Fault Detection of Markov Jumping Linear Systems

Abstract

In this paper, the fault detection (FD) problems of discrete-time Markov jumping linear systems (MJLSs) are studied. We first focus on the stationary MJLS. The proposed FD system consists of two steps: residual generation and residual evaluation. A new reference model strategy is applied to construct a residual generator, such that it is robust against disturbances and sensitive to system faults. The generated residual signals are then evaluated according to their stochastic properties, and a threshold is computed for detecting the occurrences of faults. The upper bound of the corresponding false alarm rate (FAR) is also given. For the nonstationary MJLS, similar results are also obtained. All the solutions are presented in the form of linear matrix inequalities (LMIs). Finally, a numerical example is used to illustrate the results.

1. Introduction

The complexity and automation degree of modern technical systems are continuously increasing. In order to guarantee the system safety and reliability, the model-based fault detection and isolation (FDI) technology has been developed since the early 1970s [1–3]. It is fully integrated into many industrial processes and automatic control systems, which can provide valuable information about system faults. The basic idea of model-based FDI is to generate a residual signal based on the system model and measurements and to determine a residual evaluation function to compare with a threshold regarding to all possible model uncertainties and unknown inputs. Exceeding the threshold indicates a fault in the system. Hence the FDI technology consists of two steps: residual generation and residual evaluation. As mentioned in the survey papers [4, 5] and book [2], the robustness issues have been studied in the purpose of designing of the FDI system under a cost function that expresses a tradeoff between the system robustness against unknown inputs as well as model uncertainties and the system sensitivity to faults. A unified solution for the fault detection system design was presented in [5, 6] for continuous and discrete time linear systems, respectively. In [7, 8], an 𝐻_∞-filtering formulation of the robust FDI design for uncertain system has been proposed and widely accepted, which tries to make the difference between residual and so-called weighted fault as small as possible by considering model uncertainties and unknown inputs.

Markov jump linear systems have also attracted a great deal of attention. The MJLS is one of the hybrid systems in which a state takes values in a countably finite set, referred to as the state. It can be used to represent a class of linear systems subjects to abrupt changes in their structures due to random components failures, repairs, sudden environment disturbance, change of the operation point of a linearized model of nonlinear systems, for example, electric power systems, aircraft flight control and especially networked control systems (for reasonable model of the packet delivery characteristic in communication channels). There are many theoretical works contributed in the field of MJLS. In [9, 10] the results on stability of MJLS were presented. The linear quadratic Gaussian control problem was studied in [11, 12]. The bounded real lemma for MJLS has been fully developed by [13] in the form of LMIs. The 𝐻_∞-control problems were discussed in [14, 15], where a controller stabilizing a linear system ensures that the 𝑙₂ induced norm from unknown inputs to the outputs is bounded. In [16, 17], the 𝐻_∞ filtering for MJLS was studied as the dual problem of control. For networked systems, recently there were also many new results obtained by applying the MJLS theorem, for example, in [18–20].

Although there is intensive research in FD and MJLS, the design of FD system for MJLS has just begun. In [21], the packet loss in networked control systems was modeled as Markov process, and an FD system has been designed in terms of LMIs. In [22], the fault detection system over noisy communication channels was also described via a MJLS. In [23], the design of FD for MJLS was formulated as a 𝐻_∞-filtering problem. In those works, observer-based fault detection filters were designed to minimize the influence of unknown inputs [21, 22] or to minimize the difference between residual signals and (weighted) faults [23], and the threshold is simply computed based on the expectation of the norm of residual signals. However, by only considering the unknown inputs or faults in the design, the residual generator usually cannot achieve an optimal performance in sense of the tradeoff between the system robustness and the fault sensitivity [2]. Besides, further statistic properties of residual signals were not analyzed and considered in the system design, and a proper residual evaluation scheme for MJLS is still missing to the best of the authors’ knowledge. We will show that the usual norm-based residual evaluation method, in which the norm of residual signals is used as evaluation function and compared with the threshold, cannot be directly applied in FD of MJLS. Without taking the variance of residual signals into account, the evaluation will result in a possible high FAR.

In this paper, the fault detection system design of MJLS is formulated as a set of optimization problems. The residual generator is designed to stochastically match an deterministic reference residual model which can achieve an optimal tradeoff between the system robustness and the fault sensitivity. The reference residual model is selected according to the statistic properties of the MJLS. For the stationary MJLS, in which the distributions of Markov state at different time instances remain the same, a new bounded real lemma is also derived, and the constraints of the expectation and variance of residual signals are considered in the residual generator design. The stochastic model matching problem is then solved by optimizing the 𝐻_∞-norm of an MJLS in terms of LMIs. In the residual evaluation, the absolute value of each residual signals is selected as the evaluation functions, and the corresponding threshold is computed by considering their expectations and variances. Those statistic properties of evaluated residual signals are calculated with the help of LMIs based on convex optimization problems. An upper bound of the FAR is also derived for this evaluation scheme. The FD problem of nonstationary MJLS is then addressed in a similar way. Finally a numerical example is given to illustrate the feasibility and effectiveness of the proposed design approach.

The rest of this paper is organized as follows. The problem of fault detection of MJLS is formulated in Section 2. Section 3 presents the design of residual generator and the new residual evaluation approach. In Section 4 the FD system design for nonstationary MJLS is discussed. An example is given in Section 5 and we conclude the paper in Section 6.

Notation. The notation used throughout the paper is fairly standard. The superscript “T” stands for transposition. 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space and the notation 𝑆>0 means that 𝑆 is real symmetric and positive definite. diag stands for a block-diagonal matrix. 𝐸[𝜗] means the expectation of 𝜗. For convenience we denote 𝜗=𝐸[𝜍]. The notation [⋅]_𝑗 is used to represent the 𝑗th row of a matrix or a vector. When 𝑤(𝑘)∈𝑙₂ space, its norm is given by ‖𝑤(𝑘)‖₂=^∞_𝑖=0𝑤(𝑘)^𝑇𝑤(𝑘) (1.1) and if 𝑤(𝑘) is a stochastic value and its expectation is in 𝑙₂ space, then we denote ‖𝑤(𝑘)‖_𝐸=^∞_𝑖=0𝐸𝑤(𝑘)^𝑇𝑤(𝑘). (1.2)

2. Problem Formulation

The considered MJLS is defined as follows:𝑥(𝑘+1)=𝐴𝜃_𝑘𝑥(𝑘)+𝐵𝜃_𝑘𝑢(𝑘)+𝐸_𝑑𝜃_𝑘𝑑(𝑘)+𝐸_𝑓𝜃_𝑘𝑓(𝑘),𝑦(𝑘)=𝐶𝜃_𝑘𝑥(𝑘)+𝐷𝜃_𝑘𝑢(𝑘)+𝐹_𝑑𝜃_𝑘𝑑(𝑘)+𝐹_𝑓𝜃_𝑘𝑓(𝑘), (2.1) where 𝑥∈𝐑^𝑛 denotes the state vector, 𝑢∈𝐑^𝑝 denotes the control inputs, 𝑦∈𝐑^𝑚 denotes the measured output vector, 𝑑∈𝐑^𝑛𝑑 denotes the unknown inputs, and 𝑓∈𝐑^𝑛𝑓 is the fault to be detected. {𝜃_𝑘} is a discrete homogeneous Markov chain taking values in a finite state space 𝜓={1,2,…,𝑁} with transition probability matrix Φ=[𝜆_𝑖𝑗]_{𝑖,𝑗∈𝜓}, and 𝜆_𝑖𝑗 is defined as𝜆_𝑖𝑗=Pr𝜃(𝑘+1)=𝑗∣𝜃(𝑘)=𝑖 (2.2) which are subjected to the restriction 𝜆_𝑖𝑗≥0, ∑^𝑁_𝑗=1𝜆_𝑖𝑗=1 for any 𝑖∈𝜓. For notation, define Θ(𝑘)∶={𝜃₁,…,𝜃_𝑘} as a sample of possible sequences of Markov states and 𝑃(𝑘) as the vector of state probabilities𝑃(𝑘)=𝑝₁(𝑘)𝑝₂(𝑘)⋯𝑝_𝑁(𝑘)^𝑇,𝑝_𝑖(𝑘)=Prob𝜃_𝑘=𝑖,𝑖∈𝜓. (2.3)

In this paper we consider the Markov chain with the following assumption.(A1){𝜃_𝑘} is homogeneous and 𝜆_𝑖𝑗>0.

