Fault Detection for Nonlinear Impulsive Switched Systems

Abstract

This paper is concerned with the fault detection problem for nonlinear impulsive switched systems. Fault detection filters are designed such that the augmented systems are stable, and the residual signal generated by the filters achieves the 𝐻_∞-gain for disturbances and guarantees the 𝐻₋ performance for faults. Sufficient conditions for the solvability of this problem are formulated in terms of linear matrix inequalities; furthermore, the filter gains are characterized by a convex optimization problem. A simulation on a continuous stirred tank reactor control system is given to demonstrate the effectiveness of the proposed methods.

1. Introduction

Switched systems are an important class of hybrid systems, which consist of a family of continuous-time or discrete-time subsystems and a switching law that specifies the switching between them. Study on this class of systems has attracted much attention in recent years for its theoretical significance [1–6] and engineering applications [7–9]. However, in a wide range of actual systems such as engineering, economics, and biology, there usually exist some impulses when the switched system is switching among subsystems. These special switched systems are called impulsive switched systems. As typical switched impulsive system, the circuit switching usually causes system state to change abruptly in some circuit systems [10]. The abrupt state at the time instant of switching could often lead to oscillations and instability and the lead poor performances. Recently, numbers of papers have focused on stability problem of switched impulsive systems [11–13].

On the other hand, fault detection (FD) is an important topic in system engineering from the viewpoint of the higher demands for safety and reliability of control systems [14–17]. Among these model-based approaches, the class of procedures are that observers or filters are firstly designed to construct a residual signal which is used to generate an alarm when the residual evaluation function has larger than the threshold. During the past decades, Many results, using robust 𝐻_∞ technique [18–21] and 𝐻_∞/𝐻₋ technique, investigate this important issue [22, 23].

However, the problem of FD design in switched systems schemes is still in the early stage of development and a few results have been reported in the literatures [24–28]. It is worth noticing that all the aforementioned FD approaches for switched systems do not include impulsive switched systems. Moreover, since the FD approaches for switched systems without impulsive increments are not appropriate for switched systems with impulsive increments, the new FD technique need to solve the impulsive case. Because these impulsive switched systems exist widely, the FD problem for a general class of nonlinear impulsive switched systems is significant, both theoretically and practically.

In this paper, the problem of FD filters for a general class of nonlinear impulsive switched systems is presented. Different from [10, 29] which only involve linear impulsive switched systems with linear impulsive increments, nonlinear impulsive switched systems with nonlinear impulsive increments have been considered in this paper. The main contributions of this paper include (i) the 𝐻₋ performance of nonlinear impulsive switched systems to directly reflect the effect on the residual signal from faults and obtain the better detection results and (ii) the smaller conservatism of sufficient conditions for the 𝐻₋ performance and the 𝐻_∞ performance. Finally, the filters gains are characterized in terms of the solution of a convex optimization problem.

The paper is organized as follows. Section 2 introduces the problem under consideration and presents the design objectives. Section 3 illustrates the FD filter design approach in detail. The thresholds are given in Section 4. Two examples are given in Section 5 to demonstrate the proposed method. Conclusions of this paper are given in the last section.

Notation. For a matrix 𝐴, 𝐴^𝑇 denotes its transpose. For a symmetric matrix, 𝐴>0(𝐴≥0) and 𝐴<0(𝐴≤0) denote positive definiteness (positive semidefinite matrix) and negative definiteness (negative semidefinite matrix), respectively. The Hermitian part of a square matrix 𝑀 is denoted by He(𝑀)∶=𝑀+𝑀^𝑇. The symbol ∗ within a matrix represents the symmetric entries.

2. Problem Formulation

2.1. System Model

Consider the following nonlinear impulsive switched systems: 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)1𝑑(𝑡)+𝐵_𝜎(𝑡)2𝑓(𝑡)+Υ_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐻_𝜎(𝑡)𝑥(𝑡)+Ω_𝜎(𝑡)𝑥(𝑡)(𝑡,𝑥(𝑡)),𝑡=𝑡_𝑘,𝑦(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐷_𝜎(𝑡)1𝑑(𝑡)+𝐷_𝜎(𝑡)2𝑓(𝑡),𝑥𝑡⁺₀=𝑥₀, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑦(𝑡)∈𝑅^𝑚 is the measured output, and 𝑑(𝑡)∈𝑅^𝑛𝑑 and 𝑓(𝑡)∈𝑅^𝑛𝑓 are the disturbance input and the fault input, respectively. It is assumed that both the disturbance input and the fault input are energy bounded; then it is demanded that they belong to 𝑙₂[0,∞). 𝑡_𝑘, 𝑘=0,1,2,…, are impulsive switching time points satisfying 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯<𝑡_∞=∞. The piecewise constant function 𝜎(𝑡) is a switching rule which takes its values in the finite set 𝒩={1⋯𝑁} (𝑁>1 is the number of subsystems). The index 𝜎(𝑡) denotes the sequence of the active subsystem; that is, when 𝜎(𝑡)=𝑗, the 𝑗th subsystems is activated at the time point 𝑡=𝑡_𝑘. Υ_𝜎(𝑡)(𝑡,𝑥(𝑡))∶[𝑡₀,∞)×𝑅^𝑛→𝑅^𝑛, which is globally the Lipschitz continuous, and Υ_𝜎(𝑡)(𝑡,0)≡0 for all 𝑡∈[𝑡₀,∞)Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘), with 𝑥(𝑡⁺_𝑘)=lim_𝑡→𝑡⁺_𝑘𝑥(𝑡) and 𝑥(𝑡_𝑘)=𝑥(𝑡⁻_𝑘)=lim_𝑡→𝑡⁻_𝑘𝑥(𝑡); that is, the solution 𝑥(𝑡) is left continuous. Ω_𝜎(𝑡)(𝑡,𝑥(𝑡))∶[𝑡₀,∞)×𝑅^𝑛→𝑅^𝑛 is nonlinear function, and Ω_𝜎(𝑡)(𝑡,0)≡0 for all 𝑡∈[𝑡₀,∞). The 𝑗th subsystem is denoted by the matrices 𝐴_𝑗, 𝐵_𝑗1,  𝐵_𝑗2, 𝐶_𝑗,  𝐷_𝑗1, and 𝐷_𝑗2 with appropriate dimensions.

We introduce the following assumptions for nonlinear impulsive switched systems (1).

Assumption 1. There exist nonnegative scalars 𝑔_𝑗>0, 𝑗∈𝒩, such that Υ^𝑇_𝜎(𝑡)(𝑡,𝑥(𝑡))Υ_𝜎(𝑡)(𝑡,𝑥(𝑡))≤𝑔_𝑗𝑥^𝑇(𝑡)𝑥(𝑡). (2)

Assumption 2. Denote 𝜌_𝜎(𝑡)(⋅) as the spectral radius for each subsystem. ‖Ω_𝜎(𝑡)𝑥(𝑡)(𝑡,𝑥(𝑡))‖≤𝜌_𝜎(𝑡)(𝐻_𝜎(𝑡)+1)‖𝑥(𝑡)‖ for 𝑡∈[𝑡₀,∞).

For the purpose of the fault detection, the following FD filters are designed: 𝑥̇_𝑓(𝑡)=𝐴_{𝑓𝜎(𝑡)}𝑥_𝑓(𝑡)+𝐵_{𝑓𝜎(𝑡)}𝑦(𝑡),𝑟(𝑡)=𝐶_{𝑓𝜎(𝑡)}𝑥_𝑓(𝑡)+𝐷_{𝑓𝜎(𝑡)}𝑦(𝑡), (3) where 𝑥_𝑓(𝑡) is the state of the filter and 𝑟(𝑡) is the residual signal. The matrices 𝐴_{𝑓𝜎(𝑡)},𝐵_{𝑓𝜎(𝑡)},𝐶_{𝑓𝜎(𝑡)}, and 𝐷_{𝑓𝜎(𝑡)} with appropriate dimensions are to be determined.

Combining (1) and (3), the nonlinear impulsive switched system can be written as ̇𝑥(𝑡)=𝒜_𝜎(𝑡)𝑥(𝑡)+ℬ_𝜎(𝑡)1𝑑(𝑡)+ℬ_𝜎(𝑡)2𝑓(𝑡)+𝒬_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=ℋ_𝜎(𝑡)𝑥(𝑡)+𝒪_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡=𝑡_𝑘,𝑟(𝑡)=𝒞_𝜎𝑥(𝑡)+𝒟_𝜎1𝑑(𝑡)+𝒟_𝜎2𝑓(𝑡), (4) where 𝑥(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇_𝑓(𝑡)^𝑇,Δ𝑥𝑡_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘,ℋ_𝜎(𝑡)=𝐻_𝜎(𝑡)000,𝒪_𝑗(𝑥(𝑡=Ω𝑡,))^𝑗((𝑡0𝑡,𝑥)) (5) and when 𝜎(𝑡)=𝑗, 𝒜_𝑗ℬ_𝑗1ℬ_𝑗2𝒬_𝑗(𝑡,𝑥(𝑡))𝒞_𝑗𝒟_𝑗1𝒟_𝑗20=𝐴_𝑗0𝐵_𝑗1𝐵_𝑗2Υ_𝑗((𝑡𝐵𝑡,𝑥))_𝑓𝑗𝐶_𝑗𝐴_𝑓𝑗𝐵_𝑓𝑗𝐷_𝑗1𝐵_𝑓𝑗𝐷_𝑗20𝐷_𝑓𝑗𝐶_𝑗𝐶_𝑓𝑗𝐷_𝑓𝑗𝐷_𝑗1𝐷_𝑓𝑗𝐷_𝑗20. (6) To present the purpose of this paper more precisely, the following definition is introduced.

Definition 3. Let Assumptions 1 and 2 be satisfied and 𝑑(𝑡)=0. Nonlinear impulsive switched system (4) under zero initial conditions is said to be stable with the 𝐻₋-gain 𝛽, if the condition holds that ^∞_𝑡0‖𝑟(𝑠)‖²d𝑠≥𝛽^2∞_𝑡0𝑓(𝑠)²d𝑠. (7)

2.2. Problem Formulation

The design problem of the fault detection filters to be addressed in this paper can be expressed as follows.

The frameworks of FD filter design: given nonlinear impulsive switched system (1), the FD filters (3) are designed such that nonlinear impulsive switched system (4) is stable and the fault effects on the residual signal are maximized, while the disturbance effects on the residual signals are minimized. Our design objective of the FD filters can now be formulated as the following performances: ^∞_𝑡0‖𝑟(𝑠)‖²d𝑠≥𝛽^2∞_𝑡0𝑓(𝑠)²d𝑠,_∞𝑡0‖𝑟(𝑠)‖²d𝑠≤𝛾^2∞_𝑡0‖𝑑(𝑠)‖²d𝑠. (8)(9)

Remark 4. Condition (8) is expressed to maximize the effects of the fault 𝑓(𝑡) on the residual output 𝑟(𝑡) for impulsive switched system (4). That is, the residual output 𝑟(𝑡) is sensitive for the fault 𝑓(𝑡). Condition (9) is used for the disturbance attenuation performance, which minimizes the disturbance effects on the all residual outputs and ensures that the disturbances are not disastrous.

3. The Fault Detection Filter Design

Before beginning this section, the following lemmas are needed to present our main results.

Lemma 5 (see [30]). Let 𝜖>0 be a given scalar and Ξ∈𝑅^𝑝×𝑞 be a matrix such that Ξ^𝑇Ξ≤𝐼, where 𝐼 is an identity matrix with appropriate dimension. Then 2𝑥^𝑇Ξ𝑦≤𝜖𝑥^𝑇𝑥+𝜖⁻¹𝑦^𝑇𝑦 for all 𝑥∈𝑅^𝑝 and 𝑦∈𝑅^𝑞.

Lemma 6 (see [31]). Let 𝑃∈𝑅^𝑛×𝑛 be a given symmetric positive definite matrix and let 𝑄∈𝑅^𝑛×𝑛 be a given symmetric matrix. Then 𝜆_min𝑃⁻¹𝑄𝑥(𝑡)^𝑇𝑃𝑥(𝑡)≤𝑥(𝑡)^𝑇𝑄𝑥(𝑡)≤𝜆_max𝑃⁻¹𝑄𝑥(𝑡)^𝑇𝑃𝑥(𝑡) (10) for all 𝑥(𝑡)∈𝑅^𝑛, while 𝜆_max{⋅} and 𝜆_min{⋅} denote, respectively, the largest and the smallest eigenvalues of the matrix inside the brackets.

In this section, sufficient conditions on the existence of the FD filters for nonlinear impulsive switched systems (4) would be given, and the desired filters can be obtained.

3.1. The Fault Sensitiveness Performance (8)

Considering nonlinear impulsive switched system (4) with 𝑑(𝑡)=0, we have ̇𝑥(𝑡)=𝒜_𝜎(𝑡)𝑥(𝑡)+ℬ_𝜎(𝑡)2𝑓(𝑡)+𝒬_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,Δ𝑥=ℋ_𝜎(𝑡)𝑥(𝑡)+𝒪_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡=𝑡_𝑘,𝑟(𝑡)=𝒞_𝜎(𝑡)𝑥(𝑡)+𝒟_𝜎(𝑡)2𝑓(𝑡); (11)𝐻₋ performance for nonlinear impulsive switched system (11) is given.

Lemma 7. Let 𝛼_𝑗, 𝜀 be constants satisfying 0<𝛼_𝑗<1, 𝜀>0, 𝑗∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (11) switches from 𝑖th subsystem to 𝑗th subsystem as switched time point 𝑡_𝑘. If there exists the Lyapunov functions candidate 𝑉_𝜎(𝑡)(𝑡)=𝑉_𝑗(𝑡)=𝑥(𝑡)^𝑇𝒫_𝑗𝑥(𝑡),𝑗∈𝒩, satisfying the following inequalities 𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗+𝜀⁻¹𝒫²_𝑗+𝛼_𝑗𝒫_𝑗+𝜀𝑔_𝑗𝐼−𝒞^𝑇_𝑗𝒞_𝑗𝒫_𝑗ℬ_𝑗2−𝒞^𝑇_𝑗𝒟_𝑗2ℬ^𝑇_𝑗2𝒫_𝑗−𝒟^𝑇_𝑗2𝒞_𝑗𝛽²𝐼−𝒟^𝑇_𝑗2𝒟_𝑗2<0,𝑗∈𝒩, (12) then nonlinear impulsive switched system (11) is stable with 𝐻₋-gain 𝛽 for any switching signal satisfying ln𝜇_𝑖𝑗𝑘−𝛼_𝑖𝑡_𝑘−𝑡_𝑘−1<0,𝑖∈𝒩,𝑗∈𝒩,𝑖≠𝑗,𝑘=1,2,…, (13) where 𝜇_𝑖𝑗|_𝑘=((𝛿+1)𝜆_max{(ℋ_𝑗+𝐼)^𝑇𝒫_𝑗(ℋ_𝑗+𝐼)}+(𝛿⁻¹+1)𝜌²_𝑗𝜆_max{𝒫_𝑗})/𝜆_min{𝒫_𝑖},𝛿>0.

