Admissibility Analysis and Controller Design for Discrete Singular Time-Delay Systems Embracing Uncertainties in the Difference and Systems’ Matrices

This paper mainly investigates the admissibility analysis and the admissibilizing controller design for the uncertain discrete singular system with delayed state. Based on Lyapunov–Krasovskii stability theory, an original admissibility condition for the nominal singular delay system is ﬁrst presented. By involving the uncertainties in both diﬀerence and system matrices simultaneously, we devote to analyzing the robust admissibility for the regarded uncertain discrete singular system with delayed state. Furthermore, by hiring the state feedback control law, we further discuss the admissibilizing controller design for the resulting closed-loop system. Since all the derived criteria are expressed in terms of strict linear matrix inequalities (LMIs) or parametric LMIs, we thus can handily verify them via current LMI solvers. Finally, two numerical examples are given to illustrate the eﬀectiveness and validity of the proposed approach.


Introduction
During recent years, the singular systems have received extensive attention because they can represent a wider class of physical systems, including physical models and nondynamic constraints, than traditional state-space systems [1]. ey can represent large-scale systems [2], economic systems [3,4], electrical network analysis [5], and so on. Singular systems are also referred to as descriptor systems, implicit systems, generalized state-space systems, differential-algebraic systems, or semistate systems [2,[6][7][8]. Recently, robust admissibility and robust controller synthesis for uncertain singular systems have been addressed. e corresponding stability issues are more complicated than the traditional state-space system because they require considering not only the stability but also the regularity and causality simultaneously (see, for example, [9][10][11][12][13][14][15][16][17][18] and the references therein).
Furthermore, time-delayed state often inevitably exists in many practical systems, such as remote-controlled robots, chemical processes, and electrical network grids. ey are the main sources of instability, oscillations, or degraded performances [19][20][21][22][23]. erefore, the problems of stability testing and stabilization of uncertain singular systems with delay states have aroused much attention over the recent years, e.g., [24][25][26][27][28][29][30][31][32][33][34][35][36]. For discrete-time systems, Xu et al. [24] proposed specific D-stability issues for uncertain discrete singular systems with state delay. Feng et al. [30] discussed the admissibility for discrete singular systems with timevarying delay and attained the admissibility criterion with strict linear matrix inequalities (LMIs). Also, the robust admissibility and admissibilization of uncertain discrete singular time-delay systems were further investigated [31]. In [32], novel delay-dependent sufficient conditions for the robust finite-time stability of the system are presented. en, the results can be applied to solve the finite-time stabilization problem of linear singular discrete-time control delay systems. In [33], based on the equivalent restrict decomposition form, the iterative learning control problem is studied for a class of discrete singular time-delay systems. Also, by the LMI approach, an efficient observer design method is presented for singular discrete-time systems with time delays and nonlinearity [34]. In [36], this work proposes a novel LMI-based global sliding mode control approach for uncertain discrete-time descriptor systems with multiple timevarying delays. However, in previous works, they all focus on the uncertain singular systems with a constant difference matrix or the difference matrix satisfying a prescribed form. For modeling singular systems from physical systems, if uncertainties exist in a structure or a behavior, they are inevitably cast into the systems' matrices and the derivative matrix (continuous systems) or the difference matrix (discrete systems) simultaneously [37][38][39][40]. To the best of our knowledge, the discrete singular delay systems with the uncertainties existing in both difference matrix E and the system matrices seem little to be discussed in the past literature studies.
In this work, we devote to analyzing the robust admissibility and the admissibilizing controller design for the discrete delayed singular system with the perturbed difference and system matrices simultaneously. Based on the Lyapunov stability theory, an admissibility condition is first derived for the nominal singular system with delayed state.
us, the robust admissibility criteria can be attained for the system with the allowed uncertainties in both difference and systems' matrices. Furthermore, when hiring the state feedback control law, we further address controller design criteria for the resulting closed-loop systems. For numerical evaluation, all proposed criteria are represented by strict or parametric LMIs [41], and we can readily verify them via a current LMI solver [42]. At length, two numerical examples are given to illustrate the validity and practicability of the proposed approach. Compared with previous works, the main contributions of this paper are highlighted as follows: (i) In the past, it seems that there are few results on analyzing the robust admissibility and the admissibilizing controller design for the discrete delayed singular system with embracing the uncertainties in both difference and system matrices simultaneously. is work mainly copes with the admissibility analysis and admissibilizing controller design for the regarded system with the perturbed derivative matrix.
(ii) All the proposed admissibility analyses and controller design criteria are all explicitly formulated in terms of LMIs or parametric LMIs, so we can readily verify them via LMI solvers for the admissibility assurance or systematically performing a controller design. e validity and efficiency of the proposed approach are demonstrated by illustrative examples.

