Adaptive Fault-Tolerant 𝐻_∞ Controller Design for Networked Systems with Actuator Saturation

Abstract

In this paper, an indirect adaptive fault-tolerant 𝐻_∞ controller design method is proposed for networked systems in the presence of actuator saturation. Based on the on-line estimation of eventual faults, the parameters of controller are being updated automatically to compensate the fault effects on systems. The designs are given in linear matrix inequalities (LMIs) approach, which can guarantee the disturbance tolerance level and adaptive 𝐻_∞ performances of networked systems in the cases of actuator saturation and actuator failures. An example is given to illustrate the efficiency of the design method.

1. Introduction

With the rapid developments in network technologies, more and more communication networks are used in control systems; especially, the stability analysis for systems with time delays have become an active research area. However, networked control systems with actuator saturation and time delays are often encountered in many practical systems such as electrical heaters and long transmission lines in pneumatic, hydraulic, and rolling mill systems. Since the existence of time delay and actuator saturation in a physical system often induces instability of poor performance, research on time delay systems with actuator saturation is a topic of great practical and theoretical importance. If the saturation and time delay are ignored in system analysis and design, the performance of the overall system can be degenerated. More seriously, saturation and time delay can cause instability of the overall system. Therefore, over the last several decades, many researchers have considered various control problems of disturbance rejection for linear systems subject to actuator saturation [1–11]. Papers [4, 5] carried out the 𝐿₂ gain analysis and minimization. Although there are plenty of papers that are devoted to dealing with different problems for systems with actuator saturation, the main difference and difficulty lie in their treatment of saturation nonlinearity. In paper [2], authors gave a method for maximization of an ellipsoid which is invariant under input saturation, but persistent disturbances. The works of [1, 3, 6–8] consider the situation where disturbance are bounded in energy. The works of [1, 6, 7] formulated and solved the problem of stability analysis and design. In [9, 10], authors presented LMI-based methods for regional stability and performance of linear antiwindup compensators for linear control systems. [12] presents a method for the analysis and control design of linear systems in the presence of actuator saturation and 𝐿₂ disturbances. During the last few years, problems about actuator saturation have been extended to many other fields of automatic control, such as singular systems [13], systems with parameters uncertainty [14], Markovian jump systems [15], decentralized control systems [16], and Hamiltonian systems [17].

Time delays are frequently encountered in almost all networked systems. Since the existence of a delay in a physical system often induces instability of poor performance, research on time-delay systems is a topic of great practical and theoretical importance. During the last decade, the control problem of systems with time delay has received considerable attention. The main methods can be classified into two types: delay-independent ones and delay-dependent ones.

On the other hand, fault tolerant has become a hot research area because of its importance in practical engineering [18–28]. And the design approach can be broadly classified into two types: Passive approach and Active approach. A passive fault-tolerant controller commonly has a simple structure and is easily implemented [18–22]. The system performances in normal and fault modes can be optimized. Some of these active fault-tolerant control methods may readjust controller parameters or change controller structure to compensate the fault effects on systems. Some of these methods include a strategy involving a fast subsystem for fault detection and isolation (FDI) and a supervisory system that chooses the corresponding controller for a particular type of fault. Most of the results in adaptive fault-tolerant control are based on model reference adaptive control (MRAC) [29–31], but the disturbance attenuation performances of systems have not been addressed yet within the MRAC framework. Paper [32] considered the problem of adaptive reliable controller via state feedback and dynamic output feedback, respectively, for linear time-delay systems against actuator faults. However, when actuator saturation problem is considered, the methods of [32] cannot be used.

As we all know, actuator faults and saturation always happen at the same time for networked systems. However, noting all above results, there is no work that deals with this problem. There are only a few papers that considered the problems about systems with actuator saturation and faults [13, 33, 34]. Motivated by the above observations, this paper studies the problem of designing adaptive fault-tolerant 𝐻_∞ controllers for networked systems with actuator saturation. The designs are developed in the framework of LMIs approach, which can guarantee the disturbance tolerance level and adaptive 𝐻_∞ performances of networked systems in the cases of actuator saturation and actuator failures. The difference between this paper and some existing results is that in this paper the fault tolerant and saturation are considered at same time for networked systems.

The remainder of this paper is organized as follows. Section 2 introduces notation to be used in the paper, and problem statement is given in it. An adaptive fault-tolerant 𝐻_∞ controller design method is described for networked system in Section 3. In Section 4, an example is given to illustrate the efficiency of the design method. The paper will be concluded in Section 5.

2. Problem Statement and Preliminaries

In this paper, the following LTI plant will be considered: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜏(𝑡))+𝐵₁𝜔(𝑡)+𝐵₂𝜎(𝑢),𝑧(𝑡)(𝑡)(𝑢),𝑥(𝑡)(𝑡)[],=𝐶𝑥+𝐷sat=𝜙𝑡∈−ℎ,0 (1) where 𝑥(𝑡)∈𝑅^𝑛 and 𝑥_𝑡 is the plant state at time 𝑡 defined by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑠∈[−ℎ,0], sat(𝑢)∈𝑅^𝑚 is the saturated control input, 𝑧(𝑡)∈𝑅^𝑠 is the regulated output, and 𝜔(𝑡)∈𝑅^𝑑 is an exogenous disturbance in 𝐿₂[0,∞], respectively. 𝐴,  𝐴₁,  𝐵₁, 𝐵₂, 𝐶, and 𝐷, are known constant matrices of appropriate dimensions. For simplicity only, we take single delay 𝜏(𝑡). The results of this paper can be easily applied to the case of multiple delays.

The following case for time-varying delay 𝜏(𝑡) is considered. That is, 𝜏(𝑡) is differentiable funcion 0≤𝜏(𝑡)≤ℎ,𝜏̇(𝑡)≤𝑑<1,satisfying∀𝑡≥0, (2) where 𝑑 is an upper bound on the derivative of 𝜏(𝑡).

In this paper, we formulate the fault-tolerant control problem by using the following model form which is considered in [19, 22]: 𝑢^𝐹_𝑗𝑞(𝑡)=1−𝜌^𝑞_𝑗𝜎𝑢_𝑗(𝑡),0≤𝜌^𝑞_𝑗≤𝜌^𝑞_𝑗≤𝜌^𝑞_𝑗,𝑗∈𝐈[1,𝑚],𝑞∈𝐈[1,𝐿], (3) where 𝑢^𝐹_𝑗𝑞(𝑡) represents the signal from the 𝑗th actuator which has failed in the 𝑞th fault mode, 𝜌^𝑞_𝑗 is an unknown constant, the 𝐿 is the total fault modes, and 𝜌^𝑞_𝑗 and 𝜌^𝑞_𝑗 represent the lower and upper bounds of 𝜌^𝑞_𝑗, respectively. Denote that 𝑢^𝐹_𝑞(𝑡)=𝑢^𝐹_1𝑞(𝑡),𝑢^𝐹_2𝑞(𝑡),…,𝑢^𝐹_𝑚𝑞(𝑡)^𝑇=𝐼−𝜌^𝑞𝜎(𝑢(𝑡)), (4) where 𝜌^𝑞=diag[𝜌^𝑞₁,𝜌^𝑞₂,…,𝜌^𝑞_𝑚],  𝑞∈𝐈[1,𝐿]. Considering the lower and upper bounds 𝜌^𝑞_𝑗 and 𝜌^𝑞_𝑗, the following set can be defined: 𝑁_𝜌𝑞=𝜌^𝑞𝜌^𝑞=diag𝜌^𝑞₁,𝜌^𝑞₂,…,𝜌^𝑞_𝑚,𝜌^𝑞_𝑗=𝜌^𝑞_𝑗or𝜌^𝑞_𝑗=𝜌^𝑞_𝑗. (5)

For convenience, the following uniform actuator fault model is exploited: 𝑢^𝐹(𝑡)=𝐼−𝜌sat(𝑢(𝑡)),𝜌∈𝜌¹⋯𝜌^𝐿, (6) where 𝜌 is described by 𝜌=diag[𝜌₁,𝜌₂,…,𝜌_𝑚]. The following definitions and lemmas will be used in the sequel.