This assumption means that, in such a Markov chain, it is possible to get to any state from any state.

2.1. Residual Generator Design

Residual generation is the first step of FD. We propose the following residual generator for the system (2.1):𝑥(𝑘+1)=𝐴𝜃_𝑘𝑥(𝑘)+𝐵𝜃_𝑘𝑢(𝑘)+𝐿𝜃_𝑘𝑦(𝑘)(𝑘),(𝑘)𝜃−𝑦𝑦=𝐶_𝑘𝑥(𝑘)+𝐷𝜃_𝑘𝑢(𝑘),𝑟(𝑘)=𝑊𝜃_𝑘𝑦(𝑘)−𝑦(𝑘), (2.4) where 𝑥(𝑘) and 𝑦(𝑘) are the estimated state vector and output vector, respectively. 𝑟(𝑘) is the residual vector. The matrices 𝐿(𝜃_𝑘) and 𝑊(𝜃_𝑘) are to be designed. For the convenience, we denote the matrices associated with 𝜃_𝑘=𝑖∈𝜓 by𝐴_𝑖=𝐴𝜃_𝑘,𝐵_𝑖=𝐵𝜃_𝑘,𝐸_𝑑,𝑖=𝐸_𝑑𝜃_𝑘,𝐸_𝑓,𝑖=𝐸_𝑓𝜃_𝑘,𝐶_𝑖=𝐶𝜃_𝑘,𝐷_𝑖=𝐷𝜃_𝑘,𝐹_𝑑,𝑖=𝐹_𝑑𝜃_𝑘,𝐹_𝑓,𝑖=𝐹_𝑓𝜃_𝑘,𝐿_𝑖=𝐿𝜃_𝑘,𝑊_𝑖=𝑊𝜃_𝑘. (2.5)

With 𝑒(𝑘)=𝑥(𝑘)−𝑥(𝑘), the residual dynamics of (2.4) can be written as𝑒(𝑘+1)=𝐴_𝐿𝜃_𝑘𝑒(𝑘)+𝐸_𝑑,𝐿𝜃_𝑘𝑑(𝑘)+𝐸_𝑓,𝐿𝜃_𝑘𝑓(𝑘),𝑟(𝑘)=𝑊𝜃_𝑘𝐶𝜃_𝑘𝑒(𝑘)+𝐹_𝑑𝜃_𝑘𝑑(𝑘)+𝐹_𝑓𝜃_𝑘𝑓(𝑘) (2.6) with𝐴_𝐿𝜃_𝑘=𝐴𝜃_𝑘−𝐿𝜃_𝑘𝐶𝜃_𝑘,𝐸_𝑑,𝐿𝜃_𝑘=𝐸_𝑑𝜃_𝑘−𝐿𝜃_𝑘𝐹_𝑑𝜃_𝑘,𝐸_𝑓,𝐿𝜃_𝑘=𝐸_𝑓𝜃_𝑘−𝐿𝜃_𝑘𝐹_𝑓𝜃_𝑘. (2.7)

The objective of the design is to generate the residual signals which is robust to unknown inputs and sensitive to faults. It is clear that (2.7) itself can be an MJLS instead of a deterministic one. The Markov state 𝜃_𝑘 is not known as a perquisite, while only its probability is known. Hence we propose a reference residual model which achieves an optimal tradeoff between system robustness and fault sensitivity, and then the residual generator is designed to match the reference residual model. In this approach the matrices 𝐿(𝜃_𝑘) and 𝑊(𝜃_𝑘) in (2.4) should be selected such thatmin_{𝐿(𝜃𝑘),𝑊(𝜃𝑘)}sup_𝑓,𝑑𝑟_ref−𝑟_{𝐸𝑑𝑓2} (2.8) subject to𝑟_ref−𝑟−𝐸𝑟_ref−𝑟²_𝐸=𝐸^∞_𝑘=0(𝑟(𝑘)−𝑟(𝑘))^𝑇(𝑟(𝑘)−𝑟(𝑘))<𝛼², (2.9) where 𝛼>0 and 𝑟_ref denotes the residual vector of a reference residual model in the form of𝑒_ref(𝑘+1)=𝐴_ref−𝐿_𝑜𝐶_ref𝑒_ref(𝑘)+𝐸_𝑑,ref−𝐿_𝑜𝐹_𝑑,ref𝑑(𝑘)+𝐸_𝑓,ref−𝐿_𝑜𝐹_𝑓,ref𝑓(𝑘),𝑟_ref(𝑘)=𝑊_𝑜𝐶_ref𝑒_ref(𝑘)+𝑊_𝑜𝐹_𝑑,ref𝑑(𝑘)+𝑊_𝑜𝐹_𝑓,ref𝑓(𝑘). (2.10) Here 𝐿_𝑜 and 𝑊_𝑜 are chosen by applying the unified solution proposed in [6], such thatsup_𝑓,𝑑𝑟_ref2/‖𝑑‖₂𝑟_ref2/𝑓₂. (2.11) Since 𝑟(𝑘) is stochastic vector, usually the expectation, ‖𝑟_ref−𝑟‖_𝐸, is not enough to characterize its behavior. Hence the constraint (2.9) is applied to ensure that the summation of variances of each residual signal is bounded with an expected value 𝛼². In this approach the expectation and variance of 𝑟(𝑘) are both considered in the model matching problem for MJLS, such that the stochastic 𝑟(𝑘) could approach a deterministic 𝑟_ref(𝑘).

Remark 2.1. A significant difference between the reference model for the purpose of FD adopted here and the one in most of the literatures is that unknown inputs 𝑑(𝑘) are included in our model, such that an optimal tradeoff between system robustness against unknown inputs and sensitivity to faults can be achieved. Simply reducing the influence of 𝑑(𝑘) or increasing the sensitivity to system faults 𝑓(𝑘) does not automatically lead to an optimal tradeoff between system robustness and fault sensitivity. For instance, with a residual generator which decouples 𝑑(𝑘) from 𝑟(𝑘), some 𝑓(𝑘) may also be decoupled from 𝑟(𝑘) and thus cannot be detected. Hence it is necessary to take unknown inputs into account in the reference model.

It is well known that Φ^𝑘⟶constant,𝑃(𝑘)⟶𝑃(∞),when𝑘⟶∞, (2.12) where 𝑃(∞) is a unique constant vector called the stationary state distribution of a Markov chain with the assumption (A1), and 𝑃_∞ can be computed according to𝑃_∞=Φ𝑃_∞. (2.13) This means that as time goes by, the Markov chain forgets its initial condition and converges to its stationary distribution. Hence it is reasonable to set the following matrices:𝐴_ref=^𝑁_𝑖=1𝐴_𝑖𝑝_𝑖(∞),𝐸_𝑑,ref=^𝑁_𝑖=1𝐸_𝑑,𝑖𝑝_𝑖(∞),𝐸_𝑓,ref=^𝑁_𝑖=1𝐸_𝑓,𝑖𝑝_𝑖(∞),𝐶_ref=^𝑁_𝑖=1𝐶_𝑖𝑝_𝑖(∞),𝐹_𝑑,ref=^𝑁_𝑖=1𝐹_𝑑,𝑖𝑝_𝑖(∞),𝐹_𝑓,ref=^𝑁_𝑖=1𝐹_𝑓,𝑖𝑝_𝑖(∞) (2.14) so that the reference residual model describes the optimal stationary expected behavior of the MJLS (2.7).

