#### 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.

#### 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. Section 2 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:

where is the state vector, is the control input vector, and is the output vector. , , , , and are known constant system matrices of appropriate dimensions and is a singular matrix. The corresponding state time delay , , input time delay , , and output time delay 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]). *Let and be two square constant matrices. If , for all , then is called a regular pencil.

*Definition 2 (standard pencil [21]). *Let be a regular pencil. If there exists scalars and such that , then is called a standard pencil.

Notably, for any regular pencil, can be easily transformed into a standard pencil by multiplying to and , respectively, where and are scalars such that . Therefore, the matrix coefficients of a standard pencil becomeThe modified system retains its state vector and the matrices and have the following nice properties.

Lemma 3 (see [22]). *Consider*

(a)*, meaning that and commute each other;*(b)* and 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 can be transformed into a standard pencil . Notably since is a singular matrix, which has at least one zero eigenvalue, cannot be equal to zero. Hence, multiplying (1a) by can yield the following equation: where Since , the pencil is a standard one, and has the properties that are mentioned in Lemma 3. To decompose system (3), the state is converted into by where the constant matrix is a block modal matrix of and determined by means of the extended matrix sign function [23, 24]. The matrix of state space transformation is as follows.

*Step 1. *Find using the extended matrix sign function with an adequate , where

*Step 2. *Find and .

*Step 3. *Construct the matrix , where represents the collection of linearly independent column vectors of .

Substituting (5) into (3) and multiplying by on the left yield If can be diagonalized, then (7) yields, where and . is invertible with , , and is a nilpotent matrix with dimension . Since , it is invertible. Simplifying (8) by premultiplying the block diagonal on both sides, one has where

Remarkably, since 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 ; then, . By a previously proposed method [12], the preliminary feedback gain is found and 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 The time-delay singular system (9) can be transformed into (Appendix A) where in which and is a modal matrix of with dimension such that is in the Jordan block form. The time-delay singular system in (13) is obtained by applying the linear preliminary feedback control law from (12) to the system that is given by (9). Equation (13) has the finite modes (where ) and the 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 controllable finite modes and the nondynamic equation with 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 , where and are arbitrary scalars such that is invertible. Therefore, Let where the constant matrix is determined by using the extended matrix sign function. The procedure is the same as that elucidated above for finding , except that it operates on . Substituting (17) into (16) and multiplying by yield That is, where , = . is invertible with . is a null matrix and . In (19), is assumed to be able to be diagonalized as . Then, (19) can be rewritten as

and the time-delay singular system output (1b) can be rewritten as (Appendix B)

where .

Finally, the time-delay singular system (1a) and (1b) can be decomposed as the equivalent regular time-delay system as follows:

where

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 is a piecewise constant input function: where is the sampling period. Let the state delay time be given by , where and is an integer, and let the input delay time be given by , where and 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 where in which Some terms in (25) may be combined because of the same delay, so (25) can be reduced to For the output (22b), the time-delay state for must be evaluated. System (22a) and (22b) can be rewritten as where in which Also, some terms in (29) may be combined as in (28), and (29) may be rewritten as Then, the output (22b) can be rewritten as where

Similarly, some terms in (33) can be combined, so (33) can be rewritten as 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: where is the positive semidefinite matrix, is the positive definite matrix, is the reference input vector, and the final time . To discretize the cost function , given by (36), is chosen and can be rewritten as Let be the piecewise-constant reference input vector to be determined in terms of for the tracking purpose. Then, cost function (37) can be rewritten as [27] where Construct an extended virtual state vector:The extended delay-free system that is equivalent to the original time-delay singular system (28) and (35) is obtained as We assume that the reference input is a step function with a constant magnitude, . 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 . Also, the reference input is entered in the last row of at the beginning of step . 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 whereThen, define a new virtual weighting matrix and a new virtual control input Substituting (44) and (45) into (42) results in a decoupled performance index: Substituting (45) into the extended delay-free singular system (41a) and (41b) yields where .

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 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]: where is a costate (Lagrange multiplier). Based on the well-developed optimal control theory [29, 30], the state equation is and the costate equation is with the stationary condition or and the boundary condition is Assume that can be written as follows: where 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): and (52a), (52b), and (53) are substituted into (49): or Also, substituting (56) into (54) yields or The above equation must hold for all . Hence, Equation (59) is called the Riccati equation. With reference to (52a), (52b), and (53), when at , or From (59) and (61), all for can be obtained. With reference to (53) and (56), the desired optimal virtual control input that is given by (52a) now becomes where . From (45), the original optimal controller is obtained as follows: where . 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: where , in which 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.

##### 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 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

where is the estimate of the extended state in (41a) and (41b),

Additionally, and , where and . To determine the extended observer gain 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

The algorithm for computing the analog system matrix in (67a) from the digital system matrix in (41a) via the geometric-series method [15] is as follows: where and denotes the absolute eigenspectrum of . Based on the dual concept of the digital redesign, the analog observer gain in (67a) and the digitally redesigned observer gain in (66a) can be represented, respectively, as where is the positive-definite and symmetric solution of the following Riccati equation: