Mathematical Problems in Engineering

Volume 2014, Article ID 790521, 9 pages

http://dx.doi.org/10.1155/2014/790521

## Robust Control for Singular Time-Delay Systems via Parameterized Lyapunov Functional Approach

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China

Received 22 July 2014; Revised 18 September 2014; Accepted 23 September 2014; Published 27 November 2014

Academic Editor: Mathiyalagan Kalidass

Copyright © 2014 Li-li Liu 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

A new version of delay-dependent bounded real lemma for singular systems with state delay is established by parameterized Lyapunov-Krasovskii functional approach. In order to avoid generating nonconvex problem formulations in control design, a strategy that introduces slack matrices and decouples the system matrices from the Lyapunov-Krasovskii parameter matrices is used. Examples are provided to demonstrate that the results in this paper are less conservative than the existing corresponding ones in the literature.

#### 1. Introduction

During the last decade, a considerable amount of attention has been paid to stability and control of singular systems with delay, since they can better describe and analyze physical systems than the state-space time-delay ones [1, 2]. It should be pointed out that the control problems for singular systems are much more complicated than that for state-space systems, because singular systems usually have three types of modes, namely, finite dynamic modes, impulse modes, and nondynamic modes, whereas the latter two do not appear in state-space systems [1]. Recently, the problem of control for singular systems with delay has been investigated by many researchers and various approaches have been adopted [3–15]. There are two performance indexes which are used to judge the conservatism of the derived conditions. One is the performance index while the other is admissible upper bound of the delay. For a given time-delay, the smaller the performance index, the better the conditions; for a prescribed performance level, the larger the admissible upper bound of the time-delay, the less conservative the conditions.

It is well-known that Lyapunov-Krasovskii theorems are basic theories for the study of all types of time-delay system [3–22]. The choice of appropriate Lyapunov-Krasovskii functional (LKF) is crucial for obtaining stability criteria and bounded real lemmas (BRLs) and, as a result, for obtaining solutions to various control problems. In [3–5], the simple form LKFs containing definition integral terms were used to obtain the delay-independent bounded real lemma for singular systems with state delay. Generally speaking, delay-dependent conditions are less conservative than the delay-independent ones, especially when the size of delay is small. To obtain delay-dependent conditions, many efforts have been made in the literature [6–14]. Improved LKFs containing double integral terms were given in [6–13]. Along the system trajectory, there will be definition integral term in , which causes an unwieldy question: how to deal with it to get less conservative conditions. Some strategies are usually adopted to solve the problem such as different classes of model transformations and over-bounding cross terms [6–8]; free weighting matrix approach [9–12]; delay fractioning technique [13, 14]; integral inequalities approach [15]. However, there are no obvious ways to obtain less conservative results even if one is willing to commit more computational effort to the problem and to find a more tighter bounding for cross terms. This is the serious limitation for these criteria. To overcome the limitation, one has to find some more general LKFs to handle the stability problem for singular systems.

On the other hand, LMI stability conditions via complete quadratic LKF and discretization were introduced by Gu in [17] for time delay system and appeared to be very efficient, leading in some examples to results close to analytical ones. The discretized LKF method has been extended to singular time-delay systems in [16]. An improved bounded real lemma (BRL) is presented by discretization LKF method which greatly lowers the conservatism, but the initial parameter is needed to introduce controller design. To further improve the conclusions, the control for singular time-delay systems is proceeded with studies by parameterized LKF approach.

The contribution of this paper lies in three aspects. First, a singular-type complete quadratic Lyapunov-Krasovskii functional is introduced in which the quadratic weighting matrices are defined using matrix polynomial functions. This class of LKF covers those considered in [3–15] as special cases, where the quadratic weighting matrices are defined by constant matrices. Second, a delay-dependent BRL is presented to ensure the system to be regular, impulse free, and stable with prescribed performance level, which greatly lower the conservatism. Third, the free weighting matrix approach is employed to decouple the system matrices from the weighting matrices of LKF, which urge that the robust control problem is solved and an explicit expression of the desired state-feedback control law is also derived.

*Notation*. Throughout this note, the superscript “” stands for matrix transposition, denotes the -dimensional Euclidean space with vector norm , is the set of all real matrices, and the notation means that is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by . is the space of square integrable functions with the norm .

