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

Authors: Haibin Sun; Linlin Hou
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper considers the problem ofH∞ 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.


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## Body

## 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 ofH∞ 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(1)Σ0:x(k+1)=Aσ(k)x(k)+Adσ(k)x(k-τ(k))+Bσ(k)(u(k)+d1(k))+B1σ(k)d2(k),z(k)=Cσ(k)x(k)+Cdσ(k)x(k-τ(k))+Dσ(k)u(k),x(s)=ϕ(s),s=-τ-,…,-1,0,
where x(k)∈ℝn is the states, z(k)∈ℝq 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 Ai, Adi, Bi, B1i, Ci, Cdi,Di, i∈𝒩, are constant matrices with appropriate dimensions. τ(k) is a time-varying delay of the system and satisfies
(2)τ_≤τ(k)≤τ¯,∀k∈ℤ+,
where τ_, τ¯, τ(k) are nonnegative integer numbers. ϕ(k) is the initial condition. d2(k)∈ℝp 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
(3)ω(k+1)=Wω(k),d1(k)=Vω(k),
where (W,BiV) is observable.Remark 1.
It is clear thatτ(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.
SystemΣ0 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 signalzr(k) is generated by the following system:
(4)xr(k+1)=Arxr(k)+Brr(k),zr(k)=Crxr(k),
where xr(k) is reference states, r(k)∈l2[0,∞) is reference input, and Ar is Hurwitz matrix with an appropriate dimension. Here we are interested in designing a state feedback controller by the following formula:
(5)u(k)=K1σ(k)x(k)-d^1(k)+K2σ(k)xr(k),
where d^1(k) is the estimation of the disturbances d1(k), which is obtained by the following disturbance observer:
(6)ω^(k)=ν(k)-Lx(k),ν(k+1)=W(ν(k)-Lx(k))+L((u(k)+d^1(k))Aσ(k)x(k)+Adσ(k)x(k-τ(k))+Bσ(k)(u(k)+d^1(k))),(7)d^1(k)=Vω^(k).Remark 3.
In this paper, we assume that the switching signalσ(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.Definingeω(k)=ω(k)-ω^(k) yields
(8)eω(k+1)=(W+LBσ(k)V)eω(k)+B-1σ(k)d2(k),
where B-1σ(k)=LB1σ(k).Applying controller (5) to system Σ0 and combining (3) and (7), we obtain
(9)x(k+1)=A-σ(k)x(k)+Adσ(k)x(k-τ(k))+Bσ(k)(d1(k)-d^1(k))+B1σ(k)d2(k)+B-σ(k)xr(k)=A-σ(k)x(k)+Adσ(k)x(k-τ(k))+B^σ(k)eω(k)+B1σ(k)d2(k)+B-σ(k)xr(k),
where
(10)A-σ(k)=Aσ(k)+Bσ(k)K1σ(k),B^σ(k)=Bσ(k)V,B-σ(k)=Bσ(k)K2σ(k),eω(k)=ω(k)-ω^(k).
Let
(11)e(k)=z(k)-zr(k),
then we have the following augmented switched system:
(12)Σe:ζ(k+1)=𝒜σ(k)ζ(k)+𝒜dσ(k)ζ(k-τ(k))+ℬσ(k)eω(k)+ℬ1σ(k)v(k),eω(k+1)=(W+LBσ(k)V)eω(k)+ℬ˘1σ(k)v(k),e(k)=𝒞σ(k)ζ(k)+𝒞dσ(k)ζ(k-τ(k)),
where
(13)𝒜σ(k)=[A-σ(k)B-σ(k)0Ar],𝒜dσ(k)=[Adσ(k)000],ℬσ(k)=[B^σ(k)0],ℬ1σ(k)=[B1σ(k)00Br],ζ(k)=[x(k)xr(k),]∈ℝm,v(k)=[d2(k)r(k)],𝒞σ(k)=[Cσ(k)+Dσ(k)K1σ(k)Dσ(k)K2σ(k)-Cr],𝒞dσ(k)=[Cdσ(k)0],ℬ˘1σ(k)=[B-1σ(k)0].
By defining
(14)𝒜^σ(k)=[Aσ(k)00Ar],ℬ^σ(k)=[Bσ(k)0],Kσ(k)=[K1σ(k)K2σ(k)],𝒞^σ(k)=[Cσ(k)-Cr],
then we have
(15)𝒜σ(k)=𝒜^σ(k)+ℬ^σ(k)Kσ(k),𝒞σ(k)=𝒞^σ(k)+Dσ(k)Kσ(k),
and the controller can be rewritten as
(16)u(k)=Kσ(k)ζ(k)-d^1(k).In order to prepare for a precise formulation, we introduce the following definition.Definition 4 (H∞ output tracking control problem).