Definition 1 (see [33]). Consider the following system: 𝑥̇(𝑡)=𝐴_𝑎𝜌(𝑡),𝜌𝑥(𝑡)+𝐵_𝑎𝜌(𝑡)𝜔(𝑡),𝑧(𝑡),𝜌=𝐶_𝑎𝜌(𝑡),𝜌𝑥(𝑡),𝑥(0)=0, (7) where 𝜌 is parameter vector and 𝜌(𝑡) is a time-varying parameter vector to be chosen. Let 𝑟>0 be a given constant, then system (7) is said to be with an adaptive 𝐻_∞ performance index no larger than 𝑟 if for any 𝜀>0, there exists a 𝜌(𝑡) such that the following inequality holds: ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤𝑟^2∞₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡+𝜀. (8)

Definition 2. For a matrix 𝐶_𝑐𝑙∈𝑅^𝑚×𝑛, denote the 𝑗th row of 𝐶_𝑐𝑙 as 𝐶_𝑐𝑙𝑗, and define ℘𝐶_𝑐𝑙=𝑥∈𝑅^𝑛∶𝐶_𝑐𝑙𝑗𝑥≤1,𝑗∈𝐈[1,𝑚]. (9)

Lemma 3 (see [32]). If there exists a symmetric matrix Θ with ΘΘ=¹¹Θ₁₂Θ^𝑇₁₂Θ₂₂ (10) and Θ₁₁,  Θ₂₂∈𝑅^{𝑁𝑛×𝑁𝑛} such that the following inequalities hold: Θ_22𝑗𝑗[],Θ≤0,𝑗∈𝐈1,𝑁₁₁+Θ₁₂Δ(𝛿)+Θ₁₂Δ(𝛿)^𝑇+Δ(𝛿)Θ₂₂Δ(𝛿)≥0,𝛿∈Δ_𝑣,𝐸𝑄𝐸_𝑇𝐹+𝑈^𝑇𝑈+𝐺^𝑇𝜌Θ𝐺<0,𝜌∈¹⋯𝜌^𝐿,𝜌^𝑞∈𝑁_𝜌𝑞, (11) then inequality 𝐿(𝛿)=𝑄+^𝑁_𝑗=1𝛿_𝑗𝐸_𝑗+^𝑁_𝑗=1𝛿_𝑗𝐸_𝑗^𝑇+^𝑁𝑁_𝑗=1𝑝=1𝛿_𝑗𝛿_𝑝𝐹_𝑗𝑝+𝑈₀+^𝑁_𝑗=1𝛿_𝑗𝑈_𝑗^𝑇×𝑈₀+^𝑁_𝑗=1𝛿_𝑗𝑈_𝑗<0 (12) holds for all 𝛿_𝑗∈[𝛿_𝑗𝛿_𝑗], where 𝑄=𝑄^𝑇∈𝑅^𝑛×𝑛 and 𝐹_𝑝𝑗=𝐹^𝑇_𝑝𝑗∈𝑅^𝑛×𝑛,𝐸_𝑗∈𝑅^𝑛×𝑛Δ(𝛿)=diag𝛿₁𝐼_𝑛×𝑛⋯𝛿_𝑁𝐼_𝑛×𝑛,𝐸=𝐸₁𝐸₂⋅⋅⋅𝐸_𝑁,𝑈=𝑈₀𝑈₁⋅⋅⋅𝑈_𝑁,𝐹𝐹=¹¹⋯𝐹_1𝑁𝐹⋯⋯⋯_𝑁1⋯𝐹_𝑁𝑁𝐼,𝐺=^𝑛×𝑛⋯𝐼_𝑛×𝑛00𝐼_{𝑁𝑛×𝑁𝑛}. (13)
Let 𝐃 be a set of 𝑚×𝑚 diagonal matrices whose diagonal elements are either 1 or 0. There are 2^𝑚 elements in 𝐃, and one denotes its elements as 𝐷_𝑖, 𝑖∈𝐈[0,2^𝑚−1], where for 𝑖=𝑧₁2^𝑚−1+𝑍₂2^𝑚−2+⋯+𝑧_𝑚 with 𝑧_𝑗∈{0,1}, the diagonal elements of 𝐷_𝑖 are {1−𝑧₁,1−𝑧₂,…,1−𝑧_𝑚}. Denote 𝐷⁻_𝑖=𝐼−𝐷_𝑖. It is easy to see that 𝐷⁻_𝑖∈𝐃. Then, one has the following.

Lemma 4 (see [35]). For two vectors 𝑢, 𝑣∈𝑅^𝑚. Suppose that ‖𝑣‖_∞≤1. Then, sat(𝑢)∈co𝐷_𝑖𝑢+𝐷⁻_𝑖𝑣∶𝑖∈0,2^𝑚−1, (14) where co denotes the convex hull.

For a networked system, the performance of closed-loop system can be measured by the 𝐿₂ gain. However, this gain cannot be well defined for closed-loop system, since a sufficiently large disturbance may lead to unstable closed-loop system. For this reason, we need to consider a class of disturbances whose energy is bounded by a given value; that is, 𝔐_𝛿∶=𝜔∶𝑅₊⟶𝑅^𝑑∶^∞₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡≤𝛿. (15)

In this paper, we will consider the following problems. The first question is, what is the maximal value of 𝛿 such that the state will be bounded for all 𝜔∈𝔐_𝛿 for systems with time delay? This question can be referred to as disturbance tolerance level. The system performance can be measured by the restricted 𝐿₂ gain over 𝔐_𝛿. In this paper, 𝐿₂ gain and 𝔐_𝛿 will be considered at same time for networked system.

