A Common Lyapunov Function for a Class of Switched Descriptor Systems

GU Ze-quan, LIU He-ping Information Engineering School

University of Science and Technology Beijing Beijing 100083, China guzequan@163.com

LIAO Fu-cheng School of Applied Science

University of Science and Technology Beijing Beijing 100083, China

Abstract—The existence of a common Lyapunov function for a class of linear switched descriptor systems was studied, where the system matrices were commute pairwise. We first presented work on two subsystems, a Lyapunov function was constructed and proved to be a common Lyapunov function of the system, and then we extended to the case of multi-subsytems, using the similar method. Finally, a sufficient condition for the existence of a common Lyapunov function for the linear switched descriptor system was presented, this assured the system asymptotically stable under arbitrary switching strategy.

Keywords-hybirid dynamical systems; switched descriptor systems; common Lyapunov function; asymptotically stable

I. INTRODUCTION In recent years, the scope of control theory is being

enlarged to include hybrid dynamical systems. One of the main features of such hybrid dynamical systems is the systematic application of the idea of switched systems [1-5]. In [6] the existence of the common Lyapunov function of switched systems was presented. It means that the existence of the common Lyapunov function and the stability of the switched systems under arbitrary switching strategy are equal.

As is well known, descriptor, or generalized state space systems, whose behaviors are governed by both differential equations and algebraic equations, appear often in many systems, such as social economic systems, biological systems, power systems, electrical networks, and so on. There has been extensive study on descriptor systems [7-12]. However, to our knowledge, there are only a few research results on either stability analysis or control design of switched descriptor systems up to now. One of the reasons is that both analysis and synthesis of descriptor systems are much more complicated than those of conventiona1 systems, since stability, regularity and causality should be considered at the same time in descriptor systems. In [10] the stability and stabilization of switched descriptor systems in discrete-time domain are considered, and some sufficient conditions are established under which the system is regular, causal and asymptotically stable under arbitrary switching signal. In [11] the reachability of switched linear descriptor systems was studied, and a necessary condition of the switched descriptor systems was derived. In [12] based on the traditional methods of descriptor systems, the problem of the stability of switched descriptor systems was studied, then a

sufficient condition of the stability for this system was presented under arbitrary switching strategy.

In this paper, we consider a special case of the switched descriptor systems, one in which the matrices iA commute pairwise. It is shown that a common Lyapunov function exists for this class of matrices. The main idea is given in theorem I, in which A contains only two matrices 1A

and 2A . The principal result is contained in item ii) where an explicit method of generating a common Lyapunov function is presented. This is generalized in Theorem 2 to the case where A contains a finite number of matrices.

II. PROBLEM DESCRIPTION AND PRELIMINARIES We consider a class of linear descriptor switched systems

given by ( ) ( )txAtxE i= , ( ) 00 xx = (1)

where ( ) nRtx ∈ is the descriptor state,

{ }Ni AAAA ,,, 21∈ is the subsystems of system

(1), nnRE ×∈ , nni RA ×∈ , ( )Ni ,2,1= are constant

matrices, ( ) nrErank <= . Here we first introduce some basic assumptions. Assumption 1 To the E of the system (1), with no

loss, we assume that

⎥⎦

⎤⎢⎣

⎡=

000rI

E (2)

where rI is identity matrix with rr × dimension.

Assumption 2 { }Ni AAAA ,,, 21∈ are stable matrices.

Definition 1[8] The following matrix equations about matrix V

WAVVA TT −=+ (3) 0≥= EVVE TT (4)

which are called descriptor Lyapunov equations, where 0>W .

2009 International Asia Conference on Informatics in Control, Automation and Robotics

978-0-7695-3519-7/09 $25.00 © 2009 IEEE

DOI 10.1109/CAR.2009.13

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III. MAIN RESULTS Theorem 1 Consider the linear descriptor switched

system (1) with { }21, AAAi ∈ , which are satisfied the assumption 1 and assumption 2, and the matrices { }21, AAAi ∈ are commute pairwise, such that

1221 AAAA = , then i） The system is asymptotically stable for any arbitrary

switching sequence. ii）Given a symmetric positive definite matrix 0P , let

1P and 2P be the unique symmetric definite solutions to the descriptor Lyapunov equations

01111 PAPPAT −=+ , 011 ≥= EPPET (5)

12222 PAPPAT −=+ , 022 ≥= EPPET (6)

where 1P and 2P are symmetric positive definite matrices,

which are similar to ⎥⎦

⎤⎢⎣

⎡=

GQ

P0

0, where rrRQ ×∈ ,

( ) ( )rnrnRG −×−∈ . Then the function ( ) xPExExV TT2=

is a common Lyapunov function for the system (1). iii）For a given choice of the matrix 0P , the matrices

1A and 2A can be chosen in any order in (5),(6) to yield

the same solution 2P , i.e., if

02332 PAPPAT −=+ , 033 ≥= EPPE T (7) then

31221 PAPPAT −=+ , 022 ≥= EPPET (8) Proof: i） This can be proved directly by using the fact that 1A

and 2A commute, and also follows from ii）below.

ii）Let ( ) xPExExV TT2= , where 2P is worked out

from (6), It’s obvious that 022 ≥= EPPET . If

( ) ( )txAtxE 2= , using (6) the derivative of V along the trajectories of this system is seen to be

xPExxPExV TTTT22 +=

( ) 012222 <−=+= xPxxAPPAx TTT , showing that V is a Lyapunov function for this system.