Consider systemΣe. Given a prescribed γ>0, if there exists composite controller (5) such that the following two conditions are satisfied: (R1) systemΣe is asymptotically stable when v(k)=0, for all k≥0;(R2) under the zero-initial condition, the following inequality holds:(17)∥e(k)∥2<γ∥v(k)∥2,for any nonzero v(k)∈l2[0,∞), then system Σe 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 theH∞ output tracking control problem formulated in previous section.Theorem 5 (consider systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M1, M2, N1, N2, T1, T2, K1i, K2i, Pi>0,Q1>0,Q2>0, Q3>0, R1>0, R2>0, S>0,  i∈𝒩, such that the following inequality holds: (18)Φi=[Φ11iΦ1200M1T-T1T𝒜iTΦ18iΦ19iΦ1,10Φ1,11Φ1,120*Φ22i00M2T-T2T𝒜diT𝒜diT𝒜diTΦ2,10Φ2,11Φ2,120**-S000ℬiTℬiTℬiT000Φ3,13i***-γ2I00ℬ1iTℬ1iTℬ1iT000ℬ˘1iT****-Q100000000*****-Q20000000******-Pl-1000000*******Φ8800000********Φ990000*********Φ10,10000**********Φ11,1100***********Φ12,120************-S-1]<0, where
(19)Φ11i=-Pi+Q1+Q2+(τ¯-τ_+1)Q3+N1+N1T+𝒞iT𝒞i+𝒞iT𝒞di,Φ12=-N1T+N2+T1T-M1T,Φ18i=Φ19i=(𝒜i-I)T,Φ1,10=τ¯N1T,Φ1,11=(τ¯-τ_)T1T,Φ1,13=(τ¯-τ_)M1T,Φ22i=-Q3-N2-N2T+T2+T2T-M2-M2T+𝒞diT𝒞di,Φ2,10=τ¯N2T,Φ2,11=(τ¯-τ_)T2T,Φ2,12=(τ¯-τ_)M2T,Φ3,13i=(W+LBiV)T,Φ88=-(τ¯R1)-1,Φ99=-((τ¯-τ_)R2)-1,Φ10,10=-τ¯R1,Φ11,11=-(τ¯-τ_)(R1+R2),Φ12,12=-(τ¯-τ_)R2.Proof.
Choose a Lyapunov-Krasovskii functional candidate as(20)V(k)=V1(k)+V2(k)+V3(k),
where
(21)V1(k)=ζT(k)Pσ(k)ζ(k)+eωT(k)Seω(k),V2(k)=∑θ=k-τ_k-1‍ζT(θ)Q1ζ(θ)+∑θ=k-τ¯k-1‍ζT(θ)Q2ζ(θ)+∑θ=k-τ(k)k-1‍ζT(θ)Q3ζ(θ),V3(k)=∑i=-τ¯-1‍∑θ=k+ik-1‍ρT(θ)R1ρ(θ)+∑i=-τ¯+1-τ_‍∑θ=k+i-1k-1‍ρT(θ)R2ρ(θ)+∑i=-τ¯+2-τ_+1‍∑θ=k+i-1k-1‍ζT(θ)Q3ζ(θ),
and ρ(θ)=ζ(θ+1)-ζ(θ).
Without loss of generality, we assume thatσ(k+1)=l,σ(k)=i, for all i,l∈𝒩. Then taking the forward difference yields
(22)ΔV1(k)=V1(k+1)-V1(k)=ζT(k+1)Plζ(k+1)-ζT(k)Piζ(k)+eωT(k+1)Seω(k+1)-eωT(k)Seω(k)=ξT(k)AiTPlAiξ(k)-ζT(k)Piζ(k)+ξT(k)BiTSBiξ(k)-eωT(k)Seω(k),
where
(23)ξT(k)=[ζT(k)ζT(k-τ(k))eωT(k)vT(k)],Ai=[𝒜i𝒜diℬiℬ1i],Bi=[00(W+LBiV)ℬ˘1i].
Direct computation gives
(24)ΔV2(k)=∑θ=k+1-τ_k‍ζT(θ)Q1ζ(θ)-∑θ=k-τ_k-1‍ζT(θ)Q1ζ(θ)+∑θ=k+1-τ¯k‍ζT(θ)Q2ζ(θ)-∑θ=k-τ¯k-1‍ζT(θ)Q2ζ(θ)+∑θ=k+1-τ(k+1)k‍ζT(θ)Q3ζ(θ)-∑θ=k-τ(k)k-1‍ζT(θ)Q3ζ(θ)=ζT(k)(Q1+Q2+Q3)ζ(k)-ζT(k-τ_)Q1ζ(k-τ_)-ζT(k-τ¯)Q2ζ(k-τ¯)-ζT(k-τ(k))Q3ζ(k-τ(k))+∑θ=k+1-τ(k+1)k-1‍ζT(θ)Q3ζ(θ)-∑θ=k+1-τ(k)k-1‍ζT(θ)Q3ζ(θ).