3. Main Results

The dynamics with actuator faults (6) and saturation are described by 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜏(𝑡))+𝐵₁𝜔(𝑡)+𝐵₂(𝑢(𝑡,𝑧(𝑡)(𝑡)(𝑢(𝑡,𝑥(𝑡)(𝑡)[].𝐼−𝜌sat))=𝐶𝑥+𝐷𝐼−𝜌sat))=𝜙𝑡∈−ℎ,0 (16) Rewrite (16) as 𝑥̇(𝑡)(𝑡),𝑦(𝑡)==𝑦𝐴+𝐴₁𝑥(𝑡)+𝐵₁𝜔(𝑡)+𝐵₂𝐼−𝜌sat(𝑢(𝑡))−𝐴₁^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑧(𝑡)(𝑡)(𝑢(𝑡.𝑑𝑠,=𝐶𝑥+𝐷𝐼−𝜌sat)) (17)

The controller structure is given as 𝑢(𝑡)=𝐾𝜌(𝑡)𝑥(𝑡)=𝐾₀+𝐾_𝑎𝜌(𝑡)+𝐾_𝑏𝜌(𝑡)𝑥(𝑡), (18) where 𝜌(𝑡) is used to estimate 𝜌, 𝐾_𝑎(𝜌(𝑡))=∑^𝑚_𝑗=1𝐾_𝑎𝑗𝜌_𝑗(𝑡) and 𝐾_𝑏(𝜌(𝑡))=∑^𝑚_𝑗=1𝐾_𝑏𝑗𝜌_𝑗(𝑡).

By Lemma 3, the following equality is given, with 𝑥∈℘(𝐻(𝜌)): 𝐾𝜌(𝑡)𝑥(𝑡)=sat_{2𝑚−1𝑖=0}𝜂_𝑖𝐷_𝑖𝐾𝜌(𝑡)+𝐷⁻_𝑖𝐻𝜌(𝑡)𝑥(𝑡), (19) for some scalars 0≤𝜂_𝑖≤1, 𝑖∈𝐈[0,2^𝑚−1], such that ∑^2𝑚−1_𝑖=0𝜂_𝑖=1, and the following equality holds: (𝑢(𝑡=𝐼−𝜌sat))_{2𝑚−1𝑖=0}𝜂_𝑖𝐼−𝜌𝐷_𝑖𝐾₀+𝐷_𝑖𝐾_𝑎𝜌−𝜌𝐷_𝑖𝐾_𝑎𝜌+𝐼−𝜌(𝑡)𝐷_𝑖𝐾_𝑏𝜌(𝑡)+𝐷_𝑖𝐾𝜌_𝑎^∼(𝑡)+𝜌^∼𝐷_𝑖𝐾_𝑏𝜌(𝑡)+𝐼−𝜌𝐷⁻_𝑖𝐻₀+𝐷⁻_𝑖𝐻_𝑎𝜌−𝜌𝐷⁻_𝑖𝐻_𝑎𝜌+𝐼−𝜌(𝑡)𝐷⁻_𝑖𝐻_𝑏𝜌(𝑡)+𝐷⁻_𝑖𝐻𝜌_𝑎^∼(𝑡)+𝜌^∼𝐷⁻_𝑖𝐻_𝑏𝜌(𝑡)𝑥(𝑡), (20) where 𝜌^∼(𝑡)=𝜌(𝑡)−𝜌. Denote 𝐵^𝑗=[0⋯𝑏^𝑗⋯0] with 𝐵=[𝑏¹⋯𝑏^𝑚], and Δ𝜌=𝜌=𝜌₁⋯𝜌_𝑚∶𝜌_𝑗∈min_𝑞𝜌^𝑞_𝑗,max_𝑞𝜌^𝑞_𝑗,𝑞∈𝐈[1,𝐿]. (21)

Definition 5. Let 𝑃₁∈𝑅^𝑛×𝑛 be a positive-definite matrix. Denote 𝜀𝑃₁,𝛿=𝑥∈𝑅^𝑛∶𝑥^𝑇𝑃₁,𝜀𝑥≤𝛿_∗𝑃₁,𝛿=𝑥∈𝑅^𝑛∶𝑥^𝑇𝑃₁𝑥+^𝑚∼_𝑗=1𝜌²_𝑗(𝑡)𝑙_𝑗≤𝛿. (22) Assuming that 𝑙_𝑗>0 is given, we denote 𝛿^∗=𝛿+max{∑^𝑚_𝑗=1(𝜌^∼2_𝑗(𝑡)/𝑙_𝑗)}.

The following lemma provides a method for choosing of 𝜂_𝑖's, which are Lipschitzian functions in 𝑥 and 𝜌 and thus are useful in controller design method.

Lemma 6 (see [11]). Let 𝑥∈℘(𝐻(𝜌)). For each 𝑗∈𝐈[1,𝑚], 𝜆_𝑗𝑥(𝑡),𝜌(𝑡)=1,if𝐾𝜌(𝑡)_𝑗𝑥(𝑡)=𝐻𝜌(𝑡)_𝑗𝑥(𝑡)𝜎𝐾𝜌(𝑡)_𝑗𝑥(𝑡)−𝐻𝜌(𝑡)_𝑗𝑥(𝑡)𝐾𝜌(𝑡)_𝑗−𝐻𝜌(𝑡)_𝑗𝑥(𝑡),otherwise, (23) and for each 𝑖∈𝐈[0,2^𝑚−1], let 𝑧_𝑗∈{0,1} such that 𝑖=𝑧₁2^𝑚−1+𝑧₂2^𝑚−2+⋯+𝑧_𝑚, and define 𝜂_𝑖𝑥(𝑡),𝜌(𝑡)=^𝑚_𝑗=1𝑧_𝑗1−𝜆_𝑗𝑥,𝜌+1−𝑧_𝑗𝜆_𝑗𝑥,𝜌. (24)
Then, 𝜂_𝑖's are functions Lipschitz in 𝑥 and 𝜌, such that, ∑^2𝑚−1_𝑖=0𝜂_𝑖=1, 0≤𝜂_𝑖≤1, 𝑖∈𝐈[0,2^𝑚−1]. Moreover, they satisfy relation (19).
By using the functions 𝜂_𝑖(𝑥(𝑡),𝜌(𝑡))’s and controller (18), plant (16) can be written in a quasi-LPV form as follows: 𝑦(𝑡)=𝐴+𝐴₁𝑥(𝑡)+𝐵₁𝜔(𝑡)+𝐵₂^2𝑚−1_𝑖=0𝜂_𝑖𝐼−𝜌𝐷_𝑖𝐾₀+𝐾_𝑎𝜌(𝑡)+𝐾_𝑏𝜌(𝑡)+𝐼−𝜌𝐷⁻_𝑖𝐻₀+𝐻_𝑎𝜌(𝑡)𝑥(𝑡)+𝐻_𝑏𝜌(𝑡)−𝐴₁^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑑𝑠. (25)

Lemma 7 (see [36]). For any 𝑎∈𝑅^𝑛, 𝑏∈𝑅^2𝑛, 𝑍₀∈𝑅^2𝑛×𝑛, 𝑅∈𝑅^𝑛×𝑛,𝑌∈𝑅^𝑛×2𝑛, and 𝑍∈𝑅^2𝑛×2𝑛, the following holds: −2𝑏^𝑇𝑎𝑏𝐹𝑎≤^𝑇𝑅𝑌−𝑍^𝑇₀𝑌^𝑇−𝑍₀𝑍𝑎𝑏, (26) where _𝑌^𝑅𝑌_𝑇𝑍≥0.

