Robust Stability and Stabilization for Singular Time-Delay Systems with Linear Fractional Uncertainties: A Strict LMI Approach

This paper is concerned with the problems of delay-dependent robust stability and stabilization for a class of continuous singular systems with time-varying delay in range and parametric uncertainties. The parametric uncertainties are assumed to be of a linear fractional form, which includes the norm bounded uncertainty as a special case and can describe a class of rational nonlinearities. In terms of strict linear matrix inequalities (LMIs), delay-range-dependent robust stability criteria for the unforced system are presented. Moreover, a strict LMI design approach is developed such that, when the LMI is feasible, a desired state feedback stabilizing controller can be constructed, which guarantees that, for all admissible uncertainties, the closed-loop dynamics will be regular, impulse free, and robustly asymptotically stable. Numerical examples are provided to demonstrate the effectiveness of the proposed methods.


Introduction
Singular time-delay systems, which are known as descriptor time-delay systems, implicit time-delay systems, or generalized differential-difference equations, often appear in various engineering systems, for example, aircraft attitude control, flexible arm control of robots, large-scale electric network control, chemical engineering systems, lossless transmission lines, and so forth [1,2].Since singular time-delay systems are matrix delay differential equations coupled with matrix difference equations, the study for such systems is much more complicated than that for standard state-space time-delay systems.Recently, a great deal of attention has been devoted to the study of such more general class of delay systems; see .
The existing stability criteria for singular time-delay systems can be classified into two types: delay independent [3][4][5] and delay dependent [6][7][8][9][10].Generally, delay-dependent conditions are less conservative than the delay-dependent ones, especially when the time delay is small.To obtain delaydependent conditions, many efforts have been made in the literature, among which the model transformation and bounding technology for cross-terms [8][9][10] are often used.However, it is known that the bounding technology and the model transformation are the main source of conservation [28].Recently, some improved stability conditions with less conservatism have been provided by utilizing the free weighting matrix method [11][12][13], the integral inequality [14], and the delay decomposition approach [15][16][17], in which neither the bounding technology nor model transformation is involved.However, these conditions in [6][7][8][9][10][11][12][13][14][15][16][17] were established under the assumption that the delay was time invariant.For the continuous singular systems with time-varying delay, Yue and Han investigated the delay-dependent stability condition by introducing the free weighting matrices [18].In [19], a delay-dependent stability condition was presented by using the integral inequality method.But the range of the time-varying delay considered in [18,19] is from 0 to an upper bound.In practice, a time-varying interval delay is often encountered; that is, the range of delay varies in an interval for which the lower bound is not restricted to 0. In this case, the stability criteria in [18,19] are conservative Mathematical Problems in Engineering because they do not take into account the information of the lower bound of delay.Moreover, when estimating the upper bound of the derivative of Lyapunov functional, some useful terms are ignored in [18,19].More recently, continuous singular systems with time-varying delay in a range have been extensively studied; see, for example, [20][21][22][23][24][25][26][27] and references therein.
On the other hand, in recent years, more and more attention has been devoted to derive strict LMI conditions for stability analysis and controller design; see, for example [29,30] and references therein.The strict LMI conditions, that is, definite LMIs without equality constraints, are highly tractable and reliable when checked by some recently developed algorithms for solving LMIs [31].However, it should be pointed that the stability conditions derived in [20][21][22][23][24][25][26][27] are formulated in terms of nonstrict LMIs, whose solutions are difficult to calculate since equality constraints are often fragile and usually not met perfectly.Furthermore, up to now, to the best of the authors' knowledge, for a continuous uncertain singular system with a time-varying interval delay, the problems of robust stability, stabilization, and feedback control have not been fully investigated yet [23].Particularly, strict LMI-based condition has never been reported in the published works.
In this paper, by using a strict LMI approach, we study the robust stability and stabilization problems for a class of singular systems with a time-varying interval delay and uncertainties.Different from the existing results in [13,19,21,23], first, the criteria proposed in our paper do not contain any semidefinite matrix inequality and are expressed as strict LMIs.Second, the new criteria are obtained by only using a well-known integral-inequality and do not employ any freeweighting matrix, which makes our methods more efficient.Third, a new type of uncertainty, namely, linear fractional form, is considered in this paper.Three numerical examples are given to illustrate the effectiveness of the presented method.
Definition 4 (see [3]).The singular system (6) is said to be regular and impulse free, if the pair (, ) is regular and impulse free.
The objective of this note is to develop delay-rangedependent robust stability conditions for system (2) with () = 0 and to design a state-feedback controller so that system (2) is closed-loop regular, impulse-free, and robustly asymptotically stable for admissible linear fractional form uncertainties. To this end, the following lemmas are needed.

