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

Authors: Chih-Peng Huang
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

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.


---

## Body

## 1. 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]. They 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, 6–8]. Recently, robust admissibility and robust controller synthesis for uncertain singular systems have been addressed. The 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, [9–18] 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. They are the main sources of instability, oscillations, or degraded performances [19–23]. Therefore, the problems of stability testing and stabilization of uncertain singular systems with delay states have aroused much attention over the recent years, e.g., [24–36]. 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 time-varying 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. Then, 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 time-varying 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 [37–40]. 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. Thus, 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. This 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. The validity and efficiency of the proposed approach are demonstrated by illustrative examples.


## 2. Problem Formulation and Preliminaries

Consider a discrete uncertain singular delay system described by(1)E˜xk+1=A+ΔAxk+Ad+ΔAdxk−d+B+ΔBuk,k>0,xi=ϕi,i=−d,−d+1,…,0,where x⋅∈Rn is the state vector and u⋅∈Rp is the control input; the perturbed derivative matrix E˜∈Rn×n may be singular, i.e., rankE˜=m≤n, and belongs to a polytopic set defined as(2)ΩEΔ¯¯E˜:E˜=∑ireiEi,ei≥0,∑i=1rei=1,where Ei∈Rn×n given a priori. A, Ad, and B stand for the nominal system matrices with appropriate dimensions, and ΔA, ΔAd, and ΔB represent the parameter uncertainties bounded by(3)ΔATΔAdTΔB=MDNANAdNB,where M, NA, NAd, and NB are constant matrices with compatible dimensions and D satisfies DTD≤IRemark 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.Definition 1.
(see [8, 24]).(I)
The pairE,A is said to be regular if detzE−A is not identically zero(II)
The pairE,A is said to be causal if it is regular and deg(detzE−A) = rankE(III)
LetρE,AΔ¯¯maxλ∈z|detzE−A=0λ; the pair E,A is said to be stable if ρE,A<1(IV)
The pairE,A is said to be admissible if it is regular, causal, and stableDefinition 2.
(see [8, 24]).(I)
The tripleE,A,Ad is said to be regular if detzd+1E−zdA−Ad is not identically zero(II)
The tripleE,A,Ad is said to be causal if it is regular and degznddetzE−A−z−dAd=nd+rankE(III)
The discrete singular delay systemExk+1=Axk+Adxk−d is said to be stable if ρE,A,Ad<1, where ρE,A,AdΔ¯¯maxλ∈z|detzd+1E−zdA−Ad=0λ(IV)
The discrete singular delay systemExk+1=Axk+Adxk−d (i.e., the triple E,A,Ad) is said to be admissible if it is regular, causal, and stableLemma 1.
(see [8, 24]). The nominal discrete singular delay system Exk+1=Axk+Adxk−d is regular, causal, and stable if and only if the pair E,A is regular, causal, and ρE,A,Ad<1.Lemma 2.
(see [29]). The triple E,A,Ad is admissible if and only if the corresponding transpose form ET,AT,AdT is admissible.