Definition 8. Firstly, for system (25), consider the following 𝐿𝑦𝑎𝑝𝑢𝑛𝑜𝑣-𝐾𝑟𝑎𝑠𝑜𝑣𝑠𝑘𝑖𝑖 functional 𝑉=𝑉₁+𝑉₂+𝑉₃+^𝑚∼_𝑗=1𝜌²_𝑗(𝑡)𝑙_𝑗, (27) where 𝑉₁=𝑥^𝑇(𝑡)𝐸𝑃𝑥(𝑡),𝑉₂=⁰_ℎ^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)(𝑠)𝑉𝑅𝑦𝑑𝑠𝑑𝜃,₃=^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)(𝑠)𝐼𝑆𝑥𝑑𝑠,𝐸=^𝑛×𝑛0𝑃00,𝑃=¹0𝑃₂𝑃₃,𝑃₁=𝑃^𝑇₁>0. (28)
Then, the following set can be given: 𝑀_𝑉(𝛿)=𝑥(𝑡)∈𝑅^𝑛∶𝑉=𝑉₁+𝑉₂+𝑉₃+^𝑚∼_𝑗=1𝜌²_𝑗(𝑡)𝑙_𝑗≤𝛿. (29)

Remark 9. By Definitions 5 and 8, we can draw the conclusion that 𝑀_𝑉(𝛿)⊂𝜀^∗(𝑃₁,𝛿)⊂𝜀(𝑃₁,𝛿).
By Lemma 6, we analyze the auxiliary LPV system as follows, of which the closed-loop system comprising of (25) and (18) is a special case, for all 𝑥(𝑡)∈𝑀_𝑉(𝛿)⊂℘(𝐻(𝜌)): 𝑥̇(𝑡)=𝐴𝜂𝑥(𝑡)+𝐵₁𝜔,𝜂∈Γ, (30) where 𝜂=[𝜂₀,𝜂₁,…,𝜂_2𝑚−1], and Γ=𝜂∈𝑅^2𝑚∶^2𝑚−1_𝑖=0𝜂_𝑖=1,0≤𝜂_𝑖≤1,𝑖∈𝐈0,2^𝑚,𝐴𝜂−1=𝐴+𝐴₁+𝐵₂^2𝑚−1_𝑖=0𝜂_𝑖𝐼−𝜌𝐷_𝑖𝐾₀+𝐷_𝑖𝐾_𝑎𝜌−𝜌𝐷_𝑖𝐾_𝑎𝜌+𝐼−𝜌(𝑡)𝐷_𝑖𝐾_𝑏𝜌(𝑡)+𝐷_𝑖𝐾𝜌_𝑎^∼(𝑡)+𝜌^∼𝐷_𝑖𝐾_𝑏𝜌(𝑡+𝐷)𝐼−𝜌⁻_𝑖𝐻₀+𝐷⁻_𝑖𝐻_𝑎𝜌−𝜌𝐷⁻_𝑖𝐻_𝑎𝜌+𝐼−𝜌(𝑡)𝐷⁻_𝑖𝐻_𝑏𝜌(𝑡)+𝐷⁻_𝑖𝐻𝜌_𝑎^∼(𝑡)+𝜌^∼𝐷⁻_𝑖𝐻_𝑏𝜌(𝑡)−𝐴₁^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)𝑑𝑠. (31)

