# Novel Observer-Based Suboptimal Digital Tracker for a Class of Time-Delay Singular Systems

Authors: Nien-Tsu Hu; Ter-Feng Wu; Sendren Sheng-Dun Xu; Hsu-Chih Huang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper presents a novel suboptimal digital tracker for a class of time-delay singular systems. First, some existing techniques are utilized to obtain an equivalent regular time-delay system, which has a direct transmission term from input to output. The equivalent regular time-delay system is important as it enables the optimal control theory to be conveniently combined with the digital redesign approach. The linear quadratic performance index, specified in the continuous-time domain, can be discretized into an equivalent decoupled discrete-time performance index using the newly developed extended delay-free model. Additionally, although the extended delay-free model is large, its advantage is the elimination of all delay terms (which included a new extended state vector), simplifying the proposed approach. As a result, the proposed approach can be applied to a class of time-delay singular systems. An illustrative example demonstrates the effectiveness of the proposed design methodology.


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## Body

## 1. Introduction

The singular systems naturally arise in describing large-scale systems, and there are several examples occurring in power and interconnected systems. In general, an interconnection of state variable subsystems is conveniently described as a singular system, even though an overall state space representation may not even exist. Over the past decades, much research into singular systems has solved many complex problems concerning, for example, the stability [1–4], impulsive modes [5], controllability, observability [6], and the sufficient and necessary conditions for the impulse controllability and observability of time-varying singular systems [7–11]. However, the main purpose of such work is either to stabilize the singular system or to prove its controllability and observability. Here, the key note of this paper is about tracking the issue.This investigation considers a time-delay system. The overwhelming majority of practical control systems are described by continuous-time settings with input, output, and state time delays. Those delays arise from inherent physical phenomena and are commonly encountered in various engineering systems. Several authors [12–15] have studied the linear quadratic optimal analog controllers for the analog system with input and state delays. Recently, robust control and filtering for both continuous-time and discrete-time nominal/uncertain systems with time delays have been thoroughly studied by Mahmoud [16]. Despite much progress in both analog control theory and digital control theory over the last few decades, effective digital control of analog plants with input and state delays (input-state delayed hybrid control systems) is still being developed [17, 18].The objective of this paper is to develop a novel observer-based suboptimal digital tracker for a class of time-delay singular systems. The developed digital tracker can make the outputs of the digitally controlled time-delay singular system track the desired reference signals. First, the time-delay singular system is converted into a regular time-delay system that contains a direct transmission term from input to output. Then, for effective utilization of the well-developed discrete-time optimal control theory for a regular time-delay system, it is converted into a new extended discrete delay-free model. The performance cost function is discretized using the extended discrete delay-free model. When the states of the continuous time-delay singular system are not available for measurements, a suboptimal digital observer for the original continuous time-delay singular system is constructed by using the duality of the digital redesign technique for the controller and the digital-to-analog model conversion technique [19]. As a result, the proposed novel observer-based suboptimal digital tracker is able to make the output of the digitally controlled analog time-delay system track the desired reference signals.The rest of the paper is organized as follows. Section2 presents the problem description and preliminary results. Section 3 presents the novel optimal tracker and a novel observer-based suboptimal tracker for the time-delay singular system and proposes a systematic design methodology for designing a set of high-performance trackers for a class of time-delay systems. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed approach.


## 2. Problem Description and Preliminaries

### 2.1. Problem Description

Consider the following continuous time-delay singular system:(1a)Ex˙c(t)=Axc(t)+∑i=1N1A^ixc(t-τs,i)+∑j=1N2Bjuc(t-τi,j),(1b)yc(t)=Cxc(t-τo),wherexc(t)∈ℜn is the state vector, uc(t)∈ℜm is the control input vector, and yc(t)∈ℜp is the output vector. E, A, A^i, Bj, and C are known constant system matrices of appropriate dimensions and E is a singular matrix. The corresponding state time delay τs,i, i=1,2,…,N1, input time delay τi,j, j=1,2,…,N2, and output time delay τo are assumed to be known.The continuous time-delay singular system (1a) and (1b) may be in impulsive modes. Directly designing the controller or observer for (1a) and (1b) is very difficult because impulsive modes are uncontrollable. To solve this problem, the regular pencil, the standard pencil, and the preliminary feedback control methods are used to eliminate impulsive modes and then obtain an equivalent regular time-delay system that can be applied to the original continuous time-delay singular system (1a) and (1b). The following section systematically develops the design of the novel controller and observer using the equivalent regular time-delay system.


### 2.2. Preliminaries

The regular pencil and standard pencil are defined below.Definition 1 (regular pencil [20]).
LetE and A be two square constant matrices. If det(sE-A)≠0, for all s, then (sE-A) is called a regular pencil.Definition 2 (standard pencil [21]).
Let(sEn-An) be a regular pencil. If there exists scalars α and β such that αEn+βAn=In, then (sEn-An) is called a standard pencil.
Notably, for any regular pencil,(sE-A) can be easily transformed into a standard pencil by multiplying (αE+βA)-1 to E and A, respectively, where α and β are scalars such that det(αE+βA)≠0. Therefore, the matrix coefficients of a standard pencil (sEn-An) become(2a)En=(αEr+βA)-1E,(2b)An=(αE+βA)-1A.The modified system retains its state vector xc(t) and the matrices En and An have the following nice properties.Lemma 3 (see [22]).
Consider
(a)
E
n
A
n
=
A
n
E
n, meaning that En and An commute each other;
(b)
E
n and An have the same eigenspaces.The above properties enable a singular system to be decomposed into a reduced-order regular subsystem and a nondynamic subsystem.


## 2.1. Problem Description

Consider the following continuous time-delay singular system:(1a)Ex˙c(t)=Axc(t)+∑i=1N1A^ixc(t-τs,i)+∑j=1N2Bjuc(t-τi,j),(1b)yc(t)=Cxc(t-τo),wherexc(t)∈ℜn is the state vector, uc(t)∈ℜm is the control input vector, and yc(t)∈ℜp is the output vector. E, A, A^i, Bj, and C are known constant system matrices of appropriate dimensions and E is a singular matrix. The corresponding state time delay τs,i, i=1,2,…,N1, input time delay τi,j, j=1,2,…,N2, and output time delay τo are assumed to be known.The continuous time-delay singular system (1a) and (1b) may be in impulsive modes. Directly designing the controller or observer for (1a) and (1b) is very difficult because impulsive modes are uncontrollable. To solve this problem, the regular pencil, the standard pencil, and the preliminary feedback control methods are used to eliminate impulsive modes and then obtain an equivalent regular time-delay system that can be applied to the original continuous time-delay singular system (1a) and (1b). The following section systematically develops the design of the novel controller and observer using the equivalent regular time-delay system.


## 2.2. Preliminaries

The regular pencil and standard pencil are defined below.Definition 1 (regular pencil [20]).
LetE and A be two square constant matrices. If det(sE-A)≠0, for all s, then (sE-A) is called a regular pencil.Definition 2 (standard pencil [21]).
Let(sEn-An) be a regular pencil. If there exists scalars α and β such that αEn+βAn=In, then (sEn-An) is called a standard pencil.
Notably, for any regular pencil,(sE-A) can be easily transformed into a standard pencil by multiplying (αE+βA)-1 to E and A, respectively, where α and β are scalars such that det(αE+βA)≠0. Therefore, the matrix coefficients of a standard pencil (sEn-An) become(2a)En=(αEr+βA)-1E,(2b)An=(αE+βA)-1A.The modified system retains its state vector xc(t) and the matrices En and An have the following nice properties.Lemma 3 (see [22]).
Consider
(a)
E
n
A
n
=
A
n
E
n, meaning that En and An commute each other;
(b)
E
n and An have the same eigenspaces.The above properties enable a singular system to be decomposed into a reduced-order regular subsystem and a nondynamic subsystem.