In fact, the MJLS (2.7) consists of two groups of system state: 𝑒(𝑘) and 𝜃_𝑘. In the robust control system design, the initial condition of system state is usually assumed to be zeros (so-called zero initial condition). Comparably, we make three assumptions in MJLS for the rest of the paper:(A2)𝑃(0)=𝑃_∞,(A3)𝑒(0) is deterministic and 𝑒(0)=0,(A4)𝜃_𝑘 is independent of 𝑒(0).

Assumption (A2) can be regarded as the “zeros” initial condition of 𝜃₀. It is worthy to mention that, with assumption (A2), we have 𝑃(0)=𝑃(1)=⋯=𝑃_∞. That means the state probabilities is independent of time, that is, the Markov chain is in the stationary state. We call MJLSs with assumption (2.4) as stationary MJLSs. For the FD purpose, we can generally assume that the MJLS under consideration is operating in its stationary state before a fault occurs. We denote 𝑝_𝑖(𝑘)=𝑝_𝑖 for all 𝑘 for later use.

The problem of residual generator design of a stationary MJLS is summarized as follows.

Problem RGFD [Residual Generator for Fault Detection]
Consider system (2.1). Given the reference residual model (2.10), determine the matrices 𝐿_𝑖 and 𝑊_𝑖, 𝑖∈𝜓 of the residual generator in the form of (2.4) under assumption (A1)–(A4), such that the residual dynamics described by (2.7) satisfy (2.8) and (2.9).

2.2. Residual Evaluation Design

The second step in FD is the evaluation of residual signals, where 𝑟(𝑘) is evaluated and compared with a threshold. The following logic is applied to detect the occurrences of faults‖𝑟‖_𝑒≤𝐽_th‖𝑟‖⟹fault-free,_𝑒>𝐽_th⟹faultalarm, (2.15) where ‖𝑟‖_𝑒 denotes the evaluated residual signals and 𝐽_th denotes a threshold. The residual evaluation problem of deterministic systems has been extensively studied. One important evaluation strategy is the so-called norm-based residual evaluation [2]. Assume the dynamics of a residual generator for some deterministic systems are governed by the following time invariant system𝔢(𝑘+1)=𝔄𝔢(𝑘)+𝔈_𝔡𝔡(𝑘)+𝔈_𝔣𝔣(𝑘),𝔯(𝑘)=ℭ𝔢(𝑘)+𝔉_𝔡𝔡(𝑘)+𝔉_𝔣𝔣(𝑘), (2.16) where 𝔡(𝑘) is unknown inputs, and 𝔣(𝑘) the system faults. Then the residual evaluation function is chosen as‖𝑟(𝑘)‖_𝑒=^𝑘_{𝑗=𝑘−s+1}𝑟(𝑘)^𝑇𝑟(𝑘) (2.17) with 𝑠 being the length of the evaluation window. It is clear that the residual signals 𝔯(𝑘) are corrupted with 𝔡(𝑘). Hence threshold is set to distinguish the faults from the unknown inputs. As widely accepted in the literatures, the threshold is set as𝐽_th=𝛾‖𝔡(𝑘)‖₂,𝛾>sup_𝔡∈𝑙2‖𝔯‖₂‖𝔡‖₂, (2.18) such that false fault alarms can be prevented and meanwhile missing detection of faults can be reduced as much as possible.

The residual signals of MJLS are stochastic values. Their statistic properties are associated with the Markov process, and they are determined by the dynamics of the system (2.4). It is possible to compute 𝛾 such that𝛾>sup_𝜂∈𝑙2‖𝑟(𝑘)‖_𝐸𝜂(𝑘)₂ (2.19) in a similar way as stated in [13] for deterministic systems and to set 𝐽_th=𝛾‖𝜂(𝑘)‖₂ as in many literatures. In this case only the expectation of the 𝑙₂-norm of 𝑟(𝑘) is considered for the computation of 𝐽_th. Due to the variance of 𝑟(𝑘), there could be false fault alarms and the probability of those false alarms (also called false alarm rate (FAR)) is not known. Even a fault is detected, it is difficult to say with how large percentages the fault alarm is correct. Therefore, we apply the residual evaluation approach proposed in [24]. Define a set of residual evaluation functions as follows:𝑟_𝑗(𝑘)_𝑒=∣𝑟_𝑗(𝑘)∣, (2.20) where 𝑗=1,…,𝑚 and 𝑟_𝑗(𝑘) is the 𝑗th measurement. That means the absolute value of each residual signal is selected as the evaluation function. For the evaluation function (2.20), we suggest the following threshold: 𝐽_𝑗,th=sup_𝑘∣𝑟_𝑗(𝑘)∣+𝛽sup_𝑘𝜎_𝑗(𝑟(𝑘)), (2.21) where 𝑟_𝑗(𝑘) is the absolute value of the expectation of 𝑟_𝑗(𝑘) and𝜎²_𝑗(𝑟(𝑘))=𝐸𝑟_𝑗(𝑘)−𝑟_𝑗(𝑘)^𝑇𝑟_𝑗(𝑘)−𝑟_𝑗(𝑘) (2.22) is its variance, where 𝛽>0 is some constant.

The residual evaluation problem is summarized as follows.

Problem REFD [Residual Evaluation for Fault Detection]
Consider system (2.1). Given the residual generator (2.4) and assumptions (A1)–(A4), determine the threshold 𝐽_𝑗,th, 𝑗=1,…,𝑚 for each residual signal, that is, sup_𝑘(∣𝑟_𝑗(𝑘)∣), sup_𝑘(𝜎_𝑗(𝑟(𝑘))) and 𝛽>0, such that FAR is smaller than a given constant.

3. Main Results

3.1. Residual Generation

In this section the dynamics of 𝑟(𝑘)−𝑟_ref(𝑘) are described at first. Then the existing bounded real lemma (BRL) for MJLS is reviewed, and a new BRL is derived for MJLS under assumption (A2). Based on those lemmas, the solution of RGFD is presented.

According to (2.4) and (2.10), the dynamics of 𝑟(𝑘)−𝑟_ref(𝑘) can be written as𝑥_𝑜(𝑘+1)=𝐴_𝑜𝜃_𝑘𝑥_𝑜(𝑘)+𝐸_𝑜𝜃_𝑘𝑑(𝑘),𝑟(𝑘)−𝑟_ref(𝑘)=𝐶_𝑜𝜃_𝑘𝑥_𝑜(𝑘)+𝐹_𝑜𝜃_𝑘𝑑(𝑘), (3.1) where𝑥_𝑜=𝑒𝑒_ref,𝑑𝑓𝑑=, (3.2) and we denote the matrices associated with 𝜃_𝑘=𝑖∈𝜓 by𝐴_𝑜,𝑖=𝐴_𝑖−𝐿_𝑖𝐶_𝑖00𝐴_ref−𝐿_𝑜𝐶_ref,𝐶_𝑜,𝑖=𝑊_𝑖𝐶_𝑖−𝑊_𝑜𝐶_ref,𝐸_𝑜,𝑖=𝐸_𝑑,𝑖−𝐿_𝑖𝐹_𝑑,𝑖𝐸_𝑓,𝑖−𝐿_𝑖𝐹_𝑓,𝑖𝐸_𝑑,ref−𝐿_𝑜𝐹_𝑑,ref𝐸_𝑓,ref−𝐿_𝑜𝐹_𝑓,ref,𝐹_𝑜,𝑖=𝑊_𝑖𝐹_𝑑,𝑖−𝑊_𝑜𝐹_𝑑,ref^𝑇𝑊_𝑖𝐹_𝑓,𝑖−𝑊_𝑜𝐹_𝑑,ref^𝑇𝑇. (3.3)

Before giving the solution, we first introduce the following useful lemmas. The first one is the standard BRL for MJLS given in [13].

Lemma 3.1. Consider the system 𝑥(𝑘+1)=𝐴𝜃_𝑘𝑥(𝑘)+𝐵𝜃_𝑘𝑑(𝑘),𝑦(𝑘)=𝐶𝜃_𝑘𝑥(𝑘)+𝐷𝜃_𝑘𝑑(𝑘), (3.4) for 𝑘=0,1,…, where 𝑥(𝑘), 𝑦(𝑘), 𝐴(𝜃_𝑘), 𝐵(𝜃_𝑘), 𝐶(𝜃_𝑘), 𝐷(𝜃_𝑘), and 𝜃_𝑘 are defined as in (2.1), 𝑑(𝑘)∈𝐑^𝑚 is the 𝑙₂-norm bounded input sequence. Given a constant 𝛾>0,  𝑥(0)=0 and any 𝜃₀∈Ψ, then sup_𝑓,𝑑𝐸𝑦₂‖𝑑‖₂<𝛾 (3.5) if there exist 𝑄_𝑖>0, 𝑖∈𝜓 satisfying the following LMIs: 𝐴_𝑖𝐵_𝑖𝐶_𝑖𝐷_𝑖^𝑇𝑄_𝑖0𝐴0𝐼^𝑖𝐵_𝑖𝐶_𝑖𝐷_𝑖−𝑄_𝑖00𝛾²𝐼<0, (3.6) where 𝑄_𝑖=^𝑁_𝑗=1𝜆_𝑖𝑗𝑄_𝑗. (3.7)

Proof. Here we give a short proof which is slightly different from the one in [13]. Define the function 𝑉(𝑖,𝑘)=𝑥(𝑘)^𝑇𝑄_𝑖𝑥(𝑘). (3.8) Given 𝑥(0)=0, 𝑉(𝜃₀,0)=0 for any initial mode 𝜃₀∈Ψ, we have ^∞_𝑘=0𝐸𝑉𝜃_𝑘+1,𝑘+1−𝑉𝜃_𝑘,𝑘=𝐸𝜃_∞,∞. (3.9) Then, 𝐸‖𝑟(𝑘)‖²−𝛾²‖𝑑(𝑘)‖₂≤𝐸^∞_𝑘=0𝑟(𝑘)^𝑇𝑟(𝑘)−𝛾²𝑑(𝑘)^𝑇𝑑(𝑘)+𝑉𝜃_𝑘+1,𝑘+1−𝑉𝜃_𝑘=,𝑘_∞𝑘=0𝐸𝑒(𝑘)𝑑(𝑘)^𝑇𝑅𝜃_𝑘𝑒(𝑘)𝑑(𝑘), (3.10) where 𝑅𝜃_𝑘=𝐴𝜃_𝑘𝐵𝜃_𝑘𝐶𝜃_𝑘𝐷𝜃_𝑘^𝑇𝐸𝑄_𝜃(𝑘+1)0×𝐴𝜃0𝐼^𝑘𝐵𝜃_𝑘𝐶𝜃_𝑘𝐷𝜃_𝑘−𝐸𝑄_𝜃𝑘00𝛾²𝐼. (3.11) With 𝑅(𝜃_𝑘)<0, for any 𝜃(0)∈Ψ and 𝑑(𝑘)∈𝑙₂, we have 𝐸_{𝑒(0),𝜃(0)}‖𝑦‖₂<𝛾‖𝑑‖₂.
Recall that Θ(𝑘) is any possible sequences of Markov state. By taking expectation over Θ(𝑘), we can obtain (3.6) which implies 𝑅(𝜃_𝑘)<0 for each 𝑘. Thus the lemma is proved.

Remark 3.2. Lemma 3.1 assumed that 𝜃₀ is deterministic. However, it can also be used when there is no assumption made on 𝜃₀ as shown in the proof.

The second lemma gives an equivalent expression of (3.6).

Lemma 3.3. Consider the system (3.4). Given a constant 𝛾>0, 𝑥(0)=0 and any 𝜃(0)∈𝜓, then (3.6) with 𝑄_𝑖>0,𝑖∈𝜓 are feasible, if and only if there exist matrices 𝑄_𝑖>0 and 𝐺_𝑖>0 such that the following LMIs 𝑃_𝑖−𝐺_𝑖+𝐺^𝑇_𝑖𝐺^𝑇_𝑖𝐴_𝑖𝐺^𝑇_𝑖𝐵_𝑖0∗−𝑃_𝑖0𝐶^𝑇_𝑖∗∗−𝛾²𝐼𝐷^𝑇_𝑖∗∗∗−𝐼<0 (3.12) hold for 𝑖∈𝜓.

By observing that 𝜃_𝑘 is independent of 𝜃_𝑘−1 under assumption (A2), we can obtain the following bounded real lemma.

Lemma 3.4. Consider the system (3.4), where 𝑥(𝑘), 𝑦(𝑘), 𝐴(𝜃_𝑘), 𝐵(𝜃_𝑘), 𝐶(𝜃_𝑘), 𝐷(𝜃_𝑘), and 𝜃_𝑘 are defined as in (2.1), 𝑑(𝑘)∈𝐑^𝑚 is the 𝑙₂-norm bounded input sequence. Given a constant 𝛾>0, 𝑥(0)=0 and under assumption (A2), then (3.5) is satisfied, if there exist 𝑆>0 satisfying the following LMIs: −1𝑝₁𝑆𝑆𝐴₁𝑆𝐵₁−1𝑝₁𝐼𝐶₁𝐷₁−1⋱⋮⋮𝑝_𝑁𝑆𝑆𝐴_𝑁𝑆𝐵_𝑁−1𝑝_𝑁𝐼𝐶_𝑁𝐷_𝑁∗∗∗∗∗−𝑆0∗∗∗∗∗−𝛾²𝐼<0. (3.13)

Proof. Under the assumption (A2), the expectation terms in (3.10) are 𝐸𝑄_𝜃(𝑘+1)=𝐸𝑄_𝜃(𝑘+1)=^𝑁_𝑖=1𝑝_𝑖𝑄_𝑖. (3.14) Hence 𝑅(𝜃_𝑘) in (3.10) can be rewritten as 𝑅=^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖𝐵_𝑖𝐶_𝑖𝐷_𝑖^𝑇𝑆0𝐴0𝐼^𝑖𝐵_𝑖𝐶_𝑖𝐷_𝑖−𝑆00𝛾²𝐼 (3.15) with 𝑆=∑^𝑁_𝑖=1𝑝_𝑖𝑄_𝑖,𝑖∈𝜓.
If 𝑅<0, then for any 𝜃₀∈Ψ and 𝑑(𝑘)∈𝑙₂, we have 𝐸‖𝑦‖₂<𝛾‖𝑑‖₂. Applying Shur complement and congruence transformation with diag{𝑆,𝐼,…,𝑆,𝐼,𝐼,𝐼}, 𝑅<0 can be formulated as (3.13).

Remark 3.5. When the number of modes of {𝜃_𝑘} is 1, Lemmas 3.1 and 3.4 reduce to the standard bounded real lemma for deterministic systems [25]. It is clear that inequality (3.6) in Lemma 3.1 implies (3.13). But Lemma 3.4 not only requires less computational efforts, but also provides less conservative results as shown in the following example.

Example 3.6. Given 𝜓={1,2}, 𝜆₁₁=𝜆₂₁=0.2, 𝜆₁₂=𝜆₂₂=0.8 and 𝐴₁=0.2100.1,𝐴₂=,𝐵0.8100.5₁=1001,𝐵₂=,𝐶0.5000.1₁=𝐶₂=1001,𝐷₁=𝐷₂=,0000 (3.16) we have 𝑝₁(∞)=0.2, 𝑝₂(∞)=0.8 and min_𝑄𝑖>0𝛾²=12.88 according to Lemma 3.1, and min_𝑆>0𝛾²=6.17 according to Lemma 3.4.

The fourth lemma is given to compute the bound of variances for a stationary MJLS.

Lemma 3.7. Consider the system (3.4), where 𝑥(𝑘), 𝑦(𝑘), 𝐴(𝜃_𝑘), 𝐵(𝜃_𝑘), 𝐶(𝜃_𝑘), 𝐷(𝜃_𝑘), and 𝜃_𝑘 are defined as in (2.1), 𝑑(𝑘)∈𝐑^𝑚 is the 𝑙₂-norm bounded input sequence. Given a constant 𝛾>0, 𝑥(0)=0, 𝛾>0 and under assumption (A2), then ^∞_𝑗=0𝐸𝑦𝑗−𝑦𝑗^𝑇𝑦𝑗−𝑦𝑗<𝛼²‖𝑑(𝑘)‖²₂ (3.17) if there exist 𝑆>0 satisfying the following LMI: −1𝑝₁𝑆𝑆𝐴_𝜎,1𝑆𝐵_𝜎,1−1𝑝₁𝐼𝐶_𝜎,1𝐷_𝜎,1−1⋱⋮⋮𝑝_𝑁𝑆𝑆𝐴_𝜎,𝑁𝑆𝐵_𝜎,𝑁−1𝑝_𝑁𝐼𝐶_𝜎,𝑁𝐷_𝜎,𝑁∗∗∗∗∗−𝑆0∗∗∗∗∗−𝛼²𝐼<0 (3.18) with 𝐴_𝜎,𝑖=𝐴_𝑖00^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖,𝐵_𝜎,𝑖=_{𝑖𝑁𝑖=}𝐵₁𝑝_𝑖𝐵_𝑖,𝐶_𝜎,𝑖=𝐶_𝑖^𝑁_𝑖=1𝑝_𝑖𝐶_𝑖,𝐷_𝜎,𝑖=𝐷𝜃_𝑘−^𝑁_𝑖=1𝑝_𝑖𝐷_𝑖. (3.19)

Proof. The expected behavior of (3.4) under assumption (A2) is described by 𝑥(𝑘+1)=^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖𝑥(𝑘)+^𝑁_𝑖=1𝑝_𝑖𝐵_𝑖𝑑(𝑘),𝑦(𝑘)=^𝑁_𝑖=1𝑝_𝑖𝐶_𝑖𝑥(𝑘)+^𝑁_𝑖=1𝑝_𝑖𝐷_𝑖𝑑(𝑘). (3.20) Then the dynamics of 𝑟(𝑘)−𝑟(𝑘) can be written as 𝑥(𝑘+1)𝑥(𝑘+1)=𝐴_𝜎𝜃_𝑘𝑥(𝑘)𝑥(𝑘)+𝐵_𝜎𝜃_𝑘𝑑(𝑘),𝑦(𝑘)−𝑦(𝑘)=𝐶_𝜎𝜃_𝑘𝑥(𝑘)𝑥(𝑘)+𝐷_𝜎𝜃_𝑘𝑑(𝑘). (3.21) Following the similar procedure in Lemma 3.4, the result can be obtained.

Now we are in the position to give the theorem for the solving RGFD.

Theorem 3.8. Given the system (2.1), a constant 𝛼 and under assumptions (A1)–(A4), the optimal 𝐿(𝜃_𝑘) and 𝑊(𝜃_𝑘) in the residual generator (2.4) in the sense of minimizing (2.8) and satisfying (2.9) can be obtained by solving the following optimization problem min_{𝑌𝑖,𝑊𝑖,𝑆1>0,𝑆2>0}𝛾² (3.22) subject to Π⋱⋮⋮_𝑖𝑖Π_𝑖4Π_𝑖5⋱⋮⋮∗∗∗Π_{𝑁+1,𝑁+1}0∗∗∗∗−𝛾²𝐼Π<0,⋱⋮⋮_𝑖𝑖Γ_𝑖4Γ_𝑖5⋱⋮⋮∗∗∗Π_{𝑁+1,𝑁+1}0∗∗∗∗−𝛼²𝐼<0, (3.23) where Π_𝑖𝑖=−1𝑝_𝑖𝑆₁1000−𝑝_𝑖𝑆₂0100−𝑝_𝑖𝐼,Π_𝑖4=𝑆₁𝐴_𝑖−𝑌_𝑖𝐶_𝑖00𝑆₂𝐴_ref−𝐿_𝑜𝐶_ref𝑊_𝑖𝐶_𝑖−𝑊_𝑜𝐶_ref,Π_𝑖5=𝑆₁𝐸_𝑑,𝑖−𝑌_𝑖𝐹_𝑑,𝑖𝑆₁𝐸_𝑓,𝑖−𝑌_𝑖𝐹_𝑓,𝑖𝑆₂𝐸_𝑑,ref−𝐿_𝑜𝐹_𝑑,ref𝑆₂𝐸_𝑓,ref−𝐿_𝑜𝐹_𝑓,ref𝑊_𝑖𝐹_𝑑,𝑖−𝑊_𝑜𝐹_𝑑,ref𝑊_𝑖𝐹_𝑓,𝑖−𝑊_𝑜𝐹_𝑓,ref,Γ_𝑖𝑖=−1𝑝_𝑖𝑆₁1000−𝑝_𝑖𝑆₁0100−𝑝_𝑖𝐼,Γ_𝑖4=𝑆₁𝐴_𝑖−𝑌_𝑖𝐶_𝑖00𝑆₁^𝑁_𝑗=1𝑝_𝑗𝐴_𝑗−^𝑁_𝑗=1𝑝_𝑗𝑌_𝑗𝐶_𝑗𝑊_𝑖𝐶_𝑖−^𝑁_𝑗=1𝑝_𝑗𝑊_𝑗𝐶_𝑗,Γ_𝑖5=𝑆₁𝐸_𝑑,𝑖−𝑌_𝑖𝐹_𝑑,𝑖𝑆₁𝐸_𝑓,𝑖−𝑌_𝑖𝐹_{𝑁𝑓,𝑖𝑗=1}𝑝_𝑗𝑆₁𝐸_𝑑,𝑗−𝑌_𝑗𝐹_𝑑,𝑗^𝑁_𝑗=1𝑝_𝑗𝑆₁𝐸_𝑓,𝑗−𝑌_𝑖𝐹_𝑓,𝑗𝑊_𝑖𝐹_𝑑,𝑖−^𝑁_𝑗=1𝑝_𝑗𝑊_𝑗𝐹_𝑑,𝑗𝑊_𝑖𝐹_𝑓,𝑖−^𝑁_𝑗=1𝑝_𝑗𝑊_𝑗𝐹_𝑓,𝑗 (3.24)(3.25) for 𝑖∈𝜓 and Π_{𝑁+1,𝑁+1}=−𝑆₁00−𝑆₂,Γ_{𝑁+1,𝑁+1}=−𝑆₁00−𝑆₁. (3.26) The optimal 𝐿_𝑖 is then given by 𝑆₁⁻¹𝑌_𝑖.

Proof. With Lemmas 3.4 and 3.7, the proof is straightforward, and thus omitted.

3.2. Residual Evaluation

In this section, the solution of REFD is given. The bounds of the expectation and variance of (2.20) are firstly computed by using the peak-norm and 𝑙₂-norm of unknown inputs. Then the threshold is determined, and the upper bound of the guaranteed FAR is obtained.

The following lemma gives the computation of ∣𝑟_𝑗(𝑘)∣ in terms of norm.

Lemma 3.9. Given system (2.7), a constant 𝛾_𝑗,1>0 and assumption (A1)–(A4), then 𝑟_𝑗(𝑘)<𝛾_𝑗,1‖𝑑(𝑘)‖_peak (3.27) if there exist 𝑆>0, 𝜇>0 and 0<𝜅<1 satisfying the following LMIs: −𝑆𝑆^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖−𝐿_𝑖𝐶_𝑖𝑆^𝑁_𝑖=1𝑝_𝑖𝐸_𝑑,𝑖−𝐿_𝑖𝐹_𝑑,𝑖∗(𝜅−1)𝑆0∗∗−𝜇𝐼<0,−𝛾^𝑁_𝑗,1𝑖=1𝑝_𝑖𝑊_𝑖𝐶_𝑖𝑗^𝑁_𝑖=1𝑝_𝑖𝑊_𝑖𝐹_{𝑑,𝑖𝑗}∗−𝜅𝑆0∗∗𝜇−𝛾_𝑗,1𝐼<0. (3.28)(3.29)

Proof. The expected behavior of (2.7) is just described by 𝑒(𝑘+1)=𝐴𝑒(𝑘)+𝐸𝑑(𝑘),𝑟(𝑘)=𝐶𝑒(𝑘)+𝐹𝑑(𝑘) (3.30) with 𝐴=^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖−𝐿_𝑖𝐶_𝑖,𝐸=^𝑁_𝑖=1𝑝_𝑖𝐸_𝑑,𝑖−𝐿_𝑖𝐹_𝑑,𝑖,𝐶=^𝑁_𝑖=1𝑝_𝑖𝑊_𝑖𝐶_𝑖,𝐹=^𝑁_𝑖=1𝑝_𝑖𝑊_𝑖𝐹_𝑑,𝑖. (3.31) The expected residual dynamics are governed by (3.30), which is time invariant system. The results can be easily obtained following the idea in [25]. Thus the rest of the proof is omitted.

The variance of 𝑟_𝑗(𝑘) can be computed in terms of 𝑝𝑒𝑎𝑘-norm using the following lemma.

Lemma 3.10. Given system (2.7), a constant 𝛾_𝑗,2, and assumption (A1)–(A4), then 𝜎_𝑗(𝑘)<𝛾_𝑗,2‖𝑑(𝑘)‖_peak (3.32) if there exist 𝑆>0, 𝑖∈𝜓 satisfying the following LMIs: −1𝑝₁𝑆𝑆𝐴_𝜎,1𝑆𝐸_𝑑,𝜎,1−1⋱⋮⋮𝑝_𝑁𝑆𝑆𝐴_𝜎,𝑁𝑆𝐸_{𝑑,𝜎,𝑁}()−𝛾∗∗∗𝜅−1𝑆0∗∗∗∗−𝜇𝐼<0,_𝑗,2𝑝₁𝐶_𝜎,1𝑗𝐷_𝜎,1𝑗−𝛾⋱⋮⋮_𝑗,2𝑝_𝑁𝐼𝐶_{𝜎,𝑁𝑗}𝐶_{𝜎,𝑁𝑗}∗∗∗−𝜅𝑆0∗∗∗∗𝜇−𝛾_𝑗,2𝐼<0 (3.33)(3.34) with 𝐴_𝜎,𝑖=𝐴_𝐿,𝑖00^𝑁_𝑙=1𝐴_𝐿,𝑙,𝐸_𝜎,𝑖=𝐸_{𝑑,𝐿,𝑖}−^𝑁_𝑙=1𝐸_{𝑑,𝐿,𝑙},𝐶_𝜎,𝑖=𝑊_𝑖𝐶_𝑖−^𝑁_𝑙=1𝑊_𝑙𝐶_𝑙,𝐷_𝜎,𝑖=𝑊_𝑖𝐹_𝑑,𝑖−^𝑁_𝑙=1𝑊_𝑙𝐹_𝑑,𝑙. (3.35)

Proof. From (2.7) and (3.30), we have 𝑒(𝑘+1)𝑒(𝑘+1)=𝐴_𝜎𝜃_𝑘𝑒(𝑘)𝑒(𝑘)+𝐸_𝜎𝜃_𝑘𝑑(𝑘),𝑟_𝑗(𝑘)−𝑟_𝑗(𝑘)=𝐶_𝜎𝜃_𝑘𝑒(𝑘)𝑒(𝑘)+𝐷_𝜎𝜃_𝑘𝑑(𝑘). (3.36) Define 𝜒(𝑘)=𝑒(𝑘)𝑒(𝑘),𝑉𝜒,𝑖=𝜒^𝑇𝑄_𝑖𝜒 (3.37) for some 𝑄_𝑖>0 and assume that 𝐸𝑉𝜒(𝑘),𝜃_𝑘<𝜇_𝜃𝑘𝜅 (3.38) for 0<𝜅<1 and 𝜇_𝜃𝑘>0. Note that 𝐸[𝑉(𝜒(𝑘),𝜃_𝑘)] satisfying 𝐸V𝜒(𝑘+1),𝜃_𝑘+1−(𝜅−1)𝑉𝜒(𝑘),𝜃_𝑘<𝜇_𝜃𝑘,𝑉𝜒(0),𝜃₀=0 (3.39)(3.40) is bounded by (3.38). By denoting 𝜃_𝑘=𝑖, the inequality 𝐸𝜒(𝑘)𝑑(𝑘)^𝑇𝑅₁𝜒(𝑘)𝑑(𝑘)<0 (3.41) with 𝑅₁=^𝑁_𝑖=1𝑝_𝑖𝐴^𝑇_𝜎,𝑖𝐸^𝑇_𝜎,𝑖𝑆𝐴^𝑇_𝜎,𝑖𝐸^𝑇_𝜎,𝑖^𝑇−(),1−𝜅𝑆00𝜇𝐼𝑆=^𝑁_𝑖=1𝑝_𝑖𝑄_𝑖,𝜇=^𝑁_𝑖=1𝑝_𝑖𝜇_𝑖, (3.42)(3.43) ensures (3.39) and thus (3.38). Noticing that 𝜎_𝑗(𝑘)²=𝜒()𝑑()𝑘+1𝑘+1^𝑇𝑅₂𝜒()𝑑()𝑘+1𝑘+1 (3.44) with 𝑅₂=^𝑁_𝑙=1𝑝_𝑙𝐶_𝜎^𝑇_𝑗𝐷_𝜎^𝑇_𝑗𝐶_𝜎𝑗𝐷_𝜎𝑗, (3.45) and 𝐸[𝑉(𝜒(𝑘),𝜃_𝑘)]=𝜒(𝑘)^𝑇𝑆𝜒(𝑘), then the inequality 𝛾⁻¹_𝑗,1𝑅₂<0𝛾𝜅𝑆0_𝑗,2−𝜇𝐼 (3.46) can be obtained which implies 𝜎_𝑗(𝑘)²<𝛾_𝑗,2𝛾_𝑗,2𝑑(𝑘)^𝑇𝑑(𝑘)+𝜅𝐸𝑉𝜒(𝑘+1),𝜃_𝑘+1−𝜇<𝛾²_𝑗,2‖𝑑(𝑘)‖²_peak. (3.47) Applying Shur complement and congruence transformation, (3.41) and (3.46) can be reformulated as (3.33) and (3.34), respectively.

We propose also methods to compute the expectation and variance of 𝑟(𝑘) based on 𝑙₂-norm. The following lemma gives the computation of |𝑟_𝑗(𝑘)|.

Lemma 3.11. Given system (2.7), 𝛾_𝑗,1>0, 𝛾_𝑗,2>0 and under assumption (A1)–(A4), then 𝑟_𝑗(𝑘)<𝛾²_𝑗,1^𝑘−1_𝑖=0𝑑(𝑖)^𝑇𝑑(𝑖)+𝛾²_𝑗,2𝑑(𝑘)^𝑇𝑑(𝑘) (3.48) if there exist 𝑆>0, 𝑖∈𝜓 satisfying the following LMIs: −𝑆𝑆^𝑁_𝑖=1𝑝_𝑖𝐴_𝑖−𝐿_𝑖𝐶_𝑖𝑆^𝑁_𝑖=1𝑝_𝑖𝐸_𝑑,𝑖−𝐿_𝑖𝐹_𝑑,𝑖∗−𝑆0∗∗−𝛾²_𝑗,1𝐼<0,−𝐼^𝑁_𝑖=1𝑝_𝑖𝑊_𝑖𝐶_𝑖𝑗∗−𝑆<0,−𝐼^𝑁_𝑖=1𝑝_𝑖𝑊_𝑖𝐹_{𝑑,𝑖𝑗}∗−𝛾²_𝑗,2𝐼<0. (3.49)(3.50)(3.51)

Proof. The proof is similar with Lemma 3.9.