Theorem 10. Let 𝑟_𝑓>0, 𝑟_𝑛>0, 𝑑,ℎ>0 and 𝛿>0 be given constants, then the following two conditions are satisfied.(I)The trajectories of the closed-loop system that start from the origin will remain inside the domain 𝑀_𝑉(𝛿) for every 𝜔∈𝔐_𝛿.(II)In normal case, that is, 𝜌=0, for 𝑥(0)=0, ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤𝑟²_𝑛^∞₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡+𝑟²_𝑛^𝑚∼_𝑗=1𝜌²_𝑗(0)𝑙_𝑗, (32)and in actuator failures cases, that is, 𝜌∈{𝜌¹⋯𝜌^𝐿}, for 𝑥(0)=0, ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤𝑟²_𝑓^∞₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡+𝑟²_𝑓^𝑚∼_𝑗=1𝜌²_𝑗(0)𝑙_𝑗, (33) where 𝜌^∼(𝑡)=diag{𝜌^∼₁(𝑡)⋯𝜌^∼_𝑚(𝑡)}, 𝜌^∼_𝑗(𝑡)=𝜌_𝑗(𝑡)−𝜌_𝑗 if there exist matrices 𝑄₁>0,  𝑄₂,  𝑄₃,  𝑆,  𝑅,  𝑍₁,  𝑍₂,  𝑍₃,  𝑂₀,  𝑂_𝑎𝑗,  𝑂_𝑏𝑗,  𝑌₀,  𝑌_𝑎𝑗,  𝑌_𝑏𝑗,  𝑗∈𝐈[1,𝑚] and symmetric matrixes Θ^𝑖, 𝑖∈𝐈[0,2^𝑚−1], with Θ^𝑖=Θ^𝑖₁₁Θ^𝑖₁₂Θ^𝑖𝑇₁₂Θ^𝑖₂₂ (34) and Θ^𝑖₁₁, Θ^𝑖₂₂∈𝑅^{𝑚𝑛×𝑚𝑛} such that the following inequalities hold for all 𝐷_𝑖∈𝐃 and 𝑀_𝑉(𝛿)⊂℘(𝐻(𝜌)); that is, |𝐻(𝜌)_𝑗𝑥|≤1 for all 𝑥∈𝑀_𝑉(𝛿),𝑗∈𝐈[1,𝑚]: Θ^𝑖_22𝑗𝑗≤0,𝑗∈𝐈[1,𝑚],𝑖∈𝐈0,2^𝑚Θ−1_𝑖11+Θ^𝑖₁₂Δ𝜌+Θ^𝑖₁₂Δ(𝜌)^𝑇+Δ𝜌Θ^𝑖₂₂Δ𝜌≥0,𝜌∈Δ𝜌𝑀_𝑖𝑉^𝑇_0𝑖0𝐵₁00𝐴₁(𝐼−𝜀)𝑆0𝑄₁00ℎ𝑄^𝑇₂ℎ𝑄^𝑇₃0∗−𝑟²_𝑛()𝐼0000∗∗−𝐼000∗∗∗−1−𝑑𝑆00∗∗∗∗−𝑆0∗∗∗∗∗−ℎ𝑅𝜌=0,𝑀_𝑖𝑉^𝑇_0𝑖0𝐵₁00𝐴₁(𝐼−𝜀)𝑆0𝑄₁00ℎ𝑄^𝑇₂ℎ𝑄^𝑇₃0∗−𝑟²_𝑓()𝐼0000∗∗−𝐼000∗∗∗−1−𝑑𝑆00∗∗∗∗−𝑆0∗∗∗∗∗−ℎ𝑅𝜌<0,𝜌∈¹⋯𝜌^𝐿,𝜌^𝑞∈𝑁_𝜌𝑞𝑅0𝑅𝜀𝐴^𝑇₁∗𝑍₁𝑍₂∗∗𝑍₃≥0, (35)(36)(37)(38) where 𝑀_𝑖=𝑁_0𝑖𝑈_𝑖𝑈^𝑇_𝑖Ω_𝑖+𝐺^𝑇Θ^𝑖𝑁𝐺,_0𝑖=𝑄₂+𝑄^𝑇₂+ℎ𝑍₁𝑇_1𝑖∗−𝑄₃−𝑄^𝑇₃+ℎ𝑍₃,𝑈_𝑖=𝑈_1𝑖𝑈_2𝑖⋯𝑈_𝑚𝑖,𝑉_0𝑖=𝑉_00𝑖𝑉_01𝑖⋯𝑉_0𝑚𝑖,Ω_𝑖=Ω_𝑖𝑗𝑝𝑇,𝑗,𝑝=1⋯𝑚,_1𝑖=𝑄₃−𝑄^𝑇₂+𝑄₁𝐴^𝑇+𝜀𝐴^𝑇₁+ℎ𝑍₂+𝐼−𝜌𝐷_𝑖𝑌^𝑇₀+𝐼−𝜌𝐷⁻_𝑖𝑂^𝑇₀,𝑉_00𝑖=𝐶𝑄₁+𝐷𝐼−𝜌𝐷_𝑖𝑌₀+𝐷𝐼−𝜌𝐷⁻_𝑖𝑂₀0,𝑉_0𝑗𝑖=𝐷𝐼−𝜌𝐷_𝑖𝑌_𝑎𝑗+𝑌_𝑏𝑗+𝐷𝐼−𝜌𝐷⁻_𝑖𝑂_𝑎𝑗+𝑂_𝑏𝑗0,𝑈_𝑗𝑖=0−𝑌^𝑇_𝑎𝑗𝐷_𝑖𝜌𝐵^𝑇₂+𝑌^𝑇_𝑏𝑗𝐷_𝑖𝐵^𝑇₂−𝑂^𝑇_𝑎𝑗𝐷_𝑖𝜌𝐵^𝑇₂+𝑂^𝑇_𝑏𝑗𝐷_𝑖𝐵^𝑇₂,𝐼⋮𝐼0,Ω00𝐺=0𝐼_𝑖𝑗𝑝=0−𝐵^𝑗₂𝐷_𝑖𝑌_𝑏𝑝−𝑌^𝑇_𝑏𝑗𝐷_𝑖𝐵₂^𝑝𝑇−𝐵^𝑗₂𝐷_𝑖𝑂_𝑏𝑝−𝑂^𝑇_𝑏𝑗𝐷_𝑖𝐵₂^𝑝𝑇,00𝑌_𝑎𝜌=^𝑚_𝑗=1𝑌_𝑎𝑗𝜌_𝑗, (39) and also 𝜌_𝑗(𝑡) is determined according to the adaptive law 𝜌̇_𝑗=Proj_{[min𝑞{𝜌𝑞𝑗}max𝑞{𝜌}^𝑞_𝑗}𝐿_𝑗=0,if𝜌_𝑗=min_𝑞𝜌^𝑞_𝑗,𝐿_𝑗≤0or𝜌_𝑗=max_𝑞𝜌^𝑞_𝑗,𝐿_𝑗𝐿≥0_𝑗,otherwise, (40) where 𝐿_𝑗=−𝑙_𝑗𝑥^𝑇(𝑡)Λ00_𝑗𝑖0𝑥(𝑡),Λ_𝑗𝑖=𝐵^𝑗₂^2𝑚−1_𝑖=0𝜂_𝑖𝐷_𝑖𝐾_𝑏𝜌+𝐵₂^2𝑚−1_𝑖=0𝜂_𝑖𝐷_𝑖𝐾_𝑎𝑗+𝐵^𝑗₂^2𝑚−1_𝑖=0𝜂_𝑖𝐷⁻_𝑖𝐻_𝑏𝜌+𝐵₂^2𝑚−1_𝑖=0𝜂_𝑖𝐷⁻_𝑖𝐻_𝑎𝑗,𝑄𝑄=¹0𝑄₂𝑄₃, (41) where 𝐾_𝑎𝑗=𝑌_𝑎𝑗𝑄₁⁻¹, 𝐾_𝑏𝑗=𝑌_𝑏𝑗𝑄₁⁻¹, 𝐻_𝑎𝑗=𝑂_𝑎𝑗𝑄₁⁻¹, and 𝐻_𝑏𝑗=𝑂_𝑏𝑗𝑄₁⁻¹⋅𝑙_𝑗>0  (𝑗∈𝐈[1,𝑚]) are the adaptive law gains which can be given according to practical applications. Then, the following controller can be given: 𝐾𝜌=𝑌₀𝑄₁⁻¹+^𝑚_𝑗=1𝜌_𝑗𝑌_𝑎𝑗𝑄₁⁻¹+^𝑚_𝑗=1𝜌_𝑗𝑌_𝑏𝑗𝑄₁⁻¹. (42)

Proof. See appendix.

From Theorem 10, we have the following algorithm to optimize the adaptive 𝐻_∞ performance in normal and fault cases and the disturbance tolerance level 𝛿 with considering time delay.

Algorithm 11. Suppose that 𝑟_𝑛 and 𝑟_𝑓 denote the adaptive 𝐻_∞ performance indexes for the normal case and fault cases of the closed-loop system (30), respectively. Let 𝛿 denote the disturbance tolerance level. Then, 𝑟_𝑛, 𝑟_𝑓 are minimized, and 𝛿 is maximized if the following optimization problem is solvable: min𝜂=𝛼𝜂_𝑛+𝛽𝜂_𝑓+𝛾𝜂_𝛿s.t.(a),(),(),()()(35363738b)𝑀_𝑉(𝛿)⊂℘𝐻𝜌, (43) where 𝜂_𝑛=𝑟²_𝑛, 𝜂_𝑓=𝑟²_𝑓, 𝜂_𝛿=1/𝛿^∗=1/(𝛿+max{∑^𝑚_𝑗=1(𝜌^∼2_𝑗(𝑡)/𝑙_𝑗)}), and 𝛼, 𝛽, and 𝛾 are weighting coefficients.
By Definition 2, condition (b) cannot be shown as LMIs directly. However, obviously, 𝑀_𝑉(𝛿)⊂𝜀^∗(𝑃₁,𝛿^∗)⊂𝜀(𝑃₁,𝛿^∗), which implies that (b) can be replaced with (b1) as follows: (b1)𝜀𝑃₁,𝛿^∗⊂℘𝐻𝜌. (44) Equation (44) is equivalent to 𝛿^∗ℎ𝜌_𝑗𝑃₁⁻¹ℎ𝜌^𝑇_𝑗1≤1⟺𝛿^∗ℎ𝜌_𝑗𝑃₁⁻¹∗𝑃₁⁻¹≥0, (45) for all 𝑗∈𝐈[1,𝑚], where ℎ(𝜌)_𝑗 is the 𝑗th row of 𝐻(𝜌). Then, it can be drawn that (45) is equivalent to the following inequalities: (b2)−𝜂_𝛿−𝑂_0𝑠∗−𝑄₁+^𝑚_𝑗=1𝜌_𝑗0−𝑂_𝑎𝑗𝑠−𝑂_𝑏𝑗𝑠∗0≤0, (46) where 𝜌∈Δ𝜌 and 𝑂_𝑎𝑗𝑠 is the 𝑠th row of 𝑂_𝑎𝑗,  𝑠∈𝐈[1,𝑚].