## 3. Main Results

### 3.1. Decomposition of Time-Delay Singular System

By (2a) and (2b) the regular pencil (sE-A) can be transformed into a standard pencil (sEn-An). Notably since En is a singular matrix, which has at least one zero eigenvalue, β cannot be equal to zero. Hence, multiplying (1a) by (αE+βA)-1 can yield the following equation:
(3)Enx˙c(t)=Anxc(t)+∑i=1N1A^n,ixc(t-τs,i)+∑j=1N2Bn,juc(t-τi,j),
where
(4)En=(αE+βA)-1E,A^n,i=(αE+βA)-1A^i,Bn,j=(αE+βA)-1Bj.
Since αEn+βAn=In, the pencil (sEn-An) is a standard one, and has the properties that are mentioned in Lemma 3. To decompose system (3), the state xc(t) is converted into x-c(t) by
(5)xc(t)=Mx-c(t),
where the constant matrix M is a block modal matrix of En and determined by means of the extended matrix sign function [23, 24]. The M matrix of state space transformation is as follows.Step 1.
Findsign(E~n) using the extended matrix sign function with an adequate ω, where
(6)E~n=(En-ωIn)(En+ωIn)-1.Step 2.
Findsign+(E~n)=(1/2)[In+sign(E~n)] and sign-(E~n)=(1/2)[In-sign(E~n)].Step 3.
Construct the matrixM=[ind(sign+(E~n))ind(sign-(E~n))], where ind(·) represents the collection of linearly independent column vectors of (·).Substituting (5) into (3) and multiplying by M-1 on the left yield
(7)M-1EnMx-˙c(t)=M-1AnMx-c(t)+∑i=1N1M-1A^n,iMx-c(t-τs,i)+∑j=1N2M-1Bn,juc(t-τi,j)=1β(In-αEn)x-c(t)+∑i=1N1M-1A^n,iMx-c(t-τs,i)+∑j=1N2M-1Bn,juc(t-τi,j).
If M-1A^n,iM can be diagonalized, then (7) yields,
(8)[E-1OE-2]x-˙c(t)=[1β(Ik-αE-1)OO1β(In-κ-αE-2)]x-c(t)+∑i=1N1[A-1,iOOA-2,i]x-c(t-τs,i)+∑j=1N2[B-1,jB-2,j]uc(t-τi,j),
where x-c(t)=[x-sT(t),x-fT(t)]T and M-1EnM=blockdiagonal{E-1,E-2}. E-1 is invertible with rank(E-1)=deg{det(sEr-A)}≜k, [B-1,jT,B-2,jT]T=M-1Bn,j, and E-2 is a nilpotent matrix with dimension (n-k)×(n-k). Since det(In-k-αE¯2)=1, it is invertible. Simplifying (8) by premultiplying the block diagonal {E¯1-1,β(In-k-αE¯2)-1} on both sides, one has
(9)[IkOOE-f]x-˙c(t)=[A-sOOIn-κ]x-c(t)+∑i=1N1[A~1,iOOA~2,i]x-c(t-τs,i)+∑j=1N2[B-s,jB-f,j]uc(t-τi,j),
where
(10)E-f=β(In-κ-αE-2)-1E-2,A¯s=1β(E-1-1-αIκ),A~1,i=E-1-1A-1,i,A~2,i=β(In-κ-αE¯2)-1A-2,i,B-s,j=E-1-1B-1,j,B-f,j=β(In-κ-αE-2)-1B-2,j.Remarkably, since(11)rank(E)-deg{det(sE-A)}=rank(E-f),
it is much easier to determine the number of the impulsive mode using the above equation relating to (9).For simplicity, only those singular systems that include at least one impulsive mode are discussed. First, assume that the singular system (9) has q; then, rank(E-f)=q. By a previously proposed method [12], the preliminary feedback gain Kf,j is found and Kf,j is proven to eliminate the impulsive modes. For the time-delay singular system (9), the proposed method yields a similar result (Appendix A) to that previously developed method [12] and the linear preliminary feedback control is
(12)uc(t-τi,j)=-Kf,jx^c,f(t)+vc(t-τi,j)=-[Om×k,Kf,j]x^c(t)+vc(t-τi,j).
The time-delay singular system (9) can be transformed into (Appendix A)
(13)Ekx^˙c(t)=Akx^c(t)+∑i=1N1A^k,ix^c(t-τs,i)+∑j=1N2Bk,jvc(t-τi,j),
where
(14)Ek=[IkOOE^f],Ak=[A^s-∑j=1N2B^s,jKf,jOIn-k-∑j=1N2B^f,jKf,j],A^k,i=[A~1,iOOA~2,i],Bk,j=[B^s,jB^f,j],
in which
(15)E^f=U-1E-fU,A^s=A-s,A~1,i=E-1-1A-1,i,A~2,i=β(In-κ-αE¯2)-1A-2,i,B^s,j=B-s,j,B^f,j=U-1B-f,j,
and U is a modal matrix of E-f with dimension (n-k)×(n-k) such that U-1E-fU is in the Jordan block form. The time-delay singular system in (13) is obtained by applying the linear preliminary feedback control law u(t) from (12) to the system that is given by (9). Equation (13) has the q finite modes (where q=rank(E-f)=rank(E^f)) and the k original finite modes.All of these finite modes are guaranteed to be controllable. The next task is to decompose the singular system into a reduced-order regular system with (k+q) controllable finite modes and the nondynamic equation with (n-k-q) infinite nondynamic ones. This task can be accomplished by using previously outlined steps. First, the regular form is transformed into a standard one by multiplying (13) by (γEk+ηAk)-1, where γ and η are arbitrary scalars such that (γEk+ηAk) is invertible. Therefore,
(16)(γEk+ηAk)-1Ekx^˙c(t)=(γEk+ηAk)-1Akx^c(t)+∑i=1N1(γEk+ηAk)-1A^k,ix^c(t-τs,i)+∑j=1N2(γEk+ηAk)-1Bk,jvc(t-τi,j).
Let
(17)x^c(t)=M~x~c(t),
where the constant matrix M~ is determined by using the extended matrix sign function. The procedure is the same as that elucidated above for finding M, except that it operates on (γEk+ηAk)-1Ek. Substituting (17) into (16) and multiplying by M~-1 yield
(18)M~-1(γEk+ηAk)-1EkM~x~˙c(t)=M~-1(γEk+ηAk)-1AkM~x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j)=M~-11η[In-γ(γEk+ηAk)-1Ek]M~x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j)=1η[In-γM~-1(γEk+ηAk)-1EkM~]x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j).
That is,
(19)[E-skOOE-fk]x~˙c(t)=[1η(Ik+q-γE-sk)OO1η(In-k-q-γE-fk)]x~c(t)+∑i=1N1[Λ1,iOOΛ2,i]x~c(t-τs,i)+∑j=1N2[B-sk,jB-fk,j]vc(t-τi,j),
where x~c(t)=[x~sT(t),x~fT(t)]T, M~-1(γEk+ηAk)-1EkM~=blockdiagonal{E-sk,E-fk} = blockdiagonal{E-sk,O(n-q-k)}. E-sk is invertible with rank(E-sk)=deg{det(sEk-Ak)}=(q+k). E-fk is a null matrix and [B-sk,jT,B-fk,jT]T=M~-1(γEk+ηAk)-1Bk,j. In (19), M~-1(γEk+ηAk)-1A^k,i is assumed to be able to be diagonalized as blockdiagonal{Λ1,i,Λ2,i}. Then, (19) can be rewritten as(20a)x~˙s(t)=1η(E-sk-1-γIk+q)x~s(t)+∑i=1N1E-sk-1Λ1,ix~s(t-τs,i)+∑j=1N2E-sk-1B-sk,jvc(t-τi,j),(20b)0=x~f(t)+∑i=1N1ηΛ2,ix~f(t-τs,i)+∑j=1N2ηB-fk,jvc(t-τi,j),and the time-delay singular system output (1b) can be rewritten as (Appendix B)(21a)yc(t)=Cxc(t-τo)=[C1C2][x~s(t-τo)x~f(t-τo)]=C1x~s(t-τo)+C2x~f(t-τo)=C1x~s(t-τo)-C2∑j=1N2B-fk,jvc(t-τi,j),whereCMVM~=[C1C2].Finally, the time-delay singular system (1a) and (1b) can be decomposed as the equivalent regular time-delay system as follows:(22a)x~˙s(t)=Asx~s(t)+∑i=1N1A^d,ix~s(t-τs,i)+∑j=1N2Bd,jvc(t-τi,j),(22b)yc(t)=C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j),where(23)As=1η(E-sk-1-γIk+q),A^d,i=E-sk-1Λ1,i,Bd,j=E-sk-1B-sk,j,Dj=C2B-fk,j.Following the transformation, the time-delay singular system (1a) and (1b) can be converted into a regular system (22a) and (22b) that contains a direct transmission term from input to output and the impulsive mode can be eliminated by means of the method [12]. In the next section, (22a) and (22b) will be used to develop the new optimal tracker and observer for a time-delay singular system (1a) and (1b) with a series of time-delays. The proposed approaches are more general and applicable to actual systems.


### 3.2. Based on Digital Redesign and Optimal Control to Discretize the Continuous Time-Delay Singular System and Construct the Performance Index

#### 3.2.1. Discretization of Continuous Time-Delay Singular System

Consider the continuous time-delay singular system (22a) and (22b). To discretize (22a) and (22b), assume that vc(t) is a piecewise constant input function:
(24)vc(t)=vd(kT),kT≤t<(k+1)T,
where T is the sampling period. Let the state delay time be given by τs,i=ρiT+Γi, where 0≤Γi<T and ρi≥0 is an integer, and let the input delay time be given by τi,j=ηjT+σj, where 0≤σj<T and ηj≥0 is an integer. The time-delay singular system (22a) and (22b), by both the Newton extrapolation method and the Chebyshev quadrature method [25, 26], becomes
(25)x~ds((k+1)T)=Gx~ds(kT)+∑i=1N1[G^i(1)x~ds(kT-ρiT+T)+G^i(2)x~ds(kT-ρiT)+G^i(3)x~ds(kT-ρiT-T)]+∑j=1N2[Hj(0)vd(kT-ηjT)+Hj(1)vd(kT-ηjT-T)],
where
(26)G=eAsT,G^i(1)=T2[Qi(2)+Qi(3)]A^d,i,G^i(2)=T[Qi(1)-Qi(3)]A^d,i,G^i(3)=T2[Qi(3)-Qi(2)]A^d,i,Hj(0)=[G1-γj-In]As-1Bd,j,Hj(1)=[G-G1-γj]As-1Bd,j,
in which
(27)γj=σjT,βi=ΓiT,Qi(1)=[G-In](AsT)-1,Qi(2)=[Qi(1)-(1-βi)In-βiG](AsT)-1,Qi(3)=[2Qi(2)-(1-βi)2In-βi2G](AsT)-1.
Some terms in (25) may be combined because of the same delay, so (25) can be reduced to
(28)x~ds((k+1)T)=G-x~ds(kT)+∑i=1M1G-ix~ds(kT-iT)+H-vd(kT)+∑j=1M2H-jvd(kT-jT).
For the output (22b), the time-delay state xc(t-τo) for kT≤t-τo<(k+1)T must be evaluated. System (22a) and (22b) can be rewritten as
(29)x~s(t-τo)=eAs(t-τo-kT)x~ds(kT)+∑i=1N1∫kTt-τoeAs(t-τo-λ)A^d,ix~ds(λ-τs,i)dλ+∑j=1N2∫kTt-τoeAs(t-τo-λ)Bd,jvd(λ-τi,j)dλ=δ1(t-τo-kT)x~ds(kT)+∑i=1N1[δ^i(1)(t-τo-kT)x~ds(kT-ρiT+T)xxxxxxx+δ^i(2)(t-τo-kT)x~ds(kT-ρiT)xxxxxxx+δ^i(3)(t-τo-kT)xxxxxxx×x~ds(kT-ρiT-T)δ^i(1)]+∑j=1N2[φj(0)(t-τo-kT)vd(kT-ηjT)xxxxxxx+φj(1)(t-τo-kT)xxxxxxx×vd(kT-ηjT-T)δ^i(1)],
where
(30)δ1(t-τo-kT)=eAs(t-τo-kT),δ^i(1)(t-τo-kT)=T2[qi(2)+qi(3)]A^d,i,δ^i(2)(t-τo-kT)=T[qi(1)-qi(3)]A^d,i,δ^i(3)(t-τo-kT)=T2[qi(3)-qi(2)]A^d,i,φj(0)(t-τo-kT)={On×m,t-τo<σj,[e-Asσjδ1(t-τo-kT)-In]As-1Bd,j,t-τo≥σj,φj(1)(t-τo-kT)={[δ1(t-τo-kT)-In]As-1Bd,j,t-τo<σjδ1(t-τo-kT)[In-e-Asσj]As-1Bd,j,t-τo≥σj,
in which
(31)qi(1)=[δ1(t-τo-kT)-In](AsT)-1,qi(2)=[qi(1)-(t-τo-kTT-βi)In-βiδ1(t-τo-kT)(t-τo-kTT-βi)](AsT)-1,qi(3)=[2qi(2)-(t-τo-kTT-βi)2In+βi2δ1(t-τo-kT)(t-τo-kTT-βi)2](AsT)-1.
Also, some terms in (29) may be combined as in (28), and (29) may be rewritten as
(32)x~s(t-τo)=δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+φ-0(t-τo-kT)vd(kT)+∑j=1M2φ-j(t-τo-kT)vd(kT-jT).
Then, the output (22b) can be rewritten as
(33)yc(t)=C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0(t-τo-kT)vd(kT)+∑j=1M2C1φ-j(t-τo-kT)vd(kT-jT)-∑j=1N2[Dj(0)vd(kT-ηjT)xxxxx+Dj(1)vd(kT-ηjT-T)],
where
(34)Dj(0)=Dj*(Bd,jTBd,j)-1Hj(0),Dj(1)=Dj*(Bd,jTBd,j)-1Hj(1),Dj*=[DjO]T.Similarly, some terms in (33) can be combined, so (33) can be rewritten as
(35)yc(t)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0*(t-τo-kT)vd(kT)+∑j=1M2C1φ-j*(t-τo-kT)vd(kT-jT).
Thus, the discretization of continuous time-delay singular system (22a) and (22b) is carried out using (28) and (35).


