Exponential Admissibility and Dynamic Output Feedback Control of Switched Singular Systems with Interval Time-Varying Delay

This paper is concerned with the problems of exponential admissibility and dynamic output feedback DOF control for a class of continuous-time switched singular systems with interval timevarying delay. A full-order, dynamic, synchronously switched DOF controller is considered. First, by using the average dwell time approach, a delay-range-dependent exponential admissibility criterion for the unforced switched singular time-delay system is established in terms of linear matrix inequalities LMIs . Then, based on this criterion, a sufficient condition on the existence of a desired DOF controller, which guarantees that the closed-loop system is regular, impulse free and exponentially stable, is proposed by employing the LMI technique. Finally, some illustrative examples are given to show the effectiveness of the proposed approach.


Introduction
The past decades have witnessed an enormous interest in switched systems, due to their powerful ability in modeling of event-driven systems, logic-based systems, parameter-or structure-varying systems, and so forth; for details, see 1-4 and the references therein.Switched systems are a class of hybrid systems, which consist of a collection of continuousor discrete-time subsystems and a switching rule specifying the switching between them.When focusing on the classification problems in switched systems, it is commonly recognized that there exist three basic problems 1 : i finding conditions for stability under arbitrary switching; ii identifing the limited but useful class of stabilizing switching signals, and iii construct a stabilizing switching signal.Many effective methods have been presented to tackle these three basic problems such as the multiple Lyapunov function approach 5 , the piecewise Lyapunov function approach 6 , the switched Lyapunov function approach 7 , Mathematical Problems in Engineering the convex combination technique 8 , and the dwell time or average dwell time scheme 9-12 .On the other hand, time-delay is very common in engineering systems and is frequently a source of instability and poor performance 13 .Therefore, control of switched time-delay systems has received more and more attention in the past few years; see 14-23 and the references therein.
As far as we know, singular systems known also as descriptor, implicit or differential algebraic systems also provide a natural framework for modeling of dynamic systems and describe a larger class of systems than the regular system models 24 .Switched singular systems have strong engineering background such as electrical networks 25 , economic systems 26 .Recently, many results have been obtained in the literature for switched singular systems, such as stability and stabilization 27-30 , reachability 31 , H ∞ control and filtering problems 32 .For switched singular time-delay SSTD systems, due to the coupling between the switching and the time-delay and because of the algebraic constraints in singular model, the behavior of such systems is much more complicated than that of regular switched time-delay systems or switched singular systems, and thus, to date, only a few results have been reported in the literature.In 33 , the robust stability and H ∞ control problems for discrete-time uncertain SSTD systems under arbitrary switching were discussed by using switched Lyapunov functions.In 34 , a switching signal was constructed to guarantee the asymptotic stability of continuous-time SSTD systems.However, the aforementioned results are focused on the basic problem i , see 33 , and problem iii , see 34 , for SSTD systems.Problem ii is to identify stabilizing switching signals on the premise that all the individual subsystems are stable.Basically, we will find that stability is ensured if the switching is sufficiently slow 1 , and it is well known that dwell time and average dwell time are two effective tools to define slow switching signals.In 9 , it was shown that if all the individual subsystems are exponentially stable and that the dwell time of the switching signal is not smaller than a certain lower bound, then the switched systems is exponentially stable.This result was extended to both continuous-time switched linear time-delay systems 16 and discrete-time cases 17 .Unfortunately, so far, to the best of the authors' knowledge, the problem of solving the basic problem ii for SSTD systems via the dwell time or average dwell time scheme remains open and unsolved.On the other hand, the results in 33 are derived based on the state feedback controller.In fact, in many practical systems, state variables are not always available.In this case, the design of a controller that does not require the complete access to the state vector is preferable.An important example of such controller is the dynamic output feedback DOF controller.However, little attention has been paid to the DOF controller design problem for SSTD systems.This forms the motivation of this paper.
In this paper, we are concerned with the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval timevarying delay.A full-order, dynamic, synchronously switched DOF controller is designed.First, by using the average dwell time approach and the piecewise Lyapunov function technique, a delay-range-dependent exponential admissibility criterion is derived in terms of LMIs, which guarantees the regularity, nonimpulsiveness, and exponential stability of the unforced system.A estimation of the convergence of the system is also explicitly given.Then, the corresponding solvability condition for the desired DOF controller is established by employing the LMI technique.Finally, some illustrative examples are given to show the effectiveness of the proposed approach.
Notation.Throughout this paper, R n denotes the n-dimensional Euclidean space and R n×n is the set of all n × n real matrices.P > 0 P ≥ 0 means that matrix P is positive definite semipositive definite .λ min P λ max P denotes the minimum maximum eigenvalue of symmetric matrix P .C n,d : C −d, 0 , R n denotes the Banach space of continuous vector functions mapping the interval −d, 0 to R n .Let x t ∈ C n,d be defined by x t : x t θ , θ ∈ −d, 0 .• denotes the Euclidean norm of a vector and its induced norm of a matrix and x t d sup −d≤θ≤0 x t θ .The superscript "T " represents matrix transposition, the symmetric terms in a matrix are denoted by " * ". diag{• • • } stands for a block-diagonal matrix and Sym{A} is the shorthand notation for A A T .Then, the problem to be addressed in this paper can be formulated as follows: given the SSTD system 2.1 , identify a class of switching signal σ t and design a DOF controller of the form 2.5 such that the resultant closed-loop system is exponentially admissible under the switching signal σ t .