Note that
(25)∑θ=k+1-τ(k+1)k-1‍ζT(θ)Q3ζ(θ)=∑θ=k+1-τ_k-1‍ζT(θ)Q3ζ(θ)+∑θ=k+1-τ(k+1)k-τ_‍ζT(θ)Q3ζ(θ)≤∑θ=k+1-τ(k)k-1‍ζT(θ)Q3ζ(θ)+∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
Hence we obtain
(26)ΔV2(k)≤ζT(k)(Q1+Q2+Q3)ζ(k)-ζT(k-τ_)Q1ζ(k-τ_)-ζT(k-τ¯)Q2ζ(k-τ¯)-ζT(k-τ(k))Q3ζ(k-τ(k))+∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
After some manipulations, the following inequality is satisfied:
(27)ΔV3(k)=∑i=-τ¯-1‍∑θ=k+1+ik‍ρT(θ)R1ρ(θ)-∑i=-τ¯-1‍∑θ=k+ik-1‍ρT(θ)R1ρ(θ)+∑i=-τ¯+1-τ_‍∑θ=k+ik‍ρT(θ)R2ρ(θ)-∑i=-τ¯+1-τ_‍∑θ=k+i-1k-1‍ρT(θ)R2ρ(θ)+∑i=-τ¯+2-τ_+1‍∑θ=k+ik‍ζT(θ)Q3ζ(θ)-∑i=-τ¯+2-τ_+1‍∑θ=k+i-1k-1‍ζT(θ)Q3ζ(θ)=τ¯ρT(k)R1ρ(k)-∑θ=k-τ¯k-1‍ρT(θ)R1ρ(θ)+(τ¯-τ_)ρT(k)R2ρ(k)-∑θ=k-τ¯k-τ_-1‍ρT(θ)R2ρ(θ)+(τ¯-τ_)ζT(k)Q3ζ(k)-∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
Observe that the following equalities hold naturally:
(28)2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k))-∑θ=k-τ(k)k-1‍ρ(θ))=0,2(ζT(k)T1T+ζT(k-dj(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯)-∑θ=k-τ¯k-1-τ(k)‍ρ(θ))=0,2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k))-∑θ=k-τ(k)k-1-τ_‍ρ(θ))=0.
Then
(29)-∑θ=k-τ¯k-1‍ρT(θ)R1ρ(θ)-∑θ=k-τ¯k-τ_-1‍ρT(θ)R2ρ(θ)=-∑θ=k-τ¯k-τ(k)-1‍ρT(θ)(R1+R2)ρ(θ)-∑θ=k-τ(k)k-1‍ρT(θ)R1ρ(θ)-∑θ=k-τ(k)k-τ_-1‍ρT(θ)R2ρ(θ)+2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k))-∑θ=k-τ(k)k-1‍ρ(θ))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯)-∑θ=k-τ¯k-1-τ(k)‍ρ(θ))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k))-∑θ=k-τ(k)k-1-τ_‍ρ(θ))=2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))-2(ζT(k)T1T+ζT(k-τ(k))T2T)×∑θ=k-τ¯k-1-τ(k)‍ρ(θ)-∑θ=k-τ¯k-τ(k)-1‍ρT(θ)(R1+R2)ρ(θ)-2(ζT(k)N1T+ζT(k-τ(k))N2T)×∑θ=k-τ(k)k-1‍ρ(θ)-∑θ=k-τ(k)k-1‍ρT(θ)R1ρ(θ)-2(ζT(k)M1T+ζT(k-τ(k))M2T)×∑θ=k-τ(k)k-1-τ_‍ρ(θ)-∑θ=k-τ(k)k-τ_-1‍ρT(θ)R2ρ(θ)=2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))+τ(k)(ζT(k)N1T+ζT(k-τ(k))N2T)R1-1×(N1ζ(k)+N2ζ(k-τ(k)))+(τ¯-τ(k))(ζT(k)T1T+ζT(k-τ(k))T2T)×(R1+R2)-1(T1ζ(k)+T2ζ(k-τ(k)))+(τ(k)-τ_)(ζT(k)M1T+ζT(k-τ(k))M2T)R2-1×(M1ζ(k)+M2ζ(k-τ(k)))-∑θ=k-τ(k)k-1‍ηT(k)NTR1-1Nη(k)-∑θ=k-τ¯k-1-τ(k)‍ηT(k)TT(R1+R2)-1Tη(k)-∑θ=k-τ(k)k-τ_-1‍ηT(k)MTR2-1Mη(k)≤2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))+τ¯(ζT(k)N1T+ζT(k-τ(k))N2T)R1-1×(N1ζ(k)+N2ζ(k-τ(k)))+(τ¯-τ_)(ζT(k)T1T+ζT(k-τ(k))T2T)×(R1+R2)-1(T1ζ(k)+T2ζ(k-τ(k)))+(τ¯-τ_)(ζT(k)M1T+ζT(k-τ(k))M2T)R2-1×(M1ζ(k)+M2ζ(k-τ(k))),
where
(30)ηT(k)=[ζT(k)ζT(k-τ(k))ρT(θ)],N=[N1N2R1],M=[M1M2R2],T=[T1T2R1+R2].
From (22)–(29), and by some manipulations, we obtain
(31)ΔV(k)≤λT(k)Φ1iλ(k)+ξT(k)Φ2iξ(k)+η¯T(k)Φ3η¯(k)-η¯T(k)[𝒞iT𝒞diT][𝒞i𝒞di]η¯(k),
where
(32)λT(k)=[ζT(k)ζT(k-τ(k))eωT(k)vT(k)ζT(k-τ_)ζT(k-τ¯)],η¯T(k)=[ζT(k)ζT(k-τ(k))],Φ2i=[AiTPlAi+BiTSBi+A¯iT(τ¯R1+(τ¯-τ_)R2)A¯i],A¯i=[Ai-I¯],I¯=[I000],Φ3=τ¯[N1N2]TR1-1[N1N2]+(τ¯-τ_)[T1T2]T×(R1+R2)-1[T1T2]+(τ¯-τ_)[M1M2]TR2-1[M1M2],Φ1i=[Φ11iΦ1200M1T-T1T*Φ22i00M2T-T2T**-S000***-γ2I00****-Q10*****-Q2].