Proof. When 𝑡∈(𝑡_𝑘,𝑡_𝑘+1] and 𝑗th subsystem is activated, evaluating the time derivative of 𝑉_𝑗(𝑡) along the trajectory of nonlinear impulsive switched system (11) gives 𝑉̇_𝑗(𝑡)=𝑥^𝑇(𝑡)𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝒫_𝑗ℬ_𝑗2𝑓(𝑡)+𝑓(𝑡)^𝑇ℬ^𝑇_𝑗2𝒫_𝑗𝑥(𝑡)+𝒬_𝑗(𝑡,𝑥(𝑡))^𝑇𝒫_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝒫𝒬_𝑗(𝑡,𝑥(𝑡)). (14) By Lemma 5, it is clear that 2𝒬_𝑗(𝑡,𝑥(𝑡))^𝑇𝒫_𝑗𝑥(𝑡)≤𝜀𝒬_𝑗(𝑡,𝑥(𝑡))^𝑇𝒬_𝑗(𝑡,𝑥(𝑡))+𝜀⁻¹𝑥^𝑇(𝑡)𝒫²_𝑗𝑥(𝑡). (15) Therefore, when assuming the zero input (i.e., 𝑓(𝑡)=0) and using Assumptions 1, we have the following condition from (14) and (15): 𝑉̇_{𝑗𝑓(𝑡)=0}(𝑡)≤𝑥^𝑇(𝑡)𝓠_𝑗𝑥(𝑡), (16) where 𝓠_𝑗=𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗+𝜀𝑔_𝑗𝐼+𝜀⁻¹𝒫²_𝑗. If (12) holds, then 𝓠_𝑗+𝛼_𝑗𝒫_𝑗≤0, which implies that 𝓠_𝑗≤−𝛼_𝑗𝒫_𝑗≤0. Thus, from (16), we have 𝑉̇_𝑗(𝑡)|_𝑓(𝑡)=0(𝑡)≤−𝛼_𝑗𝑉_𝑗(𝑡)≤0. Integrating this inequality gives 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗𝑡⁺_𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (17) At the impulsive switching time point 𝑡_𝑘, it has 𝑉_𝑗𝑡⁺_𝑘=Δ𝑥𝑡_𝑘+𝑥𝑡_𝑘^𝑇𝒫_𝑗Δ𝑥𝑡_𝑘+𝑥𝑡_𝑘=(ℋ_𝑗+𝐼)𝑥(𝑡_𝑘)^𝑇𝒫_𝑗ℋ_𝑗+𝐼𝑥𝑡_𝑘+2𝒪^𝑇_𝑗(𝑡,𝑥(𝑡))𝒫_𝑗ℋ_𝑗+𝐼𝑥𝑡_𝑘+𝒪^𝑇_𝑗(𝑡,𝑥(𝑡))𝒪_𝑗(𝑡,𝑥(𝑡)). (18) By Lemma 6, it has 𝑉_𝑗𝑡⁺_𝑘≤(𝛿+1)(ℋ_𝑗+𝐼)𝑥(𝑡_𝑘)^𝑇𝒫_𝑗ℋ_𝑗+𝐼𝑥𝑡_𝑘+𝛿⁻¹+1𝒪^𝑇_𝑗(𝑡,𝑥(𝑡))𝒫_𝑗𝒪_𝑗(𝑥(𝑡≤)𝜆𝑡,))(𝛿+1_maxℋ_𝑗+𝐼^𝑇𝒫_𝑗ℋ_𝑗+𝛿+𝐼₋₁+1𝜌²_𝑗𝜆_max𝒫_𝑗𝑥^𝑇𝑡_𝑘𝑥𝑡_𝑘≤𝜇_𝑖𝑗𝑘𝑉_𝑖𝑡_𝑘, (19) where 𝜇_𝑖𝑗|_𝑘=((𝛿+1)𝜆_max{(ℋ_𝑗+𝐼)^𝑇𝒫_𝑗(ℋ_𝑗+𝐼)}+(𝛿⁻¹+1)𝜌²_𝑗𝜆_max{𝒫_𝑗})/𝜆_min{𝒫_𝑖}, 𝛿>0. Therefore, from (17) and (19), we have 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗𝑡⁺_𝑘≤𝜇_𝑖𝑗𝑘𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑖𝑡_𝑘≤𝜇_𝑖𝑗𝑘𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑒^{−𝛼𝑖(𝑡𝑘−𝑡𝑘−1)}𝑉_𝑖𝑡⁺_𝑘−1≤≤⋯≤_{𝑘𝑙=1,𝑞≠𝑝𝑞∈𝒩,𝑝∈𝒩}𝜇_𝑞𝑝𝑙𝑒^{−𝛼𝑞(𝑡𝑙−𝑡𝑙−1)}𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}×𝑉_𝜎(𝑡0)𝑡₀, (20) where 𝜇_𝑞𝑝|_𝑙 means the 𝑙th switching impulsive value from 𝑞th subsystem to 𝑝th subsystems, and 𝛼_𝑞 denotes the decay rate of the Lyapunov function for 𝑞th subsystem. Since (13) holds; that is, there exists 𝜇_𝑞𝑝|_𝑙>0 such that ln(𝜇_𝑞𝑝|_𝑙)−𝛼_𝑞(𝑡_𝑙−𝑡_𝑙−1)<0, then 𝑔𝑡−𝑡₀=−𝛼_𝑗𝑡−𝑡_𝑘+^𝑘_{𝑙=1,𝑞≠𝑝𝑞∈𝒩,𝑝∈𝒩}ln𝜇_𝑞𝑝𝑙−𝛼_𝑞𝑡_𝑙−𝑡_𝑙−1<0. (21) It follows that 𝑉_𝜎(𝑡)(𝑡)≤𝑒^{𝑔(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑡₀. (22) Therefore, we conclude that 𝑉_𝜎(𝑡)(𝑡) converges to zero as 𝑡→∞ then nonlinear impulsive switched system (11) with 𝑓(𝑡)=0 is stable.
Secondly, establish the 𝐻₋ performance defined in (7) for nonlinear impulsive switched system (11). Consider the following performance index: 𝐽=^∞_𝑡0𝛽²𝑓(𝑠)²−‖𝑟(𝑠)‖²d𝑠. (23)
For any nonzero 𝑓(𝑡)∈𝑙₂[𝑡₀,∞) and zero initial condition 𝑥(𝑡₀), let Γ₁(𝑡)=𝛽²𝑓^𝑇(𝑡)𝑓(𝑡)−𝑟^𝑇(𝑡)𝑟(𝑡) and consider the Lyapunov functions as 𝑉_𝑗(𝑡)=𝑥(𝑡)^𝑇𝒫_𝑗𝑥(𝑡). For 𝑗th subsystem, it has 𝑉_𝑗(𝑥(𝑡))+𝛼_𝑗𝑉_𝑗(𝑥(𝑡))+Γ₁(𝑡)≤𝒳^𝑇₁𝓡_𝑗1𝒳₁,𝑗∈𝒩, (24) where 𝒳₁=𝑥(𝑡)^𝑇𝑓(𝑡)^𝑇𝑇,𝓡_𝑗1=𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗+𝜀⁻¹𝒫²_𝑗+𝛼_𝑗𝒫_𝑗+𝜀𝑔_𝑗𝐼−𝒞^𝑇_𝑗𝒞_𝑗𝒫_𝑗ℬ_𝑗2−𝒞^𝑇_𝑗𝒟_𝑗2ℬ^𝑇_𝑗2𝒫_𝑗−𝒟^𝑇_𝑗2𝒞_𝑗𝛽²𝐼−𝒟^𝑇_𝑗2𝒟_𝑗2. (25) Equation (12) is equivalent to 𝓡_𝑗1<0. Then, it has 𝑉_𝑗(𝑥(𝑡))+𝛼_𝑗𝑉_𝑗(𝑥(𝑡))+Γ₁(𝑡)≤0. By iteration operation on the above inequality for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we have 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗𝑡⁺_𝑘−^𝑡_𝑡𝑘Γ₁(𝑠)d𝑠. (26) Along the same lines as the impulsive switching time point 𝑡_𝑘, it has from (19) and (26) that 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗𝑡⁺_𝑘−^𝑡_𝑡𝑘Γ₁(𝑠)d𝑠≤𝜇_𝑖𝑗𝑘𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑖𝑡_𝑘−^𝑡_𝑡𝑘Γ₁(𝑠)d𝑠≤𝜇_𝑖𝑗𝑘𝑒^{−𝛼𝑗(𝑡−𝑡𝑘−1)}𝑉_𝑖𝑡⁺_𝑘−1−𝜇_𝑖𝑗𝑘𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}×^𝑡𝑘_𝑡𝑘−1Γ₁(𝑠)d𝑠−^𝑡_𝑡𝑘Γ₁(𝑠)d𝑠≤⋯≤≤𝑒^{𝑔(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑡₀−^𝑡_𝑡0𝑒^{𝑔(𝑡−𝑠)}Γ₁(𝑠)d𝑠, (27) where 𝑔(𝑡−𝑡₀) and 𝑔(𝑡−𝑠) are defined as (21).
Under the zero initial condition, (27) implies ∫^𝑡_𝑡0𝑒^{𝑔(𝑡−𝑠)}Γ₁(𝑠)d𝑠<0. Because 0<𝑔(𝑡−𝑠)<1,  𝑠∈[𝑡₀,𝑡), it has ∫^𝑡_𝑡0Γ₁(𝑠)d𝑠<∫^𝑡_𝑡0𝑒^{𝑔(𝑡−𝑠)}Γ₁(𝑠)d𝑠<0; that is, 𝛽^2𝑡_𝑡0𝑓^𝑇(𝑠)𝑓(𝑠)d𝑠≤^𝑡_𝑡0𝑟^𝑇(𝑠)𝑟(𝑠)d𝑠. (28) When 𝑡→∞, nonlinear impulsive switched system (11) has 𝐻₋ performance, which completes the proof.