Stability Issue
In this section, first of all, we will present new delay-rangedependent stability conditions that guarantee system (6) to be regular, impulse free, and asymptotically stable in terms of LMI, which will play a key role in obtaining the robust stability criterion for the uncertain system (2).
Proof.Since rank  =  ≤ , there must exist two invertible matrices  and  such that Then,  can be parameterized as where Φ ∈ R (−)×(−) is any nonsingular matrix.Similar to (13), we define Since Λ 11 < 0 and   > 0,  = 1, 2, 3, we can formulate the following inequality easily: Pre-and postmultiplying Ψ < 0 by  T and , respectively, yields where Ψ 11 and Ψ 12 represent the matrices not relevant in the following discussion.From (17), it is easy to see that which gives that  22 is nonsingular.Define ] .(19) After some algebraic manipulations, we can obtain which implies that det( − ) is not identically zero and deg(det( − )) =  = rank .Then, the pair of (, ) is regular and impulse-free, which shows that system (6) is regular and impulse-free.In the following, we will prove that system (6) is also asymptotically stable.Denote By using Schur complement and noting that   > 0,  = 1, 2,  < 1, it follows from (11) that where Pre-and post-multiplying ( 23) by diag{ HT , HT } and diag{ H, H}, respectively, yields where Substituting ( 20), ( 22) into (25), we have Pre-and post-multiplying (27 and  T , respectively, yields where Therefore, Now, let where x1 () ∈ R  and x2 () ∈ R − .Using the expressions in (20), (22), and (31), system (6) can be decomposed as or equivalently rewritten as It is easy to see that the stability of system ( 6) is equivalent to that of system (34).Construct the Lyapunov-Krasovskii functional for system (34) as By Lemma 6, the following inequalities are true x ( −  ()) ) . ( On the other hand, noticing that  T  = 0, we can deduce that where S is any matrix with appropriate dimensions. Taking the derivative of Ṽ( x ) with respect to  along the trajectory of system (34) and using ( 36) and ( 37), we have where with Mathematical Problems in Engineering It is easy to see that (17) guarantees V( x ) < 0 and where Taking into account (41), we can deduce that Therefore, where  1 = (1/ 1 ) Ṽ( x(0)) > 0, Thus, according to Definition 5, system (34) is stable.This completes the proof.
Remark 11.From the proof of Theorem 10, it is clear to see that neither model transformation nor bounding technique for cross-terms is involved.Hence, the conservatism inherited from these ideas will no longer exist in Theorem 10.
Remark 12. Free-weighting matrices in [11-13, 22, 23, 25] plays an important role to reducing the conservatism of delaydependent stability conditions.However, too many freeweighting matrices will complicate the system analysis and increase the computational demand.It is worth pointing out that no free-weighting matrix is involved in Theorem 10.
Theorem 10 presents a delay-range-dependent criterion for system (6) with time-varying delay () in a range.If we set  1 = 0 and  2 = 0, Theorem 10 yields the following delaydependent stability criterion.Corollary 14.Given scalars ℎ 2 > 0, ℎ 1 = 0 and , system (6) is regular, impulse free, and asymptotically stable if there exist positive-definite matrices ,   ,  = 2, 3,  1 , and matrix  with appropriate dimensions such that where and  ∈ R ×(−) is any matrix with full column rank and satisfies    = 0. Now, we will present the delay-range-dependent robust stability conditions for the uncertain singular time-delay system (2) with () = 0 via Theorem 10.

Control Design
On the basis of the previous stability conditions, we will present a design method of robustly stabilizing controllers in this section.For simplicity, we first consider system (6).
Theorem 16.Given scalars 0 ≤ ℎ 1 < ℎ 2 and , if there exist scalar  > 0, positive-definite matrices ,   ,  = 1, 2, 3,   ,  = 1, 2, , and matrices , ,  with appropriate dimensions such that where and  1 ∈ R ×(−) is any matrix with full column rank and satisfying    1 = 0, then there exists a state feedback controller (7) such that the resulting closed-loop system of system (6) is regular, impulse free, and asymptotically stable.In this case, a suitable controller gain is given by Proof.With the control law () = (), the resultant closed-loop system of system ( 6) is Following the same philosophy as that in [37], we represent system (56) as the following form, where () =  ẋ ().

Numerical Examples
In this section, some examples are provided to illustrate the benefits of our results.
Example 1.Consider the nominal unforced part of system (2) with The case for ℎ 1 = 0 and  = 0 has been studied in [12].We choose  = [0 1] T .The comparison among Corollary 14 in this note and those in [6-9, 11, 12, 19] is listed in Table 1 for ℎ 1 = 0.It should be pointed out that the results of [7] fail to deal with the system because the matrix describing the relationship between the fast and slow variables cannot be chosen beforehand.It can be seen that our method is less conservative than those in [6,9] and gives the same results as that in [8,11,12,19].However, when the time delay is a varying delay, our method gives better results than that in [19] for  = 0.3 and  = 0.75 since the relationship among ℎ 2 , (), ℎ 2 − (), and () − ℎ 1 has been taken into account in Corollary 14.On the other hand, for system with time-varying delay in a range, Table 1 also lists the allowable maximum upper bounds of ℎ 2 for different  with ℎ 1 = 0.3 and ℎ 1 = 0.6.
We choose  = [0 1] T .For ℎ 1 = 0, the comparison among Theorem 15 in this note and that in [19] is listed in Table 2.It is clear that the result in this note is better than that in [19].The corresponding maximum upper bounds of ℎ 2 for different ℎ 1 and  derived by Theorem 15 are also listed in Table 2.
We choose  = [−1 1 2] T .According to Theorem 17, Table 3 shows the allowed maximum upper bounds of ℎ 2 and the corresponding state feedback gain  for different ℎ 1 .

Conclusions
In this note, the delay-range-dependent robust stability and stabilization for singular time-delay systems with linear fractional uncertainty and time-varying delay in a range are studied.The results are obtained by using the strict LMI approach and constructing an appropriate Lyapunov-Krasovskii functional.Numerical examples have been given to demonstrate the effectiveness of the presented criteria and their improvement over existing results.

Example 2 .
Consider the unforced part of uncertain system (

Table 1 :
Allowable upper bounds of ℎ 2 with given ℎ 1 for different .

Table 2 :
Allowable upper bounds of ℎ 2 with given ℎ 1 for different .