Now, we develop the conclusion from two aspects. We first establish the asymptotic stability of system Σe under the condition of zero disturbances. In fact, when v(k)=0, it is verified that
(33)ΔV(k)≤λ~T(k)Φ~1iλ~(k)+ξT(k)Φ~2iξ(k)+∑j=1m‍η¯jT(k)Φ3jη¯j(k)-ζT(k)𝒢iT𝒢iζ(k),
where
(34)λ~T(k)=[ζT(k)ζT(k-τ(k))eωT(k)ζT(k-τ_)ζT(k-τ¯)],Φ~2i=[AiT~PlAi~+B^iTSB^i+AiT^(τ¯R1+(τ¯-τ_)R2)Ai^],Ai~=[𝒜i𝒜diℬ],Ai^=[Ai~-I~],I~=[I00],B^=[00W+LBiV],Φ~1i=[Φ11iΦ120M1T-T1T*Φ22i0M2T-T2T**-S00***-Q10****-Q2].
Applying Schur complement formula, we obtain ΔV(k)<0 if (18) is true. Therefore, it is easy to see that the error system Σe is asymptotically stable by the Lyapunov-Krasovskii stability theorem.
Next, we will present that under the zero-initial condition, the time-delay systemΣe satisfies (17) for all nonzero v(k)∈l2[0,∞). To this end, we introduce
(35)JN=∑k=0N‍(eT(k)e(k)-γ2vT(k)v(k)+ΔV(k))-V(N+1).
Then, by the Schur complement formula, it easily follows from (18) and (31) that
(36)JN≤∑k=0∞‍(eT(k)e(k)-γ2vT(k)v(k)+ΔV(k))<0,
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,…,PN,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 ofH∞ output tracking controller design using CCL method.Define(37)Xi=Pi-1,R-1=R1-1,R-2=R2-1,K-i=KiXi,L-=SL,Q-1i=XiQ1Xi,Q-2i=XiQ2Xi,Q-3i=XiQ3Xi,M-1=XiM1Xi,M-2=XiM2Xi,N-1=XiN1Xi,N-2=XiN2Xi,T-1=XiT1Xi,T-2=XiT2Xi,Γ=diag{Xi,Xi,I,I,Xi,Xi,I,I,I,Xi,Xi,Xi,S}.
Pre- and postmultiplying Γ and ΓT on (18), and using Schur complement formula, we obtain(38)Φ-i=[Φ-11iΦ-1200M-1T-T-1TΦ-17iΦ-18iΦ-19iΦ-1,10Φ-1,11Φ-1,120Φ-1,14i*Φ-22i00M-2T-T-2TXi𝒜diTXi𝒜diTXi𝒜diTΦ-2,10Φ-2,11Φ-2,120Φ-2,14i**-S000ℬiTℬiTℬiT000Φ-3,13i0***-γ2I00ℬ1iTℬ1iTℬ1iT000Φ-4,13i0****-Q-1i000000000*****-Q-2i00000000******-Xl0000000*******Φ-88000000********Φ-9900000*********Φ-10,10i0000**********Φ-11,11i000***********Φ-12,12i00************-S0*************-I]<0,where
(39)Φ-11i=-Xi+Q-1i+Q-2i+(τ¯-τ_+1)Q-3i+N-1+N-1T,Φ-12=-N-1T+N-2+T-1T-M-1T,Φ-17i=Xi𝒜^iT+K-iTℬ^iT,Φ-18i=Φ19i=Φ-17i-Xi,Φ-1,10=τ¯N-1T,Φ-1,11=(τ¯-τ_)T-1T,Φ-1,13=(τ¯-τ_)M-1T,Φ-1,14i=Xi𝒞^iT+K-iTDiT,Φ-22i=-Q-3i-N-2-N-2T+T-2+T-2T-M-2-M-2T,Φ-2,14i=Xi𝒞diT,Φ-2,10=τ¯N-2T,Φ-2,11=(τ¯-τ_)T-2T,Φ-2,12=(τ¯-τ_)M-2T,Φ-3,13i=WTS+(L-BiV)T,Φ-88=-1τ¯R-1,Φ-99=-1(τ¯-τ_)R-2,Φ-10,10i=-τ¯XiR1Xi,Φ-4,13i=ℬ˘1iTS,Φ-11,11i=-(τ¯-τ_)Xi(R1+R2)Xi,Φ-12,12i=-(τ¯-τ_)XiR2Xi.Theorem 7 (consider the systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M-1, M-2, N-1, N-2, T-1, T-2, K-1i, K-2i, Xi>0,Q-1>0,Q-2>0,Q-3>0, R-1>0, R-2>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 R-1, R-2, XiR1Xi and XiR2Xi. In the sequel, the CCL method is resorted to solve the desired controller gains and observer gain.Introduce two new variablesS1i and S2i such that XiR1Xi≥S1i and XiR2Xi≥S2i, then we obtain the following results.Theorem 8 (Consider systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M-1,M-2, N-1, N-2, T-1,T-2, K-1i, K-2i, Xi>0, Q-1>0, Q-2>0, Q-3>0,R-1>0, R-2>0,R1>0,R2>0, S>0, S1i,S2i, i∈𝒩, such that the following inequalities hold(40)Φ^i=[Φ-11iΦ-1200M-1T-T-1TΦ-17iΦ-18iΦ-19iΦ-1,10Φ-1,11Φ-1,120Φ-1,14i*Φ-22i00M-2T-T-2TXi𝒜diTXi𝒜diTXi𝒜diTΦ-2,10Φ-2,11Φ-2,120Φ-2,14i**-S000ℬiTℬiTℬiT000Φ-3,13i0***-γ2I00ℬ1iTℬ1iTℬ1iT000Φ-4,13i0****-Q-1000000000*****-Q-200000000******-Xl0000000*******Φ-88000000********Φ-9900000*********Φ^10,10i0000**********Φ^11,11i000***********Φ^12,12i00************-S0*************-I]<0,(41)XiR1Xi≥S1i,(42)XiR2Xi≥S2i,where Φ^10,10i=-τ-S1i,Φ^11,11i=-(τ--τ_)(S1i+S2i),Φ^12,12i=-(τ--τ_)S2i.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:
(43)MinimizeTr(∑i=1N‍(XiHi+S1iJ1i+S2iJ2i)+R-1G1+R-2G2)Subjectto(40)and[G1HiHiJ1i]≥I,[G2HiHiJ2i]≥I,[S1iIIJ1i]≥0,[S2iIIJ2i]≥0,[R-1IIG1]≥0,[R-2IIG2]≥0,[XiIIHi]≥0.
