Stability and Stabilizability of Discrete-time Switched Linear Systems With State Delay - ACC 2005

8/13/2019 Stability and Stabilizability of Discrete-time Switched Linear Systems With State Delay - ACC 2005

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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres

Stability and stabilizability of discrete-time switched

linear systems with state delay(accepted at ACC 2005)

Vincius F. Montagner

Pedro L. D. PeresSchool of Electrical and Computer Engineering

University of Campinas, Campinas, SP, Brazil

Valter J. S. Leite

UnED Divinopolis CEFET-MG

Divinopolis, MG, Brasil

Sophie Tarbouriech([email protected])

LAAS - CNRS

Toulouse, France

JUNE 9-10, 2005

Presented by: D. Arzelier

MOSAR/SAR Lille PREVIOUS NEX T 1/16


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Outline

Introduction: switched systems and time-delay systems

Problem formulation: switched linear systems with state delays

Stability: convex LMI conditions

Stabilizability: LMI design conditions (switched and robust gains)

Examples: numerical evaluation

Conclusion: final remarks



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Introduction

Switched systems

several subsystems and a switching rule

power electronics, systems with switched control laws, etc

quadratic stability: fixed matrix to assess stability and control

Lyapunov functions with switched matrices (less conservative)

Systems subject to state delays

augmented state: suitable for known or unknown but bounded

delays. Not suitable for delay independent stability

delay independent stability and control: usually based on thequadratic stability (conservative hypothesis and results)

http://goback/http://goback/http://goback/


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This paper is focused on:

discrete-time switched linear systems with state delays

all the system matrices are assumed to be switched, with arbi-trary switching rule

stability: LMI conditions to assess stability for unknown andunbounded delays

a Lyapunov-Krasovskii functional with switched matrices is usedhere, encompassing quadratic stability based results

stabilizability: LMI conditions to compute switched and robust

state feedback gains

numerical examples: less conservative results, through convexconditions



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Problem formulation

x(k+ 1) =A(k)x(k) +Ad(k)x(kd) +B(k)u((k),x(k))

+Bd(k)ud((k),x(kd)) (1)

x(k) Rn, state vector, x(k) =0 fork


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Problem 1 Determine if system (1) with u((k),x(k)) =ud((k),x(kd)) =0 (i.e. autonomous system) is stable for arbitrary switching func-

tions and irrespective of the value of the time-delay.

Problem 2Find, if possible, switched gains Ki Rm

1n

andKdi Rm

2n

,i=1, . . . ,Nyielding the linear state feedback control laws

u((k),x(k)) =K(k)x(k) , ud((k),x(kd)) =Kd(k)x(kd) (3)

such that the closed-loop system

x(k+ 1) = A(k)x(k) + Ad(k)x(kd) (4)

withA(k)=A(k)+B(k)K(k) , Ad(k)=Ad(k)+Bd(k)Kd(k) (5)

is stable for arbitrary switching functions (k), irrespective of the value d

of the time-delay.



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Stability analysis

Theorem 1System (1) is stable for any arbitrary switching function(k),irrespective of the value of the time-delay d, if there exist symmetric pos-itive definite matrices PiR

nn, i=1, . . . ,NandSRnn such that anyof the following equivalent conditions holds:

a)

PiA

i(S+ Pj)Ai A

i(S+ Pj)Adi

SAdi(S+ Pj)Adi

>0

(i,j)I I (6)

where the symbol represents symmetric blocks in the LMIs

b)

S+ Pj (S+ Pj)Ai (S+ Pj)Adi Pi 0

S

>0

(i,j)I

I (7)



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c)There exist matrices Fi Rnn, Gi R

nn and Hi Rnn such that

M

(Fi+ F

i + Pj+ S) FiAiGi FiAdiH

i

Pi+ GiAi+AiG

i GiAdi+A

iH

i

S+HiAdi+AdiH

i

> 0 ,(i,j)I I (8)

First remark: Theorem 1 provides conditions with a finite numberof LMIs for stability of system (1) with arbitrary switching functions andirrespective of the time-delays

Second remark: although the conditionsa)-c)are equivalent for thestability analysis, the extra matrix variables Fi, Gi, Hiinc)can be used to

reduce the conservatism in the control synthesis problem Third remark: the use of a set of matrices Pi, i=1, . . . ,N(instead

of only one matrix P) in the Lyapunov-Krasovskii functional reduces theconservatism of the analysis



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The quadratic stability condition can be recovered from Pi=P,Fi=F

i =(S+ P), Gi=Hi=0 inc), as stated in the next corollary

Corollary 1 If there exist symmetric positive definite matrices P Rnn

and S Rnn such that

P + S(P + S)Ai (P + S)Adi P 0

S

> 0 , i = 1, . . . ,N (9)

then the system is quadratically stable.



