(ProQuest: ... denotes non-US-ASCII text omitted.)
Jinxing Lin 1, 2 and Zhifeng Gao 1
Recommended by Jitao Sun
1, College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210023, China
2, Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing, Jiangsu 210096, China
Received 16 September 2012; Accepted 22 November 2012
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.
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 H ∞ 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]; H ∞ 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 H ∞ 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 H ∞ 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 H ∞ performance γ , 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 H ∞ performance γ . Finally, two illustrative examples are given to show the effectiveness of the proposed approach.
Notation 1.
Throughout this paper, the superscript T represents matrix transposition. R n denotes the real n -dimensional Euclidean space, and R n ×n denotes the set of all n ×n real matrices. I is an appropriately dimensioned identity matrix. P >0 ( P ...5;0 ) means that matrix P is positive definite (semi positive definite). diag { · , · , · } stands for a block diagonal matrix. λ min (P ) ( λ max (P ) ) denotes the minimum (maximum) eigenvalue of symmetric matrix P , L 2 [0 , ∞ ) is the space of square-integrable vector functions over [0 , ∞ ) , || · || denotes the Euclidean norm of a vector and its induced norm of a matrix, and Sym {A } is the shorthand notation for A + A T . 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 [figure omitted; refer to PDF] where x (t ) ∈ R n is the system state, u (t ) ∈ R m is the control input, z (t ) ∈ R q is the controlled output, y (t ) ∈ R p is the measured output, and w (t ) ∈ R l is the disturbance input that belongs to L 2 [0 , ∞ ) ; σ (t ) : [0 , + ∞ ) [arrow right] ... = {1,2 , ... ,I } with integer I >1 is the switching signal; E ∈ R n ×n is a singular matrix with rank E =r ...4;n ; for each possible value, σ (t ) =i , i ∈ ... , A i , A di , B i , B wi , C i , C di , D i , D wi , and L i are constant real matrices with appropriate dimensions; [varphi] ( θ ) is a compatible continuous vector-valued initial function on [ - d 2 ,0 ] ; d (t ) denotes interval time-varying delay satisfying [figure omitted; refer to PDF] where 0 ...4; d 1 < d 2 and μ are constants. Note that d 1 may not be equal to 0.
Since rank E =r ...4;n , there exist nonsingular matrices P , Q ∈ R n ×n such that [figure omitted; refer to PDF] In this paper, without loss of generality, let [figure omitted; refer to PDF] Corresponding to the switching signal σ (t ) , we denote the switching sequence by ...AE; : = { ( i 0 , t 0 ) , ... , ( i k , t k ) |" i k ∈ ... , k =0,1 , ... } with t 0 =0 , which means that the i k subsystem is activated when t ∈ [ t k , t k +1 ) . To present the objective of this paper more precisely, the following definitions are introduced.
Definition 2.1 (see [ 2]).
For any T 2 > T 1 ...5;0 , let N σ ( T 1 , T 2 ) denote the number of switching of σ (t ) over ( T 1 , T 2 ) . If N σ ( T 1 , T 2 ) ...4; N 0 + ( T 2 - T 1 ) / T a holds for T a >0 , N 0 ...5;0 , then T a is called average dwell time. As commonly used in the literature [ 21, 26], we choose N 0 =0 .
Definition 2.2 (see [ 21, 29, 32]).
For any delay d (t ) satisfying ( 2.2), the unforced part of system ( 2.1) with w (t ) =0 [figure omitted; refer to PDF] is said to be
(1) regular if det (sE - A i ) is not identically zero for each σ (t ) =i , i ∈ ... ,
(2) impulse if deg (det (sE - A i ) ) =rank E for each σ (t ) =i , i ∈ ... ,
(3) exponentially stable under the switching signal σ (t ) if the solution x (t ) of system ( 2.5) satisfies [figure omitted; refer to PDF] where λ >0 and ι >0 are called the decay rate and decay coefficient, respectively, and || x t 0 || c =su p - d 2 ...4; θ ...4;0 { ||x ( t 0 + θ ) || } ,
(4) exponentially admissible under the switching signal σ (t ) if it is regular, impulse free, and exponentially stable under the switching signal σ (t ) .