## 3. Admissibility Analysis and Admissibilizing Controller Design

### 3.1. Admissibility Analysis

For analyzing the robust admissibility for system (1), a sufficient condition for the unforced nominal singular system Exk+1=Axk+Adxk−d, that is, ΔA=ΔAd=0 and uk≡0 in (1) and E˜=E in (2), is first derived as follows.Lemma 3.
Unforced nominal discrete singular delay system (1) is admissible if there exist matrices P>0, R>0, and Q with appropriate dimensions such that(4)QSTA+ATSQT−ETPE+RQSTAdATPAdTSQT−RAdTPPAPAd−P<0,where S∈Rn×n−m is any matrix with full column rank and satisfies ETS=0.Proof.
Assume that matricesP>0, R>0, and Q satisfy (4). By the Schur complement [41], equation (4) leads to(5)ATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−R<0.
For matrixE with rankE=r≤n, there exist nonsingular matrices M1, N1∈Rn×n such that(6a)M1EN1=Ir000,M1AN1=A1A2A3A4,(6b)M1−TPM1−1=P1P2P2TP3,M1−TSQTN1=00S1S2,N1TRN1=R1R2R2TR3.
Using the expression in (6a), (6b) and deducing from the block (1, 1) in (5), we have(7)ATPA−ETPE+ATSQT+QSTA+R=N1−TA1TA3TA2TA4TP1P2P2TP3A1A2A3A4−Ir000P1P2P2TP3Ir000+A1TA3TA2TA4T00S1S2+0S1T0S2TA1A2A3A4+R1R2R2TR3N1−1=N1−T∗∗∗A2TP1A2+A4TP2TA2+A2TP2A4+A4TP3A4+A4TS2+S2TA4+R3N1−1=N1−T∗∗∗A4TP2TA2+S2+P2TA2+S2TA4+A2TP1A2+A4TP3A4+R3N1−1<0,where “∗” denotes the not utilized matrix. Due to P1>0, P3>0, and R3>0, the least equation implies A4TP2TA2+S2+P2TA2+S2TA4<0, i.e., the nonsingularity of P2TA2+S2TA4, and so does A4. 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(8)Vk=xTkETPExk+∑j=1dxk−jTRxk−j.
By the Lyapunov quadratic stability theory, differentiatingVk can be manipulated as(9)ΔVk=Vk+1−Vk=xTk+1ETPExk+1+∑j=1dxk+1−jTRxk+1−j−xTkETPExk−∑j=1dxk−jTRxk−j=xTk+1ETPExk+1−xTkETPExk+xTkRxk−xTk−dRxk−d.
Deducing fromΔVk along the considered system and using the property ETS=0 lead to(10)ΔVk=xTk+1ETPExk+1−xTkETPExk+xTkRxk−xTk−dRxk−d+xTk+1ETSQTxk+xTkQSTExk+1=Axk+Adxk−dTPAxk+Adxk−d−xTkETPExk+xTkRxk−xTk−dRxk−d+Axk+Adxk−dTSQTxk+xTkQSTAxk+Adxk−d=xTkATPA−ETPE+ATSQT+QSTA+Rxk+2xTkATP+QSTAdxk−d+xTk−dAdTPAd−Rxk−d=ζTkATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−Rζk,where ζkΔ¯¯xTkxTk−dT. By the Schur complement, equation (4) is equivalent to(11)ATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−R<0and leads to ΔVk<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 Lemma2, 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) with replacing E, A, and Ad with ET, AT, and AdT, respectively, and are presented as follows.Corollary 1.
Unforced nominal discrete singular delay system (1) is admissible if there exist matrices P>0, R>0, and Q with appropriate dimensions such that(12)QSTAT+ASQT−EPET+RQSTAdTAPAdSQT−RAdPPATPAdT−P<0,where S∈Rn×n−m is any matrix with full column rank and satisfies ES=0.Remark 2.
The proposed result in Lemma3 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(13)2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0,∀i≤j,where S∈Rn×n−m is any matrix with full column rank and satisfies EiS=0∀i.Proof.
Suppose that there exist matricesP>0, R>0, and Q such that (13) holds. By (13) associated with the uncertain matrix E˜ with ei≥0∀i, ∑iei=1 defined in (2), we have(14)2QSTAT+2ASQT−2E˜PE˜T+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI=2QSTAT+2ASQT−2∑ieiEiP∑ieiEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI=∑iei22QSTAT+2ASQT−2EiPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI+∑i<j2eiej2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0.
Deducing from Corollary1 associated with the uncertain matrices ΔA and ΔAd defined in (3), we have(15)2QSTA+ΔAT+2A+ΔASQT−2E˜PE˜T+2R2QSTAd+ΔAdTA+ΔAP2Ad+ΔAdSQT−2RAd+ΔAdPPA+ΔATPAd+ΔAdT−12P=2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+2QSTΔAT+2ΔASQT2QSTΔAdTΔAP2ΔAdSQT0ΔAdPPΔATPΔAdT0=2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+2QSTM0PMDNANAd0+NATNAdT0DT2MTSQ0MTP≤2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+α−12QSTM0PM2MTSQ0MTP+αNATNAdT0NANAd0.
By the Schur complement associated with the aforementioned negative definite inequality, the above is equivalent to(16)2QSTAT+2ASQT−2E˜PE˜T+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0and then ensures(17)2QSTA+ΔAT+2A+ΔASQT−2E˜PE˜T+2R2QSTAd+ΔAdTA+ΔAP2Ad+ΔAdSQT−2RAd+ΔAdPPA+ΔATPAd+ΔAdT−12P<0.
Thus, according to Corollary1, unforced uncertain discrete singular delay system (1) is asserted to be admissible.Remark 3.
The proposed conditions in Theorem1 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 Ei’s is n+C2n. Thus, the computational time is proportional to the sum of LMIs.


