#### Abstract

This paper deals with the problems of exponential admissibility and control for a class of continuous-time switched singular systems with time-varying delay. The controllers to be designed include both the state feedback (SF) and the static output feedback (SOF). First, by using the average dwell time scheme, the piecewise Lyapunov function, and the free-weighting matrix technique, an exponential admissibility criterion, which is not only delay-range-dependent but also decay-rate-dependent, is derived in terms of linear matrix inequalities (LMIs). A weighted performance criterion is also provided. Then, based on these, the solvability conditions for the desired SF and SOF controllers are established by employing the LMI technique, respectively. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

Many real-world engineering systems always exhibit several kinds of dynamic behavior in different parts of the system (e.g., continuous dynamics, discrete dynamics, jump phenomena, and logic commands) and are more appropriately modeled by hybrid systems. As an important class of hybrid systems, switched systems consist of a collection of continuous-time or discrete-time subsystems and a switching rule orchestrating the switching between them and are of great current interest; see, for example, Decarlo et al. [1], Liberzon [2], Lin and Antsaklis [3], and Sun and Ge [4] for some recent survey and monographs. Switched systems have great flexibility in modeling parameter-varying or structure-varying systems, event-driven systems, logic-based systems, and so forth. Also, multiple-controller switching technique offers an effective mechanism to cope with highly complex systems and/or systems with large uncertainties, particularly in the adaptive context [5]. Many effective methods have been developed for switched systems, for example, the multiple Lyapunov function approach [6, 7], the piecewise Lyapunov function approach [8, 9], the switched Lyapunov function method [10], convex combination technique [11], and the dwell-time or average dwell-time scheme [12–15]. Among them, the average dwell-time scheme provides a simple yet efficient tool for stability analysis of switched systems, especially when the switching is restricted and has been more and more favored [16].

On the other hand, time delay is a common phenomenon in various engineering systems and the main sources of instability and poor performance of a system. Hence, control of switched time-delay systems has been an attractive field in control theory and application in the past decade. Some of the aforementioned approaches for nondelayed switched systems have been successfully adopted to hand the switched time-delay systems; see, for example, Du et al. [17], Kim et al. [18], Mahmoud [19], Phat [20], Sun et al. [21], Sun et al. [22], Wang et al. [23], Wu and Zheng [24], Xie et al. [25], Zhang and Yu [26], and the references therein.

Recently, a more general class of switched time-delay systems described by the singular form was considered in Ma et al. [27] and Wang and Gao [28]. It is known that a singular model describes dynamic systems better than the standard state-space system model [29]. The singular form provides a convenient and natural representation of economic systems, electrical networks, power systems, mechanical systems, and many other systems which have to be modeled by additional algebraic constraints [29]. Meanwhile, it endows the aforementioned systems with several special features, such as regularity and impulse behavior, that are not found in standard state-space systems. Therefore, it is both worthwhile and challenging to investigate the stability and control problems of switched singular time-delay systems. In the past few years, some fundamental results based on the aforementioned approaches for standard state-space switched time-delay systems have been successfully extended to switched singular time-delay systems. For example, by using the switched Lyapunov function method, the robust stability, stabilization, and control problems for a class of discrete-time uncertain switched singular systems with constant time delay under arbitrary switching were investigated in Ma et al. [27]; filters were designed in Lin et al. [30] for discrete-time switched singular systems with time-varying time delay. In Wang and Gao [28], based on multiple Lyapunov function approach, a switching signal was constructed to guarantee the asymptotic stability of a class of continuous-time switched singular time-delay systems. With the help of average dwell time scheme, some initial results on the exponential admissibility (regularity, nonimpulsiveness, and exponential stability) were obtained in Lin and Fei [31] for continuous-time switched singular time-delay systems. However, to the best of our knowledge, few work has been conducted regarding the control for continuous-time switched singular time-delay systems via the dwell time or average dwell time scheme, which constitutes the main motivation of the present study.

In this paper, we aim to solve the problem of control for a class of continuous-time switched singular systems with interval time-varying delay via the average dwell time scheme. Both the state feedback (SF) control and the static output feedback (SOF) control are considered. Firstly, based on the average dwell time scheme, the piecewise Lyapunov function, as well as the free-weighting technique, a class of slow switching signals is identified to guarantee the unforced systems to be exponentially admissible with a weighted performance , and several corresponding criteria, which are not only delay-range-dependent but also decay-rate-dependent, are derived in terms of linear matrix inequalities (LMIs). Next, the LMI-based approaches are proposed to design an SF controller and an SOF controller, respectively, such that the resultant closed-loop system is exponentially admissible and satisfies a weighted performance . Finally, two illustrative examples are given to show the effectiveness of the proposed approach.

