Asymptotic Stability and L2 Gain Analysis of Distributed Parameter Switched Delay Systems

BAO Leping，WANG Pan，WANG Xiaoyong

(1.Department of Automation，Taiyuan Institute of Technology，Taiyuan，Shangxi 030008，China；2.School of Automation，Southeast University，Nanjing，jiangsu，210096，China)

The past decades has witnessed an enormous interest in switched systems due to theoretical research and practical application.A switched system can be described by a family of subsystems and a rule that orchestrates the switching between them[1].For a discussion of various issues related to switched systems，the reader is referred to see the survey article[2].A-mong them[3]，introduce multiple Lyapunov functions method as a tool for analyzing stability of switched systems.On the other hand，time-delay phenomenon often appears in engineering control systems and it is frequently a source of instability.Switched systems with time-delay have been extensively investigated by many authors[2].

Up to now，the overwhelming majority of switched systems results are available for systems governed by ordinary differential equations.Motivated by the fact that switched systems described by partial differential equations(i.e.distributed parameter switched systems)are more general，there is a real need to discuss such systems[4].There are many related works in this fields[4-8].

The issue of distributed parameter switched delay systems is essentially more complicated.The LMIs technique has been shown an effective tool for the study of distributed parameter systems[9-13].In this work，we extend the multiple Lyapunov method to distributed parameter switched delay systems.Our main contribution is to develop sufficient conditions of asymptotic stability and L2gain analysis for a class of switched distributed paramete delay systems.These conditions are given in the form of LMIs and arbitrary switching.

1 Problem Formulation

Throughout our work，we will consider the following switched distributed parameter delay systems:

Where y(x，t)∈L2(R+；L2([0，l]，Rn))is a vector-valued function representing the state of the process， ω(x，t)∈L2(R+；L2([0，l]，Rp))is the disturbance，z(x，t)∈L2(R+；L2([0，l]，Rq))is the measured output，(x，t)∈[0，l]×R+，x is the spatial coordinate，t is the time， α(t):R+→I is the switching signal mapping time to some finite index set I={1，2…，N}，the switching signalα(t)is a piecewise continuous(from the right)function depending on time or state or both.The discontinuities ofα(t)are called switching times or switching instants.Switching time t0＜t1＜…tk-1＜tk＜…with lkim∞tk=∞，→when t∈[tk，tk+1)，α(t)=α(tk)=i∈I，the i-th subsystem of the switched system is activated.Integer N is the number of subsystems of the switched system.Λi=diag(λi1，λi2，…，λin，)are known positive diagonal matrices，Bi，Ci，Di，Ei，Fi(i∈{1，2，…，N})are known constant matrices of compatible dimensions for the i-th subsystem.h is positive constant delay.In addition，we consider the following initial condition:

Whereφ(t)is continuously differentiable over[-h，0]，and the boundary condition given by:

Without loss of generality，we make the following assumptions:

Assumption 1 the state of switched system does not jump at switching instants，i.e.，the trajectory y(t)is everywhere continuous.Switching signalα(t)has finite switching number in any finite interval time.

The objective of this paper is to establish asymptotic stability and L2gain conditions for the system(1)-(3).

2 Asymptotic stability analysis

In this section，asymptotic stability analysis for system(1)-(3)without disturbance is firstly investigated.

We consider the following system:

To begin with，we demonstrate the well-posedness of the system(4).For simplicity，we at first consider the well-posedness of the non-switched system:

WhereΛ，B，C are known constant matrices of compatible dimensions.We can formulate the first equation of system(5)as an infinite dimensional system on the Hilbert space H=L2([0，l]，Rn).

Define the state function z(t)on the Hilbert space H as:

Then the first equation of the system(5)can be rewritten as the differential equation:

On the Hilbert space H with the operator A(A=Λ+B)possessing the dense domain H=D(A)={z∈H2([0，l]，Rn)∩H10([0，l]，Rn):z(0，t)=z(l，t)=0}

and the initial value:

Be given on a time interval[-h，0]in the space W

(W=C([-h，0]，D(A))∩C1([-h，0]，H)).