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 α >0 and γ >0 , system ( 2.1) is said to be exponentially admissible with a weighted H ∞ performance γ under the switching signal σ (t ) , if it is exponentially admissible with u (t ) =0 and w (t ) =0 , and under zero initial condition, that is, [varphi] ( θ ) =0 , θ ∈ [ - d 2 ,0 ] , for any nonzero w (t ) ∈ L 2 [0 , ∞ ) , it holds that [figure omitted; refer to PDF]
Remark 2.5.
For switched systems with the average dwell time switching, the Lyapunov function values at switching instants are often allowed to increase β times ( β >1 ) 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 H ∞ 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 [figure omitted; refer to PDF] and SOF control law [figure omitted; refer to PDF] where K i and F i , σ (t ) =i , i ∈ ... , 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 γ >0 , identify a class of switching signal σ (t ) 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 H ∞ performance γ under the switching signal σ (t ) .
3. Exponential Admissibility and H ∞ 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 α >0 , 0 ...4; d 1 ...4; d 2 and 0 < μ <1 , if for each i ∈ ... , there exist matrices Q il >0 , l =1,2 ,3 , Z iv >0 , M iv , N iv , S iv , v =1,2 , and P i of the following form: [figure omitted; refer to PDF] with P i11 ∈ R r , P i11 >0 , and P i22 being invertible, such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then, system ( 2.5) with d (t ) satisfying ( 2.2) is exponentially admissible for any switching sequence ...AE; with average dwell time T a ...5; T a * = (ln β ) / α , where β ...5;1 satisfies [figure omitted; refer to PDF] Moreover, an estimate on the exponential decay rate is λ = ( 1 / 2 ) ( α - (ln β ) / T a ) .
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 i ∈ ... , denote [figure omitted; refer to PDF] where A i11 ∈ R r . From ( 3.2), it is easy to see that Φ i11 <0 , i ∈ ... . Noting Q il >0 , l =1,2 ,3 , we get [figure omitted; refer to PDF] Substituting P i and E given as ( 3.1) and ( 2.4) into this inequality yields [figure omitted; refer to PDF] where * denotes a matrix which is not relevant to the discussion. This implies that A i22 , i ∈ ... , 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: [figure omitted; refer to PDF] Then, along the solution of system ( 2.5) for a fixed σ (t ) =i , i ∈ ... , we have [figure omitted; refer to PDF] From the Leibniz-Newton formula, the following equations are true for any matrices N iv , S iv , and M iv , v =1,2 , with appropriate dimensions [figure omitted; refer to PDF] On the other hand, the following equation is also true: [figure omitted; refer to PDF] By ( 3.8)-( 3.11), we have [figure omitted; refer to PDF] where η (t ) = [ x T (t ) x T (t -d (t ) ) x T (t - d 1 ) x T (t - d 2 ) ] T , A ~ i = [ A i A di 0 0 ] , and [figure omitted; refer to PDF] By Schur complement, LMI ( 3.2) implies [figure omitted; refer to PDF] Notice that the last three parts in ( 3.12) are all less than 0. So, if ( 3.14) holds, then [figure omitted; refer to PDF] For an arbitrary piecewise constant switching signal σ (t ) , and for any t >0 , we let 0 = t 0 < t 1 < ... < t k < ... , k =1,2 , ... , denote the switching points of σ (t ) over the interval (0 ,t ) . As mentioned earlier, the i k th subsystem is activated when t ∈ [ t k , t k +1 ) . Integrating ( 3.15) from t k to t k +1 gives [figure omitted; refer to PDF] Let x (t ) = [ x 1 (t ) x 2 (t ) ] , where x 1 (t ) ∈ R r and x 2 (t ) ∈ R n -r . From ( 2.4) and ( 3.1), it can be deduced that for each σ (t ) =i , i ∈ ... [figure omitted; refer to PDF] In view of this, and using ( 3.4) and ( 3.8), at switching instant t i , we have [figure omitted; refer to PDF] where t i - denotes the left limitation of t i . Therefore, it follows from ( 3.16), ( 3.18), and the relation k = N σ ( t 0 ,t ) ...4; (t - t 0 ) / T a that [figure omitted; refer to PDF] According to ( 3.8) and ( 3.19), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Considering ( 3.19) and ( 3.20) yields [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF]
(iii) Exponential stability of algebraic subsystem. Since A i22 , i ∈ ... , is nonsingular, we choose [figure omitted; refer to PDF] Then, it is easy to get [figure omitted; refer to PDF] where A ^ i11 = A i11 - A i12 A i22 -1 A i21 , A ^ i21 = A i22 -1 A i21 , P ^ i11 = P i11 , P ^ i21 = A i12 T P i11 + A i22 T P i21 , and P ^ i21 = A i22 T P i22 . According to ( 3.25), denote [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] where ξ 1 (t ) ∈ R r and ξ 2 (t ) ∈ R n -r . Then, for any σ (t ) =i , i ∈ ... , system ( 2.5) is a restricted system equivalent (r.s.e.) to [figure omitted; refer to PDF] By ( 3.2) and Schur complement, we have [figure omitted; refer to PDF] Pre- and postmultiplying this inequality by diag { H T , H T } and diag {H ,H } , respectively, noting the expressions in ( 3.25) and ( 3.26), and using Schur complement, we have [figure omitted; refer to PDF] Pre- and postmultiplying this inequality by [ - A ^ di22 T I ] and its transpose, respectively, and noting Q ^ i122 >0 , Q ^ i222 >0 , and μ ...5;0 , we obtain [figure omitted; refer to PDF] Then, according to Lemma 5 in Kharitonov et al. [ 37], we can deduce that there exist constants ... i >1 and η i ∈ (0,1 ) such that [figure omitted; refer to PDF] Define [figure omitted; refer to PDF]
Now, following similar line as in Part 3 in Theorem 1 of Lin and Fei [ 31], it can easily be obtained that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] T i k , T i k -1 , ... , T i 0 are positive finite integers, respectively, satisfying [figure omitted; refer to PDF] Combining ( 3.27), ( 3.23) and ( 3.34) yields that system ( 2.5) is exponentially stable for any switching sequence ...AE; with average dwell time T a ...5; T a * =ln β / α . This completes the proof.
Remark 3.2.
Theorem 3.1provides 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 3.1is 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, A i T Z iv A i , A di T Z iv A di , and so forth, are involved, which greatly facilitates the SF and SOF controllers design, as seen in Section 4.
Remark 3.4.
If β =1 in T a ...5; T a * = (ln β ) / α , which leads to P i11 ...1; P j11 , Q il ...1; Q jl , Z iv ...1; Z jv , l =1,2 ,3 , v =1,2 , for all i ,j ∈ ... , and T a * =0 , 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 H ∞ performance of the switched singular time-delay system ( 2.1) with u (t ) =0 .
Theorem 3.5.
For prescribed scalars α >0 , γ >0 , 0 ...4; d 1 ...4; d 2 , and 0 < μ <1 , if for each i ∈ ... , there exist matrices Q il >0 , l =1,2 ,3 , Z iv >0 , M iv , N iv , S iv , v =1,2 , and P i with the form of ( 3.1) such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and Φ i11 , Φ i12 , Φ i13 , Φ i22 , Φ i23 , Φ i33 , Φ i44 , and U i are defined in ( 3.2). Then, system ( 2.1) with u (t ) =0 is exponentially admissible with a weighted H ∞ performance γ for any switching sequence ...AE; with average dwell time T a ...5; T a * = (ln β ) / α , where β ...5;1 satisfying ( 3.4).
Proof.