The computation of 𝜎_𝑗(𝑘) based on 𝑙₂-norm is given in the following lemma.

Lemma 3.12. Given system (2.7), 𝛾_𝑗,3>0, 𝛾_𝑗,4>0 and assumption (A1)–(A4), then 𝜎_𝑗(𝑘)<𝛾²_𝑗,3^𝑘−1_𝑖=0𝑑(𝑖)^𝑇𝑑(𝑖)+𝛾²_𝑗,3𝑑(𝑘)^𝑇𝑑(𝑘) (3.52) if there exist 𝑆>0, 𝑖∈𝜓 satisfying the following LMIs: −1𝑝₁𝑆𝐴_𝜎,1𝐸_𝑑,𝜎,1−1⋱⋮⋮𝑝_𝑁𝑆𝐴_𝜎,𝑁𝐸_{𝑑,𝜎,𝑁}∗∗∗−𝑆0∗∗∗∗−𝛾²_𝑗,3𝐼−1<0,𝑝₁𝐶_𝜎,1𝑗−1⋱⋮𝑝_𝑁𝐼𝐶_{𝜎,𝑁𝑗}−1∗∗∗−𝑆<0,𝑝₁𝐶_𝜎,1𝑗−1⋱⋮𝑝_𝑁𝐼𝐶_{𝜎,𝑁𝑗}∗∗∗−𝛾²_𝑗,4𝐼<0 (3.53)(3.54)(3.55) with 𝐴_𝜎,𝑖,𝐸_{𝑑,𝜎,𝑖},𝐶_𝜎,𝑖,𝐷_𝜎,𝑖 defined in (3.35).

Proof. The dynamics of 𝑟_𝑗(𝑘)−𝑟_𝑗(𝑘)are governed by (3.36). Define 𝜒(𝑘)=𝑒(𝑘)𝑒(𝑘),𝑉𝜒=𝜒^𝑇𝑄_𝑖𝜒 (3.56) for some 𝑄_𝑖>0, 𝑖∈𝜓. Consider that 𝐸𝑉𝜒(𝑘+1),𝜃_𝑘+1−𝑉𝜒(𝑘),𝜃_𝑘<𝛾²_𝑗,3𝑑(𝑘)^𝑇𝑑(𝑘) (3.57) implies 𝐸𝑉𝜒(𝑘),𝜃_𝑘=𝐸𝜒^𝑇𝑁_𝑖=1𝑝_𝑖𝑄_𝑖𝜒<𝛾^2𝑘_𝑗,1𝑖=0𝑑(𝑖)^𝑇𝑑(𝑖). (3.58) By denoting 𝜃_𝑘=𝑖, the inequality (3.57) is equivalent with 𝐸𝑒(𝑘)𝑑(𝑘)^𝑇𝑅𝑒(𝑘)𝑑(𝑘)<0 (3.59) with 𝑅=^𝑁_𝑖=1𝑝_𝑖𝐴^𝑇_𝜎,𝑖𝐸^𝑇_𝜎,𝑖𝑆𝐴^𝑇_𝜎,𝑖𝐸^𝑇_𝜎,𝑖^𝑇−𝑆00𝛾²_𝑗,3𝐼,𝑆=∑^𝑁_𝑖=1𝑝_𝑖𝑄_𝑖. (3.60) Then (3.52) is guaranteed, if ^𝑁_𝑖=1𝑝_𝑖𝐶_𝜎,𝑖^𝑇_𝑗𝐶_{𝜎,𝑖𝑗}<𝑆,_𝑁𝑖=1𝑝_𝑖𝐶_𝜎,𝑖^𝑇_𝑗𝐶_{𝜎,𝑖𝑗}<𝛾²_𝑗,4𝐼. (3.61)(3.62) Applying Shur complement and congruence transformation, (3.59)–(3.62) can be reformulated as (3.53)–(3.55), respectively.

Based on above results, the following theorem gives the solution of REFD.

Theorem 3.13. Given system (2.7), a constant 𝛽, assumption (A1)–(A4),(i)and ‖𝑑‖_peak<𝛿_𝑑,∞, then the threshold can be set as 𝐽_𝑗,𝑡ℎ=𝛾̌_𝑗,1+𝛽𝛾̌_𝑗,2𝛿_𝑑,∞, (3.63) where 𝛾̌_𝑗,1, 𝛾̌_𝑗,2 are the optimum of the constrained optimization problem min𝛾_𝑗,1subjectto(3.28)-(3.29),min𝛾_𝑗,2subjectto(3.33)-(3.34); (3.64)(ii)‖𝑑‖₂<𝛿_𝑑,2, ‖𝑑‖_{𝑝𝑒𝑎𝑘}<𝛿_𝑑,∞, then the threshold can be set as 𝐽_𝑗,th=𝛾̌_𝑗,1+𝛽𝛾̌_𝑗,3𝛿_𝑑,2+𝛾̌_𝑗,2+𝛽𝛾̌_𝑗,4𝛿_𝑑,∞, (3.65) where 𝛾̌_𝑗,1, 𝛾̌_𝑗,2, 𝛾̌_𝑗,3 and 𝛾̌_𝑗,4 are the optimum of the constrained optimization problem: min𝛾_𝑗,1,𝛾_𝑗,2subjectto(3.49)-(3.51),min𝛾_𝑗,3,𝛾_𝑗,4subjectto(3.53)-(3.55). (3.66) The false alarm rate is upper bounded as 1FAR≤𝛽². (3.67)

Proof. The theorem can be proved in a similar way as in [24].

Now we have the threshold for each residual signal. The fault can be detected if one evaluated residual exceeds its threshold, for example, 𝑟_𝑗(𝑘)≤𝐽_𝑗,th𝑟⟹fault-free,_𝑗(𝑘)>𝐽_𝑗,th⟹fault. (3.68)A false alarm occurs when there is no fault but |𝑟_𝑗(𝑘)|>𝐽_𝑗,th. Its probability is upper bounded by (3.67).

Remark 3.14. If the number of Markov state is 1, then the computed 𝜎²_𝑗(𝑟(𝑘)) will be zero. The proposed approach reduced to the standard norm-based residual evaluation methods [2].

Remark 3.15. The evaluated residual |𝑟_𝑗(𝑘)| is a stochastic variable. By using the expectation and variance of |𝑟_𝑗(𝑘)| for the computation of the threshold, an upper bound of FAR is obtained. Such a bound is very useful in practice, as it can provide confidential information for a rising fault alarm. Without considering the variance in the residual evaluation, the FAR can be very high and no bounds of FAR can be established. Since there is unknown inputs in the system, we can only derive the upper bounds of the expectation and variance. Its higher order moments are difficult to obtain, which is lack of physical means.

4. Discussion on FD of Nonstationary MJLS

Without the assumption (A2), the MJLS could be in nonstationary state, that is, 𝑃(𝑘)≠𝑝(∞). In this case the expectation of 𝑟(𝑘) is difficult to obtain, and (2.9) cannot be established. Furthermore, the residual evaluation approach presented in last section cannot be applied. Hence we propose also an FD system for nonstationary MJLS. Firstly, the residual generator is designed to satisfy (2.8), and the solution is given in the following theorem.