If the solution of the above minimization problem is (6N+4)m, then system Σe is asymptotically stable with an H∞ performance index via controller (5) and observer (6) with gains Ki=K-iXi-1 and L=S-1L-, respectively.


## 3.1.H∞ Output Tracking Performance Analysis

In this subsection, we focus on developing delay-dependent condition to solve theH∞ output tracking control problem formulated in previous section.Theorem 5 (consider systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M1, M2, N1, N2, T1, T2, K1i, K2i, Pi>0,Q1>0,Q2>0, Q3>0, R1>0, R2>0, S>0,  i∈𝒩, such that the following inequality holds: (18)Φi=[Φ11iΦ1200M1T-T1T𝒜iTΦ18iΦ19iΦ1,10Φ1,11Φ1,120*Φ22i00M2T-T2T𝒜diT𝒜diT𝒜diTΦ2,10Φ2,11Φ2,120**-S000ℬiTℬiTℬiT000Φ3,13i***-γ2I00ℬ1iTℬ1iTℬ1iT000ℬ˘1iT****-Q100000000*****-Q20000000******-Pl-1000000*******Φ8800000********Φ990000*********Φ10,10000**********Φ11,1100***********Φ12,120************-S-1]<0, where
(19)Φ11i=-Pi+Q1+Q2+(τ¯-τ_+1)Q3+N1+N1T+𝒞iT𝒞i+𝒞iT𝒞di,Φ12=-N1T+N2+T1T-M1T,Φ18i=Φ19i=(𝒜i-I)T,Φ1,10=τ¯N1T,Φ1,11=(τ¯-τ_)T1T,Φ1,13=(τ¯-τ_)M1T,Φ22i=-Q3-N2-N2T+T2+T2T-M2-M2T+𝒞diT𝒞di,Φ2,10=τ¯N2T,Φ2,11=(τ¯-τ_)T2T,Φ2,12=(τ¯-τ_)M2T,Φ3,13i=(W+LBiV)T,Φ88=-(τ¯R1)-1,Φ99=-((τ¯-τ_)R2)-1,Φ10,10=-τ¯R1,Φ11,11=-(τ¯-τ_)(R1+R2),Φ12,12=-(τ¯-τ_)R2.Proof.
Choose a Lyapunov-Krasovskii functional candidate as(20)V(k)=V1(k)+V2(k)+V3(k),
where
(21)V1(k)=ζT(k)Pσ(k)ζ(k)+eωT(k)Seω(k),V2(k)=∑θ=k-τ_k-1‍ζT(θ)Q1ζ(θ)+∑θ=k-τ¯k-1‍ζT(θ)Q2ζ(θ)+∑θ=k-τ(k)k-1‍ζT(θ)Q3ζ(θ),V3(k)=∑i=-τ¯-1‍∑θ=k+ik-1‍ρT(θ)R1ρ(θ)+∑i=-τ¯+1-τ_‍∑θ=k+i-1k-1‍ρT(θ)R2ρ(θ)+∑i=-τ¯+2-τ_+1‍∑θ=k+i-1k-1‍ζT(θ)Q3ζ(θ),
and ρ(θ)=ζ(θ+1)-ζ(θ).
Without loss of generality, we assume thatσ(k+1)=l,σ(k)=i, for all i,l∈𝒩. Then taking the forward difference yields
(22)ΔV1(k)=V1(k+1)-V1(k)=ζT(k+1)Plζ(k+1)-ζT(k)Piζ(k)+eωT(k+1)Seω(k+1)-eωT(k)Seω(k)=ξT(k)AiTPlAiξ(k)-ζT(k)Piζ(k)+ξT(k)BiTSBiξ(k)-eωT(k)Seω(k),
where
(23)ξT(k)=[ζT(k)ζT(k-τ(k))eωT(k)vT(k)],Ai=[𝒜i𝒜diℬiℬ1i],Bi=[00(W+LBiV)ℬ˘1i].