The infinitesimal operator A generates analytical semigroup T(t).For every initial value z0∈W，nonswitched system(5)has a unique mild solution.Obviously，every subsystem of the switched system(4)has a unique mild solution.Under Assumption 1，it is easily known that every subsystem of the switched system(4)has a unique mild solution.Thus the initial problem(4)turns out to be well-posed on time interval R+(see[12]for details).

Theorem 1 under Assumption 1，if there exist constant diagonal matrices Pi＞0 and constant matrices Qi＞0(i∈I{1，2，…，N})，such that the following LMIs:

Hold.Then the system(4)is asymptotic stable under arbitrary switching.

Proof.Under arbitrary switching，for any t∈[tk，tk+1)，whenα(t)=α(tk)=i∈I，the i-th subsystem of the switched system(4)is activated.

We choose the multiple Lyapunov functions as follow:

With constant diagonal matrices Pi＞0 and constant matrices Qi＞0(i∈I≜{1，2，…，N})to be determined.

Differentiating(10)with respect to t along the state trajectory of system(4)，then，

BecauseΛi，Piare positive diagonal matrices，we find that PΛi，ΛiPiare positive diagonal matrices and PΛi=ΛiPi.By using wirtinger’s inequality[14]and taking into account the boundary condition in(4)，we have:

Substituting(12)into(11)，one can easily obtain that:

When Li＜0，we have V·i(t)＜0(∀y(x，t)≠0)

That is，under arbitrary switching，the system(4)is asymptotic stable.This completes the proof.

3 L 2 gain analysis

In this section，we consider L2gain problem for the system(1)-(3).

Theorem 2 under Assumption 1，for givenλ ＞0，if there exist constant diagonal matrices Pi＞0 and constant matrices，Qi＞0(i∈{1，2，…，N})such that the following LMIs:

Hold.Then，under arbitrary switching，the system(1)-(3)is asymptotic stable and has L2gain with disturbance levelλ.

Proof.By similar arguments of the previous section，the system(1)-(3)also has a unique mildsolution.

Under arbitrary switching，for any t∈[tk，tk+1)，whenα(t)=α(tk)=i∈I the i-th subsystem of the switched system(1)-(3)is activated.

We choose(10)asthe multiple Lyapunov functions with constant diagonal matrices Pi＞0 and constant matrices Qi＞0(i∈≜{1，2，…，N})to be determined.Differentiating Vi(t)with respect to t along the state trajectory of system(1)-(3)and using the similar method in the proof of Theorem 1，we can calculate:

Whereη(x，t)=col[y(x，t)y(x，t-h)ω(x，t)]It follows from(13)that，

Because，

It follows from(13)and(14)that，

When Mi＜0，we have，

for any t∈[tk，tk+1)，integrating(20)leads to

Repeating the above procedure，we get，

Because Vi(t)＞0，under zero initial condition，we have V(0)=0.Then，

Integrating both sides of(22)from t=0 to∞，we derive that the L2gain performance，

That is，under arbitrary switching，the system(1)-(3)has L2gain with disturbance levelλ.This completes the proof.

4 Numerical example

We now present an example to illustrate the effectiveness of the proposed method.The simulation is conducted in Matlab.

Consider the distributed parameter switched delay system(1)-(3)，supposed there are two subsystem with parameters:

Letλ=0.5，l=π ，under arbitrary switching and zero initial condition，by resolving LMIs(13)，we obtain:

Therefore，according to Theorem 2，the system(1)-(3)is asymptotic stable and has L2gain with the disturbance levelλ=0.5.

5 Conclusion

Motivated by the fact that switched systems described by partial differential equations(i.e.distributed parameter switched systems)are more general，in this work，a class of distributed parameter switched delay systems have been studied.We have obtained asymptotic stable and L2gain sufficient conditions which in the form of LMIs and arbitrary switching.The research considered L2gain problem that is different from the previous works.Numerical example illustrated the effectiveness of the method.A potential extension by using average dwell time switching signal deserves further study.