Theorem 4.1. Given the system (2.1) under assumptions (A1), (A3)–(A4), the optimals𝐿(𝜃_𝑘) and 𝑊(𝜃_𝑘) in the residual generator (2.4) in the sense of minimizing (2.8) can be obtained by solving the following optimization problem for all 𝑖∈𝜓: min_{𝑌𝑖,𝑊𝑖,𝑄𝑖>0}𝛾² (4.1) subject to 𝑁_𝑝𝑞7×7<0, (4.2) where the nonzero elements of 𝑁_𝑝𝑞 are 𝑁₁₁=𝑄_𝑖11−𝐺_𝑖11−𝐺^𝑇_𝑖11,𝑁₁₂=𝑄_𝑖12,𝑁₁₃=𝐺^𝑇_𝑖11𝐴_𝑖−𝑌_𝑖𝐶_𝑖,𝑁₁₅=𝐺^𝑇_𝑖11𝐸_𝑑,𝑖−𝑌_𝑖𝐹_𝑑,𝑖,𝑁₁₆=𝐺^𝑇_𝑖11𝐸_𝑓,𝑖−𝑌_𝑖𝐹_𝑓,𝑖,𝑁₂₂=𝑄_𝑖22−𝐺_𝑖22−𝐺^𝑇_𝑖22,𝑁₂₄=𝐺^𝑇_𝑖22𝐴_ref−𝐿_opt𝐶_ref,𝑁₂₅=𝐺^𝑇_𝑖22𝐸_𝑑,ref−𝐿_opt𝐹_𝑑,ref,N₂₆=𝐺^𝑇_𝑖22𝐸_𝑓,ref−𝐿_opt𝐹_𝑓,ref,𝑁₃₃=−𝑄_𝑖11,𝑁₃₄=−𝑄_𝑖12,𝑁₃₇=𝐶^𝑇_𝑖𝑊^𝑇_𝑖,𝑁₄₄=−𝑄_𝑖22,𝑁₄₇=𝐶^𝑇_ref𝑊^𝑇_opt,𝑁₅₅=−𝛾²𝐼_𝑛×𝑛,𝑁₅₇=𝐹^𝑇_𝑑,𝑖𝑊^𝑇_𝑖−𝐹^𝑇_𝑑,ref𝑊^𝑇_opt,𝑁₆₆=−𝛾²𝐼_𝑛×𝑛,𝑁₆₇=𝐹^𝑇_𝑓,𝑖𝑊^𝑇_𝑖−𝐹^𝑇_𝑓,ref𝑊^𝑇_opt,𝑁₇₇=−𝐼_𝑛×𝑛. (4.3) where 𝑄_𝑖=𝑄_𝑖11𝑄_𝑖12𝑄^𝑇_𝑖12𝑄_𝑖22>0,𝑄_𝑖=𝑄_𝑖11𝑄_𝑖12𝑄^𝑇_𝑖12𝑄_𝑖22>0,𝑄_𝑖=^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝑗,𝐺_𝑖=𝐺_𝑖1100𝐺_𝑖22. (4.4)(4.5) The optimal 𝐿_𝑖 is then given by 𝐺⁻¹_𝑖11𝑌_𝑖.

Proof. By applying Lemmas 3.1 and 3.3 and setting 𝑌_𝑖=𝐺^𝑇_𝑖11𝐿_𝑖, the LMI (4.2) can be easily obtained for system (3.1). Notice that 𝑄_𝑖11−𝐺_𝑖11−𝐺^𝑇_𝑖11>0 (4.6) implies that 𝐺_𝑖11 is nonsingular. Therefore, the feasibility of (4.2) always ensure the existence of optimal 𝐿_𝑖 and 𝑊_𝑖.

Then (2.17) is selected as the evaluation functions, and the threshold is suggested as𝐽_th=𝛽²sup_𝑑∈𝑙2‖𝑟(𝑘)‖_𝐸,𝛽>0, (4.7) where sup_𝑑∈𝑙2‖𝑟(𝑘)‖_𝐸 can be easily obtained by applying Lemma 3.1. According to Markov inequality [26], we haveProb‖𝑟(𝑘)‖²₂≥𝜖²≤‖𝑟(𝑘)‖²_𝐸𝜖²,𝜖>0 (4.8) which yieldsProb‖𝑟(𝑘)‖_𝑒≥𝛽sup_𝑑∈𝑙2‖𝑟(𝑘)‖_𝐸≤‖𝑟(𝑘)‖²_𝐸𝛽²sup_𝑑∈𝑙2‖𝑟(𝑘)‖²_𝐸≤1𝛽². (4.9) Hence the FAR is bounded by 1/𝛽² with the threshold (4.7). The result is summarized as the following theorem.

Theorem 4.2. Given the system (2.1) under assumptions (A1), (A3)–(A4), and the evaluation function (2.17), the threshold can be set as (4.7), and then the FAR is upper bounded by 1/𝛽².

Remark 4.3. The evaluation method suggested in Theorem 4.2 can also be used for the MJLS in stationary state. The variance of ‖𝑟(𝑘)‖_𝐸, which is difficult to obtain, is not involved in the computation of the threshold. When the variance of is very small, this evaluation method can be fairly conservative.

5. Numerical Example

To illustrate the proposed method, the following two-mode discrete-time MJLS is considered:𝐴₁=0.40.100.1,𝐴₂=,𝐸−0.10.300.6_𝑑,1=E_𝑑,2=0.200.20,𝐸_𝑓,1=𝐸_𝑓,2=,𝐵0.10.1₁=𝐵₂=0,𝐶₁=𝐶₂=1001,𝐹_𝑓,1=𝐹_𝑓,2𝐹=0,_𝑑,1=00.100.1,𝐹_𝑑,2=.00.700.7 (5.1) The transition probability matrix is given byΦ=0.40.60.50.5 (5.2) then𝑃_∞=0.45450.5455^𝑇. (5.3) Its reference residual model has𝐴_ref=0.100.2200.40,𝐹_𝑑,ref=,𝐸00.460000.4600_𝑑,ref=𝐸_𝑑,𝐸_𝑓,ref=𝐹_𝑓,𝐶_ref=𝐶,𝐹_𝑓,ref=𝐹_𝑓𝐿_opt=,𝑊−1.24561.2965−1.68201.7456_opt=.−0.9864−0.9902−68.869968.6097 (5.4) By Theorem 3.8, we have the following:𝐿₁=0.5660.2150.2350.263,𝐿₂=,𝑊−0.1110.300−0.1570.487₁=−0.654−0.545−0.223−0.179,𝑊₂=.−0.701−0.574−0.115−0.106 (5.5) During the simulation, assume that the unknown inputs are discrete-time random noises with power 0.5, the fault appears at the 500th discrete time as a step function of amplitude 15. The residual signals are evaluated as in (2.20) and the thresholds are computed based on peak-norm and 𝑙₂-norm as in (2.20). We take the second residual signal of the system as an example. The thresholds are computed according to Theorem 4.1 with 𝛽=3, that is, FAR<11.1%. The results are shown in Tables 1 and 2, respectively. The simulation results are given in the corresponding figures, where 𝐽_𝑒 is the threshold calculated only based on the expectation of residual signals. Figures show that many false alarms arise in the first 500 time steps with 𝐽_𝑒, and the number of false alarms is significantly reduced by considering the variances in the threshold computation. The fault is detected by all four thresholds.

Table 1 

Peak-norm-based residual evaluation.

Table 2 

𝑙₂-norm-based residual evaluation.

6. Conclusion

In this paper a complete design approach of FD system for stationary and nonstationary MJLS was proposed. The main focus was how to deal with the stochastic properties of MJLS. First the residual generator was designed to stochastically match an optimal reference residual model in order to achieve the best tradeoff between robustness against system unknown inputs and sensitivity to faults, where the reference residual model was selected based on the stationary expected behavior of the MJLS. Then novel residual evaluation methods based on peak-norm and 𝑙₂-norm were presented for MJLS, which can guarantee an expected FAR and meanwhile reduce missing detection of faults. In those evaluation methods, not only the expectation of evaluated residuals but also their variances were taken into account for the computation of thresholds. The proposed FD system can provide confidential information of occurring faults, which allows a practical application in real physical systems.

Figure 1 

FD of MJLS: 𝑟2(𝑘) and 𝜎2(𝑘) based on peak-norm.

Figure 2 

FD of MJLS: 𝑟2(𝑘) and 𝜎2(𝑘) based on 𝑙2-norm.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (61004071), Jiangsu Key Technologies R&D Program (BE2011046), the China Postdoctoral Science Foundation (20100471409), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.