#### 3.2.2. Establishing Performance Index for Discrete Time-Delay Singular System

The optimal state-feedback control law minimizes the following performance cost function:(36)J=∫0tf{[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]Txxxx×Q[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]xxxx+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T},
where Q is the positive semidefinite matrix, R is the positive definite matrix, r(t)∈ℜq is the reference input vector, and the final time tf<∞. To discretize the cost function J, given by (36), tf=NT is chosen and J can be rewritten as
(37)J=∑k=0N-1∫kT(k+1)T{[∑j=1N2Djvc(t-τi,j)-r(t)]TC1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T×Q[∑j=1N2Djvc(t-τi,j)C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)∑j=1N2Djvc(t-τi,j)]+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T}.
Let r*(kT)∈ℜq be the piecewise-constant reference input vector to be determined in terms of r(kT) for the tracking purpose. Then, cost function (37) can be rewritten as [27] (38)J=∑k=0N-1∫kT(k+1)T{∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)[∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]T×Q[δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]+vdT(kT)Rvd(kT)dt}=⋯=∑k=0N-1[x~dsT(kT)x~dsT(kT-T)⋯x~dsT(kT-M1T)vdT(kT-T)⋯vdT(kT-M2T)r*T(kT)vdT(kT)]×[Q1Q31⋯Q3M1M21⋯M2M2-M3M1Q31TQ211⋯Q21M1M511⋯M51M2-M61M41⋮⋮⋱⋮⋮⋱⋮⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M4M1M21TM511T⋯M5M11TR211⋯R21M2-M81R41⋮⋮⋱⋮⋮⋱⋮⋯⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2R4M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3-M7M1TM41T⋯M4M1TR41T⋯R4M2T-M7TR1]×[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)vd(kT)]T,where
(39)Q1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-0(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q2ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1δ-j(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q3i=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-i(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M2j=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M3=∫kT(k+1)T[(C1δ-0(t-τo-kT))TQ]dt,M4i=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M5ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M6i=∫kT(k+1)T[(C1δ-i(t-τo-kT))TQ]dt,M7=∫kT(k+1)T[Q(C1φ-0*(t-τo-kT))]dt,M8j=∫kT(k+1)T[(C1φ-j*(t-τo-kT))TQ]dt,R1=∫kT(k+1)T[(C1φ-0*(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))+R(C1δ-0(t-τo-kT))T]dt,R2ij=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,R3=∫kT(k+1)TQdt,R4j=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt.
Construct an extended virtual state vector:(40)Xd(kT)=[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)]T.The extended delay-free system that is equivalent to the original time-delay singular system (28) and (35) is obtained as (41a)Xd((k+1)T)=G^eXd(kT)+H^evd(kT),(41b)yd(kT)=C^eXd(kT).We assume that the reference input r(t) is a step function with a constant magnitude, r*((k+1)T)=r*(kT). Designing a system based on such a reference input can lead to predictable time-response characteristics. Although our design methodology is based on a step function, it should be pointed out that the resulting control system, if properly designed, enables to give good time responses for any arbitrary reference input r(t). Also, the reference input r*(kT) is entered in the last row of Xd(kT) at the beginning of step k. As a result, the extended new system does not have any time-delay terms and it can be utilized to simplify the representation of the cost function (38). Now, (38) can be rewritten as
(42)J=∑k=0N-1[XdT(kT)vdT(kT)][Q^M^M^TR^][Xd(kT)vd(kT)],
where(43)[Q1Q31⋯Q3M1M21⋯M2M2-M3Q31TQ211⋯Q21M1M511⋯M51M2-M61⋮⋮⋱⋮⋮⋱⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M21TM511T⋯M5M11TR211⋯R21M2-M81⋮⋮⋱⋮⋮⋱⋮⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3],M^=[M1TM41T⋯M4M1TR41T⋯R4M2T-M7T]T,R^=[R1].Then, define a new virtual weighting matrix
(44)Q~=Q^-M^R^-1M^T
and a new virtual control input
(45)S(kT)=R^-1M^TXd(kT)+vd(kT).
Substituting (44) and (45) into (42) results in a decoupled performance index:
(46)J=∑k=0N-1[XdT(kT)Q~Xd(kT)+ST(kT)R^S(kT)].
Substituting (45) into the extended delay-free singular system (41a) and (41b) yields
(47)Xd((k+1)T)=G^eXd(kT)+H^evd(kT)=G^eXd(kT)+H^e[S(kT)-R^-1M^TXd(kT)]=G~eXd(kT)+H^eS(kT),
where G~e=G^e-H^eR^-1M^T.Notably, the quadratic optimal control of the system that is given by (41a) and (41b) with the performance index that is given by (42) is equivalent to the quadratic optimal control of the system that is given by (47) with the performance index that is given by (46). The development of the desired optimal virtual control vector S(kT) that minimizes the performance index that is given by (46) can be described as follows.


### 3.3. Development of Optimal Tracker for Time-Delay Singular System with States Available

Let the Hamilton function depend on the cost function (46) [28]:
(48)Hf(kT)=[XdT(kT)Q~Xd(kT)+ST(kT)R^S(kT)]+λT((k+1)T)[G~eXd(kT)+H^eS(kT)],
where λ(kT) is a costate (Lagrange multiplier). Based on the well-developed optimal control theory [29, 30], the state equation is
(49)Xd((k+1)T)=G~eXd(kT)+H^eS(kT),
and the costate equation is
(50)λ(kT)=G~eTλ((k+1)T)+Q~Xd(kT)
with the stationary condition
(51)0=H^eTλ((k+1)T)+R^S(kT),
or(52a)S(kT)=-R^-1H^eTλ((k+1)T),
and the boundary condition is
(52b)λ(NT)=Q~Xd(NT).Assume that λ(kT) can be written as follows:
(53)λ(kT)=P(kT)Xd(kT),
where P(kT) is a real symmetric matrix of appropriate dimension. So far, the original optimal tracking problem has been transformed into an optimal regulator problem.To derive the optimal regulator, (53) is substituted into (50):
(54)P(kT)Xd(kT)=G~eTP((k+1)T)Xd((k+1)T)+Q~Xd(kT),
and (52a), (52b), and (53) are substituted into (49):
(55)Xd((k+1)T)=G~eXd(kT)+H^eR^-1H^eTλ((k+1)T)=G~eXd(kT)+H^eR^-1H^eTP((k+1)T)×Xd((k+1)T)
or
(56)Xd((k+1)T)=[I+H^eR^-1H^eTP((k+1)T)]-1G~eXd(kT).
Also, substituting (56) into (54) yields
(57)P(kT)Xd(kT)=G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1×G~eXd(kT)+Q~Xd(kT)
or
(58){[I+H^eR^-1H^eTP((k+1)T)]-1P(kT)-G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~e-Q~[I+H^eR^-1H^eTP((k+1)T)]-1}Xd(kT)=0.
The above equation must hold for all Xd(kT). Hence,
(59)P(kT)=Q~+G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~e.
Equation (59) is called the Riccati equation. With reference to (52a), (52b), and (53), when at k=N,
(60)λ(NT)=Q~Xd(NT)=P(NT)Xd(NT)
or
(61)P(NT)=Q~.
From (59) and (61), all P(kT) for 0≤k≤N can be obtained. With reference to (53) and (56), the desired optimal virtual control input that is given by (52a) now becomes
(62)S(kT)=-R^-1H^eTλ((k+1)T)=-R^-1H^eTP((k+1)T)Xd((k+1)T)=-R^-1H^eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~eXd(kT)=-R^-1H^eT[P-1((k+1)T)+H^eR^-1H^eT]-1×G~eXd(kT)=-K(kT)Xd(kT),
where K(kT)=R^-1H^eT[P-1((k+1)T)+H^eR^-1H^eT]-1G~e. From (45), the original optimal controller vd(kT) is obtained as follows:
(63)vd(kT)=S(kT)-R^-1M^TXd(kT)=-K^(kT)Xd(kT),
where K^(kT)=K(kT)+R^-1M^T. Notice that if there are no state and input time delays, the above development can be reduced to that in [30]. Equation (63) can be represented in the following form:
(64)vd(kT)=-K^(kT)Xd(kT)=-Kd(kT)×[x~dsT(kT)x~dsT(kT-T)⋯x~dsT(kT-M1T)]T-Fd(kT)[vdT(kT-T)⋯vdT(kT-M2T)]T-Ed(kT)r*(kT),
where K^(kT)=[Kd(kT)Fd(kT)Ed(kT)], in which
(65)Kd(kT)=[Kd(0)(kT)Kd(1)(kT)⋯Kd(M1)(kT)],Fd(kT)=[Fd(1)(kT)⋯Fd(M2)(kT)].
Here, the discrete optimal controller (64) for the continuous time-delay system (22a) and (22b) has been completely derived. Figure 1 presents the realization of the time-varying piecewise-constant optimal controller (64) for the digitally controlled continuous time-delay singular system.Figure 1
Digital redesign for time-delay singular system.


### 3.4. Development of Observer-Based Suboptimal Tracker for Time-Delay Singular System with States Unavailable

When the states of a continuous time-delay system (22a) and (22b) are not available for measurement, the continuous-time states can be constructed using the recently developed continuous time-delay observers [28, 31, 32]. However, the developed digital tracker (64) requires the extended discrete-time state Χd(kT) in (41a) and (41b). Using the prediction-based digital redesign [27], a new observer-based suboptimal tracker for the time-delay singular system can be designed as follows.According to the digitally redesigned observer [27] and controller [27], the extended digitally redesigned observer and controller can be represented as(66a)X^d((k+1)T)=G^oX^d(kT)+H^ovd(kT)+Ld[yd(kT)-C^eX^d(kT)],(66b)vd(kT)=-K^(kT)X^d(kT),whereΧ^d(kT)∈ℜp is the estimate of the extended state Χd(kT)∈ℜp in (41a) and (41b),
(66c)G^o=G^e-LdC^eG^e,(66d)H^o=H^e-LdC^eH^e.Additionally,G^e=eA^T and H^e=[G^e-Ip]A^-1B^, where A^=(1/T)ln(G^e) and B^=A^(G^e-I)-1H^e. To determine the extended observer gain Ld in (66a), the equivalent extended continuous-time observer (41a) and (41b) and controller (64) can be represented [19], on the basis of the analog observer and controller, as(67a)X^˙c(t)=A^X^c(t)+B^vc(t)+Lc[yc(t)-C^eX^c(t)],(67b)vc(t)=-K^cX^c(t).The algorithm for computing the analog system matrixA^ in (67a) from the digital system matrix G^e in (41a) via the geometric-series method [15] is as follows:
(67c)A^=1Tln(G^e)=2T{R^+13R^3+⋯+1nR^n[Ip-1(1+2/n)R^2]-1}xxxxxxxxxxxxxxxxxxxfor|σ(R^2)|<(1+2n)≅2TR^[Ip-13R^2]-1forn=1≅2TR^[Ip-415R^2][Ip-35R^2]-1forn=3≅⋯,where R^=[G^e-Ip][G^e+Ip]-1 and |σ(∘)| denotes the absolute eigenspectrum of (∘). Based on the dual concept of the digital redesign, the analog observer gain Lc in (67a) and the digitally redesigned observer gain Ld in (66a) can be represented, respectively, as
(68)Lc=PobC^eTR-1,(69)Ld=(G^e-Ip)A^-1Lc[I+C^e(G^e-Ip)A^-1Lc]-1,
where Pob is the positive-definite and symmetric solution of the following Riccati equation:
(70)A^Pob+PobA^T-PobC^eTR-1C^ePob+C^eTQC^e=0,
in which Q≥0 and R>0 with appropriate dimensions.Owing to the extended virtual state vector in (40), the matrix G^e in (41a) and (41b) and (67a), (67b), and (67c) is singular. The matrices A^ and B^ in (67a), (67b), and (67c) cannot be directly determined. To solve this problem, an alternative is derived via the matrix sign function method [23, 24] as follows.Following the procedures shown in Section3 [23, 24], the transformed matrix is
(71)G^b=(G^e-ζ2Ip)(G^e+ζ2Ip)-1,
where G^e∈ℜp×p and ζ2 is a radius of a circle from the origin of the coordinates. Additionally, the associated matrix sign functions are
(72)Sign(G^b)=G^b(G^b22)-1,(73a)Sign-(G^b)=12(Ip-Sign(G^b)),(73b)Sign+(G^b)=12(Ip+Sign(G^b)),respectively. A fast and stable algorithm for computing the matrix sign function [23, 24] is given as follows.For the order of the desired convergence rater=2, one has
(74)Q(l+1)=12[Q(l)+Q-1(l)],Q(0)=G^b,liml→∞Q(l)=Sign(G^b),forl=0,1,2,….
By [19, 20], a transformation matrix Tm can be found such that
(75)G^m=Tm-1G^eTm=[G^m10(p-g)×g0g×(p-g)G^m2],
where G^m1 is a nonsingular matrix and G^m2 is a singular matrix whose eigenvalues are all null. Finally, the matrix A^ is obtained by the following equation:(76a)A^=TmG^lnTm-1=Tm[G^ln100G^ln2]Tm-1,
where G^ln1=(1/T)ln(G^m13)3=(3/T)ln(G^m13) and G^ln2=νIg, in which ν is a large negative real constant. The algorithm for finding ln(G^m13) in (76a) can be found in (67c) [19]. If the matrix G^m1 has any negative real eigenvalue, then the principal third root of G^m1 is not defined for arg(λi)≠π [23, 24]. The first part G^ln1 in (76a) can be rewritten as(76b)G^ln1=3Tln(G^m13)=3Tln[(G^m1eiθ3)e-(iθ/3)],where the matrixG^m1 is rotated by a small positive real angle θ. The second part G^ln2 in (76a) is utilized to recover the property ln(0)=-∞. Matrix B^ can be evaluated as
(77)B^=A^(G^e-Ip)-1H^e=[(G^e-Ip)A^-1]-1H^e=(TIp+A^T22!+A^2T33!+⋯)-1H^e.
Substituting (76a) and (76b) into the Riccati equation (70) and solving it yield the observer gain matrices in (68) and (69). Figure 2 presents the implementation of the observer-based suboptimal tracker for the time-delay singular system.Figure 2
Observer-based suboptimal tracker for the digitally redesigned time-delay singular system.


## 3.1. Decomposition of Time-Delay Singular System

By (2a) and (2b) the regular pencil (sE-A) can be transformed into a standard pencil (sEn-An). Notably since En is a singular matrix, which has at least one zero eigenvalue, β cannot be equal to zero. Hence, multiplying (1a) by (αE+βA)-1 can yield the following equation:
(3)Enx˙c(t)=Anxc(t)+∑i=1N1A^n,ixc(t-τs,i)+∑j=1N2Bn,juc(t-τi,j),
where
(4)En=(αE+βA)-1E,A^n,i=(αE+βA)-1A^i,Bn,j=(αE+βA)-1Bj.
Since αEn+βAn=In, the pencil (sEn-An) is a standard one, and has the properties that are mentioned in Lemma 3. To decompose system (3), the state xc(t) is converted into x-c(t) by
(5)xc(t)=Mx-c(t),
where the constant matrix M is a block modal matrix of En and determined by means of the extended matrix sign function [23, 24]. The M matrix of state space transformation is as follows.Step 1.
Findsign(E~n) using the extended matrix sign function with an adequate ω, where
(6)E~n=(En-ωIn)(En+ωIn)-1.Step 2.
Findsign+(E~n)=(1/2)[In+sign(E~n)] and sign-(E~n)=(1/2)[In-sign(E~n)].Step 3.
Construct the matrixM=[ind(sign+(E~n))ind(sign-(E~n))], where ind(·) represents the collection of linearly independent column vectors of (·).Substituting (5) into (3) and multiplying by M-1 on the left yield
(7)M-1EnMx-˙c(t)=M-1AnMx-c(t)+∑i=1N1M-1A^n,iMx-c(t-τs,i)+∑j=1N2M-1Bn,juc(t-τi,j)=1β(In-αEn)x-c(t)+∑i=1N1M-1A^n,iMx-c(t-τs,i)+∑j=1N2M-1Bn,juc(t-τi,j).
If M-1A^n,iM can be diagonalized, then (7) yields,
(8)[E-1OE-2]x-˙c(t)=[1β(Ik-αE-1)OO1β(In-κ-αE-2)]x-c(t)+∑i=1N1[A-1,iOOA-2,i]x-c(t-τs,i)+∑j=1N2[B-1,jB-2,j]uc(t-τi,j),
where x-c(t)=[x-sT(t),x-fT(t)]T and M-1EnM=blockdiagonal{E-1,E-2}. E-1 is invertible with rank(E-1)=deg{det(sEr-A)}≜k, [B-1,jT,B-2,jT]T=M-1Bn,j, and E-2 is a nilpotent matrix with dimension (n-k)×(n-k). Since det(In-k-αE¯2)=1, it is invertible. Simplifying (8) by premultiplying the block diagonal {E¯1-1,β(In-k-αE¯2)-1} on both sides, one has
(9)[IkOOE-f]x-˙c(t)=[A-sOOIn-κ]x-c(t)+∑i=1N1[A~1,iOOA~2,i]x-c(t-τs,i)+∑j=1N2[B-s,jB-f,j]uc(t-τi,j),
where
(10)E-f=β(In-κ-αE-2)-1E-2,A¯s=1β(E-1-1-αIκ),A~1,i=E-1-1A-1,i,A~2,i=β(In-κ-αE¯2)-1A-2,i,B-s,j=E-1-1B-1,j,B-f,j=β(In-κ-αE-2)-1B-2,j.Remarkably, since(11)rank(E)-deg{det(sE-A)}=rank(E-f),
it is much easier to determine the number of the impulsive mode using the above equation relating to (9).For simplicity, only those singular systems that include at least one impulsive mode are discussed. First, assume that the singular system (9) has q; then, rank(E-f)=q. By a previously proposed method [12], the preliminary feedback gain Kf,j is found and Kf,j is proven to eliminate the impulsive modes. For the time-delay singular system (9), the proposed method yields a similar result (Appendix A) to that previously developed method [12] and the linear preliminary feedback control is
(12)uc(t-τi,j)=-Kf,jx^c,f(t)+vc(t-τi,j)=-[Om×k,Kf,j]x^c(t)+vc(t-τi,j).
The time-delay singular system (9) can be transformed into (Appendix A)
(13)Ekx^˙c(t)=Akx^c(t)+∑i=1N1A^k,ix^c(t-τs,i)+∑j=1N2Bk,jvc(t-τi,j),
where
(14)Ek=[IkOOE^f],Ak=[A^s-∑j=1N2B^s,jKf,jOIn-k-∑j=1N2B^f,jKf,j],A^k,i=[A~1,iOOA~2,i],Bk,j=[B^s,jB^f,j],
in which
(15)E^f=U-1E-fU,A^s=A-s,A~1,i=E-1-1A-1,i,A~2,i=β(In-κ-αE¯2)-1A-2,i,B^s,j=B-s,j,B^f,j=U-1B-f,j,
and U is a modal matrix of E-f with dimension (n-k)×(n-k) such that U-1E-fU is in the Jordan block form. The time-delay singular system in (13) is obtained by applying the linear preliminary feedback control law u(t) from (12) to the system that is given by (9). Equation (13) has the q finite modes (where q=rank(E-f)=rank(E^f)) and the k original finite modes.All of these finite modes are guaranteed to be controllable. The next task is to decompose the singular system into a reduced-order regular system with (k+q) controllable finite modes and the nondynamic equation with (n-k-q) infinite nondynamic ones. This task can be accomplished by using previously outlined steps. First, the regular form is transformed into a standard one by multiplying (13) by (γEk+ηAk)-1, where γ and η are arbitrary scalars such that (γEk+ηAk) is invertible. Therefore,
(16)(γEk+ηAk)-1Ekx^˙c(t)=(γEk+ηAk)-1Akx^c(t)+∑i=1N1(γEk+ηAk)-1A^k,ix^c(t-τs,i)+∑j=1N2(γEk+ηAk)-1Bk,jvc(t-τi,j).
Let
(17)x^c(t)=M~x~c(t),
where the constant matrix M~ is determined by using the extended matrix sign function. The procedure is the same as that elucidated above for finding M, except that it operates on (γEk+ηAk)-1Ek. Substituting (17) into (16) and multiplying by M~-1 yield
(18)M~-1(γEk+ηAk)-1EkM~x~˙c(t)=M~-1(γEk+ηAk)-1AkM~x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j)=M~-11η[In-γ(γEk+ηAk)-1Ek]M~x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j)=1η[In-γM~-1(γEk+ηAk)-1EkM~]x~c(t)+∑i=1N1M~-1(γEk+ηAk)-1A^k,iM~x~c(t-τs,i)+∑j=1N2M~-1(γEk+ηAk)-1Bk,jvc(t-τi,j).
That is,
(19)[E-skOOE-fk]x~˙c(t)=[1η(Ik+q-γE-sk)OO1η(In-k-q-γE-fk)]x~c(t)+∑i=1N1[Λ1,iOOΛ2,i]x~c(t-τs,i)+∑j=1N2[B-sk,jB-fk,j]vc(t-τi,j),
where x~c(t)=[x~sT(t),x~fT(t)]T, M~-1(γEk+ηAk)-1EkM~=blockdiagonal{E-sk,E-fk} = blockdiagonal{E-sk,O(n-q-k)}. E-sk is invertible with rank(E-sk)=deg{det(sEk-Ak)}=(q+k). E-fk is a null matrix and [B-sk,jT,B-fk,jT]T=M~-1(γEk+ηAk)-1Bk,j. In (19), M~-1(γEk+ηAk)-1A^k,i is assumed to be able to be diagonalized as blockdiagonal{Λ1,i,Λ2,i}. Then, (19) can be rewritten as(20a)x~˙s(t)=1η(E-sk-1-γIk+q)x~s(t)+∑i=1N1E-sk-1Λ1,ix~s(t-τs,i)+∑j=1N2E-sk-1B-sk,jvc(t-τi,j),(20b)0=x~f(t)+∑i=1N1ηΛ2,ix~f(t-τs,i)+∑j=1N2ηB-fk,jvc(t-τi,j),and the time-delay singular system output (1b) can be rewritten as (Appendix B)(21a)yc(t)=Cxc(t-τo)=[C1C2][x~s(t-τo)x~f(t-τo)]=C1x~s(t-τo)+C2x~f(t-τo)=C1x~s(t-τo)-C2∑j=1N2B-fk,jvc(t-τi,j),whereCMVM~=[C1C2].Finally, the time-delay singular system (1a) and (1b) can be decomposed as the equivalent regular time-delay system as follows:(22a)x~˙s(t)=Asx~s(t)+∑i=1N1A^d,ix~s(t-τs,i)+∑j=1N2Bd,jvc(t-τi,j),(22b)yc(t)=C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j),where(23)As=1η(E-sk-1-γIk+q),A^d,i=E-sk-1Λ1,i,Bd,j=E-sk-1B-sk,j,Dj=C2B-fk,j.Following the transformation, the time-delay singular system (1a) and (1b) can be converted into a regular system (22a) and (22b) that contains a direct transmission term from input to output and the impulsive mode can be eliminated by means of the method [12]. In the next section, (22a) and (22b) will be used to develop the new optimal tracker and observer for a time-delay singular system (1a) and (1b) with a series of time-delays. The proposed approaches are more general and applicable to actual systems.