*Notation 1. *Throughout this paper, the superscript represents matrix transposition. denotes the real -dimensional Euclidean space, and denotes the set of all real matrices. is an appropriately dimensioned identity matrix. () means that matrix is positive definite (semi positive definite). stands for a block diagonal matrix. () denotes the minimum (maximum) eigenvalue of symmetric matrix , is the space of square-integrable vector functions over , denotes the Euclidean norm of a vector and its induced norm of a matrix, and is the shorthand notation for . In symmetric block matrices, we use an asterisk () to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

#### 2. Preliminaries and Problem Formulation

Consider a class of switched singular time-delay system of the form where is the system state, is the control input, is the controlled output, is the measured output, and is the disturbance input that belongs to ; 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 continuous vector-valued initial function on ; denotes interval time-varying delay satisfying where and are constants. Note that may not be equal to 0.

Since , there exist nonsingular 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 2.1 (see [2]). *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 [21, 26], we choose .

*Definition 2.2 (see [21, 29, 32]). *For any delay satisfying (2.2), the unforced part of system (2.1) with
is said to be(1)regular if is not identically zero for each , , (2)impulse if for each , , (3)exponentially stable under the switching signal if the solution of system (2.5) satisfies
where and are called the decay rate and decay coefficient, respectively, and , (4)exponentially admissible under the switching signal if it is regular, impulse free, and exponentially stable under the switching signal .

*Remark 2.3. *The regularity and nonimpulsiveness of the switched singular time-delay system (2.5) ensure that its every subsystem has unique solution for any compatible initial condition. However, even if a switched singular system is regular and causal, it still has inevitably finite jumps due to the incompatible initial conditions caused by subsystem switching [33]. For more details about the impulsiveness effects on the stability of systems, we refer readers to Chen and Sun [34], Li et al. [35], and the references therein. In this paper, without loss of generality, we assume that such jumps cannot destroy the stability of system (2.1). Nevertheless, how to suppress or eliminate the finite jumps in switched singular systems is a challenging problem which deserves further investigation.

*Definition 2.4. *For the given and , system (2.1) is said to be exponentially admissible with a weighted performance under the switching signal , if it is exponentially admissible with and , and under zero initial condition, that is, , , for any nonzero , it holds that

*Remark 2.5. *For switched systems with the average dwell time switching, the Lyapunov function values at switching instants are often allowed to increase times () to reduce the conservatism in system stability analysis, which will lead to the normal disturbance attenuation performance hard to compute or check, even in linear setting [15, 36]. Therefore, the weighted performance criterion (2.7) [15, 21, 24] is adopted here to evaluate disturbance attenuation while obtaining the expected exponential stability.

This paper considers both SF control law
and SOF control law
where 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 switched singular time-delay system (2.1) and a prescribed scalar , identify a class of switching signal and design an SF controller of the form (2.8) and an SOF controller of the form (2.9) such that the resultant closed-loop system is exponentially admissible with a weighted performance under the switching signal .

#### 3. Exponential Admissibility and Performance Analysis

First, we apply the average dwell time approach and the piecewise Lyapunov function technique to investigate the exponential admissibility for the switched singular time-delay system (2.5) 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
**
Then, system (2.5) 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; (iii) to show the exponential stability of the algebraic subsystem.

(i) Regularity and nonimpulsiveness. According to (2.4), for each , denote
where . From (3.2), it is easy to see that , . Noting , , we get
Substituting and given as (3.1) and (2.4) into this inequality yields
where denotes a matrix which is not relevant to the discussion. This implies that , , is nonsingular. Then, by Dai [29] and Definition 2.1, system (2.5) is regular and impulse free.

(ii) Exponential stability of differential subsystem. Define the piecewise Lyapunov functional candidate for system (2.5) as the following:
Then, along the solution of system (2.5) for a fixed , , we have
From the Leibniz-Newton formula, the following equations are true for any matrices , , and , , with appropriate dimensions
On the other hand, the following equation is also true:
By (3.8)–(3.11), we have
where , , and
By Schur complement, LMI (3.2) implies
Notice that the last three parts in (3.12) are all less than 0. So, if (3.14) holds, then
For an arbitrary piecewise constant switching signal , and for any , we let , , denote the switching points of over the interval . As mentioned earlier, the subsystem is activated when . Integrating (3.15) from to gives
Let , where and . From (2.4) and (3.1), it can be deduced that for each ,
In view of this, and using (3.4) and (3.8), at switching instant , we have
where denotes the left limitation of . Therefore, it follows from (3.16), (3.18), and the relation that
According to (3.8) and (3.19), we obtain
where
Considering (3.19) and (3.20) yields
which implies