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Stabilizability

Theorem 2 If there exist symmetric positive definite matrices Pi Rnn

and S Rnn, and matrices Fi Rnn, Zi Rm1n and Zdi Rm2n,i=1, . . . ,N, such that

(Fi+ F

i

+ Pj+ S) FiA

i

+Zi

B

i

FiA

di

+Zdi

B

di Pi 0

S

> 0 , (i,j) II(10)

then the switched state feedback gains Ki and Kdi given by

Ki=Zi(F

i)1

, Kdi=Zdi(F

i)1

, i=1, . . . ,N (11)

are such that the closed-loop system (4) is stable for any arbitrary switch-

ing function (k), irrespective of the value dof the time delay.

Theorem 2 allows to determine switched stabilizing gains, irrespec-tive of the value ofd, assuming that the state vectors x(k), x(kd)can

be used in the control action



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Corollary 2 If there exist symmetric positive definite matrices P Rnn

and S Rnn and matrices F Rnn, Z Rm1n and Zd Rm2n such

that

(F+ F + P + S) FAi+Z

Bi FAdi+Z

dB

di

P 0 S

> 0 , i = 1, . . . ,N

(12)

then the robust state feedback gains Kand Kdgiven by

K=Z(F)1 , Kd=Zd(F)1 (13)

are such that the closed-loop system

x(k+ 1) = (A(k)+ B(k)K)x(k)) + (Ad(k)+Bd(k)Kd)x(k d) (14)

is quadratically stable for any arbitrary switching function (k), irrespec-tive of the value of the dof the time-delay.



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Examples

Example 1: autonomous system (1) (i.e. u((k),x(k)) = u(kd) =0) with four randomly generated subsystems

A1=

0.1663 0.2088

0.2731 0.0005

, Ad1=

0.1475 0.15030.3507 0.0981

(15)

A2=

0.1230 0.2059

0.2690 0.0126

, Ad2=

0.2097 0.12250.2091 0.0547

(16)

A3= 0.4302 0.4653

0.1198 0.0028 , Ad3=

0.5369 0.4610

0.3242 0.3083 (17)

A4=

0.2100 0.35910.1068 0.0884

, Ad4=

0.2154 0.23500.1310 0.1565

(18)

This system is not quadratically stable through Corollary 1

Theorem 1 has a solution assuring system stability for any arbitraryswitching function (k), irrespective of the value of the time-delay d



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Example 2: from Xu et al., S&CL, 2001, with matrices

A1= 0.545 0.43

0.185 0.61

, Ad1= 0.24 0.070.12 0.09

(19)

A2=

0.455 0.37

0.215 0.59

, Ad2=

0.36 0.13

0.08 0.11

(20)

representing here two subsystems of a discrete-time switched system withdelay

Corollary 1 yields

P=

0.5682 0.04110.0411 0.9707

, S=

0.3729 0.02030.0203 0.4023

(21)

implying that the uncertain system is quadratically stable irrespective ofthe value dof the time-delay (not only for d=2 as in Xu et al.) and forany arbitrary switching function


S bili d bili bili f di i i h d li i h d l M L i T b i h P


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Control problem: subsystems perturbed by 1, such that(A1,A2,Ad1,Ad2), and input matrices given by

B1=B2=Bd1=Bd2=

0 1

(22)

Control strategy max()T2(Ki, Kdi) 4.36T2(Ki) 3.07T2(Kdi) 1.24

C2(K, Kd) 3.93C2(K) 2.84C2(Kd) 1.22

The switching control strategies provide larger bounds of stabiliz-ability

When both state vectors are available for feedback, Theorem 2 andCorollary 2 provide their respective largest bounds of stabilizability


Stability and stabilizability of discrete time switched linear systems with state delay Montagner Leite Tarbouriech Peres


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Conclusion

Convex delay independent LMI conditions have been given forstability and stabilizability of discrete-time switched linear systems witharbitrary switching functions and with unknown and unbounded state de-lays

The use of switched matrices in the Lyapunov-Krasovskii func-tional and of extra matrices in the conditions provide less conservativeevaluations of stability domains for this class of switched systems

LMI tests allow to design switched state feedback stabilizinggains encompassing previous results based on quadratic stability

Structural constraints can be imposed in the extra matricesused to compute the feedback gains without imposing constraints in thematrices of the Lyapunov-Krasovskii functional


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Stability and Stabilizability of Discrete-time Switched Linear Systems With State Delay - ACC 2005