Direct computation gives
(24)ΔV2(k)=∑θ=k+1-τ_k‍ζT(θ)Q1ζ(θ)-∑θ=k-τ_k-1‍ζT(θ)Q1ζ(θ)+∑θ=k+1-τ¯k‍ζT(θ)Q2ζ(θ)-∑θ=k-τ¯k-1‍ζT(θ)Q2ζ(θ)+∑θ=k+1-τ(k+1)k‍ζT(θ)Q3ζ(θ)-∑θ=k-τ(k)k-1‍ζT(θ)Q3ζ(θ)=ζT(k)(Q1+Q2+Q3)ζ(k)-ζT(k-τ_)Q1ζ(k-τ_)-ζT(k-τ¯)Q2ζ(k-τ¯)-ζT(k-τ(k))Q3ζ(k-τ(k))+∑θ=k+1-τ(k+1)k-1‍ζT(θ)Q3ζ(θ)-∑θ=k+1-τ(k)k-1‍ζT(θ)Q3ζ(θ).
Note that
(25)∑θ=k+1-τ(k+1)k-1‍ζT(θ)Q3ζ(θ)=∑θ=k+1-τ_k-1‍ζT(θ)Q3ζ(θ)+∑θ=k+1-τ(k+1)k-τ_‍ζT(θ)Q3ζ(θ)≤∑θ=k+1-τ(k)k-1‍ζT(θ)Q3ζ(θ)+∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
Hence we obtain
(26)ΔV2(k)≤ζT(k)(Q1+Q2+Q3)ζ(k)-ζT(k-τ_)Q1ζ(k-τ_)-ζT(k-τ¯)Q2ζ(k-τ¯)-ζT(k-τ(k))Q3ζ(k-τ(k))+∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
After some manipulations, the following inequality is satisfied:
(27)ΔV3(k)=∑i=-τ¯-1‍∑θ=k+1+ik‍ρT(θ)R1ρ(θ)-∑i=-τ¯-1‍∑θ=k+ik-1‍ρT(θ)R1ρ(θ)+∑i=-τ¯+1-τ_‍∑θ=k+ik‍ρT(θ)R2ρ(θ)-∑i=-τ¯+1-τ_‍∑θ=k+i-1k-1‍ρT(θ)R2ρ(θ)+∑i=-τ¯+2-τ_+1‍∑θ=k+ik‍ζT(θ)Q3ζ(θ)-∑i=-τ¯+2-τ_+1‍∑θ=k+i-1k-1‍ζT(θ)Q3ζ(θ)=τ¯ρT(k)R1ρ(k)-∑θ=k-τ¯k-1‍ρT(θ)R1ρ(θ)+(τ¯-τ_)ρT(k)R2ρ(k)-∑θ=k-τ¯k-τ_-1‍ρT(θ)R2ρ(θ)+(τ¯-τ_)ζT(k)Q3ζ(k)-∑θ=k+1-τ¯k-τ_‍ζT(θ)Q3ζ(θ).
Observe that the following equalities hold naturally:
(28)2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k))-∑θ=k-τ(k)k-1‍ρ(θ))=0,2(ζT(k)T1T+ζT(k-dj(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯)-∑θ=k-τ¯k-1-τ(k)‍ρ(θ))=0,2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k))-∑θ=k-τ(k)k-1-τ_‍ρ(θ))=0.
Then
(29)-∑θ=k-τ¯k-1‍ρT(θ)R1ρ(θ)-∑θ=k-τ¯k-τ_-1‍ρT(θ)R2ρ(θ)=-∑θ=k-τ¯k-τ(k)-1‍ρT(θ)(R1+R2)ρ(θ)-∑θ=k-τ(k)k-1‍ρT(θ)R1ρ(θ)-∑θ=k-τ(k)k-τ_-1‍ρT(θ)R2ρ(θ)+2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k))-∑θ=k-τ(k)k-1‍ρ(θ))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯)-∑θ=k-τ¯k-1-τ(k)‍ρ(θ))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k))-∑θ=k-τ(k)k-1-τ_‍ρ(θ))=2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))-2(ζT(k)T1T+ζT(k-τ(k))T2T)×∑θ=k-τ¯k-1-τ(k)‍ρ(θ)-∑θ=k-τ¯k-τ(k)-1‍ρT(θ)(R1+R2)ρ(θ)-2(ζT(k)N1T+ζT(k-τ(k))N2T)×∑θ=k-τ(k)k-1‍ρ(θ)-∑θ=k-τ(k)k-1‍ρT(θ)R1ρ(θ)-2(ζT(k)M1T+ζT(k-τ(k))M2T)×∑θ=k-τ(k)k-1-τ_‍ρ(θ)-∑θ=k-τ(k)k-τ_-1‍ρT(θ)R2ρ(θ)=2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))+τ(k)(ζT(k)N1T+ζT(k-τ(k))N2T)R1-1×(N1ζ(k)+N2ζ(k-τ(k)))+(τ¯-τ(k))(ζT(k)T1T+ζT(k-τ(k))T2T)×(R1+R2)-1(T1ζ(k)+T2ζ(k-τ(k)))+(τ(k)-τ_)(ζT(k)M1T+ζT(k-τ(k))M2T)R2-1×(M1ζ(k)+M2ζ(k-τ(k)))-∑θ=k-τ(k)k-1‍ηT(k)NTR1-1Nη(k)-∑θ=k-τ¯k-1-τ(k)‍ηT(k)TT(R1+R2)-1Tη(k)-∑θ=k-τ(k)k-τ_-1‍ηT(k)MTR2-1Mη(k)≤2(ζT(k)N1T+ζT(k-τ(k))N2T)×(ζ(k)-ζ(k-τ(k)))+2(ζT(k)T1T+ζT(k-τ(k))T2T)×(ζ(k-τ(k))-ζ(k-τ¯))+2(ζT(k)M1T+ζT(k-τ(k))M2T)×(ζ(k-τ_)-ζ(k-τ(k)))+τ¯(ζT(k)N1T+ζT(k-τ(k))N2T)R1-1×(N1ζ(k)+N2ζ(k-τ(k)))+(τ¯-τ_)(ζT(k)T1T+ζT(k-τ(k))T2T)×(R1+R2)-1(T1ζ(k)+T2ζ(k-τ(k)))+(τ¯-τ_)(ζT(k)M1T+ζT(k-τ(k))M2T)R2-1×(M1ζ(k)+M2ζ(k-τ(k))),
where
(30)ηT(k)=[ζT(k)ζT(k-τ(k))ρT(θ)],N=[N1N2R1],M=[M1M2R2],T=[T1T2R1+R2].