(iii) Exponential stability of algebraic subsystem. Since , , is nonsingular, we choose
Then, it is easy to get
where , , , , and . According to (3.25), denote
and let
where and . Then, for any , , system (2.5) is a restricted system equivalent (r.s.e.) to
By (3.2) and Schur complement, we have
Pre- and postmultiplying this inequality by and , respectively, noting the expressions in (3.25) and (3.26), 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 5 in Kharitonov et al. [37], we can deduce that there exist constants and such that
Define

Now, following similar line as in Part 3 in Theorem 1 of Lin and Fei [31], it can easily be obtained that
where
are positive finite integers, respectively, satisfying
Combining (3.27), (3.23) and (3.34) yields that system (2.5) is exponentially stable for any switching sequence with average dwell time . This completes the proof.

*Remark 3.2. *Theorem 3.1 provides a sufficient condition of the exponential admissibility for the switched singular time-delay system (2.5). Note that due to the existence of algebraic constraints in system states, the stability analysis of switched singular time-delay systems is much more complicated than that for switched state-space time-delay systems [21–23, 25, 38]. Note also that the condition established in Theorem 3.1 is not only delay-range-dependent but also decay-rate-dependent. The delay-range-dependence makes the result less conservative, while the decay-rate-dependence enables one to control the transient process of differential and algebraic subsystems with a unified performance specification.

*Remark 3.3. *Different from the integral inequality method used in our previous work [31], the free-weighting matrix method [39] is adopted when deriving Theorem 3.1, and thus no three-product terms, for example, , , and so forth, are involved, which greatly facilitates the SF and SOF controllers design, as seen in Section 4.

*Remark 3.4. *If in , which leads to , , , , , for all , and , then system (2.5) possesses a common Lyapunov function, and the switching signals can be arbitrary.

Now, the following theorem presents a sufficient condition on exponential admissibility with a weighted performance of the switched singular time-delay system (2.1) with .

Theorem 3.5. *For prescribed scalars , , , and , if for each , there exist matrices , , , , , , , and with the form of (3.1) such that
**
where
**
and , , , , , , , and are defined in (3.2). Then, system (2.1) with is exponentially admissible with a weighted performance for any switching sequence with average dwell time , where satisfying (3.4).*

* Proof. *Choose the piecewise Lyapunov function defined by (3.8). Since (3.37) implies (3.2), system (2.1) with and is exponentially admissible by Theorem 3.1. On the other hand, similar to the proof of Theorem 3.1, from (3.37), we have that for ,
where . This implies that
By induction, we have
Under zero initial condition, (3.41) gives
Multiplying both sides of (3.42) by yields
Noting that and , we get . Then, it follows from (3.43) that . Integrating both sides of this inequality from to leads to inequality (2.7). This completes the proof of Theorem 3.5.

*Remark 3.6. *Note that when , which is a trivial case, system (2.1) with achieves the normal performance under arbitrary switching.

#### 4. Controller Design

In this section, based on the results of the previous section, we are to deal with the design problems of both SF and SOF controllers for the switched singular time-delay system (2.1).

##### 4.1. SF Controller Design

Applying the SF controller (2.8) to system (2.1) gives the following closed-loop system: where The following theorem presents a sufficient condition for solvability of the SF controller design problem for system (2.1).

Theorem 4.1. *For prescribed scalars , , , and , if for each , and given scalars , , , and , there exist matrices , , , , and of the following form:
**
with , , and being invertible, such that
**
where
**
Then, there exists an SF controller (2.8) such that the closed-loop system (4.1) with satisfying (2.2) is exponentially admissible with a weighted performance for any switching sequence with average dwell time , where satisfies
**
Moreover, the feedback gain of the controller is
*

* Proof. *According to Theorem 3.5, the closed-loop system (4.1) is exponentially admissible with a weighted performance if for each , there exist matrices , , , , , , , and with the form of (3.1) such that inequality (3.37) holds with and instead of and , respectively. By Schur complement, (3.37) is equivalent to
where , , , , , and are defined in (3.2), and
Since and is invertible, then is invertible. Let
By (3.1), has the form of (4.3). Pre- and postmultiplying (4.8) by and its transpose, respectively, and noting (4.10), we obtain
where
Now, introducing change of variables