Preliminaries and Problem Formulation
Before ending this section, we introduce the following lemma, which is essential for the development of our main results.Lemma 2.4.For any constant matrix Z ∈ R n×n , Z Z T > 0, positive scalar α, and vector function ẋ : −τ, ∞ → R n such that the following integration is well defined, then where 0 ≤ d t ≤ τ.
Proof.The proof is almost the same as Lemma 1 in 20 .Using Schur complement, we have Integrating it from t − d t to t, we get Using Schur complement again, we find that Lemma 2.4 holds.

Main Results
In this section, we first apply the average dwell time approach to investigate the exponential admissibility for SSTD system 2.4 , and give the following result.
, and P i of the following form with P i11 ∈ R r , P i11 > 0, and P i22 being invertible, such that

3.3
Moreover, an estimate on the exponential decay rate is λ Proof.The proof is divided into three parts: i to show the regularity and nonimpulsiveness; ii to show the exponential stability of the differential subsystem; and iii to show the exponential stability of the algebraic subsystem.Part i regularity and nonimpulsiveness.According to 2.3 , for each i ∈ I, denote where and E given as 3.1 , 3.4 and 2.3 into this inequality and using Schur complement, we have Sym{A T i22 P i22 } < 0, which implies that A i22 , i ∈ I, is nonsingular.Then by 24 and Definition 2.2, system 2.4 is regular and impulse free.
Part ii exponential stability of the differential subsystem.Define the piecewise Lyapunov functional candidate for system 2.4 as follows

3.5
As mentioned earlier, the i k th subsystem is activated when t ∈ t k , t k 1 .Then, along the solution of system 2.4 under the switching sequence S, for t ∈ t k , t k 1 , we have

3.6
By , where x 1 t ∈ R r and x 2 t ∈ R n−r .From 2.3 and 3.1 , it can be seen that for each i, i ∈ I, x T t E T P i x t x T 1 t P i11 x 1 t .Noting this, and using 3.3 and 3.5 , at switching instant t k , we have where t − k denotes the left limitation of t k .Therefore, it follows from 3.8 , 3.9 and the relation

3.10
According to 3.5 and 3.10 , we obtain where λ 1 min ∀i∈I λ min P i11 , and Then, combining 3.10 with 3.11 yields

3.12
Part iii exponential stability of the algebraic subsystem.Since and H I r 0 0 I n−r .Then, it is easy to get

3.13
where A T i12 P i11 A T i22 P i21 , and P i22 A T i22 P i22 .According to 3.13 , denote

3.14
Let where ξ 1 t ∈ R r and ξ 2 t ∈ R n−r .Then, for any fixed σ t i, i ∈ I, system 2.4 is restricted system equivalent r.s.e. to 3.17 By 3.2 and Schur complement, we have Pre-and postmultiplying this inequality by diag{H T , H T } and diag{H, H}, respectively, noting the expressions in 3.13 and 3.14 , and using Schur complement, we have Pre-and postmultiplying this inequality by − A T di22 I and its transpose, respectively, and noting Q i122 > 0 and 0 ≤ μ < 1, we obtain Then, according to Lemma 7 in 38 , we can deduce that there exist constants i > 1 and η i > 0 such that 3.19 Define t 0 t, t j t j−1 − d t j−1 , j 1, 2, . . ., 3.20