From (22)–(29), and by some manipulations, we obtain
(31)ΔV(k)≤λT(k)Φ1iλ(k)+ξT(k)Φ2iξ(k)+η¯T(k)Φ3η¯(k)-η¯T(k)[𝒞iT𝒞diT][𝒞i𝒞di]η¯(k),
where
(32)λT(k)=[ζT(k)ζT(k-τ(k))eωT(k)vT(k)ζT(k-τ_)ζT(k-τ¯)],η¯T(k)=[ζT(k)ζT(k-τ(k))],Φ2i=[AiTPlAi+BiTSBi+A¯iT(τ¯R1+(τ¯-τ_)R2)A¯i],A¯i=[Ai-I¯],I¯=[I000],Φ3=τ¯[N1N2]TR1-1[N1N2]+(τ¯-τ_)[T1T2]T×(R1+R2)-1[T1T2]+(τ¯-τ_)[M1M2]TR2-1[M1M2],Φ1i=[Φ11iΦ1200M1T-T1T*Φ22i00M2T-T2T**-S000***-γ2I00****-Q10*****-Q2].
Now, we develop the conclusion from two aspects. We first establish the asymptotic stability of system Σe under the condition of zero disturbances. In fact, when v(k)=0, it is verified that
(33)ΔV(k)≤λ~T(k)Φ~1iλ~(k)+ξT(k)Φ~2iξ(k)+∑j=1m‍η¯jT(k)Φ3jη¯j(k)-ζT(k)𝒢iT𝒢iζ(k),
where
(34)λ~T(k)=[ζT(k)ζT(k-τ(k))eωT(k)ζT(k-τ_)ζT(k-τ¯)],Φ~2i=[AiT~PlAi~+B^iTSB^i+AiT^(τ¯R1+(τ¯-τ_)R2)Ai^],Ai~=[𝒜i𝒜diℬ],Ai^=[Ai~-I~],I~=[I00],B^=[00W+LBiV],Φ~1i=[Φ11iΦ120M1T-T1T*Φ22i0M2T-T2T**-S00***-Q10****-Q2].
Applying Schur complement formula, we obtain ΔV(k)<0 if (18) is true. Therefore, it is easy to see that the error system Σe is asymptotically stable by the Lyapunov-Krasovskii stability theorem.
Next, we will present that under the zero-initial condition, the time-delay systemΣe satisfies (17) for all nonzero v(k)∈l2[0,∞). To this end, we introduce
(35)JN=∑k=0N‍(eT(k)e(k)-γ2vT(k)v(k)+ΔV(k))-V(N+1).
Then, by the Schur complement formula, it easily follows from (18) and (31) that
(36)JN≤∑k=0∞‍(eT(k)e(k)-γ2vT(k)v(k)+ΔV(k))<0,
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,…,PN,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 ofH∞ output tracking controller design using CCL method.Define(37)Xi=Pi-1,R-1=R1-1,R-2=R2-1,K-i=KiXi,L-=SL,Q-1i=XiQ1Xi,Q-2i=XiQ2Xi,Q-3i=XiQ3Xi,M-1=XiM1Xi,M-2=XiM2Xi,N-1=XiN1Xi,N-2=XiN2Xi,T-1=XiT1Xi,T-2=XiT2Xi,Γ=diag{Xi,Xi,I,I,Xi,Xi,I,I,I,Xi,Xi,Xi,S}.