### 3.2. Admissibilizing Controller Design

Consider uncertain discrete singular delay system (1) associated with a state feedback control law, ukΔ¯¯Kxk. The resulting closed-loop system can be formed as(18)E˜xk+1=A+ΔA+B+ΔBKxk+Ad+ΔAdxk−d,where the uncertainties’ terms are given in (2) and (3).In the sequel, we focus on determining a gain matrixK to admissibilize the resulting closed-loop system (18).Theorem 2.
The 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 al such that(19)Ψ2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM02AdSQT+αNAdTNA−2R+αNAdTNAdAdP00PAT+P∑lalKlTBTPAdT−12PPMP∑lalKlTNBT2MTSQT0MTP−αI000NB∑lalKlP0−βI<0,∀l,i≤j,where ΨΔ¯¯2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA+βMMT and S∈Rn×n−m is any matrix with full column rank and satisfies EiS=0, ∀i, and an admissibilizing gain matrix is denoted as KΔ¯¯∑lalKl, KlS=0, ∀l.Proof.
Based on Theorem1, the admissible criterion for the resulting closed-loop system (18) with KΔ¯¯∑lalKl, KlS=0, ∀l, can be constructed by(20)2QSTA+B+ΔBKT+2A+B+ΔBKSQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdA+B+ΔBKP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PA+B+ΔBKTPAdT−12PPM2MTSQT0MTP−αI=2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+00ΔB∑lalKlP00000P∑lalKlTΔBT0000000=2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+M000D00NB∑lalKlP0+00P∑lalKlTNBT0DTMT000≤2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+βM000MT000+β−100P∑lalKlTNBT000NB∑lalKlP0<0.
Using the Schur complement in (19), it can directly meet the above. Thus, the resulting closed-loop singular delay system with K=∑lalKl, KlS=0, ∀l, is asserted to be admissible according to Theorem 1.