Remark 8. For switched systems, the Lyapunov function is discontinuous at the time instant of switching; that is to say, this Lyapunov function does not have its derivative in the whole time domain. Then the piecewise Lyapunov function was utilized in Lemma 7. Each subsystem has one corresponding Lyapunov function, and each Lyapunov function has its derivative in the corresponding subsystem time domain.

Based on Lemma 7, the following theorem is given to obtain the fault sensitiveness performance conditions in terms of linear matrix inequalities.

Theorem 9. Let 𝛼_𝑗, 𝛽, and 𝜀 be constants satisfying 0<𝛼_𝑗<1, 𝛽>0, 𝜀>0 and Assumptions 1 and 2 hold. If there exist matrix variables 𝑌_𝑎𝑗, 𝑋_𝑎𝑗, 𝑀_𝑎𝑗, 𝑆_𝑎𝑗, 𝑍_𝑎𝑗, 𝑁_𝑗, 𝑅_𝑗, 𝐴_𝑓𝑗, 𝐵_𝑓𝑗, 𝐶_𝑓𝑗, 𝐷_𝑓𝑗, and symmetric positive-definite matrices 𝒫_𝑎𝑗=𝑃_𝑎𝑗1𝑃_𝑎𝑗2∗𝑃_𝑎𝑗3>0,𝑗∈𝒩, (29) satisfying the following inequalities Θ_𝑎1Θ_𝑎2Θ_𝑎300∗Θ_𝑎4Θ_𝑎5𝒫_𝑎𝑗0∗∗Θ_𝑎60𝑅^𝑇_𝑗∗∗∗−𝜀⁻¹𝐼0∗∗∗∗−𝐼<0,𝑗∈𝒩, (30) whereΘ_𝑎1=−He𝑋_𝑎𝑗−𝑌_𝑎𝑗−𝑎₁𝑁^𝑇_𝑗∗−He𝑎₂𝑁_𝑗,Θ_𝑎3=−𝑍_𝑎𝑗+𝑋^𝑇_𝑎𝑗𝐵_𝑗2+𝑎₁𝐵_𝑓𝑗𝐷_𝑗2−𝑎₅𝑁_𝑗𝑣_𝑗+𝑌^𝑇_𝑎𝑗𝐵_𝑗2+𝑎₂𝐵_𝑓𝑗𝐷_𝑗2,Θ_𝑎2=𝑃_𝑎𝑗1−𝑀_𝑎𝑗+𝑋^𝑇_𝑎𝑗𝐴_𝑗+𝑎₁𝐵_𝑓𝑗𝐶_𝑗𝑃_𝑎𝑗2−𝑆_𝑎𝑗+𝑎₁𝐴_𝑓𝑗𝑃^𝑇_𝑎𝑗2−𝑎₃𝑁_𝑗+𝑌^𝑇_𝑎𝑗𝐴_𝑗+𝑎₂𝐵_𝑓𝑗𝐶_𝑗𝑃_𝑎𝑗3−𝑎₄𝑁_𝑗+𝑎₂𝐴_𝑓𝑗,Θ_𝑎4=𝛼_𝑗𝑃_𝑎𝑗1+𝜀𝑔_𝑗𝐼+He𝑀^𝑇_𝑎𝑗𝐴_𝑗+𝑎₃𝐵_𝑓𝑗𝐶_𝑗𝛼_𝑗𝑃_𝑎𝑗2+𝑎₄𝐴_𝑓𝑗+𝐴^𝑇_𝑗𝑆_𝑎𝑗+𝑎₄𝐶^𝑇_𝑗𝐵^𝑇_𝑓𝑗∗𝛼_𝑗𝑃_𝑎𝑗3+𝜀𝑔_𝑗𝐼+He𝑎₄𝐴_𝑓𝑗,Θ_𝑎5=𝑀^𝑇_𝑎𝑗𝐵_𝑗2+𝑎₃𝐵_𝑓𝑗𝐷_𝑗2+𝐴^𝑇_𝑗𝑍_𝑎𝑗+𝑎₅𝐶^𝑇_𝑗𝐵^𝑇_𝑓𝑗𝑣_𝑗−𝐶^𝑇_𝑗𝐷^𝑇_𝑓𝑗𝑆^𝑇_𝑎𝑗𝐵_𝑗2+𝑎₄𝐵_𝑓𝑗𝐷_𝑗2+𝑎₅𝐴^𝑇_𝑓𝑗𝑣_𝑗−𝐶^𝑇_𝑓𝑗,Θ_𝑎6=𝛽²𝐼+He𝐵^𝑇_𝑗2𝑍_𝑎𝑗+𝑎₅𝐷^𝑇_𝑗2𝐵^𝑇_𝑓𝑗𝑣_𝑗−𝐷_𝑓𝑗𝐷_𝑗2, (31)then, for any switching signal satisfying (13), nonlinear impulsive switched system (11) is stable and guarantees the 𝐻₋ performance (8). Moreover, if (30) is feasible, then the FD filter gains in form of (3) can be given by 𝐴_𝑓𝑗𝐵_𝑓𝑗𝐶_𝑓𝑗𝐷_𝑓𝑗=𝑁_𝑗00𝑅_𝑗^−𝑇𝐴_𝑓𝑗𝐵_𝑓𝑗𝐶_𝑓𝑗𝐷_𝑓𝑗. (32)