Pre- and postmultiplying Γ and ΓT on (18), and using Schur complement formula, we obtain(38)Φ-i=[Φ-11iΦ-1200M-1T-T-1TΦ-17iΦ-18iΦ-19iΦ-1,10Φ-1,11Φ-1,120Φ-1,14i*Φ-22i00M-2T-T-2TXi𝒜diTXi𝒜diTXi𝒜diTΦ-2,10Φ-2,11Φ-2,120Φ-2,14i**-S000ℬiTℬiTℬiT000Φ-3,13i0***-γ2I00ℬ1iTℬ1iTℬ1iT000Φ-4,13i0****-Q-1i000000000*****-Q-2i00000000******-Xl0000000*******Φ-88000000********Φ-9900000*********Φ-10,10i0000**********Φ-11,11i000***********Φ-12,12i00************-S0*************-I]<0,where
(39)Φ-11i=-Xi+Q-1i+Q-2i+(τ¯-τ_+1)Q-3i+N-1+N-1T,Φ-12=-N-1T+N-2+T-1T-M-1T,Φ-17i=Xi𝒜^iT+K-iTℬ^iT,Φ-18i=Φ19i=Φ-17i-Xi,Φ-1,10=τ¯N-1T,Φ-1,11=(τ¯-τ_)T-1T,Φ-1,13=(τ¯-τ_)M-1T,Φ-1,14i=Xi𝒞^iT+K-iTDiT,Φ-22i=-Q-3i-N-2-N-2T+T-2+T-2T-M-2-M-2T,Φ-2,14i=Xi𝒞diT,Φ-2,10=τ¯N-2T,Φ-2,11=(τ¯-τ_)T-2T,Φ-2,12=(τ¯-τ_)M-2T,Φ-3,13i=WTS+(L-BiV)T,Φ-88=-1τ¯R-1,Φ-99=-1(τ¯-τ_)R-2,Φ-10,10i=-τ¯XiR1Xi,Φ-4,13i=ℬ˘1iTS,Φ-11,11i=-(τ¯-τ_)Xi(R1+R2)Xi,Φ-12,12i=-(τ¯-τ_)XiR2Xi.Theorem 7 (consider the systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M-1, M-2, N-1, N-2, T-1, T-2, K-1i, K-2i, Xi>0,Q-1>0,Q-2>0,Q-3>0, R-1>0, R-2>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 R-1, R-2, XiR1Xi and XiR2Xi. In the sequel, the CCL method is resorted to solve the desired controller gains and observer gain.Introduce two new variablesS1i and S2i such that XiR1Xi≥S1i and XiR2Xi≥S2i, then we obtain the following results.Theorem 8 (Consider systemΣe).
For scalarsτ_>0,τ¯>0,γ>0, the error system Σe is asymptotically stable with an H∞ performance index, if there exist matrices M-1,M-2, N-1, N-2, T-1,T-2, K-1i, K-2i, Xi>0, Q-1>0, Q-2>0, Q-3>0,R-1>0, R-2>0,R1>0,R2>0, S>0, S1i,S2i, i∈𝒩, such that the following inequalities hold(40)Φ^i=[Φ-11iΦ-1200M-1T-T-1TΦ-17iΦ-18iΦ-19iΦ-1,10Φ-1,11Φ-1,120Φ-1,14i*Φ-22i00M-2T-T-2TXi𝒜diTXi𝒜diTXi𝒜diTΦ-2,10Φ-2,11Φ-2,120Φ-2,14i**-S000ℬiTℬiTℬiT000Φ-3,13i0***-γ2I00ℬ1iTℬ1iTℬ1iT000Φ-4,13i0****-Q-1000000000*****-Q-200000000******-Xl0000000*******Φ-88000000********Φ-9900000*********Φ^10,10i0000**********Φ^11,11i000***********Φ^12,12i00************-S0*************-I]<0,(41)XiR1Xi≥S1i,(42)XiR2Xi≥S2i,where Φ^10,10i=-τ-S1i,Φ^11,11i=-(τ--τ_)(S1i+S2i),Φ^12,12i=-(τ--τ_)S2i.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:
(43)MinimizeTr(∑i=1N‍(XiHi+S1iJ1i+S2iJ2i)+R-1G1+R-2G2)Subjectto(40)and[G1HiHiJ1i]≥I,[G2HiHiJ2i]≥I,[S1iIIJ1i]≥0,[S2iIIJ2i]≥0,[R-1IIG1]≥0,[R-2IIG2]≥0,[XiIIHi]≥0.
If the solution of the above minimization problem is (6N+4)m, then system Σe is asymptotically stable with an H∞ performance index via controller (5) and observer (6) with gains Ki=K-iXi-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:(44)A1=[0.200-0.035],Ad1=[0.0200.01-0.03],B1=[87],B11=[-0.030.05],C1=[1.20.8],Cd1=[1-0.6],D1=16,A2=[-0.300.01-0.015],Ad2=[0.01-0.010.02-0.05],B2=[1010],B12=[-0.010.07],C2=[0.61.6],Cd2=[0.8-0.4],D2=14.
The reference model is given by the following parameters:
(45)Ar=[0.5-10.3-0.035],Br=[0.20.8],Cr=[0.50.5].
The disturbance model is presented by the following parameters:
(46)W=[0.87760.4794-0.47940.8776],V=[100].
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:
(47)K11=[-0.0477-0.0308],K21=[-0.00110.0432],L=[-0.0012-0.00040.00050.0002],K12=[-0.0159-0.0103],K22=[-0.00040.0144].
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 xr=[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(48)r(k)={10,80≤k<150,0,elsek,d2(k)={30k+1,80≤k<150,0,elsek.
Curves of zk and zr 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 ofz(k) and zr(k) under Case 1.Figure 2
Curves of disturbances and disturbances estimation under Case1.Case 2 (sinusoidal reference input).
Let(49)r(k)={10sin(k),80≤k<150,0,elsek,d2(k)={30k+1,80≤k<150,0,elsek.
Figure 3 presents the curves of zk and zr 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 ofz(k) and zr(k) under Case 2.Figure 4
Curves of disturbances and disturbances estimation under Case2.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 ofH∞ 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.



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*Source: 698935-2013-05-02.xml*