## 3.1. Admissibility Analysis

For analyzing the robust admissibility for system (1), a sufficient condition for the unforced nominal singular system Exk+1=Axk+Adxk−d, that is, ΔA=ΔAd=0 and uk≡0 in (1) and E˜=E in (2), is first derived as follows.Lemma 3.
Unforced nominal discrete singular delay system (1) is admissible if there exist matrices P>0, R>0, and Q with appropriate dimensions such that(4)QSTA+ATSQT−ETPE+RQSTAdATPAdTSQT−RAdTPPAPAd−P<0,where S∈Rn×n−m is any matrix with full column rank and satisfies ETS=0.Proof.
Assume that matricesP>0, R>0, and Q satisfy (4). By the Schur complement [41], equation (4) leads to(5)ATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−R<0.
For matrixE with rankE=r≤n, there exist nonsingular matrices M1, N1∈Rn×n such that(6a)M1EN1=Ir000,M1AN1=A1A2A3A4,(6b)M1−TPM1−1=P1P2P2TP3,M1−TSQTN1=00S1S2,N1TRN1=R1R2R2TR3.
Using the expression in (6a), (6b) and deducing from the block (1, 1) in (5), we have(7)ATPA−ETPE+ATSQT+QSTA+R=N1−TA1TA3TA2TA4TP1P2P2TP3A1A2A3A4−Ir000P1P2P2TP3Ir000+A1TA3TA2TA4T00S1S2+0S1T0S2TA1A2A3A4+R1R2R2TR3N1−1=N1−T∗∗∗A2TP1A2+A4TP2TA2+A2TP2A4+A4TP3A4+A4TS2+S2TA4+R3N1−1=N1−T∗∗∗A4TP2TA2+S2+P2TA2+S2TA4+A2TP1A2+A4TP3A4+R3N1−1<0,where “∗” denotes the not utilized matrix. Due to P1>0, P3>0, and R3>0, the least equation implies A4TP2TA2+S2+P2TA2+S2TA4<0, i.e., the nonsingularity of P2TA2+S2TA4, and so does A4. 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(8)Vk=xTkETPExk+∑j=1dxk−jTRxk−j.
By the Lyapunov quadratic stability theory, differentiatingVk can be manipulated as(9)ΔVk=Vk+1−Vk=xTk+1ETPExk+1+∑j=1dxk+1−jTRxk+1−j−xTkETPExk−∑j=1dxk−jTRxk−j=xTk+1ETPExk+1−xTkETPExk+xTkRxk−xTk−dRxk−d.
Deducing fromΔVk along the considered system and using the property ETS=0 lead to(10)ΔVk=xTk+1ETPExk+1−xTkETPExk+xTkRxk−xTk−dRxk−d+xTk+1ETSQTxk+xTkQSTExk+1=Axk+Adxk−dTPAxk+Adxk−d−xTkETPExk+xTkRxk−xTk−dRxk−d+Axk+Adxk−dTSQTxk+xTkQSTAxk+Adxk−d=xTkATPA−ETPE+ATSQT+QSTA+Rxk+2xTkATP+QSTAdxk−d+xTk−dAdTPAd−Rxk−d=ζTkATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−Rζk,where ζkΔ¯¯xTkxTk−dT. By the Schur complement, equation (4) is equivalent to(11)ATPA−ETPE+ATSQT+QSTA+RPA+SQTTAdAdTPA+SQTAdTPAd−R<0and leads to ΔVk<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 Lemma2, 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) with replacing E, A, and Ad with ET, AT, and AdT, respectively, and are presented as follows.Corollary 1.
Unforced nominal discrete singular delay system (1) is admissible if there exist matrices P>0, R>0, and Q with appropriate dimensions such that(12)QSTAT+ASQT−EPET+RQSTAdTAPAdSQT−RAdPPATPAdT−P<0,where S∈Rn×n−m is any matrix with full column rank and satisfies ES=0.Remark 2.
The proposed result in Lemma3 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(13)2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0,∀i≤j,where S∈Rn×n−m is any matrix with full column rank and satisfies EiS=0∀i.Proof.
Suppose that there exist matricesP>0, R>0, and Q such that (13) holds. By (13) associated with the uncertain matrix E˜ with ei≥0∀i, ∑iei=1 defined in (2), we have(14)2QSTAT+2ASQT−2E˜PE˜T+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI=2QSTAT+2ASQT−2∑ieiEiP∑ieiEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI=∑iei22QSTAT+2ASQT−2EiPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI+∑i<j2eiej2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0.
Deducing from Corollary1 associated with the uncertain matrices ΔA and ΔAd defined in (3), we have(15)2QSTA+ΔAT+2A+ΔASQT−2E˜PE˜T+2R2QSTAd+ΔAdTA+ΔAP2Ad+ΔAdSQT−2RAd+ΔAdPPA+ΔATPAd+ΔAdT−12P=2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+2QSTΔAT+2ΔASQT2QSTΔAdTΔAP2ΔAdSQT0ΔAdPPΔATPΔAdT0=2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+2QSTM0PMDNANAd0+NATNAdT0DT2MTSQ0MTP≤2QSTAT+2ASQT−2E˜PE˜T+2R2QSTAdTAP2AdSQT−2RAdPPATPAdT−12P+α−12QSTM0PM2MTSQ0MTP+αNATNAdT0NANAd0.
By the Schur complement associated with the aforementioned negative definite inequality, the above is equivalent to(16)2QSTAT+2ASQT−2E˜PE˜T+2R+αNATNA2QSTAdT+αNATNAdAP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PATPAdT−12PPM2MTSQT0MTP−αI<0and then ensures(17)2QSTA+ΔAT+2A+ΔASQT−2E˜PE˜T+2R2QSTAd+ΔAdTA+ΔAP2Ad+ΔAdSQT−2RAd+ΔAdPPA+ΔATPAd+ΔAdT−12P<0.
Thus, according to Corollary1, unforced uncertain discrete singular delay system (1) is asserted to be admissible.Remark 3.
The proposed conditions in Theorem1 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 Ei’s is n+C2n. Thus, the computational time is proportional to the sum of LMIs.


