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

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.


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 statespace 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][4][5][6][7][8][9][10][11][12][13][14][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.
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][4][5][6][7][8][9][10][11][12][13][14][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, R  denotes the -dimensional Euclidean space with vector norm ‖ ⋅ ‖, R × is the set of all  ×  real matrices, and the notation  > 0 means that  is symmetric and positive definite.The symmetric elements of the symmetric matrix will be denoted by * .

Preliminaries and Problem Formulation
Consider the following linear singular system with state delay and parameter uncertainties described by where () ∈ R  is the state vector; () ∈ R  is the control input vector; () ∈ R  is the disturbance input which belongs to L 2 [0, ∞), and () ∈ R  is the controlled output.The matrix  ∈ R × may be singular.We will assume that rank() =  ≤ .The matrices  0 ,  1 , ,  0 ,  1 , , and  are known real constant matrices with appropriate dimensions. > 0 is a constant time delay.() is a compatible vector valued initial function.Δ 0 , Δ 1 , Δ, Δ 0 , Δ 1 , and Δ are time-invariant matrices representing norm-bounded parameter uncertainties and are assumed to be of the form where  1 ,  2 ,  1 ,  2 , and  3 are constant matrices and  is an unknown real matrix satisfying The parameter uncertainties Δ 0 , Δ 1 , Δ, Δ 0 , Δ 1 , and Δ are said to be admissible if both (2) and (3) hold.
The nominal singular time-delay system of Σ 1 with () = 0 can be written as Throughout the paper, we will adopt the following definition.
(1) The pair (,  0 ) is said to be regular if det( −  0 ) is not identically zero.
Lemma 2 (see [3]).Suppose the pair (,  0 ) is regular and impulsive free, then the solution to unforced systems Σ 0 exists and is impulse free and unique on [0, ∞).
Remark 3. The regularity of pair (,  0 ) guarantees the existence and uniqueness of solution for system Σ 0 ; 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 Σ 0 .
Definition 4 (see [3]).(1) The singular time-delay system Σ 0 is said to be regular and impulse free, if the pair (,  0 ) is regular and impulse free.
(2) The singular time-delay system Σ 0 is said to be stable if, for any  > 0, there exists a scalar () > 0, such that for any compatible initial function () satisfies sup −≤≤0 ‖()‖ ≤ (), and the solutions () of system Σ 0 satisfy ‖()‖ ≤  for  ≥ 0. Furthermore, () → 0 as  → ∞.Definition 5.The uncertain singular time-delay system Σ 1 is said to be robustly stable if the unforced system of Σ 1 is regular, impulse free, and stable for all admissible uncertainties.Definition 6.For a given scalar  > 0, the uncertain singular time-delay system Σ 1 with () = 0 is said to be robustly stable with  ∞ performance , if it is robustly stable and under zero initial condition ‖()‖ 2 < ‖()‖ 2 is satisfied for any nonzero () ∈ L  2 [0, ∞) and all admissible uncertainties.The aim of this paper is for prescribed scalars  > 0 and  > 0 to develop a state feedback controller such that closed-loop system is robustly stable with  ∞ performance , where  0 and  1 are matrices to be determined.
We conclude this section by presenting the following lemma, which is extensively used in uncertain system research.

Main Results
First of all, we present delay-dependent result that assures the nominal unforced singular system Σ 0 to be regular, impulse free, and stable with  ∞ performance , which will play a key role in solving the  ∞ control problem.

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 Ẇ() = D(), where . (11) 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  > 0, the system Σ 0 is regular, impulse free, and stable with  ∞ performance ; suppose that there exist symmetric positive definite matrices  ∈ R × ,  ∈ R × , T ∈ R × , and R ∈ R × and matrices hold, where  ∈ R ×(−) is any matrix with full column rank and satisfies    = 0 and Proof.Since rank() =  ≤ , there exist nonsingular matrices  and  such that  =  = [   0 0 0 ].Noting this and    = 0, rank() =  − , we can show that  can be parameterized as  =   [ 0  ], where  ∈ R (−)×(−) is any nonsingular matrix.Accordingly, we define the following transformations: Then, pre-and postmultiplying Ψ 11 by   and  get which obviously implies that  04 is nonsingular.Then we can deduce system Σ 0 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.
Remark 10.When  is nonsingular, the singular system Σ 0 reduces to a state-space system.In this case, if we choose  = 0 in the proof of Theorem 8, we can derive a new version of bounded real lemma for state-space system.

Robust 𝐻 ∞ 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 closedloop system is regular, impulse free, and stable with  ∞ performance .
Remark 14.In the case where (−) is available for feedback, the use of  0 and  1 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  1 = 0 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 Σ 1 .The closed-loop singular system of Σ 1 can be written as Theorem 15.For prescribed scalar  > 0, the closed-loop singular system Σ 0 is regular, impulse free, and stable with  ∞ performance ; suppose that there exist symmetric positive definite matrices  ∈ R × , T ∈  × , R ∈  × and matrices  ∈ R × , Q ∈  × ,   ∈ R × ,  = 0, 1 and scalars   > 0, ( = 1, 2),  > 0 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.For given  > 0, 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.

Examples
Tables 2 and 3 give the comparison results on the maximum allowed delay  ⋆ for given  and the minimum allowed  for given  > 0, 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 Σ 0 with the following parameters:  For a given delay  = 1.2, 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 (45) 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.

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][20][21][22].

Figure 1 :
Figure 1: The state response of the closed-loop system in Example 3.

Table 1 :
Comparisons of maximum allowed delay  ⋆ for Example 1.

Table 2 :
Comparisons of maximum allowed delay  ⋆ for Example 2.

Table 3 :
Comparisons of minimum allowed  for Example 2.

Table 4 :
Comparisons of minimum  ∞ performance index  ⋆ for Example 3.