stability and stabilizability of discrete-time switched linear systems with state delay - acc 2005
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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(tarbour@laas.fr)
LAAS - CNRS
Toulouse, France
JUNE 9-10, 2005
Presented by: D. Arzelier
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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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)
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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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 and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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
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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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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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
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Stability and stabilizability of discrete time switched linear systems with state delay Montagner Leite Tarbouriech Peres
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Stability and stabilizability of discrete-time switched linear systems with state delay Montagner, Leite, Tarbouriech, Peres
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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