Proof. Now, to establish the convex condition, (12) can be rewritten as 𝒰^𝑇ℳ_𝑎𝑗𝒰<0, (33) where 𝒜𝒰=^𝑗ℬ_𝑗2,ℳ𝐼00𝐼_𝑎𝑗=0𝒫_𝑎𝑗0𝒫_𝑎𝑗𝛼_𝑗𝒫_𝑎𝑗+𝜀⁻¹𝒫²_𝑎𝑗+𝜀𝑔_𝑗𝐼−𝒞^𝑇_𝑗𝒞_𝑗−𝒞^𝑇_𝑗𝒟_𝑗20−𝒟^𝑇_𝑗2𝒞_𝑗𝛽²𝐼−𝒟^𝑇_𝑗2𝒟_𝑗2. (34) On the other hand, we also can have −𝐼𝒜_𝑗ℬ_𝑗2𝒰=0. (35)
Based on Finsler’s Lemma, it has ℳ_𝑎𝑗+He𝐴−𝐼_𝑇𝑗𝐵^𝑇_𝑗2𝒲_𝑎𝑗<0, (36) where 𝒲_𝑎𝑗 introduced by Finsler’s Lemma is the matrix variable of appropriate dimensions. Partition 𝒲_𝑎𝑗 as 𝒲_𝑎𝑗=[𝒲_𝑎𝑗1𝒲_𝑎𝑗2𝒲_𝑎𝑗3], and 𝒲_𝑎𝑗1=𝑋_𝑎𝑗𝑌_𝑎𝑗𝑎₁𝑁_𝑗𝑎₂𝑁_𝑗,𝒲_𝑎𝑗2=𝑀_𝑎𝑗𝑆_𝑎𝑗𝑎₃𝑁_𝑗𝑎₄𝑁_𝑗,𝒲_𝑎𝑗3=𝑍_𝑎𝑗𝑎₅𝑁_𝑗𝑣_𝑗. (37) Due to −𝒞^𝑇_𝑗𝒞_𝑗, −𝒞^𝑇_𝑗𝒟_𝑗2 and −𝒟^𝑇_𝑗2𝒟_𝑗2 in ℳ_𝑎𝑗, then the condition (36) is non-convex. To establish the convex condition, (36) can be rewritten as 𝒰^𝑇ℳ_𝑎𝑗𝒰<0, (38) where𝒰=𝐼000𝐼000𝐼0𝒞_𝑗𝒟_𝑗2,ℳ_𝑎𝑗=−He𝒲_𝑎𝑗1𝒫_𝑎𝑗−𝒲_𝑎𝑗2+𝒲^𝑇_𝑎𝑗1𝒜_𝑗−𝒲_𝑎𝑗3+𝒲^𝑇_𝑎𝑗1ℬ_𝑗20∗𝛼_𝑗𝒫_𝑎𝑗+𝜀𝑔_𝑗𝐼+𝜀⁻¹𝒫²_𝑎𝑗+He𝒜^𝑇_𝑗𝒲_𝑎𝑗2𝒜^𝑇_𝑗𝒲_𝑎𝑗3+𝒲^𝑇_𝑎𝑗2ℬ_𝑗20∗∗𝛽²𝐼+Heℬ^𝑇_𝑗2𝒲_𝑎𝑗30.∗∗∗0∗∗∗−𝐼 (39)Using Finsler’s Lemma again, (38) holds if ℳ_𝑎𝑗+He(ℛℒ)<0 (40) holds for any ℛ with appropriate dimension and ℒ=[0,−𝒞_𝑗,−𝒟_𝑗2,𝐼]. Let ℛ=[0,0,𝑅_𝑗,0]^𝑇 and define 𝐴_𝑓𝑗=𝑁^𝑇_𝑗𝐴_𝑓𝑗, 𝐵_𝑓𝑗=𝑁^𝑇_𝑗𝐵_𝑓𝑗, 𝐶_𝑓𝑗=𝑅^𝑇_𝑗𝐶_𝑓𝑗, 𝐷_𝑓𝑗=𝑅^𝑇_𝑗𝐷_𝑓𝑗; then (30) is reached by the Schur complement on (40). Hence if the conditions (30) holds, nonlinear impulsive switched system (11) is stable and guarantees the 𝐻₋ performance (8), which completes the proof.

Remark 10. Scalars 𝑎₁∼𝑎₅ are used to reduce conservatism for the same variable 𝑁_𝑗 in matrix variables 𝒲_𝑎𝑗1∼𝒲_𝑎𝑗3. On the other hand, the coefficient matrices 𝑣_𝑗 are given to adjust the variable dimension. When we set 𝑎₁∼𝑎₅, 𝑣_𝑗 as the fixed parameters, the conditions in Theorem 9 become convex.

3.2. The Disturbance Attenuation Performance (8)

Consider nonlinear impulsive switched system (4) with 𝑓(𝑡)=0, we have ̇𝑥(𝑡)=𝒜_𝜎(𝑡)𝑥(𝑡)+ℬ_𝜎1𝑑(𝑡)+𝒬_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,Δ𝑥=ℋ_𝜎(𝑡)𝑥(𝑡)+𝒪_𝜎(𝑡)(𝑡,𝑥(𝑡)),𝑡=𝑡_𝑘,𝑟(𝑡)=𝒞_𝜎𝑥(𝑡)+𝒟_𝜎1𝑑(𝑡); (41)

𝐻_∞ performance for nonlinear impulsive switched system (41) is given.

Lemma 11. Let 𝛼_𝑗, 𝜀 be constants satisfying 0<𝛼_𝑗<1, 𝜀>0, 𝑗∈𝒩, and Assumptions 1 and 2 hold. Furthermore, suppose that nonlinear impulsive switched system (41) switches from 𝑖th subsystem to 𝑗th subsystem as switched time point 𝑡_𝑘. If there exists the Lyapunov functions candidate 𝑉_𝜎(𝑡)(𝑡)=𝑉_𝑗(𝑡)=𝑥(𝑡)^𝑇𝒫_𝑗𝑥(𝑡), 𝑗∈𝒩, satisfying 𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗+𝜀⁻¹𝒫²_𝑗+𝛼_𝑗𝒫_𝑗+𝜀𝑔_𝑗𝐼+𝒞^𝑇_𝑗𝒞_𝑗𝒫_𝑗ℬ_𝑗1+𝒞^𝑇_𝑗𝒟_𝑗1ℬ^𝑇_𝑗1𝒫_𝑗+𝒟^𝑇_𝑗1𝒞_𝑗−𝛾²𝐼+𝒟^𝑇_𝑗1𝒟_𝑗1<0,𝑗∈𝒩, (42) then nonlinear impulsive switched system (41) is stable with 𝐻_∞-gain 𝛾 for any switching signal satisfying (13).

Proof. Following the same lines as those for Lemma 7, nonlinear impulsive switched system (41) is stable, for any nonzero 𝑑(𝑡)∈𝑙₂[0,∞) and zero initial condition 𝑥(𝑡₀). Let Γ₂(𝑡)=𝑟^𝑇(𝑡)𝑟(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡) and consider the Lyapunov function as Lemma 7; we can have 𝑉_𝑗(𝑥(𝑡))+𝛼_𝑗𝑉_𝑗(𝑥(𝑡))+Γ₂(𝑡)≤𝒳^𝑇₂𝓡_𝑗2𝒳₂,𝑗∈𝒩, (43) where 𝒳₂=𝑥(𝑡)^𝑇𝑑(𝑡)^𝑇𝑇,𝓡_𝑗2=𝒜^𝑇_𝑗𝒫_𝑗+𝒫_𝑗𝒜_𝑗+𝜀⁻¹𝒫²_𝑗+𝛼_𝑗𝒫_𝑗+𝜀𝑔_𝑗𝐼+𝒞^𝑇_𝑗𝒞_𝑗𝒫_𝑗ℬ_𝑗1+𝒞^𝑇_𝑗𝒟_𝑗1ℬ^𝑇_𝑗1𝒫_𝑗+𝒟^𝑇_𝑗1𝒞_𝑗−𝛾²𝐼+𝒟^𝑇_𝑗1𝒟_𝑗1. (44) Equation (42) is equivalent to 𝓡_𝑗2<0. Thus, it has 𝑉_𝑗(𝑥(𝑡))+𝛼_𝑗𝑉_𝑗(𝑥(𝑡))+Γ₂(𝑡)≤0. (45) By iteration operation on the above inequality for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we have 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗(𝑡⁺_𝑘)−∫^𝑡_𝑡𝑘Γ₂(𝑠)d𝑠. As the same lines as those for Lemma 7, it has 𝑉_𝑗(𝑡)≤𝑒^{−𝛼𝑗(𝑡−𝑡𝑘)}𝑉_𝑗𝑡⁺_𝑘−^𝑡_𝑡𝑘Γ₂(𝑠)d𝑠≤⋯≤≤𝑒^{𝑔(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑡₀−^𝑡_𝑡0𝑒^{𝑔(𝑡−𝑠)}Γ₂(𝑠)d𝑠≤𝑒^{𝑔(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑡₀−^𝑡_𝑡0Γ₁(𝑠)d𝑠. (46)
Under the zero initial condition, (46) implies ∫^𝑡_𝑡0𝑒^{𝑔(𝑡−𝑠)}Γ₂(𝑠)d𝑠<0. Because 0<𝑔(𝑡−𝑠)<1, 𝑠∈[𝑡₀,𝑡), it can further have ^𝑡_𝑡0𝑟^𝑇(𝑠)𝑟(𝑠)d𝑠≤𝛾^2𝑡_𝑡0𝑑^𝑇(𝑠)𝑑(𝑠)d𝑠. (47)
When 𝑡→∞, the nonlinear impulsive switched system (41) has 𝐻_∞ performance, which completes the proof.

Based on Lemma 11, the following theorem is given to obtain sufficient conditions by linear matrix inequalities.

Theorem 12. Let 𝛼_𝑗, 𝛾, and 𝜀 be constants satisfying 0<𝛼_𝑗<1, 𝛾>0, and 𝜀>0 and Assumptions 1 and 2 hold. If there exist matrix variables 𝑌_𝑏𝑗, 𝑋_𝑏𝑗, 𝑀_𝑏𝑗, 𝑆_𝑏𝑗, 𝑁_𝑗, 𝑅_𝑗, 𝐴_𝑓𝑗, 𝐵_𝑓𝑗, 𝐶_𝑓𝑗, and 𝐷_𝑓𝑗 and symmetric positive-definite matrices 𝒫_𝑏𝑗=𝑃_𝑏𝑗1𝑃_𝑏𝑗2∗𝑃_𝑏𝑗3>0,𝑗∈𝒩, (48) satisfying the following inequalities Θ_𝑏1Θ_𝑏2Θ_𝑏300∗Θ_𝑏4Θ_𝑏5𝒫_𝑏𝑗Θ_𝑏6∗∗−𝛾²𝐼0𝐷^𝑇_𝑗1𝐷^𝑇_𝑓𝑗∗∗∗−𝜀⁻¹𝑅𝐼0∗∗∗∗𝐼−He_𝑗<0,𝑗∈𝒩, (49) whereΘ_𝑏1=−He𝑋_𝑏𝑗−𝑌_𝑏𝑗−𝑏₁𝑁^𝑇_𝑗∗−He𝑏₂𝑁_𝑗,Θ_𝑏3=𝑋^𝑇_𝑏𝑗𝐵_𝑗1+𝑏₁𝐵_𝑓𝑗𝐷_𝑗1𝑌^𝑇_𝑏𝑗𝐵_𝑗1+𝑏₂𝐵_𝑓𝑗𝐷_𝑗1,Θ_𝑏2=𝑃_𝑏𝑗1−𝑀_𝑏𝑗+𝑋^𝑇_𝑏𝑗𝐴_𝑗+𝑏₁𝐵_𝑓𝑗𝐶_𝑗𝑃_𝑏𝑗2−𝑆_𝑏𝑗+𝑏₁𝐴_𝑓𝑗𝑃^𝑇_𝑏𝑗2−𝑏₃𝑁_𝑗+𝑌^𝑇_𝑏𝑗𝐴_𝑗+𝑏₂𝐵_𝑓𝑗𝐶_𝑗𝑃_𝑏𝑗3−𝑏₄𝑁_𝑗+𝑏₂𝐴_𝑓𝑗,Θ_𝑏4=𝛼_𝑗𝑃_𝑏𝑗1+He𝑀^𝑇_𝑏𝑗𝐴_𝑗+𝑏₃𝐵_𝑓𝑗𝐶_𝑗+𝜀𝑔_𝑗𝐼𝛼_𝑗𝑃_𝑏𝑗2+𝑏₃𝐴_𝑓𝑗+𝐴^𝑇_𝑗𝑆_𝑏𝑗+𝑏₄𝐶^𝑇_𝑗𝐵^𝑇_𝑓𝑗∗𝛼_𝑗𝑃_𝑏𝑗3+𝜀𝑔_𝑗𝐼+He𝑏₄𝐴_𝑓𝑗,Θ_𝑏5=𝑀^𝑇_𝑏𝑗𝐵_𝑗1+𝑏₃𝐵_𝑓𝑗𝐷_𝑗1𝑆^𝑇_𝑏𝑗𝐵_𝑗1+𝑏₄𝐵_𝑓𝑗𝐷_𝑗1,Θ_𝑏6=𝐶^𝑇_𝑗𝐷^𝑇_𝑓𝑗𝐶^𝑇_𝑓𝑗, (50)then, for any switching signal satisfying (13), nonlinear impulsive switched system (41) is stable and guarantees the 𝐻_∞ performance (9). Moreover, if (49) is feasible, then the FD filter gains in form of (3) can be given by 𝐴_𝑓𝑗𝐵_𝑓𝑗𝐶_𝑓𝑗𝐷_𝑓𝑗=𝑁_𝑗00𝑅_𝑗^−𝑇𝐴_𝑓𝑗𝐵_𝑓𝑗𝐶_𝑓𝑗𝐷_𝑓𝑗. (51)