#### 2. Preliminaries and Problem Formulation

Consider the following linear singular system with state delay and parameter uncertainties described by where is the state vector; is the control input vector; is the disturbance input which belongs to , and is the controlled output. The matrix may be singular. We will assume that . The matrices , , , , , , and are known real constant matrices with appropriate dimensions. is a constant time delay. is a compatible vector valued initial function. , , , , , and are time-invariant matrices representing norm-bounded parameter uncertainties and are assumed to be of the form where , , , , and are constant matrices and is an unknown real matrix satisfying The parameter uncertainties , , , , , and are said to be admissible if both (2) and (3) hold.

The nominal singular time-delay system of with can be written as Throughout the paper, we will adopt the following definition.

*Definition 1 (see [1]). * The pair is said to be regular if is not identically zero.

The pair is said to be impulse free if .

Lemma 2 (see [3]). *Suppose the pair is regular and impulsive free, then the solution to unforced systems exists and is impulse free and unique on .*

*Remark 3. *The regularity of pair guarantees the existence and uniqueness of solution for system ; impulse terms are generally not expected to appear since strong impulse behavior may stop the system from work or even destroy it.

In view of this, we introduce the following definition for singular time-delay system .

*Definition 4 (see [3]). * The singular time-delay system is said to be regular and impulse free, if the pair is regular and impulse free.

The singular time-delay system is said to be stable if, for any , there exists a scalar , such that for any compatible initial function satisfies , and the solutions of system satisfy for . Furthermore, as .

*Definition 5. *The uncertain singular time-delay system is said to be robustly stable if the unforced system of is regular, impulse free, and stable for all admissible uncertainties.

*Definition 6. *For a given scalar , the uncertain singular time-delay system with is said to be robustly stable with performance , if it is robustly stable and under zero initial condition is satisfied for any nonzero and all admissible uncertainties.

The aim of this paper is for prescribed scalars and to develop a state feedback controller such that closed-loop system is robustly stable with performance , where and are matrices to be determined.

We conclude this section by presenting the following lemma, which is extensively used in uncertain system research.

Lemma 7 (see [23]). *For appropriate dimensional matrices , and symmetric matrix , all the satisfied ,
**
if and only if there exists a constant such that
*

*3. Main Results*

*First of all, we present delay-dependent result that assures the nominal unforced singular system to be regular, impulse free, and stable with performance , which will play a key role in solving the control problem.*

*3.1. A New Version of Delay-Dependent Bounded Real Lemma*

*We introduce a singular-type complete quadratic Lyapunov-Krasovskii functional
where and are matrix polynomial functions in the form of
where
*

*Noting that the differentiation of the partitioned matrix , where
*

*Based on it, the differentiation of the functions , , and is given by
A new version of bounded real lemma is obtained via the singular-type complete quadratic Lyapunov-Krasovskii functional (8) with polynomial parameter (9).*

*Theorem 8. For a given scalar , the system is regular, impulse free, and stable with performance ; suppose that there exist symmetric positive definite matrices , , , and and matrices , , , , and such that
hold, where is any matrix with full column rank and satisfies and
*

*Proof. *Since , there exist nonsingular matrices and such that . Noting this and , , we can show that can be parameterized as , where is any nonsingular matrix. Accordingly, we define the following transformations:
Then, pre- and postmultiplying by and get
which obviously implies that is nonsingular. Then we can deduce system is regular and impulse free. Here, the terms denoted are irrelevant to the results of the above discussion, so the real expression of these variables is omitted.

Next, we will show the nominal singular system is stable and has the performance . Consider the functional (8) with the functions , , and as in (9), and define the vector ; then
Denote ; the functional (8) satisfies
If , , and , then is positive definite. The derivation of along system gives
As , Jensen’s inequality ensures that

On the other hand, for any appropriate dimensional matrices , , and , the following equation is true:

Hence
where . Under the zero initial condition, it can be shown that, for any nonzero , the following index is
where

Thus, implies that ; that is, for any nonzero . Moreover, similar to [16], the feasibility of LMIs (13), (14), and
implies ; then the system with is stable. This completes the proof.

*Remark 9. *In Theorem 8, matrices are introduced to decouple the system matrices , from the LKF weighting matrices and , which provides convenience for controller design.

*Remark 10. *When is nonsingular, the singular system reduces to a state-space system. In this case, if we choose in the proof of Theorem 8, we can derive a new version of bounded real lemma for state-space system.