Remark 12. Theorem 10 prevents a condition for the existence of an adaptive fault tolerant 𝐻_∞ controller. In Theorem 10, if set 𝑌_𝑎𝑗=0, 𝑌_𝑏𝑗=0, 𝑂_𝑎𝑗=0, and 𝑂_𝑏𝑗=0, 𝑗∈𝐈[1,𝑚], the condition of Theorem 10 is reduced to fixed gains condition. By the following example, we can get that the adaptive controller can guarantee better effect.

4. Examples

Example 13. Consider the system of the form (1) with 𝐴=32340,𝐴₁=,𝐵−100−2₁=1010,𝐵₂=,400040𝐶=400000^𝑇,𝐷=00.50001^𝑇,𝜏(𝑡)=1+sin(𝑡)4, (47) and the following two possible fault modes.

Fault mode 1: both of the two actuators are normal; that is 𝜌¹₁=𝜌¹₂=0. Fault mode 2: the first actuator is outage, and the second actuator may be normal or loss of effectiveness, described by 𝜌²₁=1, 0≤𝜌²₂≤𝑎, where 𝑎=0.5 denotes the maximal loss of effectiveness for the second actuator.

Let 𝛼=10,  𝛽=1, and 𝛾=10, and let the optimal indexes with fixed controller gains are 𝜂_𝑛=0.1963, 𝜂_𝑓=9.8933, 𝜂_𝛿=20.5385, and 𝜂=217.2408. By solving (43), the optimal indexes can be given as 𝜂_𝑛=0.5881, 𝜂_𝑓=9.1236, 𝜂_𝛿=9.6701, and  𝜂=111.7048. For getting smaller number for every optimal index, we may revise 𝛼=110,𝛽=0.2, and 𝛾=0.5. Then, the indexes can be drawn that 𝜂_𝑛=0.1676, 𝜂_𝑓=7.1242, and 𝜂_𝛿=18.6399. For illustrating the efficiency of the design method, the following simulations is given.

During the following simulation, fault case is considered as follows. At 0 second, the first actuator is outage. Here, we choose 𝑙₁=𝑙₂=100.

Firstly, we consider the 𝐻_∞ performance. The disturbance is given as 𝜔₁(𝑡)=𝜔₂(𝑡)=cos(𝑡),4.2≤𝑡≤6.90,otherwise. (48) Figures 1 and 2 show the responses curves of the first state in normal and fault case, respectively.

Figure 1 

Response curve of the first state in normal case with the adaptive controller (solid) and the fixed gain controller (dashed).

Figure 2 

Response curve of the first state in fault case with the adaptive controller (solid) and the fixed gain controller (dashed).

Then, we consider the disturb tolerance problem. The disturbance is given as 𝜔₁(𝑡)=𝜔₂(𝑡)=27.2,4≤𝑡≤50,otherwise. (49) Figures 3 and 4 show the responses curves of the states in normal case.

Figure 3 

Responses curves of the states with adaptive controller in normal case.

Figure 4 

Responses curves of the states with fixed gain controller in normal case.

5. Conclusions

In this paper, an adaptive fault-tolerant 𝐻_∞ controllers design method was given for networked systems with actuator saturation. The designs were proposed in LMIs approach, which could guarantee the disturbance tolerance ability and adaptive 𝐻_∞ performances of networked systems in the cases of actuator saturation and actuator failures. An example, has been given to illustrate the efficiency of the design method.