## 3.2. Admissibilizing Controller Design

Consider uncertain discrete singular delay system (1) associated with a state feedback control law, ukΔ¯¯Kxk. The resulting closed-loop system can be formed as(18)E˜xk+1=A+ΔA+B+ΔBKxk+Ad+ΔAdxk−d,where the uncertainties’ terms are given in (2) and (3).In the sequel, we focus on determining a gain matrixK to admissibilize the resulting closed-loop system (18).Theorem 2.
The 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 al such that(19)Ψ2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM02AdSQT+αNAdTNA−2R+αNAdTNAdAdP00PAT+P∑lalKlTBTPAdT−12PPMP∑lalKlTNBT2MTSQT0MTP−αI000NB∑lalKlP0−βI<0,∀l,i≤j,where ΨΔ¯¯2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA+βMMT and S∈Rn×n−m is any matrix with full column rank and satisfies EiS=0, ∀i, and an admissibilizing gain matrix is denoted as KΔ¯¯∑lalKl, KlS=0, ∀l.Proof.
Based on Theorem1, the admissible criterion for the resulting closed-loop system (18) with KΔ¯¯∑lalKl, KlS=0, ∀l, can be constructed by(20)2QSTA+B+ΔBKT+2A+B+ΔBKSQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdA+B+ΔBKP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PA+B+ΔBKTPAdT−12PPM2MTSQT0MTP−αI=2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+00ΔB∑lalKlP00000P∑lalKlTΔBT0000000=2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+M000D00NB∑lalKlP0+00P∑lalKlTNBT0DTMT000≤2QSTAT+2ASQT−EiPEjT−EjPEiT+2R+αNATNA2QSTAdT+αNATNAdAP+B∑lalKlP2QSTM2AdSQT+αNAdTNA−2R+αNAdTNAdAdP0PAT+P∑lalKlTBTPAdT−12PPM2MTSQT0MTP−αI+βM000MT000+β−100P∑lalKlTNBT000NB∑lalKlP0<0.
Using the Schur complement in (19), it can directly meet the above. Thus, the resulting closed-loop singular delay system with K=∑lalKl, KlS=0, ∀l, is asserted to be admissible according to Theorem 1.


## 4. 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(21)ε1100ε20000xk+1=A+ΔAxk+Ad+ΔAdxk−1,where ε1∈0.8,1, ε2∈1,1.2, A=0.40.200.30.30.40.20.30.5, Ad=0.20.100.10.30.30.10.20.1, and ΔATΔAdT=MDNANAd with M=111T, NA=0.20.30.2, and NAd=0.10.10.2. Since perturbed difference matrix E˜ has two uncertain parameters mainly in the diagonal elements, according to (2), we can denote matrices as(22)E1=0.810010000,E2=110010000,E3=0.810010000,E4=0.81001.20000.
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 [24–36] with a constant E cannot be applied for admissibility verification. However, by Theorem 1 with the given matrix S=001T satisfying EiS=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(23)P=42.8072−21.1936−9.9868−21.193630.1259−11.4180−9.9868−11.418026.9956>0,R=9.41314.19401.03564.194017.57232.35381.03562.35383.8766>0,Q=−2.1728−7.6921−14.5536,α=83.2608>0.
Thus, the considered system ensures to be admissible for all the allowable uncertainties according to Theorem1.Example 2.
Consider an uncertain discrete singular system with delayed state and control input represented by(24)ε1100ε20000xk+1=A+ΔAxk+Ad+ΔAdxk−1+B+ΔBuk,where ε1∈1,1.5, ε2∈1,1.1,(25)A=1.210.60.21.30.2−0.10.50.8,Ad=0.20.300.20.20.1−0.20.10.2,B=110,and ΔATΔAdTΔB=MDNANAdNB with M=0.20.20.1T, NA=0.20.20.2, NAd=0.20.10.2, and NB=0.2.
Since perturbed difference matrixE˜ has two uncertain parameters, according to (2), we can denote matrices Ei as(26)E1=110010000,E2=1.510010000,E3=11001.10000,E4=1.51001.10000.
For the unforced singular delay system, i.e., the considered system withutΔ¯¯0, the states’ variables with a given initial condition x0=5−4−1.5T 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 [24–36] are not suitable. However, according to the result in Theorem 2 with giving a priori matrix S=001T satisfying EiS=0, ∀i, and K1=100, K2=010, 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 a1∈−5,5 and a2∈−5,5, we thus attain a set of feasible solutions as(27)a1=1.1,a2=−1.9,α=0.7415>0,β=0.7392>0,P=30.769232.5080−24.652632.508060.2108−25.2156−24.6526−25.215639.3956×10−2>0,R=35.766622.41456.101522.414517.2951−0.25996.1015−0.259919.3578×10−2>0,Q=16.483715.3854−37.8544×10−2.
Then, an admissibilizing controller gain can be consequently determined by(28)K=∑lalKl=a1K1+a2K2=1.1−1.90.
Thus, 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 Figures2 and 3, respectively. It is shown that all the controlled states xik, ∀i, are surely convergent to zero in a short period.Figure 1
Unforced states’ responses for Example2.Figure 2
States’ responses with the control input for Example2.Figure 3
State feedback control input for Example2.


## 5. 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 closed-loop 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. The 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.



---

*Source: 8845558-2021-01-20.xml*