Problem Formulation and Preliminaries
Consider a discrete uncertain singular delay system described by where x(·) ∈ R n is the state vector and u(·) ∈ R p is the control input; the perturbed derivative matrix E ∈ R n×n may be singular, i.e., rank(E) � m ≤ n, and belongs to a polytopic set defined as where E i ∈ R n×n given a priori. A, A d , and B stand for the nominal system matrices with appropriate dimensions, and ΔA, ΔA d , and ΔB represent the parameter uncertainties bounded by where M, N A , N Ad , and N B are constant matrices with compatible dimensions and D satisfies D T D ≤ I Remark 1 (see [40]). For system's model (1), the presented uncertainties in the derivative matrix and the system matrices are formulated by different forms. For mathematical modeling from a physical system, if there exists parametric perturbation within its inner structures or behaviors, the main parametric uncertainties can usually be cast into system matrices, and the remaining few uncertainties are thus cast into the difference matrix. So, we can reasonably formulate the system's matrices with whole uncertainties by the norm bound in equation (3) and the difference matrix with individual uncertainties by the polytopic form in equation (2). A circuit example can be found in Example 1 in the previous work [40].
To investigate the stability and design issues for system (1), definitions for the discrete singular system are given as follows.
is said to be admissible if it is regular, causal, and stable Lemma 1 (see [8,24]). e nominal discrete singular delay system

Admissibility Analysis and Admissibilizing
Controller Design

Admissibility Analysis.
For analyzing the robust admissibility for system (1), a sufficient condition for the unforced nominal singular system (2), is first derived as follows. (1) is admissible if there exist matrices P > 0, R > 0, and Q with appropriate dimensions such that

Lemma 3. Unforced nominal discrete singular delay system
where S ∈ R n×(n− m) is any matrix with full column rank and satisfies E T S � 0.
Proof. Assume that matrices P > 0, R > 0, and Q satisfy (4). By the Schur complement [41], equation (4) leads to For matrix E with rank(E) � r ≤ n, there exist nonsingular matrices M 1 , N 1 ∈ R n×n such that Using the expression in (6a), (6b) and deducing from the block (1, 1) in (5), we have where " * " denotes the not utilized matrix. Due to P 1 > 0, P 3 > 0, and R 3 > 0, the least equation implies , the nonsingularity of (P T 2 A 2 + S 2 ) T A 4 , and so does A 4 . From Definition 1 associated with Lemma 1, the nominal discrete singular delay system is asserted to be regular and causal.
Choose a Lyapunov-Krasovskii functional candidate for the regarded system as

Mathematical Problems in Engineering
By the Lyapunov quadratic stability theory, differentiating V(k) can be manipulated as Deducing from ΔV(k) along the considered system and using the property E T S � 0 lead to where By the Schur complement, equation (4) is equivalent to and leads to ΔV(k) < 0, and the considered system is stable. Associated with regularity and causality, the system is thus concluded to be admissible according to Definition 2 associated with Lemma 1. Based on Lemma 2, the stability criteria in Lemma 3 for unforced nominal discrete singular delay system (1) can be transformed into an alternative matrices' transpose form from (5) where S ∈ R n×(n− m) is any matrix with full column rank and satisfies ES � 0.

Remark 2.
e proposed result in Lemma 3 for the nominal discrete delayed system is directly deduced from matrix algebraic manipulation associated with the Lyapunov-Krasovskii stability theory, where based on this form, we can further investigate the regarded systems in (1) embracing uncertainties in both difference and systems' matrices. However, a distinct result in [8] for the nominal discrete delayed system can also be attained by using the argument system approach.
A robust admissibility criterion for unforced uncertain discrete singular system (1) is mainly presented as follows.
Theorem 1. Unforced uncertain discrete singular delay system (1) is admissible if there exist matrices P > 0, R > 0, and Q with appropriate dimensions and a scalar α > 0 such that where S ∈ R n×(n− m) is any matrix with full column rank and satisfies E i S � 0 ∀i.
Proof. Suppose that there exist matrices P > 0, R > 0, and Q such that (13) holds. By (13) associated with the uncertain matrix E with e i ≥ 0 ∀i, i e i � 1 defined in (2), we have Mathematical Problems in Engineering Deducing from Corollary 1 associated with the uncertain matrices ΔA and ΔA d defined in (3), we have By the Schur complement associated with the aforementioned negative definite inequality, the above is equivalent to and then ensures us, according to Corollary 1, unforced uncertain discrete singular delay system (1) is asserted to be admissible.