*Based on Theorem 8, we obtain the following result on performance analysis for the uncertain singular system with .*

*Theorem 11. For a given scalar , the system is regular, impulse free, and stable with performance ; suppose that there exist symmetric positive definite matrices , , , and and matrices , , , , and and scalar such that (13), (14), and
hold, where , , and .*

*Proof. *Suppose that there exist symmetric positive definite matrices , , , and and matrices , , , , and and scalar such that (13), (14), and (28) hold. Then, by Schur complement, it follows from (28) that
where
By Lemma 7, , which is in the form of by replacing , , , and with , , , and , respectively. Thus, by Theorem 8, we have that the uncertain singular system with is robustly stable with performance .

*3.2. Robust Controller Design*

*3.2. Robust Controller Design*

*In this subsection, we will apply the bounded real lemma obtained above to design the state feedback controller (5) such that the resultant closed-loop system is regular, impulse free, and stable with performance .*

*The nominal closed-loop singular system of can be written as
*

*Theorem 12. For prescribed scalar , the closed-loop singular system is regular, impulse free, and stable with performance ; suppose that there exist symmetric positive definite matrices , , and and matrices , , and , and scalars , such that
hold. Then a suitable state-feedback control law is given by
where
*

*Proof. *According to Theorem 8, the closed-loop singular system is regular, impulse free, and stable with performance , if there exist symmetric positive definite matrices , , and and matrices , , , , , and such that (13), (14), and hold, where is in the form of with replaced by , , , and . Let , , , and . Pre- and postmultiply by and its transpose; pre- and postmultiply by and its transpose; pre- and postmultiply by and its transpose. We introduce the following:
After some manipulation, we can obtain (32), (33), and (34), and the desired controller gains are given by , .

*Remark 13. *In this case, we can assume that the matrix is nonsingular. If this is not the case, we can choose some such that is nonsingular and satisfies (32) and (34), in which is any nonsingular matrix satisfying .

*Remark 14. *In the case where is available for feedback, the use of and can lead to a significative reduction on the values of . On the other hand, when is not available for feedback, a memoryless control law is required; it is enough to set in LMI (34).

*Now we are in a position to present the result on the problem of delay-dependent robust control for the uncertain singular system . The closed-loop singular system of can be written as
*

*Theorem 15. For prescribed scalar , the closed-loop singular system is regular, impulse free, and stable with performance ; suppose that there exist symmetric positive definite matrices , , and matrices , , , and scalars , , such that (32), (33) and
hold. Then a suitable state-feedback control law is given by
where
*

*
The proof can be carried out by resorting to Theorem 12 and following a similar line as in the proof of Theorem 11 and is thus omitted.*

*4. Examples*

*4. Examples**Example 1. *Consider a singular time-delay system in the form of with

*
For given , we can calculate the maximum allowed delay satisfying the LMIs in Theorem 8. To show the low conservativeness of the result, we compare ours with the criteria of [6, 11, 16] in Table 1. It is clear that the characterization of bounded realness in this paper is an improvement over the previous ones.*

*Example 2. *Consider the uncertain singular time-delay system with , where

*Tables 2 and 3 give the comparison results on the maximum allowed delay for given and the minimum allowed for given , respectively. It is clear that the results of Theorem 11 are significantly better than those in [10].*

*Example 3. *We consider the problem of control for the singular time-delay system with the following parameters:

*
For a given delay , Table 4 provides the comparison results on the minimum performance index for given delay via the methods in [6, 11, 12, 15, 16] and Theorem 12 in this paper, which shows that Theorem 12 in this paper can lower the performance index. The state feedback controller achieving the minimum performance level can be obtained as
The state response of the closed-loop system is shown in Figure 1. We can see that the state responses are converging. The controlled output of the closed-loop system is shown in Figure 2.*

*5. Conclusions*

*5. Conclusions**The problem of delay-dependent robust control for singular time-delay system with admissible uncertainties has been investigated by parameterized LKF method. A new version of bounded real lemma is presented to ensure the system to be regular, impulse free, and robustly stable with performance condition. The slack matrices are suitably introduced to decouple the systems matrices from the LKF parameter, so that the controller law can be derived directly. Three examples are given to illustrate the effectiveness of our method and the improvement over some existing ones. As a future research direction, it would be of interest to apply the parameterized LKF method in passivity and filtering for neutral systems and singularly perturbed systems [19–22].*

*Conflict of Interests*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments**This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant GK201102023 and the NSFC under Grant 11201277.*

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