3.21
As mentioned earlier, under the switching sequence S, for t ∈ t k , t k 1 , the i k th subsystem is activated.Then, from 3.17 and 3.20 , we have Similarly, it can be obtained that Continuing in the same manner and noting that t j < t j−1 , then there exists a finite positive integer T i k such that where Then, following a similar procedure as the above, there exists a finite positive integer

3.24
where After k-times iterative manipulations, t belongs to t 0 , t 1 , and there exists a finite positive integer T i 0 such that

3.25
where By a simple induction, we have

3.30
Using 3.19 and the relation T a ≥ T * a ln β /α, the first term in 3.28 can be estimated as

3.32
Then, the second term in 3.28 can be estimated as

3.33
Similarly, the third term in 3.28 can be bounded by

3.34
In addition, following a similar deduction as that in 3.32 , we obtain

3.35
Then, considering this and 3.19 , the fourth term in 3.28 can be estimated as

3.36
Similarly, the fifth term in 3.28 can be bounded by 3.37 Therefore, using 3.31 and 3.33 -3.37 , ξ 2 t can be estimated as for all i, j ∈ I, and T * a 0, then system 2.4 possesses a common Lyapunov function and the switching signals can be arbitrary.
In the following, we are to deal with the design problem of DOF controller for the SSTD system 2.1 .Applying the DOF controller 2.5 to system 2.1 gives the following closed-loop system where η t x T t x T c t T , and

3.40
The following Theorem presents a sufficient condition for solvability of the DOF controller design problem for system 2.1 .
Theorem 3.5.For prescribed scalars α > 0, γ > 0, d 1 ≥ 0, d 2 > 0 and 0 ≤ μ < 1, if for each i ∈ I, and given scalars ξ i1 and ξ i2 , there exist matrices , l 1, 2, and R i and U i of the following form Then, there exists a DOF controller in the form of 2.5 , such that system 3.39 is exponentially admissible for any switching sequence S with average dwell time T a ≥ T * a ln β/α, where β ≥ 1 satisfies

3.45
Proof.From Theorem 3.1, we known that system 3.39 is exponentially admissible for any switching sequence S with average dwell time T a ≥ T * a ln β /α, where β ≥ 1 satisfying 3.3 , if for each i ∈ I, there exist matrices Q il > 0, Z il > 0, l 1, 2, and P i with the form of 3.1 such that the inequality 3.2 with E, A i and A di instead of E, A i and C i , respectively, holds.By decomposing Φ i in 3.2 , we obtain that for each i ∈ I where J i is any invertible matrix with compatible dimension, and and Let E diag{E, E}.For each i ∈ I, define

3.49
By 2.3 and 3.41 , we have we can easily obtain 3.42 .This completes the proof.
Remark 3.6.Note that condition Φ i of Theorem 3.1 involves some product terms between the Lyapunov matrices and the system matrices, which complicates the DOF control synthesis problem.To solve this problem, in the proof of Theorem 3.5, we have made a decoupling between the Lyapunov matrices and the system matrices by introducing a slack matrix J i in condition Λ i .Compared with the variable change method used in 39, 40 , the decoupling technique proposed here simplifies the DOF controller design problem greatly, which decreases the conservatism in some sense.
Remark 3.7.Scalars ξ i1 and ξ i2 , i ∈ I, in Theorem 3.5 are tuning parameters which need to be specified first.The optimal values of these parameters can be found by applying some optimization algorithms such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm 41 .
Remark 3.8.It is noted that in this paper, the derivative matrix E is assumed to be switchmode-independent.If E is also switch-mode-dependent, then E is changed to E i , i ∈ I.In this case, the transformation matrices P and Q should become P i and Q i so that P i E i Q i diag{I r i , 0}, and the state of the transformed system becomes x t Q −1 i t x T i1 t x T i2 t T with x T i1 t ∈ R r i and x T i1 t ∈ R n−r i , which implies that there does not exist one common state space coordinate basis for all subsystems.Then, some assumptions for E i e.g., E i , i ∈ I, have the same right zero subspace 27 should be made so that Q i remains the same; in this case, the method presented in this paper is also valid.How to investigate the general SSTD system with E being switch-mode-dependent is an interesting problem for future work via other approaches.

