Mathematical Problems in Engineering

Volume 2010, Article ID 680382, 21 pages

http://dx.doi.org/10.1155/2010/680382

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

College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210046, China

Received 2 August 2010; Accepted 25 October 2010

Academic Editor: Cristian Toma

Copyright © 2010 Jinxing Lin and Chunxia Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 time-varying 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.

#### 1. 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 continuous- or 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], 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], 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 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 time-varying 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 1. *Throughout this paper, denotes the -dimensional Euclidean space and is the set of all real matrices. () means that matrix is positive definite (semipositive definite). () denotes the minimum (maximum) eigenvalue of symmetric matrix . denotes the Banach space of continuous vector functions mapping the interval to . Let be defined by , . denotes the Euclidean norm of a vector and its induced norm of a matrix and . The superscript “” represents matrix transposition, the symmetric terms in a matrix are denoted by “”. stands for a block-diagonal matrix and is the shorthand notation for .

#### 2. Preliminaries and Problem Formulation

Consider a class of SSTD systems of the form where is the system state, is the control input, and is the measured output; with integer is the switching signal; is a singular matrix with ; for each possible value , , , , , and are constant real matrices with appropriate dimensions; is a compatible vector valued initial function; is an interval time-varying delay satisfying where , and are constants.

*Remark 2.1. *Model (2.1) can describe many practical time-delay systems (e.g., chemical engineering systems, lossless transmission lines, partial element equivalent circuit, etc.) with time-varying parameters or structures, which may be caused by random failures and repairs of the components, sudden environment changes, and varying of the operating point of a system [13, 35]. In real application, the importance of the study of controller design problem for model (2.1) also arises from the extensive applications in networked control [36].

Since , there exist nonsingular transformation matrices , such that . In this paper, without loss of generality, let Corresponding to the switching signal , we denote the switching sequence by , with , which means that the subsystem is activated when . To present the objective of this paper more precisely, the following definitions are introduced.

*Definition 2.2 (see [16, 37]). *For the switching signal and any delay satisfying (2.2), the unforced part of system (2.1)
is said to be(1)regular if is not identically zero for each , , (2)impulse free if for each , , (3)exponentially stable under the switching signal if the solution of system (2.4) satisfies , for all , where and are called the decay rate and decay coefficient, respectively;(4)exponentially admissible if it is regular, impulse free and exponentially stable under the switching signal .

*Definition 2.3 (see [9]). *For any , let denote the number of switching of over . If holds for , , then is called average dwell time. As commonly used in the literature, we choose .

This paper considers the full-order DOF controller of the following form: where is the controller state vector, and , , , , and , , , are appropriately dimensioned constant matrices to be determined.

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 and design a DOF controller of the form (2.5) such that the resultant closed-loop system is exponentially admissible under the switching signal .

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 , , positive scalar , and vector function such that the following integration is well defined, then
**
where .*

*Proof. * The proof is almost the same as Lemma 1 in [20]. Using Schur complement, we have
Integrating it from to , we get
Using Schur complement again, we find that Lemma 2.4 holds.

#### 3. 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.

Theorem 3.1. *For prescribed scalars , , , and , if for each , there exist matrices , , , and of the following form
**
with , , and being invertible, such that
**
where , , , , and . Then, system (2.4) with satisfying (2.2) is exponentially admissible for any switching sequence with average dwell time , where satisfies
**
Moreover, an estimate on the exponential decay rate is .*

*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 , denote
where and . From (3.2), it is easy to see that , . Noting and , , we get . Substituting , , and given as (3.1), (3.4) and (2.3) into this inequality and using Schur complement, we have , which implies that , , 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
As mentioned earlier, the th subsystem is activated when . Then, along the solution of system (2.4) under the switching sequence , for , we have
By replacing with and using Lemma 2.4 and Schur complement, LMI (3.2) yields
Integrating (3.7) from to gives
Let , where and . From (2.3) and (3.1), it can be seen that for each , , . Noting this, and using (3.3) and (3.5), at switching instant , we have
where denotes the left limitation of . Therefore, it follows from (3.8), (3.9) and the relation that
According to (3.5) and (3.10), we obtain
where , and . Then, combining (3.10) with (3.11) yields