Appendix

Proof of Theorem 10. Item (II) will be proved firstly. By Definition 8, since 𝑥^𝑇(𝑡)𝐸𝑃𝑥(𝑡)=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡), then 𝑑𝑑𝑡𝑥^𝑇(𝑡)𝐸𝑃𝑥(𝑡)=2𝑥^𝑇(𝑡)𝑃₁𝑥̇(𝑡)=2𝑥^𝑇(𝑡)𝑃^𝑇𝑥̇(𝑡)0. (A.1) From the derivative of 𝑉(𝑡) along the closed-loop system (30), it follows that 𝑉̇=𝑥^𝑇(𝑡)^2𝑚−1_𝑖=0𝜂_𝑖Φ_1𝑖𝑥(𝑡)(𝑡)−()𝑥+𝜒1−𝑑^𝑇((𝑡((𝑡−𝑡−𝜏))𝑆𝑥𝑡−𝜏))_{𝑡𝑡−ℎ}𝑦^𝑇(𝑠)𝑅𝑦(𝑠)𝑑𝑠+2^𝑚∼_𝑖=1𝜌_𝑖(𝑡)^∼̇𝜌_𝑖(𝑡)𝑙_𝑖+2𝑥^𝑇(𝑡)𝑃^𝑇0𝐵₁𝜔(𝑡), (A.2) where Φ_1𝑖=𝑃^𝑇Δ_0𝑖+Δ^𝑇_0𝑖,𝜒(𝑡)𝑃+𝑆00ℎ𝑅=−2^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑡)𝑃^𝑇0𝐴₁𝑦(𝑠)Δ𝑑𝑠,_0𝑖=0𝐼𝐴+𝐴₁+𝐵₂𝐼−𝜌𝐷_𝑖𝐾𝜌+𝐷⁻_𝑖𝐻𝜌.−𝐼 (A.3) By Lemma 7, taking 𝑍₀=𝑃^𝑇0_𝐴1𝑎=𝑦(𝑠), 𝑏=𝑥(𝑡), it follows that 𝜒(𝑡)≤^𝑡_{𝑡−𝜏(𝑡)}𝑦^𝑇(𝑠)𝑥^𝑇(𝑠)𝑊₁𝑦(𝑠)𝑥(𝑠)=𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}𝑦^𝑇(𝑠)𝑅𝑦(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑡)𝑍𝑥(𝑡)𝑑𝑠+2^𝑡_{(𝑡)𝑡−𝜏}𝑦^𝑇(𝑠)𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)𝑑𝑠=^𝑡_{(𝑡)𝑡−𝜏}𝑦^𝑇(𝑠)𝑅𝑦(𝑠)𝑑𝑠+𝜏(𝑡)𝑥^𝑇(𝑡)𝑍𝑥(𝑡)+2^𝑡_{(𝑡)𝑡−𝜏}𝑥̇^𝑇(𝑠)𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)≤𝑑𝑠_{𝑡𝑡−ℎ}𝑦^𝑇(𝑠)𝑅𝑦(𝑠)𝑑𝑠+2𝑥^𝑇(𝑡)𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)+ℎ𝑥^𝑇(𝑡)𝑍𝑥(𝑡), (A.4) where 𝑊₁=^𝑅𝑌−0𝐴^𝑇₁^𝑃_∗𝑍 and 𝑅, 𝑌, 𝑍 satisfying ^𝑅𝑌_∗𝑍≥0.
Furthermore, by (20) it follows that 𝑉̇+1𝛾²_𝑓𝑧^𝑇(𝑡)𝑧(𝑡)−𝑤^𝑇(𝑡)𝑤(𝑡)=𝑥^𝑇(𝑡)^2𝑚−1_𝑖=0𝜂_𝑖Φ_2𝑖𝑥(𝑡)−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)+2^𝑚∼_𝑖=1𝜌_𝑖(𝑡)^∼̇𝜌𝑖+1𝑙𝑖𝛾²_𝑓𝑥^𝑇(𝑡)×𝐶+𝐷(𝐼−𝜌)^2𝑚−1_𝑖=0𝜂_𝑖𝐷_𝑖𝐾𝜌+𝐷⁻_𝑖𝐻𝜌^𝑇×𝐶+𝐷𝐼−𝜌^2𝑚−1_𝑖=0𝜂_𝑖𝐷_𝑖𝐾𝜌+𝐷⁻_𝑖𝐻𝜌𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃^𝑇0𝐵^𝑇₁^𝑇0𝐵^𝑇₁𝑃𝑥(𝑡)−(1−𝑑)𝑥^𝑇((𝑡((𝑡−𝜔𝑡−𝜏))𝑆𝑥𝑡−𝜏))_𝑇−𝑥^𝑇(𝑡)𝑃^𝑇0𝐵^𝑇₁^𝑇𝜔−0𝐵^𝑇₁𝑃𝑥+2𝑥^𝑇(𝑡)𝑃^𝑇𝐵00_{22𝑚−1𝑖}∑₌₀𝜂_𝑖𝐷_𝑖𝐾𝜌_𝑎^∼+𝜌^∼𝐾_𝑏𝜌0+𝐷⁻_𝑖𝐻𝜌_𝑎^∼+𝜌^∼𝐻_𝑏𝜌𝑥(𝑡), (A.5) where Φ_2𝑖=𝑃^𝑇Δ_1𝑖+Δ^𝑇_1𝑖+𝑌𝑃+𝑆00ℎ𝑅+ℎ𝑍_𝑇0^𝑇+𝑌^𝑇0 (A.6) with Δ_1𝑖=_𝑊^0𝐼_2𝑖−𝐼,  𝑊_2𝑖=𝐴+𝐵₂∑^2𝑚−1_𝑖=0𝜂_𝑖[(𝐼−𝜌)𝐷_𝑖𝐾₀+𝐷_𝑖𝐾_𝑎(𝜌)−𝜌𝐷_𝑖𝐾_𝑎(𝜌)+(𝐼−𝜌)𝐷_𝑖𝐾_𝑏(𝜌)+(𝐼−𝜌)𝐷⁻_𝑖𝐻₀+𝐷⁻_𝑖𝐻_𝑎(𝜌)−𝜌𝐷⁻_𝑖𝐻_𝑎(𝜌)+(𝐼−𝜌)𝐷⁻_𝑖𝐻_𝑏(𝜌)].
Then, 𝑉̇+1𝛾²_𝑓𝑧^𝑇(𝑡)𝑧(𝑡)−𝑤^𝑇(𝑡)𝑤(𝑡)≤−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑌−0𝐴^𝑇₁𝑃𝑥(𝑡)+2^𝑚∼_𝑖=1𝜌_𝑖(𝑡)^∼̇𝜌_𝑖(𝑡)𝑙_𝑖+𝑥^{𝑇2𝑚−1}_𝑖=0𝜂_𝑖Φ_3𝑖𝑥+2𝑥^𝑇(𝑡)𝑃^𝑇Φ00₄0𝑥(𝑡), (A.7) where Φ_3𝑖=𝑃^𝑇Δ_1𝑖+Δ^𝑇_1𝑖𝑃+𝑆00ℎ𝑅+𝑃^𝑇0𝐵^𝑇₁^𝑇0𝐵^𝑇₁𝑃+ℎ𝑍+𝑌^𝑇0+𝑌^𝑇0^𝑇+1𝛾²_𝑓Φ^𝑇₅0Φ₅0,Φ₄=𝐵₂^2𝑚−1_𝑖=0𝜂_𝑖𝐷_𝑖𝐾𝜌_𝑎^∼+𝜌^∼𝐾_𝑏𝜌+𝐷⁻_𝑖𝐻𝜌_𝑎^∼+𝜌^∼𝐻_𝑏𝜌,Φ₅=𝐶+𝐷𝐼−𝜌𝐷_𝑖𝐾𝜌+𝐷⁻_𝑖𝐻𝜌. (A.8)
Let 𝐵₂=[𝑏¹⋯𝑏^𝑚]𝐵^𝑗₂=[0⋯𝑏^𝑗⋯0], then we have 𝐵𝜌₂^∼𝐷_𝑖𝐾_𝑏𝜌=^𝑚∼_𝑗=1𝜌_𝑗𝐵^𝑗₂𝐷_𝑖𝐾_𝑏𝜌,𝐵𝜌₂^∼𝐷⁻_𝑖𝐻_𝑏𝜌=^𝑚∼_𝑗=1𝜌_𝑗𝐵^𝑗₂𝐷⁻_𝑖𝐻_𝑏𝜌,𝐵₂𝐷_𝑖𝐾𝜌_𝑎^∼=^𝑚∼_𝑗=1𝜌_𝑗𝐵₂𝐷_𝑖𝐾_𝑎𝑗,𝐵₂𝐷⁻_𝑖𝐻𝜌_𝑎^∼=^𝑚∼_𝑗=1𝜌_𝑗𝐵₂𝐷⁻_𝑖𝐻_𝑎𝑗. (A.9)
In fact, 𝜌_𝑖 is an unknow constant which denotes the loss of effectiveness of the 𝑖th actuator. So, from 𝜌^∼_𝑗(𝑡)=𝜌_𝑗(𝑡)−𝜌, it follows that ̇‌^∼𝜌_𝑗(𝑡)=𝜌̇. Now, if the adaptive laws are chosen as (40), then 2𝑥^𝑇𝑃^𝑇𝐵00_{22𝑚−1𝑖}∑₌₀𝜂_𝑖𝐷_𝑖𝐾𝜌_𝑎^∼+𝜌^∼𝐾_𝑏𝜌0+𝐷⁻_𝑖𝐻𝜌_𝑎^∼+𝜌^∼𝐻_𝑏𝜌𝑥+2^𝑚∼_𝑗=1𝜌_𝑖̇‌^∼𝜌_𝑖𝑙_𝑗≤0. (A.10)