## 3.2. Based on Digital Redesign and Optimal Control to Discretize the Continuous Time-Delay Singular System and Construct the Performance Index

### 3.2.1. Discretization of Continuous Time-Delay Singular System

Consider the continuous time-delay singular system (22a) and (22b). To discretize (22a) and (22b), assume that vc(t) is a piecewise constant input function:
(24)vc(t)=vd(kT),kT≤t<(k+1)T,
where T is the sampling period. Let the state delay time be given by τs,i=ρiT+Γi, where 0≤Γi<T and ρi≥0 is an integer, and let the input delay time be given by τi,j=ηjT+σj, where 0≤σj<T and ηj≥0 is an integer. The time-delay singular system (22a) and (22b), by both the Newton extrapolation method and the Chebyshev quadrature method [25, 26], becomes
(25)x~ds((k+1)T)=Gx~ds(kT)+∑i=1N1[G^i(1)x~ds(kT-ρiT+T)+G^i(2)x~ds(kT-ρiT)+G^i(3)x~ds(kT-ρiT-T)]+∑j=1N2[Hj(0)vd(kT-ηjT)+Hj(1)vd(kT-ηjT-T)],
where
(26)G=eAsT,G^i(1)=T2[Qi(2)+Qi(3)]A^d,i,G^i(2)=T[Qi(1)-Qi(3)]A^d,i,G^i(3)=T2[Qi(3)-Qi(2)]A^d,i,Hj(0)=[G1-γj-In]As-1Bd,j,Hj(1)=[G-G1-γj]As-1Bd,j,
in which
(27)γj=σjT,βi=ΓiT,Qi(1)=[G-In](AsT)-1,Qi(2)=[Qi(1)-(1-βi)In-βiG](AsT)-1,Qi(3)=[2Qi(2)-(1-βi)2In-βi2G](AsT)-1.
Some terms in (25) may be combined because of the same delay, so (25) can be reduced to
(28)x~ds((k+1)T)=G-x~ds(kT)+∑i=1M1G-ix~ds(kT-iT)+H-vd(kT)+∑j=1M2H-jvd(kT-jT).
For the output (22b), the time-delay state xc(t-τo) for kT≤t-τo<(k+1)T must be evaluated. System (22a) and (22b) can be rewritten as
(29)x~s(t-τo)=eAs(t-τo-kT)x~ds(kT)+∑i=1N1∫kTt-τoeAs(t-τo-λ)A^d,ix~ds(λ-τs,i)dλ+∑j=1N2∫kTt-τoeAs(t-τo-λ)Bd,jvd(λ-τi,j)dλ=δ1(t-τo-kT)x~ds(kT)+∑i=1N1[δ^i(1)(t-τo-kT)x~ds(kT-ρiT+T)xxxxxxx+δ^i(2)(t-τo-kT)x~ds(kT-ρiT)xxxxxxx+δ^i(3)(t-τo-kT)xxxxxxx×x~ds(kT-ρiT-T)δ^i(1)]+∑j=1N2[φj(0)(t-τo-kT)vd(kT-ηjT)xxxxxxx+φj(1)(t-τo-kT)xxxxxxx×vd(kT-ηjT-T)δ^i(1)],
where
(30)δ1(t-τo-kT)=eAs(t-τo-kT),δ^i(1)(t-τo-kT)=T2[qi(2)+qi(3)]A^d,i,δ^i(2)(t-τo-kT)=T[qi(1)-qi(3)]A^d,i,δ^i(3)(t-τo-kT)=T2[qi(3)-qi(2)]A^d,i,φj(0)(t-τo-kT)={On×m,t-τo<σj,[e-Asσjδ1(t-τo-kT)-In]As-1Bd,j,t-τo≥σj,φj(1)(t-τo-kT)={[δ1(t-τo-kT)-In]As-1Bd,j,t-τo<σjδ1(t-τo-kT)[In-e-Asσj]As-1Bd,j,t-τo≥σj,
in which
(31)qi(1)=[δ1(t-τo-kT)-In](AsT)-1,qi(2)=[qi(1)-(t-τo-kTT-βi)In-βiδ1(t-τo-kT)(t-τo-kTT-βi)](AsT)-1,qi(3)=[2qi(2)-(t-τo-kTT-βi)2In+βi2δ1(t-τo-kT)(t-τo-kTT-βi)2](AsT)-1.
Also, some terms in (29) may be combined as in (28), and (29) may be rewritten as
(32)x~s(t-τo)=δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+φ-0(t-τo-kT)vd(kT)+∑j=1M2φ-j(t-τo-kT)vd(kT-jT).
Then, the output (22b) can be rewritten as
(33)yc(t)=C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0(t-τo-kT)vd(kT)+∑j=1M2C1φ-j(t-τo-kT)vd(kT-jT)-∑j=1N2[Dj(0)vd(kT-ηjT)xxxxx+Dj(1)vd(kT-ηjT-T)],
where
(34)Dj(0)=Dj*(Bd,jTBd,j)-1Hj(0),Dj(1)=Dj*(Bd,jTBd,j)-1Hj(1),Dj*=[DjO]T.Similarly, some terms in (33) can be combined, so (33) can be rewritten as
(35)yc(t)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0*(t-τo-kT)vd(kT)+∑j=1M2C1φ-j*(t-τo-kT)vd(kT-jT).
Thus, the discretization of continuous time-delay singular system (22a) and (22b) is carried out using (28) and (35).