Remark 3.
e proposed conditions in eorem 1 are explicitly expressed by LMIs, and we can handily verify them by the current LMI solver [42]. For evaluating feasible solutions, we need to previously construct a set of LMIs by (13), and the total of LMIs with n numbers of E i 's is n + C n 2 . us, the computational time is proportional to the sum of LMIs.

Admissibilizing Controller Design.
Consider uncertain discrete singular delay system (1) associated with a state feedback control law, u(k) Δ Kx(k). e resulting closedloop system can be formed as where the uncertainties' terms are given in (2) and (3).
In the sequel, we focus on determining a gain matrix K to admissibilize the resulting closed-loop system (18).

Theorem 2.
e resulting closed-loop discrete singular delay system (18) is admissible if there exist matrices P > 0, R > 0, and Q with appropriate dimensions and scalars α > 0, β > 0, and a l such that Mathematical Problems in Engineering is any matrix with full column rank and satisfies E i S � 0, ∀i, and an admissibilizing gain matrix is denoted as K Δ l a l K l , K l S � 0, ∀l.
Proof. Based on eorem 1, the admissible criterion for the resulting closed-loop system (18) with K Δ l a l K l , K l S � 0, ∀l, can be constructed by Using the Schur complement in (19), it can directly meet the above.
us, the resulting closed-loop singular delay system with K � l a l K l , K l S � 0, ∀l, is asserted to be admissible according to eorem 1.

Illustrative Examples
In this section, we give two illustrative examples to demonstrate the effectiveness and feasibility of the proposed methods.
Example 1. Consider a discrete uncertain singular system with delayed state described by where and N Ad � 0.1 0.1 0.2 . Since perturbed difference matrix E has two uncertain parameters mainly in the diagonal elements, according to (2), we can denote matrices as Due to the regarded discrete singular uncertain system with delayed state includes the uncertainties in both the difference matrix and the systems' matrices, past results [24][25][26][27][28][29][30][31][32][33][34][35][36] with a constant E cannot be applied for admissibility verification. However, by eorem 1 with the given matrix S � 0 0 1 T satisfying E i S � 0, ∀i, the admissibility analysis criteria for the considered system can be represented as a set of strict LMI conditions by (13). When hiring the LMI tool [42] for solving, a set of feasible solutions can be attained and presented as us, the considered system ensures to be admissible for all the allowable uncertainties according to eorem 1.

Example 2.
Consider an uncertain discrete singular system with delayed state and control input represented by where ε 1 ∈ [1,  Since perturbed difference matrix E has two uncertain parameters, according to (2), we can denote matrices E i as For the unforced singular delay system, i.e., the considered system with u(t) Δ 0, the states' variables with a given initial condition x(0) � 5 −4 −1.5 T are first simulated and depicted in Figure 1. It shows that the states' trajectories are unstable, and a control law needs to be implemented. Since the regarded system embraces two uncertain parameters in difference matrix E, all the past works [24][25][26][27][28][29][30][31][32][33][34][35][36] are not suitable. However, according to the result in eorem 2 with giving a priori matrix S � 0 0 1 T satisfying E i S � 0, ∀i, and K 1 � 1 0 0 , K 2 � 0 1 0 , a design procedure with the state feedback control can be started up and performed by a set of parametric LMIs by (19). By hiring the LMI solver with given parametric intervals as a 1 ∈ [−5, 5] and a 2 ∈ [−5, 5], we thus attain a set of feasible solutions as en, an admissibilizing controller gain can be consequently determined by K � l a l K l � a 1 K 1 + a 2 K 2 � 1.1 −1.9 0 .
us, when denoting the same initial condition as the above, the considered system which equipped the state feedback control with the obtained gain matrix is simulated again. Also, the states' trajectories and the control input are depicted in Figures 2 and 3, respectively. It is shown that all the controlled states x i (k), ∀i, are surely convergent to zero in a short period.

Conclusions
In this work, we have investigated the admissibility analysis and the admissibilizing controller design for the uncertain discrete singular system with delayed state. First, based on Lyapunov-Krasovskii stability theory, an original admissibility condition for the nominal singular delay system was presented. In the sequel, we dealt with the admissibility analysis by hiring the LMI method. Furthermore, by involving the state feedback control law, we further discussed the admissibilizing controller design for the resulting closedloop system which can contain parametric uncertainties in the difference and systems' matrices simultaneously. Since all the presented criteria are represented in terms of strict LMIs or parametric LMIs, we can handily evaluate feasible solutions via the current LMI programming tools. Finally, the proposed result was demonstrated to be effective and practicable by the two examples given. However, the proposed analysis and controller design criteria are proposed for admissibility assurance of the considered systems. e future work will be suggested forward to involving some control objectives of specific admissibility assurance with the specific decay rate, damping, settle time, and so on.

Data Availability
No data were used to support this study.    Mathematical Problems in Engineering 11