Numerical Examples
In this section, some numerical examples are presented to demonstrate the effectiveness of the proposed methods.
Example 4.1.Consider the switched system 2.4 with E I, N 2 e.g., there are two subsystems and the following parameters, which are borrowed from 21 : For μ 0.4, α 0.  1 shows the values of the upper bound for various d 1 and the number of involved variables by using different methods.It is easily seen from Table 1 that Theorem 3.1 of this paper not only provides better results than those criteria in 21, 22 but also reduces the computational overhead to some extent.Example 4.2.Consider the switched system 2.4 with N 2 and the related parameters are given as follows:  1 and 2, respectively, with the initial condition given by φ t 1 2 T , t ∈ −0.5, 0 .In view of this, our goal is to design a DOF control u t in the form of 2.5 such that the closed-loop system is exponentially admissible.Set α 0. To show the effectiveness of the obtained DOF controller, giving a random switching signal with the average dwell time T a ≥ 0.13 as shown in Figure 3, we get the state trajectories of the closed-loop system as shown in Figure 4, for the given initial condition φ t 1 2 T , t ∈ −0.5, 0 .It is clear that the designed controller is feasible and ensures the stability of the closed-loop system despite the switching and the time-varying delay.

Conclusions
In this paper, the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval time-varying delay have been investigated.A class of switching signals has been identified for the switched singular timedelay systems to be exponentially admissible under the average dwell time scheme.The DOF controller has been designed, and the corresponding solvability condition has been established by using the LMI technique.Numerical examples have been provided to illustrate the effectiveness of the proposed methods.

Figure 1 :
Figure 1: State trajectories of the open-loop subsystem 1.

Figure 2 :
Figure 2: State trajectories of the open-loop subsystem 2.

Figure 3 :
Figure 3: Switching signal with the average dwell time T a > 0.13.
A di , B i , C i and C di are constant real matrices with appropriate dimensions; φ t ∈ C n,d 1 d 2 is a compatible vector valued initial function; d t is an interval time-varying delay satisfying then T a is called average dwell time.As commonly used in the literature, we choose N 0 0.
c , A ci , B ci , C ci and D ci , σ t i, i ∈ I, are appropriately dimensioned constant matrices to be determined.
. Then, system 2.4 with d t satisfying 2.2 is exponentially admissible for any switching sequence S with average dwell time T a ≥ T * a ln β /α, where β ≥ 1 satisfies replacing E ẋ t with A i k x t A di k x t−d t and using Lemma 2.4 and Schur complement, LMI 3.2 yields −t 0 x t 0 d 1 d 2 .3.38Combining 3.15 , 3.12 and 3.38 yields that system 2.4 is exponentially stable for any switching sequence S with average dwell time T a ≥ T * Remark 3.2.In terms of LMIs, Theorem 3.1 presents a delay-range-dependent exponential admissibility condition for the switched singular systems with interval time-varying delay.It is noted that this condition is obtained by using the integral inequality Lemma 2.4 ; no additional free-weighting matrices are introduced to deal with the cross-term.Therefore, the condition proposed here involves much less decision variables than those obtained by using the free-weighting matrices method 16, 19, 21, 22 if the same Lyapunov function is chosen.
a ln β /α.This completes the proof.Remark 3.3.Equation 3.26 plays an important role in analyzing the exponential stability of the algebraic subsystem, which can be seen as a generalization of the iterative equation in 37 for nonswitched singular time-delay system to switched case.a ln β /α, which leads to P i11

Table 1 :
Comparison of allowable upper bound d 2 for different d 1 in Example 4.1.
5 and β 1.1, employing the LMIs in 21, 22 and those in Theorem 3.1 yields an allowable upper bound d 2 in this paper d 2 d 1 d 2 of the delay d t that guarantees the stability of system 2.4 .Table

Figure 4 :
State trajectories of the closed-loop system under DOF control.It can be verified that both of the above two subsystems are stable.Let β 1; it can be found that there is no feasible solution to this case, which implies that there is no common Lyapunov function for the above two subsystems see Remark 3.4 .Now, we consider the average dwell time scheme, and set β 1.2.Solving the LMIs 3.2 gives the following solutions: Example 4.3.Consider the switched system 2.1 with N 2 and By simulation, it can be checked that both of the above two subsystems with u t 0 are unstable, and the state responses of the corresponding open-loop systems are shown in Figures which means that the above switched system is exponentially admissible.Moreover, by further analysis, it can be found that the allowable minimum of β is β min 1.046 when α 0.5; in this case T * a ln β min /α 0.0899.