### 3.2.2. Establishing Performance Index for Discrete Time-Delay Singular System

The optimal state-feedback control law minimizes the following performance cost function:(36)J=∫0tf{[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]Txxxx×Q[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]xxxx+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T},
where Q is the positive semidefinite matrix, R is the positive definite matrix, r(t)∈ℜq is the reference input vector, and the final time tf<∞. To discretize the cost function J, given by (36), tf=NT is chosen and J can be rewritten as
(37)J=∑k=0N-1∫kT(k+1)T{[∑j=1N2Djvc(t-τi,j)-r(t)]TC1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T×Q[∑j=1N2Djvc(t-τi,j)C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)∑j=1N2Djvc(t-τi,j)]+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T}.
Let r*(kT)∈ℜq be the piecewise-constant reference input vector to be determined in terms of r(kT) for the tracking purpose. Then, cost function (37) can be rewritten as [27] (38)J=∑k=0N-1∫kT(k+1)T{∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)[∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]T×Q[δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]+vdT(kT)Rvd(kT)dt}=⋯=∑k=0N-1[x~dsT(kT)x~dsT(kT-T)⋯x~dsT(kT-M1T)vdT(kT-T)⋯vdT(kT-M2T)r*T(kT)vdT(kT)]×[Q1Q31⋯Q3M1M21⋯M2M2-M3M1Q31TQ211⋯Q21M1M511⋯M51M2-M61M41⋮⋮⋱⋮⋮⋱⋮⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M4M1M21TM511T⋯M5M11TR211⋯R21M2-M81R41⋮⋮⋱⋮⋮⋱⋮⋯⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2R4M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3-M7M1TM41T⋯M4M1TR41T⋯R4M2T-M7TR1]×[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)vd(kT)]T,where
(39)Q1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-0(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q2ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1δ-j(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q3i=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-i(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M2j=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M3=∫kT(k+1)T[(C1δ-0(t-τo-kT))TQ]dt,M4i=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M5ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M6i=∫kT(k+1)T[(C1δ-i(t-τo-kT))TQ]dt,M7=∫kT(k+1)T[Q(C1φ-0*(t-τo-kT))]dt,M8j=∫kT(k+1)T[(C1φ-j*(t-τo-kT))TQ]dt,R1=∫kT(k+1)T[(C1φ-0*(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))+R(C1δ-0(t-τo-kT))T]dt,R2ij=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,R3=∫kT(k+1)TQdt,R4j=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt.
Construct an extended virtual state vector:(40)Xd(kT)=[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)]T.The extended delay-free system that is equivalent to the original time-delay singular system (28) and (35) is obtained as (41a)Xd((k+1)T)=G^eXd(kT)+H^evd(kT),(41b)yd(kT)=C^eXd(kT).We assume that the reference input r(t) is a step function with a constant magnitude, r*((k+1)T)=r*(kT). Designing a system based on such a reference input can lead to predictable time-response characteristics. Although our design methodology is based on a step function, it should be pointed out that the resulting control system, if properly designed, enables to give good time responses for any arbitrary reference input r(t). Also, the reference input r*(kT) is entered in the last row of Xd(kT) at the beginning of step k. As a result, the extended new system does not have any time-delay terms and it can be utilized to simplify the representation of the cost function (38). Now, (38) can be rewritten as
(42)J=∑k=0N-1[XdT(kT)vdT(kT)][Q^M^M^TR^][Xd(kT)vd(kT)],
where(43)[Q1Q31⋯Q3M1M21⋯M2M2-M3Q31TQ211⋯Q21M1M511⋯M51M2-M61⋮⋮⋱⋮⋮⋱⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M21TM511T⋯M5M11TR211⋯R21M2-M81⋮⋮⋱⋮⋮⋱⋮⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3],M^=[M1TM41T⋯M4M1TR41T⋯R4M2T-M7T]T,R^=[R1].Then, define a new virtual weighting matrix
(44)Q~=Q^-M^R^-1M^T
and a new virtual control input
(45)S(kT)=R^-1M^TXd(kT)+vd(kT).
Substituting (44) and (45) into (42) results in a decoupled performance index:
(46)J=∑k=0N-1[XdT(kT)Q~Xd(kT)+ST(kT)R^S(kT)].
Substituting (45) into the extended delay-free singular system (41a) and (41b) yields
(47)Xd((k+1)T)=G^eXd(kT)+H^evd(kT)=G^eXd(kT)+H^e[S(kT)-R^-1M^TXd(kT)]=G~eXd(kT)+H^eS(kT),
where G~e=G^e-H^eR^-1M^T.Notably, the quadratic optimal control of the system that is given by (41a) and (41b) with the performance index that is given by (42) is equivalent to the quadratic optimal control of the system that is given by (47) with the performance index that is given by (46). The development of the desired optimal virtual control vector S(kT) that minimizes the performance index that is given by (46) can be described as follows.


## 3.2.1. Discretization of Continuous Time-Delay Singular System

Consider the continuous time-delay singular system (22a) and (22b). To discretize (22a) and (22b), assume that vc(t) is a piecewise constant input function:
(24)vc(t)=vd(kT),kT≤t<(k+1)T,
where T is the sampling period. Let the state delay time be given by τs,i=ρiT+Γi, where 0≤Γi<T and ρi≥0 is an integer, and let the input delay time be given by τi,j=ηjT+σj, where 0≤σj<T and ηj≥0 is an integer. The time-delay singular system (22a) and (22b), by both the Newton extrapolation method and the Chebyshev quadrature method [25, 26], becomes
(25)x~ds((k+1)T)=Gx~ds(kT)+∑i=1N1[G^i(1)x~ds(kT-ρiT+T)+G^i(2)x~ds(kT-ρiT)+G^i(3)x~ds(kT-ρiT-T)]+∑j=1N2[Hj(0)vd(kT-ηjT)+Hj(1)vd(kT-ηjT-T)],
where
(26)G=eAsT,G^i(1)=T2[Qi(2)+Qi(3)]A^d,i,G^i(2)=T[Qi(1)-Qi(3)]A^d,i,G^i(3)=T2[Qi(3)-Qi(2)]A^d,i,Hj(0)=[G1-γj-In]As-1Bd,j,Hj(1)=[G-G1-γj]As-1Bd,j,
in which
(27)γj=σjT,βi=ΓiT,Qi(1)=[G-In](AsT)-1,Qi(2)=[Qi(1)-(1-βi)In-βiG](AsT)-1,Qi(3)=[2Qi(2)-(1-βi)2In-βi2G](AsT)-1.
Some terms in (25) may be combined because of the same delay, so (25) can be reduced to
(28)x~ds((k+1)T)=G-x~ds(kT)+∑i=1M1G-ix~ds(kT-iT)+H-vd(kT)+∑j=1M2H-jvd(kT-jT).
For the output (22b), the time-delay state xc(t-τo) for kT≤t-τo<(k+1)T must be evaluated. System (22a) and (22b) can be rewritten as
(29)x~s(t-τo)=eAs(t-τo-kT)x~ds(kT)+∑i=1N1∫kTt-τoeAs(t-τo-λ)A^d,ix~ds(λ-τs,i)dλ+∑j=1N2∫kTt-τoeAs(t-τo-λ)Bd,jvd(λ-τi,j)dλ=δ1(t-τo-kT)x~ds(kT)+∑i=1N1[δ^i(1)(t-τo-kT)x~ds(kT-ρiT+T)xxxxxxx+δ^i(2)(t-τo-kT)x~ds(kT-ρiT)xxxxxxx+δ^i(3)(t-τo-kT)xxxxxxx×x~ds(kT-ρiT-T)δ^i(1)]+∑j=1N2[φj(0)(t-τo-kT)vd(kT-ηjT)xxxxxxx+φj(1)(t-τo-kT)xxxxxxx×vd(kT-ηjT-T)δ^i(1)],
where
(30)δ1(t-τo-kT)=eAs(t-τo-kT),δ^i(1)(t-τo-kT)=T2[qi(2)+qi(3)]A^d,i,δ^i(2)(t-τo-kT)=T[qi(1)-qi(3)]A^d,i,δ^i(3)(t-τo-kT)=T2[qi(3)-qi(2)]A^d,i,φj(0)(t-τo-kT)={On×m,t-τo<σj,[e-Asσjδ1(t-τo-kT)-In]As-1Bd,j,t-τo≥σj,φj(1)(t-τo-kT)={[δ1(t-τo-kT)-In]As-1Bd,j,t-τo<σjδ1(t-τo-kT)[In-e-Asσj]As-1Bd,j,t-τo≥σj,
in which
(31)qi(1)=[δ1(t-τo-kT)-In](AsT)-1,qi(2)=[qi(1)-(t-τo-kTT-βi)In-βiδ1(t-τo-kT)(t-τo-kTT-βi)](AsT)-1,qi(3)=[2qi(2)-(t-τo-kTT-βi)2In+βi2δ1(t-τo-kT)(t-τo-kTT-βi)2](AsT)-1.
Also, some terms in (29) may be combined as in (28), and (29) may be rewritten as
(32)x~s(t-τo)=δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+φ-0(t-τo-kT)vd(kT)+∑j=1M2φ-j(t-τo-kT)vd(kT-jT).
Then, the output (22b) can be rewritten as
(33)yc(t)=C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0(t-τo-kT)vd(kT)+∑j=1M2C1φ-j(t-τo-kT)vd(kT-jT)-∑j=1N2[Dj(0)vd(kT-ηjT)xxxxx+Dj(1)vd(kT-ηjT-T)],
where
(34)Dj(0)=Dj*(Bd,jTBd,j)-1Hj(0),Dj(1)=Dj*(Bd,jTBd,j)-1Hj(1),Dj*=[DjO]T.Similarly, some terms in (33) can be combined, so (33) can be rewritten as
(35)yc(t)=C1δ-0(t-τo-kT)x~ds(kT)+∑i=1M1C1δ-i(t-τo-kT)x~ds(kT-iT)xxxx+C1φ-0*(t-τo-kT)vd(kT)+∑j=1M2C1φ-j*(t-τo-kT)vd(kT-jT).
Thus, the discretization of continuous time-delay singular system (22a) and (22b) is carried out using (28) and (35).


## 3.2.2. Establishing Performance Index for Discrete Time-Delay Singular System

The optimal state-feedback control law minimizes the following performance cost function:(36)J=∫0tf{[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]Txxxx×Q[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]xxxx+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T},
where Q is the positive semidefinite matrix, R is the positive definite matrix, r(t)∈ℜq is the reference input vector, and the final time tf<∞. To discretize the cost function J, given by (36), tf=NT is chosen and J can be rewritten as
(37)J=∑k=0N-1∫kT(k+1)T{[∑j=1N2Djvc(t-τi,j)-r(t)]TC1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T×Q[∑j=1N2Djvc(t-τi,j)C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)∑j=1N2Djvc(t-τi,j)]+vcT(t)Rvc(t)[C1x~s(t-τo)-∑j=1N2Djvc(t-τi,j)-r(t)]T}.
Let r*(kT)∈ℜq be the piecewise-constant reference input vector to be determined in terms of r(kT) for the tracking purpose. Then, cost function (37) can be rewritten as [27] (38)J=∑k=0N-1∫kT(k+1)T{∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)[∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]T×Q[δ-0(t-τo-kT)x~ds(kT)+∑i=1M1δ-i(t-τo-kT)x~ds(kT-iT)+φ-0*(t-τo-kT)vd(kT)+∑j=1M2φ-j*(t-τo-kT)vd(kT-jT)-r*(kT)]+vdT(kT)Rvd(kT)dt}=⋯=∑k=0N-1[x~dsT(kT)x~dsT(kT-T)⋯x~dsT(kT-M1T)vdT(kT-T)⋯vdT(kT-M2T)r*T(kT)vdT(kT)]×[Q1Q31⋯Q3M1M21⋯M2M2-M3M1Q31TQ211⋯Q21M1M511⋯M51M2-M61M41⋮⋮⋱⋮⋮⋱⋮⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M4M1M21TM511T⋯M5M11TR211⋯R21M2-M81R41⋮⋮⋱⋮⋮⋱⋮⋯⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2R4M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3-M7M1TM41T⋯M4M1TR41T⋯R4M2T-M7TR1]×[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)vd(kT)]T,where
(39)Q1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-0(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q2ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1δ-j(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,Q3i=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1δ-i(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M1=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M2j=∫kT(k+1)T[(C1δ-0(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M3=∫kT(k+1)T[(C1δ-0(t-τo-kT))TQ]dt,M4i=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M5ij=∫kT(k+1)T[(C1δ-i(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,M6i=∫kT(k+1)T[(C1δ-i(t-τo-kT))TQ]dt,M7=∫kT(k+1)T[Q(C1φ-0*(t-τo-kT))]dt,M8j=∫kT(k+1)T[(C1φ-j*(t-τo-kT))TQ]dt,R1=∫kT(k+1)T[(C1φ-0*(t-τo-kT))T×Q(C1φ-0*(t-τo-kT))+R(C1δ-0(t-τo-kT))T]dt,R2ij=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt,R3=∫kT(k+1)TQdt,R4j=∫kT(k+1)T[(C1φ-i*(t-τo-kT))T×Q(C1φ-j*(t-τo-kT))(C1δ-0(t-τo-kT))T]dt.
Construct an extended virtual state vector:(40)Xd(kT)=[x~ds(kT)x~ds(kT-T)⋯x~ds(kT-M1T)vd(kT-T)⋯vd(kT-M2T)r*(kT)]T.The extended delay-free system that is equivalent to the original time-delay singular system (28) and (35) is obtained as (41a)Xd((k+1)T)=G^eXd(kT)+H^evd(kT),(41b)yd(kT)=C^eXd(kT).We assume that the reference input r(t) is a step function with a constant magnitude, r*((k+1)T)=r*(kT). Designing a system based on such a reference input can lead to predictable time-response characteristics. Although our design methodology is based on a step function, it should be pointed out that the resulting control system, if properly designed, enables to give good time responses for any arbitrary reference input r(t). Also, the reference input r*(kT) is entered in the last row of Xd(kT) at the beginning of step k. As a result, the extended new system does not have any time-delay terms and it can be utilized to simplify the representation of the cost function (38). Now, (38) can be rewritten as
(42)J=∑k=0N-1[XdT(kT)vdT(kT)][Q^M^M^TR^][Xd(kT)vd(kT)],
where(43)[Q1Q31⋯Q3M1M21⋯M2M2-M3Q31TQ211⋯Q21M1M511⋯M51M2-M61⋮⋮⋱⋮⋮⋱⋮⋮Q3M1TQ2M11⋯Q2M1M1M5M11⋯M5M1M2-M6M1M21TM511T⋯M5M11TR211⋯R21M2-M81⋮⋮⋱⋮⋮⋱⋮⋮M2M2TM51M2T⋯M5M1M2TR2M21⋯R2M2M2-M8M2-M3T-M61T⋯-M6M1T-M81T⋯-M8M2TR3],M^=[M1TM41T⋯M4M1TR41T⋯R4M2T-M7T]T,R^=[R1].Then, define a new virtual weighting matrix
(44)Q~=Q^-M^R^-1M^T
and a new virtual control input
(45)S(kT)=R^-1M^TXd(kT)+vd(kT).
Substituting (44) and (45) into (42) results in a decoupled performance index:
(46)J=∑k=0N-1[XdT(kT)Q~Xd(kT)+ST(kT)R^S(kT)].
Substituting (45) into the extended delay-free singular system (41a) and (41b) yields
(47)Xd((k+1)T)=G^eXd(kT)+H^evd(kT)=G^eXd(kT)+H^e[S(kT)-R^-1M^TXd(kT)]=G~eXd(kT)+H^eS(kT),
where G~e=G^e-H^eR^-1M^T.Notably, the quadratic optimal control of the system that is given by (41a) and (41b) with the performance index that is given by (42) is equivalent to the quadratic optimal control of the system that is given by (47) with the performance index that is given by (46). The development of the desired optimal virtual control vector S(kT) that minimizes the performance index that is given by (46) can be described as follows.


## 3.3. Development of Optimal Tracker for Time-Delay Singular System with States Available

Let the Hamilton function depend on the cost function (46) [28]:
(48)Hf(kT)=[XdT(kT)Q~Xd(kT)+ST(kT)R^S(kT)]+λT((k+1)T)[G~eXd(kT)+H^eS(kT)],
where λ(kT) is a costate (Lagrange multiplier). Based on the well-developed optimal control theory [29, 30], the state equation is
(49)Xd((k+1)T)=G~eXd(kT)+H^eS(kT),
and the costate equation is
(50)λ(kT)=G~eTλ((k+1)T)+Q~Xd(kT)
with the stationary condition
(51)0=H^eTλ((k+1)T)+R^S(kT),
or(52a)S(kT)=-R^-1H^eTλ((k+1)T),
and the boundary condition is
(52b)λ(NT)=Q~Xd(NT).Assume that λ(kT) can be written as follows:
(53)λ(kT)=P(kT)Xd(kT),
where P(kT) is a real symmetric matrix of appropriate dimension. So far, the original optimal tracking problem has been transformed into an optimal regulator problem.To derive the optimal regulator, (53) is substituted into (50):
(54)P(kT)Xd(kT)=G~eTP((k+1)T)Xd((k+1)T)+Q~Xd(kT),
and (52a), (52b), and (53) are substituted into (49):
(55)Xd((k+1)T)=G~eXd(kT)+H^eR^-1H^eTλ((k+1)T)=G~eXd(kT)+H^eR^-1H^eTP((k+1)T)×Xd((k+1)T)
or
(56)Xd((k+1)T)=[I+H^eR^-1H^eTP((k+1)T)]-1G~eXd(kT).
Also, substituting (56) into (54) yields
(57)P(kT)Xd(kT)=G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1×G~eXd(kT)+Q~Xd(kT)
or
(58){[I+H^eR^-1H^eTP((k+1)T)]-1P(kT)-G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~e-Q~[I+H^eR^-1H^eTP((k+1)T)]-1}Xd(kT)=0.
The above equation must hold for all Xd(kT). Hence,
(59)P(kT)=Q~+G~eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~e.
Equation (59) is called the Riccati equation. With reference to (52a), (52b), and (53), when at k=N,
(60)λ(NT)=Q~Xd(NT)=P(NT)Xd(NT)
or
(61)P(NT)=Q~.
From (59) and (61), all P(kT) for 0≤k≤N can be obtained. With reference to (53) and (56), the desired optimal virtual control input that is given by (52a) now becomes
(62)S(kT)=-R^-1H^eTλ((k+1)T)=-R^-1H^eTP((k+1)T)Xd((k+1)T)=-R^-1H^eTP((k+1)T)×[I+H^eR^-1H^eTP((k+1)T)]-1G~eXd(kT)=-R^-1H^eT[P-1((k+1)T)+H^eR^-1H^eT]-1×G~eXd(kT)=-K(kT)Xd(kT),
where K(kT)=R^-1H^eT[P-1((k+1)T)+H^eR^-1H^eT]-1G~e. From (45), the original optimal controller vd(kT) is obtained as follows:
(63)vd(kT)=S(kT)-R^-1M^TXd(kT)=-K^(kT)Xd(kT),
where K^(kT)=K(kT)+R^-1M^T. Notice that if there are no state and input time delays, the above development can be reduced to that in [30]. Equation (63) can be represented in the following form:
(64)vd(kT)=-K^(kT)Xd(kT)=-Kd(kT)×[x~dsT(kT)x~dsT(kT-T)⋯x~dsT(kT-M1T)]T-Fd(kT)[vdT(kT-T)⋯vdT(kT-M2T)]T-Ed(kT)r*(kT),
where K^(kT)=[Kd(kT)Fd(kT)Ed(kT)], in which
(65)Kd(kT)=[Kd(0)(kT)Kd(1)(kT)⋯Kd(M1)(kT)],Fd(kT)=[Fd(1)(kT)⋯Fd(M2)(kT)].
Here, the discrete optimal controller (64) for the continuous time-delay system (22a) and (22b) has been completely derived. Figure 1 presents the realization of the time-varying piecewise-constant optimal controller (64) for the digitally controlled continuous time-delay singular system.Figure 1
Digital redesign for time-delay singular system.


## 3.4. Development of Observer-Based Suboptimal Tracker for Time-Delay Singular System with States Unavailable

When the states of a continuous time-delay system (22a) and (22b) are not available for measurement, the continuous-time states can be constructed using the recently developed continuous time-delay observers [28, 31, 32]. However, the developed digital tracker (64) requires the extended discrete-time state Χd(kT) in (41a) and (41b). Using the prediction-based digital redesign [27], a new observer-based suboptimal tracker for the time-delay singular system can be designed as follows.According to the digitally redesigned observer [27] and controller [27], the extended digitally redesigned observer and controller can be represented as(66a)X^d((k+1)T)=G^oX^d(kT)+H^ovd(kT)+Ld[yd(kT)-C^eX^d(kT)],(66b)vd(kT)=-K^(kT)X^d(kT),whereΧ^d(kT)∈ℜp is the estimate of the extended state Χd(kT)∈ℜp in (41a) and (41b),
(66c)G^o=G^e-LdC^eG^e,(66d)H^o=H^e-LdC^eH^e.Additionally,G^e=eA^T and H^e=[G^e-Ip]A^-1B^, where A^=(1/T)ln(G^e) and B^=A^(G^e-I)-1H^e. To determine the extended observer gain Ld in (66a), the equivalent extended continuous-time observer (41a) and (41b) and controller (64) can be represented [19], on the basis of the analog observer and controller, as(67a)X^˙c(t)=A^X^c(t)+B^vc(t)+Lc[yc(t)-C^eX^c(t)],(67b)vc(t)=-K^cX^c(t).The algorithm for computing the analog system matrixA^ in (67a) from the digital system matrix G^e in (41a) via the geometric-series method [15] is as follows:
(67c)A^=1Tln(G^e)=2T{R^+13R^3+⋯+1nR^n[Ip-1(1+2/n)R^2]-1}xxxxxxxxxxxxxxxxxxxfor|σ(R^2)|<(1+2n)≅2TR^[Ip-13R^2]-1forn=1≅2TR^[Ip-415R^2][Ip-35R^2]-1forn=3≅⋯,where R^=[G^e-Ip][G^e+Ip]-1 and |σ(∘)| denotes the absolute eigenspectrum of (∘). Based on the dual concept of the digital redesign, the analog observer gain Lc in (67a) and the digitally redesigned observer gain Ld in (66a) can be represented, respectively, as
(68)Lc=PobC^eTR-1,(69)Ld=(G^e-Ip)A^-1Lc[I+C^e(G^e-Ip)A^-1Lc]-1,
where Pob is the positive-definite and symmetric solution of the following Riccati equation:
(70)A^Pob+PobA^T-PobC^eTR-1C^ePob+C^eTQC^e=0,
in which Q≥0 and R>0 with appropriate dimensions.Owing to the extended virtual state vector in (40), the matrix G^e in (41a) and (41b) and (67a), (67b), and (67c) is singular. The matrices A^ and B^ in (67a), (67b), and (67c) cannot be directly determined. To solve this problem, an alternative is derived via the matrix sign function method [23, 24] as follows.Following the procedures shown in Section3 [23, 24], the transformed matrix is
(71)G^b=(G^e-ζ2Ip)(G^e+ζ2Ip)-1,
where G^e∈ℜp×p and ζ2 is a radius of a circle from the origin of the coordinates. Additionally, the associated matrix sign functions are
(72)Sign(G^b)=G^b(G^b22)-1,(73a)Sign-(G^b)=12(Ip-Sign(G^b)),(73b)Sign+(G^b)=12(Ip+Sign(G^b)),respectively. A fast and stable algorithm for computing the matrix sign function [23, 24] is given as follows.For the order of the desired convergence rater=2, one has
(74)Q(l+1)=12[Q(l)+Q-1(l)],Q(0)=G^b,liml→∞Q(l)=Sign(G^b),forl=0,1,2,….
By [19, 20], a transformation matrix Tm can be found such that
(75)G^m=Tm-1G^eTm=[G^m10(p-g)×g0g×(p-g)G^m2],
where G^m1 is a nonsingular matrix and G^m2 is a singular matrix whose eigenvalues are all null. Finally, the matrix A^ is obtained by the following equation:(76a)A^=TmG^lnTm-1=Tm[G^ln100G^ln2]Tm-1,
where G^ln1=(1/T)ln(G^m13)3=(3/T)ln(G^m13) and G^ln2=νIg, in which ν is a large negative real constant. The algorithm for finding ln(G^m13) in (76a) can be found in (67c) [19]. If the matrix G^m1 has any negative real eigenvalue, then the principal third root of G^m1 is not defined for arg(λi)≠π [23, 24]. The first part G^ln1 in (76a) can be rewritten as(76b)G^ln1=3Tln(G^m13)=3Tln[(G^m1eiθ3)e-(iθ/3)],where the matrixG^m1 is rotated by a small positive real angle θ. The second part G^ln2 in (76a) is utilized to recover the property ln(0)=-∞. Matrix B^ can be evaluated as
(77)B^=A^(G^e-Ip)-1H^e=[(G^e-Ip)A^-1]-1H^e=(TIp+A^T22!+A^2T33!+⋯)-1H^e.
Substituting (76a) and (76b) into the Riccati equation (70) and solving it yield the observer gain matrices in (68) and (69). Figure 2 presents the implementation of the observer-based suboptimal tracker for the time-delay singular system.Figure 2
Observer-based suboptimal tracker for the digitally redesigned time-delay singular system.


## 4. An Illustrative Example

Consider a continuous time-delay singular system, described in (1a) and (1b), with
(78)E=[1211-3-20221-3-31211-3-21213-5-40211-2-21000-10],A=I6,A^1=[0.4470000.447000.2236-0.11180-0.4470000.22360000000.447-0.8944000000.44700000-0.89440.447],B1=[10000-100-1100]T,C=[101000010100],N1=N2=1,τs1=0,τi1=τo=0.5×T.
Let the sampling period T=0.01 (s) and apply the reference input r(t)=[0.5sin(t)0.5cos(t)]T to the system. The initial condition is xc(0)=(MVM~)[x~sT(0)x~fT(0)]T=[000000]T, x~s(0)=[0000]T, and x~f=[00]T.Since0×E+A=I6, and according to the definition of the standard form, {E,A} is in standard form. If α=0 and β=1 are set, then En=E, An=A, A^n,1=A^1, and Bn,1=B1. Since En is singular, En includes some zero eigenvalues, and the bilinear transform must be performed to find the similarity transformation matrix M of En. Assume ω=0.5; the algorithm that was described in Section 3 yields,
(79)E~n=[0.33331.6-2.40.160.90672.2400.61.60.16-1.76-1.761.33331.6-3.40.160.90672.241.33331.6-2.40.76-0.69330.6401.6-2.40.161.242.241.3333000-1.3333-1],sign(E~n)=[1220-4-20120-2-22210-4-22221-6-40220-3-22000-2-1],M=[1100-1-101000-1110-1-10111-1-1-10100-1-1100-100].
Based on Section 3.1 and Appendices, the time-delay singular system can be decomposed as follows:(80a)x~˙s(t)=Asx~s(t)+A^d,1x~s(t-τs,1)+Bd,1vc(t-τi,1),(80b)yc(t)=C1x~s(t-τo)-D1vc(t-τi,1),where(81)As=[100-0.500.5-0.25-0.5000.500000.5],A^d,1=[0.447200-0.223600.2236-0.1118-0.2236000.223600000.2236],Bd,1=[0.50.5-0.25-0.250.50.50.5-0.5],C1=[10101000],D1=[020.51.5],
and the other parameters are listed below:
(82)M~=[1000000100000010000001100000.501000100],γ=2,η=-1,Λ1,1=[0.447200-0.298200.1491-0.0994-0.1988000.149100000.1491],Λ2,1=[-0.447200-0.4472],Bsk,1=[0.33330.6667-0.3333-0.11110.33330.33330.3333-0.3333],V=I6,Kf,1=[0100-10].
Following the proposed method in Section 3, the schemes of Figures 1 and 2 are implemented. For simplification, the numerical analysis is not presented and Figures 3 and 4 show the results of the simulation.Figure 3
Output responses of time-delay singular system with states available by the new digital redesign approach.Figure 4
Output responses of time-delay singular system with states unavailable by new observer-based suboptimal approach.Comparing with the offline observer/Kalman filter identification (OKID) method, the advantages of the proposed approach can be shown in [33, 34]. Following [33, 34], let the unknown system (80a) and (80b) be excited by the white-noise control force u(t)=[u1(t)u2(t)]T with a zero mean and covariance diag[cov(u1(t)),cov(u2(t))]=diag[0.20.2]. The input-output sampled data is given in Figure 5.Figure 5
SystemI/O data for identification.The identified system (G^, H^, C^) and observer gain (F) matrices for the unknown system (80a) and (80b) are given as
(83)G^=[8.7538-4.37468.9478-0.5128010.29565.53611.4682-3.67300-19.57699.2613-19.74219.53770-0.08210.6656-1.0711-359.391900000-112.9831],H^=[-0.2470-0.2116-0.8808-0.8879-0.5713-0.500722.247821.604800],C^=[-1.0867-0.58710.7911-0.057900.5676-0.5710-0.4333-0.04880],F=[1.4216-2.06150.77401.05790.5564-0.8225-0.0043-0.004100].
Then, the observer-based outputs by OKID compared with the actual system outputs for the unknown system (80a) and (80b) are shown in Figure 6.(a) The comparison between the system outputys1(kT) and its observer-based output yokid1(kT) by OKID. (b) The comparison between the system output ys2(kT) and its observer-based output yokid2(kT) by OKID.
(a)
(b)To overcome the effect of modeling error, an improved observer (69) with the high-gain property has been proposed in this paper, where the observer gain matrices are given as(84)Ld=[-0.3231-0.57080.3103-0.10510-0.06280.4008-1.0446-0.4013-0.164600.1289-0.0833-0.11196.440701.10060.1373-0.1585-0.404710.242201.64360.3760-16.6049228.32440.00030.01240.011900-25.8086363.61770.00010.01910.018400]T.Then, the comparisons between the actual outputs and the proposed method outputs for the unknown system (80a) and (80b) are shown in Figure 7.(a) The comparison between the system outputys1(kT) and its observer-based output yid1(kT) by the proposed method. (b) The comparison between the system output ys2(kT) and its observer-based output yid2(kT) by the proposed method.
(a)
(b)Obviously, the proposed method is better than OKID method on the tracking error performance from Figures6 and 7.


## 5. Conclusion

This paper presents a systematic methodology for developing novel observer-based suboptimal digital trackers for a class of time-delay singular systems. The time-delay property and singular system have been attracting more attention in recent years. The proposed controller and observer depend on the concepts of optimal control and the digital redesign with high-gain property to ensure effective tracking and favorable state matching performance. In future works, we will pay more attention to the online application and the real-time implementation of fault tolerant control system with performance optimization by using the proposed methods.



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