Proof. To establish the convex condition, (42) can be rewritten as follows: 𝒜_𝑗ℬ_𝑗1𝐼0^𝑇0𝒫_𝑏𝑗𝒫_𝑏𝑗𝛼𝒫_𝑏𝑗𝒜_𝑗ℬ_𝑗1+𝜀𝐼0₋₁𝒫²_𝑏𝑗+𝜀𝑔_𝑗𝐼+𝒞^𝑇_𝑗𝒞_𝑗𝒞^𝑇_𝑗𝒟_𝑗1𝒟^𝑇_𝑗1𝒞_𝑗−𝛾²𝐼+𝒟^𝑇_𝑗1𝒟_𝑗1<0. (52) Denote ℳ_𝑏𝑗=0𝒫_𝑏𝑗0𝒫_𝑏𝑗𝛼𝒫_𝑏𝑗+𝜀⁻¹𝒫²_𝑏𝑗+𝜀𝑔_𝑗𝐼+𝒞^𝑇_𝑗𝒞_𝑗𝒞^𝑇_𝑗𝒟_𝑗10𝒟^𝑇_𝑗1𝒞_𝑗−𝛾²𝐼+𝒟^𝑇_𝑗1𝒟_𝑗1. (53) From (52), we have 𝒜_𝑗ℬ_𝑗1𝐼00𝐼^𝑇ℳ_𝑏𝑗𝒜_𝑗ℬ_𝑗1𝐼00𝐼<0. (54) On the other hand, 00𝐼^𝑇ℳ_𝑏𝑗00𝐼=−𝛾²𝐼+𝒟^𝑇_𝑗1𝒟_𝑗1<0. (55) Based on Projection Lemma, it follows from (54) and (55) that ℳ_𝑏𝑗+He𝐴−𝐼_𝑇𝑗𝐵^𝑇_𝑗1𝒲_𝑏𝑗𝐼000𝐼0<0, (56) where 𝒲_𝑏𝑗 introduced by Projection Lemma is the matrix variable of appropriate dimensions. Partition 𝒲_𝑏𝑗 as 𝒲_𝑏𝑗=[𝒲_𝑏𝑗1𝒲_𝑏𝑗2]. By the Schur complement, (56) is equivalent to−He𝒲_𝑏𝑗1𝒫_𝑏𝑗−𝒲_𝑏𝑗2+𝒲^𝑇_𝑏𝑗1𝒜_𝑗𝒲^𝑇_𝑏𝑗1ℬ_𝑗100∗𝛼_𝑗𝒫_𝑏𝑗+𝜀𝑔_𝑗𝐼+He𝒜_𝑗𝒲_𝑏𝑗2𝒲^𝑇_𝑏𝑗2ℬ_𝑗1𝒫_𝑏𝑗𝒞^𝑇_𝑗∗∗−𝛾²𝐼0𝒟^𝑇_𝑗1∗∗∗−𝜀⁻¹𝐼0∗∗∗∗−1<0. (57)Then pre- and postmultiply diag{𝐼,𝐼,𝐼,𝐼,𝑅^𝑇_𝑗} to (57); one obtains−He𝒲_𝑏𝑗1𝒫_𝑏𝑗−𝒲_𝑏𝑗2+𝒲^𝑇_𝑏𝑗1𝒜_𝑗𝒲^𝑇_𝑏𝑗1ℬ_𝑗100∗𝛼_𝑗𝒫_𝑏𝑗+𝜀𝑔_𝑗𝐼+He𝒜^𝑇_𝑗𝒲_𝑏𝑗2𝒲^𝑇_𝑏𝑗2ℬ_𝑗1𝒫_𝑏𝑗𝒞^𝑇_𝑗𝑅_𝑗∗∗−𝛾²𝐼0𝒟^𝑇_𝑗1𝑅_𝑗∗∗∗−𝜀⁻¹𝐼0∗∗∗∗−𝑅^𝑇_𝑗𝑅_𝑗<0. (58)we partition 𝒲_𝑏𝑗1, 𝒲_𝑏𝑗2, respectively, as 𝒲_𝑏𝑗1=𝑋_𝑏𝑗𝑌_𝑏𝑗𝑎₁𝑁_𝑗𝑎₂𝑁_𝑗,𝒲_𝑏𝑗2=𝑀_𝑏𝑗𝑆_𝑏𝑗𝑎₃𝑁_𝑗𝑎₄𝑁_𝑗 (59) and 𝑏₁∼𝑏₄ are set to be fixed parameters. Define 𝐴_𝑓𝑗=𝑁^𝑇_𝑗𝐴_𝑓𝑗, 𝐵_𝑓𝑗=𝑁^𝑇_𝑗𝐵_𝑓𝑗, 𝐶_𝑓𝑗=𝑅^𝑇_𝑗𝐶_𝑓𝑗, and 𝐷_𝑓𝑗=𝑅^𝑇_𝑗𝐷_𝑓𝑗; −𝑅^𝑇_𝑗𝑅_𝑗≤𝐼−He(𝑅_𝑗) then (58) becomes (49). Hence if the condition (30) holds, nonlinear impulsive switched system (41) is stable and guarantees the 𝐻_∞ performance (8), which completes the proof.

Remark 13. Scalars 𝑏₁∼𝑏₄ are used to reduce conservatism for the same variable 𝑁_𝑗 in matrix variables 𝒲_𝑏𝑗1 and 𝒲_𝑏𝑗2. When we set 𝑏₁∼𝑏₄ as the fixed parameters, the conditions in Theorem 12 become convex.

3.3. Algorithm

In the previous sections, Theorems 9 and 12 have formulated the inequality conditions for the performances (8)-(9), respectively. Summarily, we have the following algorithm.

It is noted that conditions (30) and (49) are all convex. Hence, the problem of FD filter design can directly translate into the following optimization problem: ()()max𝛽s.t.30and49,𝑗∈𝒩. (60)

Remark 14. The 𝐻₋ performance and 𝐻_∞ performance are considered to describe the fault sensitiveness performance and the disturbance attenuation performance, respectively. Due to the multiobjective optimization problem, each performance has some LMI conditions. However, by analyzing the theorems and their proofs, it can be discovered that 𝒲_𝑎𝑗1, 𝒲_𝑎𝑗2 in Theorem 9 and 𝒲_𝑏𝑗1, 𝒲_𝑏𝑗2 in Theorem 12 can be defined as the same variable to decrease the computational complexity.

Finally, the gain matrices 𝐴_𝑓𝑗, 𝐵_𝑓𝑗, 𝐶_𝑓𝑗, and 𝐷_𝑓𝑗 of the filters can be derived as 𝐴_𝑓𝑗=𝑁_𝑗^−𝑇𝐴_𝑓𝑗,𝐵_𝑓𝑗=𝑁_𝑗^−𝑇𝐵_𝑓𝑗,𝐶_𝑓𝑗=𝑅_𝑗^−𝑇𝐶_𝑓𝑗,𝐷_𝑓𝑗=𝑅_𝑗^−𝑇𝐷_𝑓𝑗. (61)

4. Thresholds Computation

After the gain matrices of the filters 𝐴_𝑓𝑗, 𝐵_𝑓𝑗, 𝐶_𝑓𝑗, and 𝐷_𝑓𝑗 are designed, similarly to that proposed in [32], the residual evaluation function 𝐽_𝑟(𝜏) can be chosen as 𝐽_𝑟(𝜏)=1𝜏^𝑠𝑘_𝑠𝑙𝑟^𝑇(𝑠)𝑟(𝑠)d𝑠, (62) where 𝐽_𝑟(𝜏) is square value which means the average energy of residual signal over a time interval (𝑠_𝑙,𝑠_𝑘), 𝑠_𝑙 denotes the initial evaluation time instant, and 𝑠_𝑘 stands for the evaluation time.