Part (iii) exponential stability of the algebraic subsystem. Since , , is nonsingular, set and . Then, it is easy to get
where , , , , and . According to (3.13), denote
Let
where and . Then, for any fixed , , system (2.4) is restricted system equivalent (r.s.e.) to
By (3.2) and Schur complement, we have , where and . Pre- and postmultiplying this inequality by and , respectively, noting the expressions in (3.13) and (3.14), and using Schur complement, we have
Pre- and postmultiplying this inequality by and its transpose, respectively, and noting and , we obtain . Then, according to Lemma 7 in [38], we can deduce that there exist constants and such that

Define
As mentioned earlier, under the switching sequence , for , the subsystem is activated. Then, from (3.17) and (3.20), we have
Similarly, it can be obtained that . Substituting this into (3.22), we get . Continuing in the same manner and noting that , then there exists a finite positive integer such that
where and . When , the subsystem is activated. Then, following a similar procedure as the above, there exists a finite positive integer such that
where and . After - iterative manipulations, belongs to , and there exists a finite positive integer such that
where and . By a simple induction, we have
where
Therefore, from (3.15), (3.21), and (3.26), and noting , we obtain
where
Note
Using (3.19) and the relation , the first term in (3.28) can be estimated as
By (3.15), (3.12), (3.21), (3.20) and (3.19), we get
Then, the second term in (3.28) can be estimated as
Similarly, the third term in (3.28) can be bounded by
In addition, following a similar deduction as that in (3.32), we obtain
Then, considering this and (3.19), the fourth term in (3.28) can be estimated as
Similarly, the fifth term in (3.28) can be bounded by
Therefore, using (3.31) and (3.33)–(3.37), can be estimated as
Combining (3.15), (3.12) and (3.38) yields that system (2.4) is exponentially stable for any switching sequence with average dwell time . This completes the proof.

*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.

*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.

*Remark 3.4. *If in , which leads to , , , , for all , and , 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 , and 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 , , , and , if for each , and given scalars and , there exist matrices , , , , , , , , , , , , and and of the following form
**
with , , , , , , and and being invertible, such that
**
where , , , , , , , , , , , , , , , , , . Then, there exists a DOF controller in the form of (2.5), such that system (3.39) is exponentially admissible for any switching sequence with average dwell time , where satisfies
**
Moreover, a desired DOF controller realisation is given by
*

*Proof. * From Theorem 3.1, we known that system (3.39) is exponentially admissible for any switching sequence with average dwell time , where satisfying (3.3), if for each , there exist matrices , , , and with the form of (3.1) such that the inequality (3.2) with , and instead of , and , respectively, holds. By decomposing in (3.2), we obtain that for each
where is any invertible matrix with compatible dimension, and
with , , and . Hence, holds if

Let . For each , define
By (2.3) and (3.41), we have
which combining (3.49) yields
Then, from (3.41), (3.44) and (3.51), it can be deduced that

Denote
Substituting (3.43), (3.49) and (3.53) into (3.48), and defining
we can easily obtain (3.42). This completes the proof.

*Remark 3.6. *Note that condition 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 in condition . 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 and , , 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 is assumed to be switch-mode-independent. If is also switch-mode-dependent, then is changed to , . In this case, the transformation matrices and should become and so that , and the state of the transformed system becomes with and , which implies that there does not exist one common state space coordinate basis for all subsystems. Then, some assumptions for (e.g., , , have the same right zero subspace [27]) should be made so that remains the same; in this case, the method presented in this paper is also valid. How to investigate the general SSTD system with being switch-mode-dependent is an interesting problem for future work via other approaches.

#### 4. 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.g., there are two subsystems) and the following parameters, which are borrowed from [21]:
For , and , employing the LMIs in [21, 22] and those in Theorem 3.1 yields an allowable upper bound (in this paper ) of the delay that guarantees the stability of system (2.4). Table 1 shows the values of the upper bound for various 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 and the related parameters are given as follows:
and , , and . It can be verified that both of the above two subsystems are stable. Let ; 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 . Solving the LMIs (3.2) gives the following solutions: