Composite Disturbance Observer-Based Control and 𝐻_∞ Output Tracking Control for Discrete-Time Switched Systems with Time-Varying Delay

Abstract

This paper considers the problem of 𝐻_∞ output tracking control for discrete-time switched systems with time-varying delay and external disturbances. The control scheme combining disturbance observer-based control (DOBC) and 𝐻_∞ control is proposed. The disturbances are assumed to include two parts. One part is generated by an exogenous system, which imposes on system with control inputs in the same channel. The other part is supposed to have the bounded 𝐻₂ norm. A new disturbance observer is developed to estimate and reject the first case disturbances for switched system with time-varying delay, and the second case disturbances are attenuated by 𝐻_∞ control scheme. The stability analysis of the closed-loop system is developed by switched Lyapunov function, and a solvable delay-dependent sufficient condition is presented in terms of linear matrix inequalities (LMIs) and cone complement linearization (CCL) methods. A numerical example is given to demonstrate the effectiveness of the proposed composite control scheme.

1. Introduction

Switched systems are a special class of hybrid control systems, which are composed of a family of subsystems described by continuous or discrete time dynamics and a switched rule among them [1, 2]. Switched systems can be employed to represent many practical systems, for example, power electronics, embedded systems, chemical processes, computer-controlled systems, automotive industries, and so on. Hence, switched systems have attracted considerable attention during the last decade, and many meaningful results have been reported on stability analysis and controller design for switched systems (see, e.g., [3–14]). By far, there are a number of methodologies on dealing with the stability analysis and control synthesis of switched systems, such as common Lyapunov function [1, 11], multiple Lyapunov function [3, 12], dwell time and average dwell time method [7, 13], and switched Lyapunov function [4, 14].

On the one hand, time delay always encounters in various engineering systems, such as long transmission lines in pneumatic, hydraulic, and chemical processes, economic and rolling mill systems, to name a few, which usually leads to instability and poor performance of the closed-loop system. Recently, many researches have paid more and more attention to stabilization and performance analysis of systems or switched systems with delays [15–20]. On the other hand, disturbances widely exist in systems, which may degrade the closed-loop system performance. Usually, some performance indexes, that is, 𝐻_∞ and 𝐿₂−𝐿_∞ (𝑙₂−𝑙_∞), are employed to deal with external disturbances. Many results have been developed with these performance indexes [10, 13, 18, 19, 21–25]. However, disturbances need to satisfy the bounded 𝐻₂ norm. In reality, many disturbances do not satisfy this condition, for example, constant disturbances or harmonics disturbances. To improve the ability of disturbance attenuation and rejection, disturbance observer-based control (DOBC) scheme has been studied from 1980s and applied in many control fields [26–37], which can be regarded as an active disturbance rejection method. In this method, disturbances are estimated by disturbance observer (DOB), and the disturbance estimation is employed to feedforward compensation.

The problem of output tracking control is important and numerous results have been developed for kinds of systems in the literature [38–40]. Along with the development of switched system theory, increased attention has been paid to studying output tracking controller design for switched systems. In [41–43], the tracking control problems have been considered for kinds of continuous switched systems. Exponential 𝑙₂−𝑙_∞ output tracking control problem has been solved in [13] for a class of discrete-time switched systems. However, in [13], disturbances are assumed to be the bounded 𝐻₂ norm.

In this paper, a composite control scheme is developed to solve the problem of 𝐻_∞ output tracking control for discrete-time switched systems with time-varying delay and external disturbances. Here, external disturbances can be divided into two parts. One part is generated by exogenous system, which imposes on system in the same channel with control inputs. The other part satisfies bounded 𝐻₂ norm. A new disturbance observer is designed to estimate and reject the first case disturbances for switched system with time-varying delay. 𝐻_∞ control is employed to analyze attenuation performance with respect to the second case disturbances. Hence, a composite control method, consisting of DOBC and 𝐻_∞ control, is proposed. By resorting to the switched Lyapunov function approach and inspired by [44, 45], some delay-dependent conditions for the problem of 𝐻_∞ output tracking control are presented. In order to obtain the desired controller and observer gains, a cone complement linearization (CCL) method is used to transform the nonconvex feasibility problem to some sequential optimization problem subject to linear matrix inequalities (LMIs) constraints. Finally, a numerical example is provided to demonstrate the effectiveness of the main result.

2. Problem Formulation and Preliminaries

Consider the following discrete-time switched systems with time-varying delays described by Σ₀∶𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐵_𝜎(𝑘)𝑢(𝑘)+𝑑₁(𝑘)+𝐵_1𝜎(𝑘)𝑑₂(𝑘),𝑧(𝑘)=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐶_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐷_𝜎(𝑘)𝑢(𝑘),𝑥(𝑠)=𝜙(𝑠),𝑠=−𝜏,…,−1,0, (1) where 𝑥(𝑘)∈ℝ^𝑛 is the states, 𝑧(𝑘)∈ℝ^𝑞 is the signal to be estimated, and 𝑢(𝑘) is the control input. 𝜎(𝑘)∶ℤ⁺→𝒩={1,2,…,𝑁} is the switching signal, which specifies which subsystem will be activated at a certain discrete time instant and 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐵_1𝑖, 𝐶_𝑖, 𝐶_𝑑𝑖,𝐷_𝑖, 𝑖∈𝒩, are constant matrices with appropriate dimensions. 𝜏(𝑘) is a time-varying delay of the system and satisfies 𝜏≤𝜏(𝑘)≤𝜏,∀𝑘∈ℤ⁺, (2) where 𝜏, 𝜏, 𝜏(𝑘) are nonnegative integer numbers. 𝜙(𝑘) is the initial condition. 𝑑₂(𝑘)∈ℝ^𝑝 is the external disturbances, which is assumed to belong to 𝑙₂[0,∞). 𝑑₁(𝑘) is also the external disturbances, which is generated by the exogenous system 𝜔(𝑘+1)=𝑊𝜔(𝑘),𝑑₁(𝑘)=𝑉𝜔(𝑘), (3) where (𝑊,𝐵_𝑖𝑉) is observable.

Remark 1. It is clear that 𝜏(𝑘) is an interval-like time-varying delay. When 𝜏=0, it is reduced to 0≤𝜏(𝑘)≤𝜏, for all 𝑘∈ℤ⁺, which was recently discussed in [23]. Therefore, this note can be viewed as some extensions of their results.

Remark 2. System Σ₀ contains two kinds of external disturbances: matched external disturbances and mismatched external disturbances. Two different methods are employed to deal with these disturbances. First, a disturbance observer is introduced to estimate matched disturbances; then the estimation values are used to feedforward compensation. Then, a performance index, that is, 𝐻_∞ performance index, is presented to reject and attenuate mismatched disturbances.

Assume that the reference signal 𝑧_𝑟(𝑘) is generated by the following system: 𝑥_𝑟(𝑘+1)=𝐴_𝑟𝑥_𝑟(𝑘)+𝐵_𝑟𝑟(𝑘),𝑧_𝑟(𝑘)=𝐶_𝑟𝑥_𝑟(𝑘), (4) where 𝑥_𝑟(𝑘) is reference states, 𝑟(𝑘)∈𝑙₂[0,∞) is reference input, and 𝐴_𝑟 is Hurwitz matrix with an appropriate dimension. Here we are interested in designing a state feedback controller by the following formula: 𝑢(𝑘)=𝐾_1𝜎(𝑘)𝑥(𝑘)−𝑑₁(𝑘)+𝐾_2𝜎(𝑘)𝑥_𝑟(𝑘), (5) where 𝑑₁(𝑘) is the estimation of the disturbances 𝑑₁(𝑘), which is obtained by the following disturbance observer: 𝜔(𝑘)(𝑘)(𝑘),𝜈()(𝜈(𝑘)(𝑘𝐴=𝜈−𝐿𝑥𝑘+1=𝑊−𝐿𝑥))+𝐿_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐵_𝜎(𝑘)𝑢(𝑘)+𝑑₁(𝑘),𝑑₁(𝑘)=𝑉𝜔(𝑘). (6)(7)

Remark 3. In this paper, we assume that the switching signal 𝜎(𝑘) is not known a priori but its instantaneous value is available in real time [4]. Here we only consider the case of synchronous switching; that is, the controller switches just as the system Σ₀ does.

Defining 𝑒_𝜔(𝑘)=𝜔(𝑘)−𝜔(𝑘) yields 𝑒_𝜔(𝑘+1)=𝑊+𝐿𝐵_𝜎(𝑘)𝑉𝑒_𝜔(𝑘)+𝐵_1𝜎(𝑘)𝑑₂(𝑘), (8) where 𝐵_1𝜎(𝑘)=𝐿𝐵_1𝜎(𝑘).

Applying controller (5) to system Σ₀ and combining (3) and (7), we obtain 𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐵_𝜎(𝑘)𝑑₁(𝑘)−𝑑₁(𝑘)+𝐵_1𝜎(𝑘)𝑑₂(𝑘)+𝐵_𝜎(𝑘)𝑥_𝑟(𝑘)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐵_𝜎(𝑘)𝑒_𝜔(𝑘)+𝐵_1𝜎(𝑘)𝑑₂(𝑘)+𝐵_𝜎(𝑘)𝑥_𝑟(𝑘), (9) where 𝐴_𝜎(𝑘)=𝐴_𝜎(𝑘)+𝐵_𝜎(𝑘)𝐾_1𝜎(𝑘),𝐵_𝜎(𝑘)=𝐵_𝜎(𝑘)𝑉,𝐵_𝜎(𝑘)=𝐵_𝜎(𝑘)𝐾_2𝜎(𝑘),𝑒_𝜔(𝑘)=𝜔(𝑘)−𝜔(𝑘). (10) Let 𝑒(𝑘)=𝑧(𝑘)−𝑧_𝑟(𝑘), (11) then we have the following augmented switched system: Σ_𝑒∶𝜁(𝑘+1)=𝒜_𝜎(𝑘)𝜁(𝑘)+𝒜_{𝑑𝜎(𝑘)}𝜁(𝑘−𝜏(𝑘))+ℬ_𝜎(𝑘)𝑒_𝜔(𝑘)+ℬ_1𝜎(𝑘)𝑣(𝑘),𝑒_𝜔(𝑘+1)=𝑊+𝐿𝐵_𝜎(𝑘)𝑉𝑒_𝜔(𝑘)+ℬ̆_1𝜎(𝑘)𝑣(𝑘),𝑒(𝑘)=𝒞_𝜎(𝑘)𝜁(𝑘)+𝒞_{𝑑𝜎(𝑘)}𝜁(𝑘−𝜏(𝑘)), (12) where 𝒜_𝜎(𝑘)=𝐴_𝜎(𝑘)𝐵_𝜎(𝑘)0𝐴_𝑟,𝒜_{𝑑𝜎(𝑘)}=𝐴_{𝑑𝜎(𝑘)}0,ℬ00_𝜎(𝑘)=𝐵_𝜎(𝑘)0,ℬ_1𝜎(𝑘)=𝐵_1𝜎(𝑘)00𝐵_𝑟,𝜁(𝑘)=𝑥(𝑘)𝑥_𝑟(𝑘),∈ℝ^𝑚(𝑘)=𝑑,𝑣²(𝑘)𝑟(𝑘),𝒞_𝜎(𝑘)=𝐶_𝜎(𝑘)+𝐷_𝜎(𝑘)𝐾_1𝜎(𝑘)𝐷_𝜎(𝑘)𝐾_2𝜎(𝑘)−𝐶_𝑟,𝒞_{𝑑𝜎(𝑘)}=𝐶_{𝑑𝜎(𝑘)}0,ℬ̆_1𝜎(𝑘)=𝐵_1𝜎(𝑘)0. (13) By defining 𝒜_𝜎(𝑘)=𝐴_𝜎(𝑘)00𝐴_𝑟,ℬ_𝜎(𝑘)=𝐵_𝜎(𝑘)0,𝐾_𝜎(𝑘)=𝐾_1𝜎(𝑘)𝐾_2𝜎(𝑘),𝒞_𝜎(𝑘)=𝐶_𝜎(𝑘)−𝐶_𝑟, (14) then we have 𝒜_𝜎(𝑘)=𝒜_𝜎(𝑘)+ℬ_𝜎(𝑘)𝐾_𝜎(𝑘),𝒞_𝜎(𝑘)=𝒞_𝜎(𝑘)+𝐷_𝜎(𝑘)𝐾_𝜎(𝑘), (15) and the controller can be rewritten as 𝑢(𝑘)=𝐾_𝜎(𝑘)𝜁(𝑘)−𝑑₁(𝑘). (16)

In order to prepare for a precise formulation, we introduce the following definition.

Definition 4 (𝐻_∞ output tracking control problem). Consider system Σ_𝑒. Given a prescribed 𝛾>0, if there exists composite controller (5) such that the following two conditions are satisfied:  (R1) system Σ_𝑒 is asymptotically stable when 𝑣(𝑘)=0, for all 𝑘≥0;  (R2) under the zero-initial condition, the following inequality holds: ‖𝑒(𝑘)‖₂<𝛾‖𝑣(𝑘)‖₂, (17)for any nonzero 𝑣(𝑘)∈𝑙₂[0,∞), then system Σ_𝑒 is asymptotically stable with an 𝐻_∞ performance index 𝛾.

3. Main Results

3.1. 𝐻_∞ Output Tracking Performance Analysis

In this subsection, we focus on developing delay-dependent condition to solve the 𝐻_∞ output tracking control problem formulated in previous section.

Theorem 5 (consider system Σ_𝑒). For scalars 𝜏>0,𝜏>0,𝛾>0, the error system Σ_𝑒 is asymptotically stable with an 𝐻_∞ performance index, if there exist matrices 𝑀₁, 𝑀₂, 𝑁₁, 𝑁₂, 𝑇₁, 𝑇₂, 𝐾_1𝑖, 𝐾_2𝑖, 𝑃_𝑖>0,𝑄₁>0,𝑄₂>0, 𝑄₃>0, 𝑅₁>0, 𝑅₂>0, 𝑆>0, 𝑖∈𝒩, such that the following inequality holds: Φ_𝑖=Φ_11𝑖Φ₁₂00𝑀^𝑇₁−𝑇^𝑇₁𝒜^𝑇_𝑖Φ_18𝑖Φ_19𝑖Φ_1,10Φ_1,11Φ_1,120∗Φ_22𝑖00𝑀^𝑇₂−𝑇^𝑇₂𝒜^𝑇_𝑑𝑖𝒜^𝑇_𝑑𝑖𝒜^𝑇_𝑑𝑖Φ_2,10Φ_2,11Φ_2,120∗∗−𝑆000ℬ^𝑇_𝑖ℬ^𝑇_𝑖ℬ^𝑇_𝑖000Φ_3,13𝑖∗∗∗−𝛾²𝐼00ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖000ℬ̆^𝑇_1𝑖∗∗∗∗−𝑄₁00000000∗∗∗∗∗−𝑄₂0000000∗∗∗∗∗∗−𝑃_𝑙⁻¹000000∗∗∗∗∗∗∗Φ₈₈00000∗∗∗∗∗∗∗∗Φ₉₉0000∗∗∗∗∗∗∗∗∗Φ_10,10000∗∗∗∗∗∗∗∗∗∗Φ_11,1100∗∗∗∗∗∗∗∗∗∗∗Φ_12,120∗∗∗∗∗∗∗∗∗∗∗∗−𝑆⁻¹<0, (18) where Φ_11𝑖=−𝑃_𝑖+𝑄₁+𝑄₂+𝜏−𝜏+1𝑄₃+𝑁₁+𝑁^𝑇₁+𝒞^𝑇_𝑖𝒞_𝑖+𝒞^𝑇_𝑖𝒞_𝑑𝑖,Φ₁₂=−𝑁^𝑇₁+𝑁₂+𝑇^𝑇₁−𝑀^𝑇₁,Φ_18𝑖=Φ_19𝑖=(𝒜_𝑖−𝐼)^𝑇,Φ_1,10=𝜏𝑁^𝑇₁,Φ_1,11=𝜏−𝜏𝑇^𝑇₁,Φ_1,13=𝜏−𝜏𝑀^𝑇₁,Φ_22𝑖=−𝑄₃−𝑁₂−𝑁^𝑇₂+𝑇₂+𝑇^𝑇₂−𝑀₂−𝑀^𝑇₂+𝒞^𝑇_𝑑𝑖𝒞_𝑑𝑖,Φ_2,10=𝜏𝑁^𝑇₂,Φ_2,11=𝜏−𝜏𝑇^𝑇₂,Φ_2,12=𝜏−𝜏𝑀^𝑇₂,Φ_3,13𝑖=(𝑊+𝐿𝐵_𝑖𝑉)^𝑇,Φ₈₈=−(𝜏𝑅₁)⁻¹,Φ₉₉=−((𝜏−𝜏)𝑅₂)⁻¹,Φ_10,10=−𝜏𝑅₁,Φ_11,11=−𝜏−𝜏𝑅₁+𝑅₂,Φ_12,12=−𝜏−𝜏𝑅₂. (19)

Proof. Choose a Lyapunov-Krasovskii functional candidate as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘), (20) where 𝑉₁(𝑘)=𝜁^𝑇(𝑘)𝑃_𝜎(𝑘)𝜁(𝑘)+𝑒^𝑇_𝜔(𝑘)𝑆𝑒_𝜔(𝑘),𝑉₂(𝑘)=^𝑘−1_{𝜃=𝑘−𝜏}𝜁^𝑇(𝜃)𝑄₁𝜁(𝜃)+^𝑘−1_{𝜃=𝑘−𝜏}𝜁^𝑇(𝜃)𝑄₂𝜁(𝜃)+^𝑘−1_{𝜃=𝑘−𝜏(𝑘)}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃),𝑉₃(𝑘)=⁻¹_𝑖=−𝜏^𝑘−1_{𝜃=𝑘+𝑖}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)+^−𝜏_{𝑖=−𝜏+1}^𝑘−1_{𝜃=𝑘+𝑖−1}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)+^−𝜏+1_{𝑖=−𝜏+2}^𝑘−1_{𝜃=𝑘+𝑖−1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃), (21) and 𝜌(𝜃)=𝜁(𝜃+1)−𝜁(𝜃).
Without loss of generality, we assume that 𝜎(𝑘+1)=𝑙,𝜎(𝑘)=𝑖, for all 𝑖,𝑙∈𝒩. Then taking the forward difference yields Δ𝑉₁(𝑘)=𝑉₁(𝑘+1)−𝑉₁(𝑘)=𝜁^𝑇(𝑘+1)𝑃_𝑙𝜁(𝑘+1)−𝜁^𝑇(𝑘)𝑃_𝑖𝜁(𝑘)+𝑒^𝑇_𝜔(𝑘+1)𝑆𝑒_𝜔(𝑘+1)−𝑒^𝑇_𝜔(𝑘)𝑆𝑒_𝜔(𝑘)=𝜉^𝑇(𝑘)𝓐^𝑇_𝑖𝑃_𝑙𝓐_𝑖𝜉(𝑘)−𝜁^𝑇(𝑘)𝑃_𝑖𝜁(𝑘)+𝜉^𝑇(𝑘)𝓑^𝑇_𝑖𝑆𝓑_𝑖𝜉(𝑘)−𝑒^𝑇_𝜔(𝑘)𝑆𝑒_𝜔(𝑘), (22) where 𝜉^𝑇(𝑘)=𝜁^𝑇(𝑘)𝜁^𝑇(𝑘−𝜏(𝑘))𝑒^𝑇_𝜔(𝑘)𝑣^𝑇(𝑘),𝓐_𝑖=𝒜_𝑖𝒜_𝑑𝑖ℬ_𝑖ℬ_1𝑖,𝓑_𝑖=00𝑊+𝐿𝐵_𝑖𝑉ℬ̆_1𝑖. (23) Direct computation gives Δ𝑉₂(𝑘)=^𝑘_{𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₁𝜁(𝜃)−^𝑘−1_{𝜃=𝑘−𝜏}𝜁^𝑇(𝜃)𝑄₁𝜁(𝜃)+^𝑘_{𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₂𝜁(𝜃)−^𝑘−1_{𝜃=𝑘−𝜏}𝜁^𝑇(𝜃)𝑄₂𝜁(𝜃)+^𝑘_{()𝜃=𝑘+1−𝜏𝑘+1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)−^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)=𝜁^𝑇(𝑘)𝑄₁+𝑄₂+𝑄₃𝜁(𝑘)−𝜁^𝑇𝑘−𝜏𝑄₁𝜁𝑘−𝜏−𝜁^𝑇(𝑘−𝜏)𝑄₂𝜁(𝑘−𝜏)−𝜁^𝑇(𝑘−𝜏(𝑘))𝑄₃𝜁((𝑘+𝑘−𝜏))_{𝑘−1()𝜃=𝑘+1−𝜏𝑘+1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)−^𝑘−1_{𝜃=𝑘+1−𝜏(𝑘)}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃). (24) Note that ^𝑘−1_{()𝜃=𝑘+1−𝜏𝑘+1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)=^𝑘−1_{𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)+^𝑘−𝜏_{()𝜃=𝑘+1−𝜏𝑘+1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)≤^𝑘−1_{(𝑘)𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)+^𝑘−𝜏_{𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃). (25) Hence we obtain Δ𝑉₂(𝑘)≤𝜁^𝑇(𝑘)𝑄₁+𝑄₂+𝑄₃𝜁(𝑘)−𝜁^𝑇𝑘−𝜏𝑄₁𝜁𝑘−𝜏−𝜁^𝑇(𝑘−𝜏)𝑄₂𝜁(𝑘−𝜏)−𝜁^𝑇(𝑘−𝜏(𝑘))𝑄₃𝜁((𝑘+𝑘−𝜏))_{𝑘−𝜏𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃). (26) After some manipulations, the following inequality is satisfied: Δ𝑉₃(𝑘)=⁻¹_𝑖=−𝜏^𝑘_{𝜃=𝑘+1+𝑖}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)−⁻¹_𝑖=−𝜏^𝑘−1_{𝜃=𝑘+𝑖}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)+^−𝜏_𝑖=−^𝑘_{𝜏+1𝜃=𝑘+𝑖}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)−^−𝜏_{𝑖=−𝜏+1}^𝑘−1_{𝜃=𝑘+𝑖−1}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)+^−𝜏+1_𝑖=−^𝑘_{𝜏+2𝜃=𝑘+𝑖}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)−^−𝜏+1_{𝑖=−𝜏+2}^𝑘−1_{𝜃=𝑘+𝑖−1}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃)=𝜏𝜌^𝑇(𝑘)𝑅₁𝜌(𝑘)−^𝑘−1_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)+𝜏−𝜏𝜌^𝑇(𝑘)𝑅₂𝜌(𝑘)−^{𝑘−𝜏−1}_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)+𝜏−𝜏𝜁^𝑇(𝑘)𝑄₃𝜁(𝑘)−^𝑘−𝜏_{𝜃=𝑘+1−𝜏}𝜁^𝑇(𝜃)𝑄₃𝜁(𝜃). (27) Observe that the following equalities hold naturally: 2𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×𝜁(𝑘)−𝜁(𝑘−𝜏(𝑘))−^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜌(𝜃2𝜁)=0,_𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇𝑘−𝑑_𝑗(𝑘)𝑇^𝑇₂×𝜁(𝑘−𝜏(𝑘))−𝜁(𝑘−𝜏)−^{𝑘−1−𝜏(𝑘)}_{𝜃=𝑘−𝜏}𝜌(𝜃)2𝜁=0,_𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×𝜁𝑘−𝜏−𝜁(𝑘−𝜏(𝑘))−^{𝑘−1−𝜏}_{𝜃=𝑘−𝜏(𝑘)}𝜌(𝜃)=0. (28) Then −^𝑘−1_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)−^{𝑘−𝜏−1}_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)=−^{𝑘−𝜏(𝑘)−1}_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁+𝑅₂𝜌(𝜃)−^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)−^{𝑘−𝜏−1}_{(𝑘)𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)+2𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×𝜁(𝑘)−𝜁(𝑘−𝜏(𝑘))−^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜌(𝜃)+2𝜁^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×𝜁(𝑘−𝜏(𝑘))−𝜁(𝑘−𝜏)−^{𝑘−1−𝜏(𝑘)}_{𝜃=𝑘−𝜏}𝜌(𝜃)+2𝜁^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×𝜁𝑘−𝜏−𝜁(𝑘−𝜏(𝑘))−^{𝑘−1−𝜏}_{(𝑘)𝜃=𝑘−𝜏}𝜌(𝜃)=2𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×(𝜁(𝑘)((𝑘𝜁−𝜁𝑘−𝜏)))+2^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×(𝜁(𝑘−𝜏(𝑘))−𝜁(𝑘−𝜏𝜁))+2^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×𝜁𝑘−𝜏((𝑘𝜁−𝜁𝑘−𝜏))−2^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×^{𝑘−1−𝜏(𝑘)}_{𝜃=𝑘−𝜏}𝜌(𝜃)−^{𝑘−𝜏(𝑘)−1}_{𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁+𝑅₂𝜌(𝜃)−2𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜌(𝜃)−^𝑘−1_{(𝑘)𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₁𝜌(𝜃)−2𝜁^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×^{𝑘−1−𝜏}_{(𝑘)𝜃=𝑘−𝜏}𝜌(𝜃)−^{𝑘−𝜏−1}_{(𝑘)𝜃=𝑘−𝜏}𝜌^𝑇(𝜃)𝑅₂𝜌(𝜃)=2𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×(𝜁(𝑘)((𝑘𝜁−𝜁𝑘−𝜏)))+2^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×(𝜁(𝑘−𝜏(𝑘))−𝜁(𝑘−𝜏𝜁))+2^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×𝜁𝑘−𝜏((𝑘(𝑘)𝜁−𝜁𝑘−𝜏))+𝜏^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂𝑅₁⁻¹×𝑁₁𝜁(𝑘)+𝑁₂𝜁((𝑘+(𝑘−𝜏))𝜏−𝜏(𝑘))𝜁^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×(𝑅₁+𝑅₂)⁻¹𝑇₁𝜁(𝑘)+𝑇₂𝜁((𝑘+𝜏(𝑘)𝑘−𝜏))−𝜏𝜁^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂𝑅₂⁻¹×𝑀₁𝜁(𝑘)+𝑀₂𝜁((𝑘−𝑘−𝜏))_{𝑘−1(𝑘)𝜃=𝑘−𝜏}𝜂^𝑇(𝑘)𝑁^𝑇𝑅₁⁻¹(𝑘)−𝑁𝜂_{𝑘−1−𝜏(𝑘)𝜃=𝑘−𝜏}𝜂^𝑇(𝑘)𝑇^𝑇(𝑅₁+𝑅₂)⁻¹(𝑘)−𝑇𝜂_{𝑘−𝜏−1(𝑘)𝜃=𝑘−𝜏}𝜂^𝑇(𝑘)𝑀^𝑇𝑅₂⁻¹(𝑘)𝜁𝑀𝜂≤2^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂×(𝜁(𝑘)((𝑘𝜁−𝜁𝑘−𝜏)))+2^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×(𝜁(𝑘−𝜏(𝑘))−𝜁(𝑘−𝜏𝜁))+2^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂×𝜁𝑘−𝜏((𝑘+−𝜁𝑘−𝜏))𝜏𝜁^𝑇(𝑘)𝑁^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑁^𝑇₂𝑅₁⁻¹×𝑁₁𝜁(𝑘)+𝑁₂𝜁((𝑘+𝑘−𝜏))𝜏−𝜏𝜁^𝑇(𝑘)𝑇^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑇^𝑇₂×𝑅₁+𝑅₂⁻¹𝑇₁𝜁(𝑘)+𝑇₂𝜁((𝑘+𝑘−𝜏))𝜏−𝜏𝜁^𝑇(𝑘)𝑀^𝑇₁+𝜁^𝑇(𝑘−𝜏(𝑘))𝑀^𝑇₂𝑅₂⁻¹×𝑀₁𝜁(𝑘)+𝑀₂𝜁(𝑘−𝜏(𝑘)), (29) where 𝜂^𝑇(𝑘)=𝜁^𝑇(𝑘)𝜁^𝑇(𝑘−𝜏(𝑘))𝜌^𝑇(𝜃),𝑁=𝑁₁𝑁₂𝑅₁,𝑀=𝑀₁𝑀₂𝑅₂,𝑇=𝑇₁𝑇₂𝑅₁+𝑅₂. (30) From (22)–(29), and by some manipulations, we obtain Δ𝑉(𝑘)≤𝜆^𝑇(𝑘)Φ_1𝑖𝜆(𝑘)+𝜉^𝑇(𝑘)Φ_2𝑖𝜉(𝑘)+𝜂^𝑇(𝑘)Φ₃𝜂(𝑘)−𝜂^𝑇(𝑘)𝒞^𝑇_𝑖𝒞^𝑇_𝑑𝑖𝒞_𝑖𝒞_𝑑𝑖𝜂(𝑘), (31) where 𝜆^𝑇(𝑘)=𝜁^𝑇(𝑘)𝜁^𝑇(𝑘−𝜏(𝑘))𝑒^𝑇_𝜔(𝑘)𝑣^𝑇(𝑘)𝜁^𝑇𝑘−𝜏𝜁^𝑇(𝑘−𝜏),𝜂^𝑇(𝑘)=𝜁^𝑇(𝑘)𝜁^𝑇((𝑘,Φ𝑘−𝜏))_2𝑖=𝓐^𝑇_𝑖𝑃_𝑙𝓐_𝑖+𝓑^𝑇_𝑖𝑆𝓑_𝑖+𝓐^𝑇_𝑖𝜏𝑅₁+𝜏−𝜏𝑅₂𝓐_𝑖,𝓐_𝑖=𝓐_𝑖−𝐼,,Φ𝐼=𝐼000₃=𝜏[𝑁₁𝑁₂]^𝑇𝑅₁⁻¹𝑁₁𝑁₂+𝜏−𝜏𝑇₁𝑇₂^𝑇×(𝑅₁+𝑅₂)⁻¹𝑇₁𝑇₂+𝜏−𝜏𝑀₁𝑀₂^𝑇𝑅₂⁻¹𝑀₁𝑀₂,Φ_1𝑖=Φ_11𝑖Φ₁₂00𝑀^𝑇₁−𝑇^𝑇₁∗Φ_22𝑖00𝑀^𝑇₂−𝑇^𝑇₂∗∗−𝑆000∗∗∗−𝛾²𝐼00∗∗∗∗−𝑄₁0∗∗∗∗∗−𝑄₂. (32) Now, we develop the conclusion from two aspects. We first establish the asymptotic stability of system Σ_𝑒 under the condition of zero disturbances. In fact, when 𝑣(𝑘)=0, it is verified that Δ𝑉(𝑘)≤𝜆^∼𝑇(𝑘)Φ^∼∼_1𝑖𝜆(𝑘)+𝜉^𝑇(𝑘)Φ^∼_2𝑖𝜉(𝑘)+^𝑚_𝑗=1𝜂^𝑇_𝑗(𝑘)Φ_3𝑗𝜂_𝑗(𝑘)−𝜁^𝑇(𝑘)𝒢^𝑇_𝑖𝒢_𝑖𝜁(𝑘), (33) where 𝜆^∼𝑇(𝑘)=𝜁^𝑇(𝑘)𝜁^𝑇(𝑘−𝜏(𝑘))𝑒^𝑇_𝜔(𝑘)𝜁^𝑇𝑘−𝜏𝜁^𝑇(𝑘−𝜏),_∼Φ_2𝑖=𝓐^∼𝑇_𝑖𝑃𝓐_𝑙^∼_𝑖+𝓑^𝑇_𝑖𝑆𝓑_𝑖+𝓐^𝑇_𝑖𝜏𝑅₁+𝜏−𝜏𝑅₂𝓐_𝑖,_∼𝓐_𝑖=𝒜_𝑖𝒜_𝑑𝑖ℬ,𝓐_𝑖=𝓐^∼_𝑖−𝐼^∼,𝐼^∼=𝐼00,𝓑=00𝑊+𝐿𝐵_𝑖𝑉,_∼Φ_1𝑖=Φ_11𝑖Φ₁₂0𝑀^𝑇₁−𝑇^𝑇₁∗Φ_22𝑖0𝑀^𝑇₂−𝑇^𝑇₂∗∗−𝑆00∗∗∗−𝑄₁0∗∗∗∗−𝑄₂. (34) Applying Schur complement formula, we obtain Δ𝑉(𝑘)<0 if (18) is true. Therefore, it is easy to see that the error system Σ_𝑒 is asymptotically stable by the Lyapunov-Krasovskii stability theorem.
Next, we will present that under the zero-initial condition, the time-delay system Σ_𝑒 satisfies (17) for all nonzero 𝑣(𝑘)∈𝑙₂[0,∞). To this end, we introduce 𝐽_𝑁=^𝑁_𝑘=0𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)+Δ𝑉(𝑘)−𝑉(𝑁+1). (35) Then, by the Schur complement formula, it easily follows from (18) and (31) that 𝐽_𝑁≤^∞_𝑘=0𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)+Δ𝑉(𝑘)<0, (36) which implies that ‖𝑒(𝑘)‖₂<𝛾‖𝑣(𝑘)‖₂ holds under the zero-initial condition. This completes the proof.

Remark 6. It is obvious that the condition in (18) is not an LMI with respect to the parameters 𝑃₁,…,𝑃_𝑁,𝑅₁, 𝑅₂, 𝑆. In order to solve the controller in the form of (5), we will cast the 𝐻_∞ output tracking control problem into an LMI framework.

3.2. 𝐻_∞ Output Tracking Controller Design

In this subsection, we try to obtain a solvable condition for the problem of 𝐻_∞ output tracking controller design using CCL method.

Define 𝑋_𝑖=𝑃_𝑖⁻¹,𝑅₁=𝑅₁⁻¹,𝑅₂=𝑅₂⁻¹,𝐾_𝑖=𝐾_𝑖𝑋_𝑖,𝐿=𝑆𝐿,𝑄_1𝑖=𝑋_𝑖𝑄₁𝑋_𝑖,𝑄_2𝑖=𝑋_𝑖𝑄₂𝑋_𝑖,𝑄_3𝑖=𝑋_𝑖𝑄₃𝑋_𝑖,𝑀₁=𝑋_𝑖𝑀₁𝑋_𝑖,𝑀₂=𝑋_𝑖𝑀₂𝑋_𝑖,𝑁₁=𝑋_𝑖𝑁₁𝑋_𝑖,𝑁₂=𝑋_𝑖𝑁₂𝑋_𝑖,𝑇₁=𝑋_𝑖𝑇₁𝑋_𝑖,𝑇₂=𝑋_𝑖𝑇₂𝑋_𝑖,Γ=diag𝑋_𝑖,𝑋_𝑖,𝐼,𝐼,𝑋_𝑖,𝑋_𝑖,𝐼,𝐼,𝐼,𝑋_𝑖,𝑋_𝑖,𝑋_𝑖,𝑆. (37) Pre- and postmultiplying Γ and Γ^𝑇 on (18), and using Schur complement formula, we obtainΦ_𝑖=Φ_11𝑖Φ₁₂00𝑀^𝑇₁−𝑇^𝑇₁Φ_17𝑖Φ_18𝑖Φ_19𝑖Φ_1,10Φ_1,11Φ_1,120Φ_1,14𝑖∗Φ_22𝑖00𝑀^𝑇₂−𝑇^𝑇₂𝑋_𝑖𝒜^𝑇_𝑑𝑖𝑋_𝑖𝒜^𝑇_𝑑𝑖𝑋_𝑖𝒜^𝑇_𝑑𝑖Φ_2,10Φ_2,11Φ_2,120Φ_2,14𝑖∗∗−𝑆000ℬ^𝑇_𝑖ℬ^𝑇_𝑖ℬ^𝑇_𝑖000Φ_3,13𝑖0∗∗∗−𝛾²𝐼00ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖000Φ_4,13𝑖0∗∗∗∗−𝑄_1𝑖000000000∗∗∗∗∗−𝑄_2𝑖00000000∗∗∗∗∗∗−𝑋_𝑙0000000∗∗∗∗∗∗∗Φ₈₈000000∗∗∗∗∗∗∗∗Φ₉₉00000∗∗∗∗∗∗∗∗∗Φ_10,10𝑖0000∗∗∗∗∗∗∗∗∗∗Φ_11,11𝑖000∗∗∗∗∗∗∗∗∗∗∗Φ_12,12𝑖00∗∗∗∗∗∗∗∗∗∗∗∗−𝑆0∗∗∗∗∗∗∗∗∗∗∗∗∗−𝐼<0, (38)where Φ_11𝑖=−𝑋_𝑖+𝑄_1𝑖+𝑄_2𝑖+𝜏−𝜏+1𝑄_3𝑖+𝑁₁+𝑁^𝑇₁,Φ₁₂=−𝑁^𝑇₁+𝑁₂+𝑇^𝑇₁−𝑀^𝑇₁,Φ_17𝑖=𝑋_𝑖𝒜^𝑇_𝑖+𝐾^𝑇_𝑖ℬ^𝑇_𝑖,Φ_18𝑖=Φ_19𝑖=Φ_17𝑖−𝑋_𝑖,Φ_1,10=𝜏𝑁^𝑇₁,Φ_1,11=𝜏−𝜏𝑇^𝑇₁,Φ_1,13=𝜏−𝜏𝑀^𝑇₁,Φ_1,14𝑖=𝑋_𝑖𝒞^𝑇_𝑖+𝐾^𝑇_𝑖𝐷^𝑇_𝑖,Φ_22𝑖=−𝑄_3𝑖−𝑁₂−𝑁^𝑇₂+𝑇₂+𝑇^𝑇₂−𝑀₂−𝑀^𝑇₂,Φ_2,14𝑖=𝑋_𝑖𝒞^𝑇_𝑑𝑖,Φ_2,10=𝜏𝑁^𝑇₂,Φ_2,11=𝜏−𝜏𝑇^𝑇₂,Φ_2,12=𝜏−𝜏𝑀^𝑇₂,Φ_3,13𝑖=𝑊^𝑇𝑆+(𝐿𝐵_𝑖𝑉)^𝑇,Φ₈₈1=−𝜏𝑅₁,Φ₉₉1=−𝜏−𝜏𝑅₂,Φ_10,10𝑖=−𝜏𝑋_𝑖𝑅₁𝑋_𝑖,Φ_4,13𝑖=ℬ̆^𝑇_1𝑖𝑆,Φ_11,11𝑖=−𝜏−𝜏𝑋_𝑖𝑅₁+𝑅₂𝑋_𝑖,Φ_12,12𝑖=−𝜏−𝜏𝑋_𝑖𝑅₂𝑋_𝑖. (39)

Theorem 7 (consider the system Σ_𝑒). For scalars 𝜏>0,𝜏>0,𝛾>0, the error system Σ_𝑒 is asymptotically stable with an 𝐻_∞ performance index, if there exist matrices 𝑀₁, 𝑀₂, 𝑁₁, 𝑁₂, 𝑇₁, 𝑇₂, 𝐾_1𝑖, 𝐾_2𝑖, 𝑋_𝑖>0,𝑄₁>0,𝑄₂>0,𝑄₃>0, 𝑅₁>0, 𝑅₂>0, 𝑅₁>0, 𝑅₂>0,𝑆>0, 𝑖∈𝒩, such that inequality (38) holds.

It is clear that (38) is a nonlinear matrix inequality due to the existence of terms 𝑅₁, 𝑅₂, 𝑋_𝑖𝑅₁𝑋_𝑖 and 𝑋_𝑖𝑅₂𝑋_𝑖. In the sequel, the CCL method is resorted to solve the desired controller gains and observer gain.

Introduce two new variables 𝑆_1𝑖 and 𝑆_2𝑖 such that 𝑋_𝑖𝑅₁𝑋_𝑖≥𝑆_1𝑖 and 𝑋_𝑖𝑅₂𝑋_𝑖≥𝑆_2𝑖, then we obtain the following results.

Theorem 8 (Consider system Σ_𝑒). For scalars 𝜏>0,𝜏>0,𝛾>0, the error system Σ_𝑒 is asymptotically stable with an 𝐻_∞ performance index, if there exist matrices 𝑀₁,𝑀₂, 𝑁₁, 𝑁₂, 𝑇₁,𝑇₂, 𝐾_1𝑖, 𝐾_2𝑖, 𝑋_𝑖>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0,𝑅₁>0, 𝑅₂>0,𝑅₁>0,𝑅₂>0, 𝑆>0, 𝑆_1𝑖,𝑆_2𝑖, 𝑖∈𝒩, such that the following inequalities holdΦ_𝑖=Φ_11𝑖Φ₁₂00𝑀^𝑇₁−𝑇^𝑇₁Φ_17𝑖Φ_18𝑖Φ_19𝑖Φ_1,10Φ_1,11Φ_1,120Φ_1,14𝑖∗Φ_22𝑖00𝑀^𝑇₂−𝑇^𝑇₂𝑋_𝑖𝒜^𝑇_𝑑𝑖𝑋_𝑖𝒜^𝑇_𝑑𝑖𝑋_𝑖𝒜^𝑇_𝑑𝑖Φ_2,10Φ_2,11Φ_2,120Φ_2,14𝑖∗∗−𝑆000ℬ^𝑇_𝑖ℬ^𝑇_𝑖ℬ^𝑇_𝑖000Φ_3,13𝑖0∗∗∗−𝛾²𝐼00ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖ℬ^𝑇_1𝑖000Φ_4,13𝑖0∗∗∗∗−𝑄₁000000000∗∗∗∗∗−𝑄₂00000000∗∗∗∗∗∗−𝑋_𝑙0000000∗∗∗∗∗∗∗Φ₈₈000000∗∗∗∗∗∗∗∗Φ₉₉Φ00000∗∗∗∗∗∗∗∗∗_10,10𝑖Φ0000∗∗∗∗∗∗∗∗∗∗_11,11𝑖Φ000∗∗∗∗∗∗∗∗∗∗∗_12,12𝑖00∗∗∗∗∗∗∗∗∗∗∗∗−𝑆0∗∗∗∗∗∗∗∗∗∗∗∗∗−𝐼<0,𝑋_𝑖𝑅₁𝑋_𝑖≥𝑆_1𝑖,𝑋_𝑖𝑅₂𝑋_𝑖≥𝑆_2𝑖, (40)(41)(42)where Φ_10,10𝑖=−𝜏𝑆_1𝑖,Φ_11,11𝑖=−(𝜏−𝜏)(𝑆_1𝑖+𝑆_2𝑖),Φ_12,12𝑖=−(𝜏−𝜏)𝑆_2𝑖.

From (41) and (42), we know that the conditions in Theorem 8 are not strict linear matrix inequalities. By the assistance of the CCL method [46], the nonconvex feasibility problem formulated by (40)–(42) can be transformed into the following nonlinear minimization problem: MinimizeTr^𝑁_𝑖=1𝑋_𝑖𝐻_𝑖+𝑆_1𝑖𝐽_1𝑖+𝑆_2𝑖𝐽_2𝑖+𝑅₁𝐺₁+𝑅₂𝐺₂Subjectto(40)and𝐺₁𝐻_𝑖𝐻_𝑖𝐽_1𝑖𝐺≥𝐼,²𝐻_𝑖𝐻_𝑖𝐽_2𝑖𝑆≥𝐼,_1𝑖𝐼𝐼𝐽_1𝑖𝑆≥0,^2𝑖𝐼𝐼𝐽_2𝑖≥0,𝑅₁𝐼𝐼𝐺₁≥0,𝑅₂𝐼𝐼𝐺₂𝑋≥0,_𝑖𝐼𝐼𝐻_𝑖≥0. (43) If the solution of the above minimization problem is (6𝑁+4)𝑚, then system Σ_𝑒 is asymptotically stable with an 𝐻_∞ performance index via controller (5) and observer (6) with gains 𝐾_𝑖=𝐾_𝑖𝑋_𝑖⁻¹ and 𝐿=𝑆⁻¹𝐿, respectively.

4. A Numerical Example

Now, we provide an example to show the effectiveness of the main result in this paper.

Consider discrete-time switched system with parameters as follows: 𝐴₁=0.200−0.035,𝐴_𝑑1=,𝐵0.0200.01−0.03₁=87,𝐵₁₁=−0.030.05,𝐶₁=,𝐶1.20.8_𝑑1=1−0.6,𝐷₁𝐴=16,₂=−0.300.01−0.015,𝐴_𝑑2=,𝐵0.01−0.010.02−0.05₂=1010,𝐵₁₂=−0.010.07,𝐶₂=,𝐶0.61.6_𝑑2=0.8−0.4,𝐷₂=14. (44) The reference model is given by the following parameters: 𝐴_𝑟=0.5−10.3−0.035,𝐵_𝑟=,𝐶0.20.8_𝑟=0.50.5. (45) The disturbance model is presented by the following parameters: 𝑊=.0.87760.4794−0.47940.8776,𝑉=100 (46) Set 𝜏=1 and 𝜏=4. Here, we suppose the disturbance attenuation level 𝛾=0.8. Then, using Matlab Control Toolbox to solve Theorem 8, we obtain controller gains and observer gain as follows: 𝐾₁₁=−0.0477−0.0308,𝐾₂₁=,,𝐾−0.00110.0432𝐿=−0.0012−0.00040.00050.0002₁₂=−0.0159−0.0103,𝐾₂₂=−0.00040.0144. (47) Suppose the switching sequence as 121212⋯. The initial value of the states is chosen as 𝜙(𝑠)=[−0.2𝑒^−𝑠0.1𝑒^−𝑠]^𝑇, and the reference model of initial condition is selected as 𝑥_𝑟=[0.1−0.2]^𝑇. In the sequel, two kinds of reference inputs, that is, step reference input and sinusoidal reference input, are considered to demonstrate the effectiveness of the proposed method.

Case 1 ( step reference input). Let 𝑟(𝑘)=10,80≤𝑘<150,0,else𝑑𝑘,₂(𝑘)=30𝑘+1,80≤𝑘<150,0,else𝑘. (48) Curves of 𝑧_𝑘 and 𝑧_𝑟 are depicted in Figure 1 under input signal (48). From Figure 1, we can see that the system output can effectively track the reference model output in presence of matched disturbances and mismatched disturbances, which demonstrates the effectiveness of the proposed method. In order to evaluate the effectiveness of the DOB, curves of disturbances and disturbances estimation are shown in Figure 2. It illustrates that DOB can effectively estimate disturbances.

Figure 1 

Curves of 𝑧(𝑘) and 𝑧𝑟(𝑘) under Case 1.

Figure 2 

Curves of disturbances and disturbances estimation under Case 1.

Case 2 (sinusoidal reference input). Let 𝑟(𝑘)=10sin(𝑘),80≤𝑘<150,0,else𝑑𝑘,₂(𝑘)=30𝑘+1,80≤𝑘<150,0,else𝑘. (49) Figure 3 presents the curves of 𝑧_𝑘 and 𝑧_𝑟 under input signal (49), which demonstrates that the proposed method obtains good tracking performance in spite of matched disturbances and mismatched disturbances. Figure 4 shows the disturbances estimation results, which depicts that the DOB can effectively estimate disturbances.

Figure 3 

Curves of 𝑧(𝑘) and 𝑧𝑟(𝑘) under Case 2.

Figure 4 

Curves of disturbances and disturbances estimation under Case 2.

Remark 9. In this paper, a composite tracking controller is designed for a class of discrete-time switched systems with time-varying delay. In reality, many physical systems can be modelled as switched system, for example, flight control systems [47, 48] and inverted-pendulum system [49]. Take flight control system; for example, the flight control system can be modelled as switched system corresponding to finite operating points within the flight envelope, where 𝑑₂(𝑡) is denoted as wind gust and 𝑑₁(𝑡) is regarded as unknown harmonic disturbances. In order to better serve engineering, we will pay attention to studying a tracking controller design for a practical system based on our method in future.

5. Conclusions

The problem of 𝐻_∞ output tracking control for discrete-time switched systems subject to time-varying delay and disturbances has been studied. 𝐻_∞ control has achieved the attenuation performance with respect to norm bounded disturbances. DOBC has been employed to reject the disturbances with some known information. In this paper, a composite control scheme, that is, consisting of 𝐻_∞ control method and DOBC technique, has been proposed, which can effectively attenuate and reject the external disturbances. A numerical example has been provided to show the effectiveness of the proposed algorithm.

Acknowledgments

The authors thank Beihang University Lei Guo Professor for helpful discussions and the anonymous associate editor and reviewers for constructive suggestions. This work was supported in part by Key Project of Chinese National Programs for Fundamental Research and Development (973 program) under Grant (2012CB720003) and the National Natural Science Foundation of China (nos. 61273123, 61203011, and 61203013).