We propose to use the following threshold: 𝐽_th=sup_{𝑑(𝑡)∈𝑙2[)0,∞𝑓(𝑡)=0,𝑗∈𝒩}𝐽_𝑟. (63) Consequently, the occurrence of faults can be detected by the following logic rule: 𝐽_𝑟(𝜏)≤𝐽_th𝐽⟹noalarm_𝑟(𝜏)>𝐽_th⟹alarm. (64)

5. Examples

In this section, we present examples derived from a liquid level control system to illustrate the effectiveness of FD design approach.

Consider a continuous stirred tank reactor (CSTR) control system [33] shown in Figure 1. It consists of a constant-volume CSTR fed by a single inlet stream through a selector valve which is connected to two different source streams.

Figure 1 

Schematic diagram of the process.

Assuming constant liquid volume, negligible heat losses, perfectly mixing, and a first-order reaction in reactant 𝐴, the dynamical equations of the CSTR at each operating mode are described by 𝐶̇_𝐴=𝑞_𝑖𝑉𝐶_𝐴𝑓𝑖−𝐶_𝐴−𝑎₀−𝐸exp𝐶𝑅𝑇^𝐴,𝑇̇=𝑞_𝑖𝑉𝑇_𝑓𝑖−𝑇−𝑎₁−𝐸exp𝐶𝑅𝑇^𝐴+𝑎₂𝑇_𝑎−𝑇, (65) where 𝐶_𝐴 is the reactant 𝐴 concentration (moL/L), 𝑞_𝑖 is the 𝑖th mode’s feed flow rate (L/min), 𝐶_𝐴𝑓𝑖 is the 𝑖th mode’s concentration of the feed stream (moL/L), 𝑉 is the volume of the reactor (𝐿), 𝐸 is the activation energy, 𝑅 is the gas constant (J/(moL·K)), 𝑇 is the reactor temperature (K), 𝑇_𝑓𝑖 is the 𝑖th mode’s reactor temperature of the feed stream (𝐾), and 𝑇_𝑎 is the coolant temperature (K). These nominal values of the parameters are described in [34], and we choose the nominal operating conditions corresponding to an unstable equilibrium point as 𝑇^∗_𝐴=300K, 𝐶^∗_𝐴=0.5moL/L, and 𝑇^∗=350K for both modes.

Since the objective in the example is to testify the FD filter design techniques for the switched system and sustain the main theoretical results, we make design as [33] and the closed-loop system can be obtained with matrices𝐴_𝑎1=0.51−3−3,𝐴_𝑎2=21−8−3. (66) Suppose other system matrices in (1) as 𝐵₁₁=0.24−0.10,𝐵₁₂=,𝐵0.95−1.29₂₁=0.1−0.31,𝐵₂₂=,Υ−0.521.34₁((𝑡=𝑥𝑡,𝑥))sin₁(𝑡)𝑒^−0.05𝑡0,Υ₂((𝑡=𝑥𝑡,𝑥))sin₁(𝑡)𝑒^−0.05𝑡sin𝑥₂(𝑡)𝑒^−0.05𝑡,𝐻₁=𝐻₂,Ω=−0.58000.58₁(𝑡,𝑥(𝑡))=Ω₂((𝑡=𝑥𝑡,𝑥))0.3sin₁𝑡_𝑖0.3sin𝑥₂𝑡_𝑖,𝐶₁=𝐶₂=[1,1],𝐷₁₁=𝐷₂₁𝐷=0.1,₁₂=𝐷₁₂=0.94. (67) Let 𝑞₁=𝑞₃=−1, 𝑞₂=𝑞₄=1, 𝑞∈{𝑎,𝑏}, 𝑣_𝑗=[1,1]^𝑇, 𝑗∈𝒩. By giving 𝛾=1, 𝛼_𝑗=0.5, and 𝜀=0.02and obtaining 𝑔_𝑗=1, 𝜌_𝑗=0.32, we solve the convex optimization problem (60) and get the optimal sensitivity performance gain 𝛽=1.0884. The gain matrices of the fault detection filters and the matrix in the Lyapunov function are 𝐴_𝑓1𝐵_𝑓1𝐶_𝑓1𝐷_𝑓1=−0.9011−0.26861.2677−0.3316−0.7128−1.52270.2561−0.4982,𝐴3.2893_𝑓2𝐵_𝑓2𝐶_𝑓2𝐷_𝑓2=2.90381.6812−0.6576−10.1086−4.66031.6406−1.9562−1.3863,𝒫1.4977₁=,𝒫35.267624.1501−6.6667−1.726024.150122.2633−3.1844−4.7123−6.6667−3.184419.791211.8843−1.7260−4.712311.884312.3490₂=.28.415110.2415−22.8368−8.196910.24154.8161−9.0122−4.0940−22.8368−9.012224.47909.1073−8.1969−4.09409.10734.1803 (68) Then by (13), the dwell time for each subsystem is obtained with 𝛿=1: dwelltimeforsubsystem1⟹𝜏₁>ln𝜇_12𝑘1𝛼₁=9.6813,dwelltimeforsubsystem2⟹𝜏₂>ln𝜇_21𝑘2𝛼₂𝑘=13.8568₁,𝑘₂∈{1,2,…},𝑘₁≠𝑘₂. (69)

To illustrate the simulation results of the FD objective, the two cases which include the fault for subsystem 1 and subsystem 2, respectively, are considered. The disturbance is assumed to be 𝑑(𝑡)=0.8cos(0.53𝑡)exp(−0.05𝑡). The switching signal in this paper is shown in Figure 2.

Figure 2 

Switch signal.

Case 1. The fault for subsystem 1 with the unit amplitude occurs from 100 sec to 200 sec. The generated residual 𝑟(𝑡) and the evolution of the residual evaluation function 𝐽_𝑟(𝑡) are shown in Figure 3. The simulation results show that when the fault for subsystem 1 occurs, the residual signal varies sharply, and 𝐽_𝑟(𝑡)>𝐽_th at 109 sec, which means that the fault for subsystem 1 can be detected 9 sec after the fault of subsystem 1 occurs. Hence, the fault for subsystem 1 can be detected.

(a)

(b)

Figure 3 

(a) The residual signal. (b) The residual evaluation function and threshold when the fault for subsystem 1 occurs from 100 sec to 200 sec.

Case 2. The fault for subsystem 2 with the unit amplitude also occurs from 200 sec to 300 sec. The generated residual 𝑟(𝑡) and the evolution of the residual evaluation function 𝐽_𝑟(𝑡) are shown in Figure 4. It can be seen that when subsystem 2 are activated at 210 sec, the residual signal is changed sharply and 𝐽_𝑟(𝑡)>𝐽_th at 234 sec. Thus, the fault for subsystem 2 can be detected.

(a)

(b)

Figure 4 

(a) The residual signal. (b) The residual evaluation function and threshold when the fault for subsystem 2 occurs from 200 sec to 300 sec.

From Cases 1 and 2, we see that both the faults for subsystem 1 and subsystem 2, respectively, can be detected, and they can demonstrate the effectiveness of the proposed design method.

6. Conclusion

In this paper, the problem of FD filter design for nonlinear impulsive switched systems has been investigated. Firstly, the 𝐻₋ performance and the 𝐻_∞ performance are presented for nonlinear impulsive switched systems, and sufficient conditions to characterize given performances have been obtained. Subsequently, the design of FD filters is formulated as a multiobjective optimization problem, and the filter gains are characterized in term of the solution of LMI conditions. Finally, an example has been given to illustrate the effectiveness of the proposed method.

Future works on fault detection for nonlinear impulsive switched systems are to focus on how to cope with stochastic switched systems as in [35, 36] and switched systems with time delay as in [37–39].