Choose the piecewise Lyapunov function defined by ( 3.8). Since ( 3.37) implies ( 3.2), system ( 2.1) with u (t ) =0 and w (t ) =0 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 t ∈ [ t k , t k +1 ) , [figure omitted; refer to PDF] where Γ (t ) = z T (t )z (t ) - γ 2 w T (t )w (t ) . This implies that [figure omitted; refer to PDF] By induction, we have [figure omitted; refer to PDF] Under zero initial condition, ( 3.41) gives [figure omitted; refer to PDF] Multiplying both sides of ( 3.42) by e - N α (0 ,t )ln β yields [figure omitted; refer to PDF] Noting that N α (0 ,s ) ...4;s / T a and T a ...5; T a * = (ln β ) / α , we get N α (0 ,s )ln β ...4; αs . Then, it follows from ( 3.43) that ∫ 0 t ... e - α (t -s ) - αs z T (s )z (s )ds ...4; γ 2 ∫ 0 t ... e - α (t -s ) w T (s )w (s )ds . Integrating both sides of this inequality from t =0 to ∞ leads to inequality ( 2.7). This completes the proof of Theorem 3.5.
Remark 3.6.
Note that when β =1 , which is a trivial case, system ( 2.1) with u (t ) =0 achieves the normal H ∞ performance γ 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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 α >0 , γ >0 , 0 ...4; d 1 ...4; d 2 , and 0 < μ <1 , if for each i ∈ ... , and given scalars ... if , f =1,2 , ... ,6 , ... i7 >0 , and ... i8 >0 , there exist matrices R il >0 , l =1,2 ,3 , Z iv >0 , T i , and X i of the following form: [figure omitted; refer to PDF] with X i11 ∈ R r , X i11 >0 , and X i22 being invertible, such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then, there exists an SF controller ( 2.8) such that the closed-loop system ( 4.1) with d (t ) satisfying ( 2.2) is exponentially admissible with a weighted H ∞ performance γ for any switching sequence ...AE; with average dwell time T a ...5; T a * =ln β / α , where β ...5;1 satisfies [figure omitted; refer to PDF] Moreover, the feedback gain of the controller is [figure omitted; refer to PDF]
Proof.
According to Theorem 3.5, the closed-loop system ( 4.1) is exponentially admissible with a weighted H ∞ performance γ if for each i ∈ ... , there exist matrices Q il >0 , l =1,2 ,3 , Z iv >0 , M iv , N iv , S iv , v =1,2 , and P i with the form of ( 3.1) such that inequality ( 3.37) holds with A i and C i instead of A - i and C - i , respectively. By Schur complement, ( 3.37) is equivalent to [figure omitted; refer to PDF] where Φ i12 , Φ i13 , Φ i22 , Φ i23 , Φ i33 , and Φ i44 are defined in ( 3.2), and [figure omitted; refer to PDF] Since P i11 >0 and P i22 is invertible, then P i is invertible. Let [figure omitted; refer to PDF] By ( 3.1), X i has the form of ( 4.3). Pre- and postmultiplying ( 4.8) by diag { X i T , R i3 , R i1 , R i2 ,I ,I ,I ,I ,I ,I ,I } and its transpose, respectively, and noting ( 4.10), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Now, introducing change of variables [figure omitted; refer to PDF] where ... if ,f =1,2 , ... ,6 are scalars, noting the fact that [figure omitted; refer to PDF] where ... i7 and ... i8 are positive scalars, and using Schur complement on ( 4.11), we can easy obtain ( 4.4). In addition, by ( 3.4) and ( 4.10), it is easily to verify that the condition ( 4.6) is equivalent to ( 3.4). This completes the proof.
Remark 4.2.
Scalars ... ih , h =1,2 , ... ,8 , i ∈ ... , in Theorem 4.1are tuning parameters which need to be specified first; such tuning parameters are frequently encountered when dealing with the SF control problem of singular time-delay systems; see, for example, Ma et al. [ 27], Shu and Lam [ 40], and Wu et al. [ 38]. A simple way to choose these tuning parameters is using the trial-and-error method. In fact, ( 4.4) for fixed ... ih , is bilinear matrix inequality (BMI) regarding these tuning parameters. Therefore, if one can accept more computation burden, better results can be obtained by directly applying some existing optimization algorithms, such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm [ 41], and the branch-and-cut algorithm [ 42].
4.2. SOF Controller Design
Connecting the SOF controller ( 2.9) to system ( 2.1) yields the closed-loop system [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The following theorem presents a sufficient condition for solvability of the SOF controller design problem for system ( 2.1).
Theorem 4.3.
For prescribed scalars α >0 , γ >0 , 0 ...4; d 1 ...4; d 2 , and 0 < μ <1 , if for each i ∈ ... , and a given matrix J i , there exist matrices Q il >0 , l =1,2 ,3 , Z iv >0 , and P i of the form ( 3.2) such that [figure omitted; refer to PDF] where c 1 , c 12 , and U i are defined in ( 3.2), and [figure omitted; refer to PDF] Then, there exists an SOF controller ( 2.9) such that the closed-loop system ( 4.15) with d (t ) satisfying ( 2.2) is exponentially admissible with a weighted H ∞ performance γ for any switching sequence ...AE; with average dwell time T a ...5; T a * =ln β / α , where β ...5;1 satisfying ( 3.4).
Proof.
From Theorem 3.5, we know that system ( 4.15) is exponentially admissible with a weighted H ∞ performance γ for any switching sequence ...AE; with average dwell time T a ...5; T a * = (ln β ) / α , where β ...5;1 satisfying ( 3.4), if for each i ∈ ... , there exist matrices Q il >0 , l =1,2 ,3 , Z iv >0 , M iv , N iv , S iv , v =1,2 , and P i with the form of ( 3.1) such that the inequality ( 4.10) with A i and C i instead of A - i [variant prime] and C - i [variant prime] , respectively, holds. By decomposing Φ i in ( 4.10), we obtain that for each i ∈ ... , [figure omitted; refer to PDF] where Λ i is exactly the left half of the inequality ( 4.17), and [figure omitted; refer to PDF] Hence, the condition ( 4.17) implies Φ i <0 . This completes the proof.
Remark 4.4.
Note that there exist product terms between the Lyapunov and system matrices in inequality ( 3.37) of Theorem 3.5, which will bring some difficulties in solving the SOF controller design problem. To resolve this problem, in the proof of Theorem 4.3, we have made a decoupling between the Lyapunov and system matrices by introducing a slack matrix variable J i and then obtained a new inequality ( 4.17). It should be pointed that in Haidar et al. [ 32], a sufficient condition for solvability of the SOF controller design problem for the deterministic singular time-delay system has been proposed. However, the controller gain was computed by using an iterative LMI algorithm, which was complex. Although the new inequality ( 4.17) may be conservative mainly due to the introduction of matrix variable J i , the introduced decoupling technique enables us to obtain a more easily tractable condition for the synthesis of SOF controller.
Remark 4.5.
Matrices J i , i ∈ ... , in Theorem 4.3can be specified by the algorithm stated in Remark 3.6.
Remark 4.6.
In this paper, we have only discussed a special case of the derivative matrix E having no switching modes. If E also has switching modes, then E is changed to E i , i ∈ ... . In this case, the transformation matrices P and Q should become P i and Q i , and we have P i E i Q i =diag { I r i ,0 } . Accordingly, the state of the transformed system becomes x ~ (t ) = [ x ~ i1 T (t ) x ~ i2 T (t ) ] T with x ~ i1 T (t ) ∈ R r i , which means that there does not exist one common state space coordinate basis for all subsystems, and thus it is complicated to discuss the transformed system. Hence, some assumptions for the matrices E i (e.g., E i , i ∈ ... , have the same right zero subspace [ 43]) should be given so that the matrices Q i remain the same; in this case, the method presented here is also valid. However, the general case of E with switching modes is an interesting problem for future investigation via other methods.
5. Numerical Examples
In this section, we present two illustrative examples to demonstrate the applicability and effectiveness of the proposed approach.
Example 5.1.
Consider the switched system ( 2.5) with I =2 (i.e., there are two subsystems) and the related parameters are given as follows: [figure omitted; refer to PDF] and d 1 =0.1 , d 2 =0.3 , μ =0.4 , and α =0.5 . It can be checked that the previous two subsystems are both stable independently. Consider the quadratic approach (see Remark 3.3, β =1 , and we know that it requires a common Lyapunov functional for all subsystems); by simulation, it can be found that there is no feasible solution to this case, that is to say, there is no common Lyapunov functional for all subsystems. Now, we consider the average dwell time scheme, and set β =1.25 , and solving the LMIs ( 3.2) gives the following solutions: [figure omitted; refer to PDF] which means that the aforementioned switched system is exponentially admissible. Moreover, by further analysis, we find that the allowable minimum of β is β min =1.046 when α =0.5 is fixed; in this case, T a * = (ln β min ) / α =0.0899 . By the previous analysis, we know that the average dwell time approach proposed in this paper is less conservative than the quadratic approach.
