‘06)

Axelsson & Boccadoro (‘06)

Patino & Riedinger (’07)

Spinelli (’06), Das (’06),

Wei & Bengea & Decarlo (’07)

Bemporad & Morari (’95, ‘98)

Egerstedt (’03)

Bengea & DeCarlo (’05)

P. Riedinger (’00)

Real time applications Autonomous

case withuncertains

Convex Embedded solution –Numerical

applic.

Convexification-feedback switched

affine

66

LITERATURE REVIEW (4)Recent Alternative Approaches

Garavello & Piccoli (’05)

Hybrid Necessary Principle

Strong variations algorithm

Alamir & Attia (’04,’05)

Shaikh & Caines(’07)

Algorithms for Hybrid Optimal Control

Vinter (’03)

Mancilla-Aguilar, García, Sontag (’05)

Goebel & Teel (’06)

Differentialinclusions

67

A Generalization for Nonlinear SwitchedSystemsIn the same way we obtain the SOS conditions in the next proposition

Documents

and Nonlinear Systems - imag.fr fileand Nonlinear Systems A Defense Committee: DiReviewers, (Icommittee) Dr. Alain GAUTHIER, Dr. Jean JacqueANO, and s s 2009