Let 𝜉(𝑡)=col𝑥(𝑡)𝑦(𝑡)𝑥(𝑡−𝜏(𝑡)), then 𝑉̇(𝑡)+1𝛾²_𝑓𝑧^𝑇(𝑡)𝑧(𝑡)−𝜔^𝑇(𝑡)𝜔(𝑡)≤𝜉^𝑇(𝑡)^2𝑚−1_𝑖=1𝜂_𝑖Ψ_𝑖𝜉(𝑡), (A.11) where Ψ_𝑖=Φ_3𝑖𝑃^𝑇0𝐴₁−𝑌^𝑇().∗−𝑆1−𝑑 (A.12)
Furthermore, the problem 𝑉̇(𝑡)+(1/𝛾²_𝑓)𝑧^𝑇(𝑡)𝑧(𝑡)−𝜔^𝑇(𝑡)𝜔(𝑡)≤0 reduces to Ψ_𝑖<0,𝑅𝑌∗𝑍≥0,𝑖∈𝐈0,2^𝑚−1. (A.13)
It is obvious from the requirement of 0<𝑃₁ and the fact that in (A.13) −(𝑃₃+𝑃^𝑇₃) must be negative and 𝑃 is nonsingular.
Defining 𝑃⁻¹𝑄=𝑄=¹0𝑄₂𝑄₃,Π=diag{𝑄,𝐼}. (A.14) We multiply Ψ_𝑖 by Π^𝑇 and Π, on the left and the right, respectively. Applying Fisher's lemma to the emerging quadratic term in 𝑄, denoting 𝑆=𝑆⁻¹𝑍=^𝑍1𝑍2_{𝑍𝑇2𝑍3}=𝑄^𝑇𝑍𝑄, 𝑅=𝑅⁻¹, and choosing 𝑌₁𝑌₂=𝜀𝐴^𝑇₁𝑃₂𝑃₃, where 𝜀∈𝑅^𝑛×𝑛 is a diagonal matrix, we obtain the following: Ψ_𝑖<0 is equivalent to Ξ_0𝑖+𝑄₁𝑆𝑄₁+ℎ𝑄^𝑇₂𝑅𝑄₂Ξ_1𝑖+ℎ𝑄^𝑇₂𝑅𝑄₃∗Ξ₂<0 (A.15) with Ξ_0𝑖=1𝛾²_𝑓Ξ^𝑇_3𝑖Ξ_3𝑖+𝑄₂+𝑄^𝑇₂+ℎ𝑍₁,Ξ_1𝑖=𝑄₃−𝑄^𝑇₂+𝑄₁𝐴^𝑇+𝜀𝐴^𝑇₁+ℎ𝑍₂+(𝐼−𝜌)(𝐷_𝑖𝑌₀+𝐷⁻_𝑖𝑂₀)^𝑇𝐵^𝑇₂+𝐷_𝑖𝑌_𝑎(𝜌)+𝐷⁻_𝑖𝑂_𝑎(𝜌)^𝑇𝐵^𝑇₂−𝜌(𝐷_𝑖𝑌_𝑎(𝜌)+𝐷⁻_𝑖𝑂_𝑎(𝜌))^𝑇𝐵^𝑇₂+𝐼−𝜌𝐷_𝑖𝑌_𝑏𝜌+𝐷⁻_𝑖𝑂_𝑏𝜌^𝑇𝐵^𝑇₂,Ξ₂=−𝑄₃−𝑄^𝑇₃+ℎ𝑍₃+Υ₀+ℎ𝑄^𝑇₃𝑅𝑄₃,Ξ_3𝑖=𝐶𝑄₁+𝐷𝐼−𝜌𝐷_𝑖𝑌𝜌+𝐷⁻_𝑖𝑂𝜌,Υ₀=𝐴₁𝐼_𝑛−𝜀(1−𝑑)⁻¹𝐼_𝑛−𝜀𝐴^𝑇₁+𝐵₁𝐵^𝑇₁,𝑌₀=𝐾₀𝑄₁,𝑌_𝑎𝑗=𝐾_𝑎𝑗𝑄₁,𝑌_𝑏𝑗=𝐾_𝑏𝑗𝑄₁,𝑌_𝑎𝜌=^𝑚_𝑗=1𝑌_𝑎𝑗𝜌_𝑗,𝑌_𝑎𝜌=^𝑚_𝑗=1𝑌_𝑎𝑗𝜌_𝑗,𝑌_𝑏𝜌=^𝑚_𝑗=1𝑌_𝑏𝑗𝜌_𝑗,𝑌𝜌=𝑌₀+𝑌_𝑎𝜌+𝑌_𝑏𝜌,𝑂₀=𝐻₀𝑄₁,𝑂_𝑎𝑗=𝐻_𝑎𝑗𝑄₁,𝑂_𝑏𝑗=𝐻_𝑏𝑗𝑄₁,𝑂_𝑎𝜌=^𝑚_𝑗=1𝑂_𝑎𝑗𝜌_𝑗,𝑂_𝑎𝜌=^𝑚_𝑗=1𝑂_𝑎𝑗𝜌_𝑗,𝑂_𝑏𝜌=^𝑚_𝑗=1𝑂_𝑏𝑗𝜌_𝑗,𝑂𝜌=𝑂₀+𝑂_𝑎𝜌+𝑂_𝑏𝜌. (A.16) Furthermore, (A.15) can be described by 𝑀_𝑖𝜌=𝑁_1𝑖+^𝑚_𝑗=1𝜌_𝑗𝑈_𝑗𝑖+^𝑚_𝑗=1𝜌_𝑗𝑈_𝑗𝑖^𝑇+^𝑚𝑚_𝑗=1𝑝=1𝜌_𝑗𝜌_𝑝Ω_𝑖𝑗𝑝+𝑉_00𝑖+^𝑚_𝑗=1𝜌_𝑗𝑉_0𝑗𝑖^𝑇𝑉_00𝑖+^𝑚_𝑝=1𝜌_𝑗𝑉_0𝑗𝑖<0, (A.17) where 𝑁_1𝑖=𝑁_0𝑖+𝑄₁𝑆𝑄₁+ℎ𝑄^𝑇₂𝑅𝑄₂ℎ𝑄^𝑇₂𝑅𝑄₃∗Υ₀+ℎ𝑄^𝑇₃𝑅𝑄₃ (A.18) and 𝑈_𝑗𝑖, Ω_𝑖𝑗𝑝, 𝑉_00𝑖, 𝑉_0𝑗𝑖, 𝑗=1⋯𝑚 are defined in (39).
If we multiply ^𝑅𝑌_∗𝑍≥0, on the left and on the right, by diag{𝑅⁻¹,𝑄^𝑇} and diag{𝑅⁻¹,𝑄}, then it follows that ^{𝑅0𝑅𝜀𝐴𝑇1∗𝑍}₁^𝑍_2∗∗𝑍3≥0. By Lemmas 3 and 7, it is easy to see if conditions (34), (37), and (38) hold, then (A.17) and ^𝑅𝑌_∗𝑍≥0 are satisfied, which implies that 𝑉̇(𝑡)≤0. Furthermore, by Lemma 3 and (37), it follows that 𝑉̇(𝑡)+(1/𝛾²_𝑓)𝑧^𝑇(𝑡)𝑧(𝑡)−𝜔^𝑇(𝑡)𝜔(𝑡)≤0 holds for any 𝑥∈℘(𝐻(𝜌)), 𝜌∈{𝜌¹⋯𝜌^𝐿}, 𝜌^𝑞∈𝑁_𝜌𝑞, and 𝜌 satisfying (40). The proofs for the normal case of closed-loop system (16) are similar and omitted here.
To prove item (I): by the proof of item (II), we have 𝑉̇≤𝜔^𝑇𝜔, which implies that 𝑉(𝑥(𝑡))≤^∞₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡+^𝑚∼_𝑗=1𝜌²_𝑗(0)𝑙_𝑗≤𝛿^∗,for𝑥(0)=0. (A.19)
Then, the conclusion can be drawn that trajectories of the closed-loop system that start from the origin will remain inside 𝜀^∗(𝑃,𝛿^∗) for every 𝜔∈𝔐_𝛿.

Acknowledgments

This work was supported in part by Program for the Funds of National Science of China (Grant nos. 61203087 and 61074090), the Liaoning Education Department Plan Project of China (no. L2010426), and the Liaoning Students Innovation Training Plan Project of China (nos. 201210143012 and 2012106).