Example 5.2.
Consider the switched system ( 2.1) with I =2 and [figure omitted; refer to PDF] and d (t ) =0.3 +0.2sin (1.5t ) . A straightforward calculation gives d 1 =0.1 , d 2 =0.5 , and μ =0.3 . By simulation, it can be checked that the previous two subsystems with u (t ) =0 are both unstable, and the state responses of the corresponding open-loop systems are shown in Figures 1and 2, respectively, with the initial condition given by [varphi] (t ) = [ 1 2 ] T , t ∈ [ -0.5 ,0 ] . In view of this, our goal is to design an SF control u (t ) in the form of ( 2.8) and an SOF control u (t ) in the form of ( 2.9), such that the closed-loop system is exponentially admissible with a weighted H ∞ performance γ =1.5 .
For SF control law, set α =0.4 , β =1.05 (thus T a ...5; T a * = (ln β ) / α =0.122 ), and choose ... 11 =0.2 , ... 12 =0.1 , ... 13 =0.1 , ... 14 =0.02 , ... 15 =0.004 , ... 16 =0.03 , ... 17 =1.9 , ... 18 =1 , ... 21 =0.06 , ... 22 =0.1 , ... 23 =0.14 , ... 24 =0.17 , ... 25 =0.1 , ... 26 =0.1 , ... 27 =0.4 , and ... 28 =0.1 . Solving the LMIs ( 4.4), we obtain the following solutions: [figure omitted; refer to PDF] Therefore, from ( 4.7), the gain matrices of an SF controller can be obtained as [figure omitted; refer to PDF]
For SOF control law, let α , β be the same as in the SF control case, and choose [figure omitted; refer to PDF] By solving the LMIs ( 4.17), we obtain the following solutions: [figure omitted; refer to PDF] To show the effectiveness of the designed SF and SOF controllers, giving a random switching signal with the average dwell time T a ...5;0.13 as shown in Figure 3, we get the state responses using the SF and SOF controllers for the system as shown in Figures 4and 5, respectively, for the given initial condition [varphi] (t ) = [ 1 2 ] T , t ∈ [ -0.5 ,0 ] . It is obvious that the designed controllers are feasible and ensure the stability of the closed-loop systems despite the interval time-varying delays.
Figure 1: The state trajectories of the open-loop subsystem 1.
[figure omitted; refer to PDF]
Figure 2: The state trajectories of the open-loop subsystem 2.
[figure omitted; refer to PDF]
Figure 3: Switching signal with the average dwell time T a >0.13 .
[figure omitted; refer to PDF]
Figure 4: The state trajectories of the closed-loop system under SF control.
[figure omitted; refer to PDF]
Figure 5: The state trajectories of the closed-loop system under SOF control.
[figure omitted; refer to PDF]
6. Conclusions
In this paper, the problems of exponential admissibility and H ∞ control for a class of continuous-time switched singular systems with interval time-varying delay have been investigated. A class of switching signals specified by the average dwell time has been identified for the unforced systems to be exponentially admissible with a weighted H ∞ performance. The state feedback and static output feedback controllers have been designed, and their corresponding solvability conditions have been established by using the LMI technique. Simulation results have demonstrated the effectiveness of the proposed design method.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 60904020, Natural Science Foundation of Jiangsu Province of China (no. BK2011253), Open Fund of Key Laboratory of Measurement and Control of CSE (no. MCCSE2012A06), Ministry of Education of China, Southeast University, and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (no. NY210080).
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Copyright © 2012 Jinxing Lin and Zhifeng Gao. Jinxing Lin et al. 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 deals with the problems of exponential admissibility and [subscript] H ∞[/subscript] control for a class of continuous-time switched singular systems with time-varying delay. The [subscript] H ∞[/subscript] 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 [subscript] H ∞[/subscript] 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.
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