Now, the derivative of V along the trajectories of the system ( ) ( )txAtxE 1= is given by

xPExxPExV TTTT22 +=

( )xAPPAx TT1221 += .

It remains to show that this is negative definite. Substituting for 1P from (6) to (5) and using the

commutativity of 1A and 2A , we get

( ) ( ) 12222222210 AAPPAAPPAAP TTT +++=

( ) ( ) 2122112212 AAPPAAPPAA TTT +++= (9)

Since 2A is stable and 00 >P , this shows

that 01221 <+ APPAT , as required. Finally, the derivative of V along the trajectories of the

switched descriptor system (1) is given by xPExxPExV TTTT

22 +=

( )( )⎩

⎨⎧

<+<−=+

=,xAPPAx

,xPxxAPPAxTT

TTT

00

1221

12222

1

2

AAAA

i

i

==

and hence V is also a common Lyapunov function for the switched descriptor system.

iii）Since 3P is the solution to the descriptor Lyapunov equation (7), it is positive definite. Hence there is a unique positive definite solution 4P , that is similar to P , to the descriptor Lyapunov equation

31441 PAPPAT −=+ , 044 ≥= EPPET (10)

The statement will be proved if we show that 42 PP = . Using (7) and (10) and the commutativity of 1A and 2A , it follows that

( ) ( ) 12442244210 AAPPAAPPAAP TTT +++= (11)

From (6) and (8), 2222 APPAT + and 2442 APPAT + are the unique solutions of the same Lyapunov equation. Hence ( ) ( ) 0242422 =−+− APPPPAT . Since 2A is

stable, 42 PP = . Theorem 2 Consider the switched descriptor system (1)

with { }Ni A,,A,AA 21∈ ,which are satisfied the assumption 1 and assumption 2, and the matrices

{ }Nq A,,A,AA 21∈ are commute pairwise, such that

pqqp AAAA = , qp ≠ ,then i） The system is asymptotically stable for any arbitrary

switching sequence. ii）Given a symmetric positive definite matrix 0P ,

let NP,,P,P 21 be the unique symmetric definite solutions to the descriptor Lyapunov equations

1−−=+ iiiiTi PAPPA , 0≥= EPPE ii

T , N,,,i 21= (12)

where iP , N,,,i 21= are similar to P . Then the

function ( ) xPExExV NTT= is a common Lyapunov

function for the switched descriptor system (1).

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iii）For a given choice of the matrix 0P , the matrices

NA,,A,A 21 can be chosen in any order in (12) to yield

the same solution NP . Proof: The proof proceeds along the same lines as in theorem 1

and hence only the basic steps are given below. i） Follows from part ii） of the theorem. ii ） If xAxE i= , the derivative of V along the

trajectories of this system is given by ( )xAPPAxV iNN

Ti

T += and hence we have to show

that 0<+ iNNTi APPA , N,,,i 21= .

For this purpose, define the matrices ijj

Tiij APPAP +Δ . If we show that 0<ijP

for N,,,i 21= , N,,i,ij 1+= , then the result

follows by choosing Nj = for each i . Hence,

let { }N,,,i 21∈ . From (12), 01 <−= −iii PP . Now

assume that 0<ijP for some { }11 −+∈ N,,i,ij .

Then, using (12), the commutativity of iA and 1+jA and

the stability of 1+jA we find that

( ) 01111 >−=+−=+ ++++ ijijjTijj,ij,i

Tj PAPPAAPPA ,

showing that 01 <+j,iP also, proving the claim induction. iii）This can be shown as in Theorem 1 by taking some

permutation of NA,,A,A 21 in (12), assuming that the

resulting solution, say NR , is not equal to NP , and using the uniqueness of solutions to the Lyapunov equations to show that NN PR = .

IV. CONCLUSIONS In this paper, the existence of a common Lyapunov

function for a class of switched descriptor systems was studied. With some assumptions, a sufficient condition for the existence of a common Lyapunov function for the linear switched descriptor system was presented and the

construction of this common Lyapunov function was given, that assures the system asymptotically stable under arbitrary switching strategy. Considered the respective features of the switched systems and descriptor systems, we extended the result of [6] to the switched descriptor systems.

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