Adaptive Observer-Based Fault Estimation for Stochastic Markovian Jumping Systems

Abstract

This paper studies the adaptive fault estimation problems for stochastic Markovian jump systems (MJSs) with time delays. With the aid of the selected Lyapunov-Krasovskii functional, the adaptive fault estimation algorithm based on adaptive observer is proposed to enhance the rapidity and accuracy performance of fault estimation. A sufficient condition on the existence of adaptive observer is presented and proved by means of linear matrix inequalities techniques. The presented results are extended to multiple time-delayed MJSs. Simulation results illustrate that the validity of the proposed adaptive faults estimation algorithms.

1. Introduction

Fault detection and isolation (FDI) [1, 2] has been the subject of extensive research since the 1970s and becomes one of the hotspots in control theory presently. With the rising demands of product quality, effectiveness, and safety in modern industries, people expect that they can get the failure information before the fault damages the system. Many techniques have been proposed especially for sensor and actuator failures with application to a wide range of engineering fields. Among these, the most commonly used schemes for fault detection relate to observer-based approaches [2–5]. It should be pointed out that the observer-based approach, which uses a parametric design technique to perform both detection and diagnosis, only works for a small number of sensor faults. In some cases, fault estimation strategies [6–8] are needed to carry on controlling the faulty system. Compared with FDI, fault estimation is a more challenging task because it requires an estimation of the location after the alarm has been set, and the size of the fault should be made. Recently, some results based on adaptive or robust observers [2–8] for fault estimation have been obtained. However, very few results in the literature consider the fault detection and estimation problem for stochastic systems.

In fact, as a special class of stochastic systems [9, 10] that involves both time-evolving and event-driven mechanisms, Markovian jump systems (MJSs) [11] have received considerable attention. This class of systems includes two components which are the mode and the state and the dynamics of jumping modes and continuous states, which are, respectively, modeled by finite-state Markov chains and differential equations. It can be used to model a variety of physical systems, which may experience abrupt changes in structures and parameters due to, for example, sudden environment changes, subsystem switching, system noises, and failures occurred in components or interconnections and executor faults. Some illustrative applications of MJSs can be found for examples in [3, 4, 6, 12–18] and the references therein. In recent years, the FDI problems for MJSs have regained increasing interest, and some results are also available [3, 4, 16–18]. However, very little is known on the problem of fault estimation for time-delay nonlinear MJSs. This problem forms the main purpose of this paper, where an adaptive technique [1, 19–21] is proposed and modified for the estimation of actuator faults.

In this paper, we studied the problem of fault estimation for a class of time-delay MJSs. By comparing with the presented results of time-delay and linear dynamic systems, it can be shown that a derivative term is added on the basis of fault estimation equation in our design, which renders that the conventional adaptive fault estimation algorithm [21] can be treated as a special case of the fast adaptive fault estimation algorithm. The introduction of the derivative term plays a major role in improving the rapidity of fault estimation. The augmented dynamic system is firstly constructed based on the adaptive fault estimation observer, and the observer parameters are designed on the system modes. Sufficient conditions are subsequently established on the existence of the mode-dependent adaptive fault estimation observer. The design criterions are presented in the form of linear matrix inequalities (LMIs) [22], which can be easily checked. The presented results are then extended to multiple time-delayed MJSs case. Finally, a numerical example is included to illustrate the effectiveness of the developed techniques.

Let us introduce some notations. The symbols ℜ^𝑛 and ℜ^𝑛×𝑚 stand for an 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively, 𝐴^T and 𝐴⁻¹ denote the matrix transpose and matrix inverse, diag{𝐴𝐵} represents the block-diagonal matrix of 𝐴 and 𝐵, 𝜎_max(𝐶) denote the maximal eigenvalue of a positive-define matrix 𝐶, ∑^𝑁_𝑖<𝑗 denotes, for example, for 𝑁=3, ∑^𝑁_𝑖<𝑗𝑎_𝑖𝑗⇔𝑎₁₂+𝑎₁₃+𝑎₂₃, ‖∗‖ denotes the Euclidean norm of vectors, 𝐄{∗} denotes the mathematics statistical expectation of the stochastic process or vector, 𝐿^𝑛₂(0∞) is the space of 𝑛 dimensional square integrable function vector over (0∞), 𝑃>0 (or 𝑃≥0) stands for a positive-definite (or nonnegative-definite) matrix, 𝐼 is the unit matrix with appropriate dimensions, 0 is the zero matrix with appropriate dimensions, ∗ means that the symmetric terms in a symmetric matrix.

2. Problem Formulation

Given a probability space (Ω,𝐹,𝜌), where Ω is the sample space, 𝐹 is the algebra of events, and 𝜌 is the probability measure defined on 𝐹. Let the random form process {𝑟_𝑡,𝑡≥0} be the continuous-time discrete-state Markov stochastic process taking values in a finite set Λ={1,2,…,𝑁} with transition probabilities given by 𝑃_𝑟𝑟_𝑡+Δ𝑡=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ𝑡+𝑜(Δ𝑡),𝑖≠𝑗,1+𝜋_𝑖𝑖Δ𝑡+𝑜(Δ𝑡),𝑖=𝑗, (2.1) where Δ𝑡>0 and lim_Δ𝑡↓0𝑜(Δ𝑡)/Δ𝑡→0. 𝜋_𝑖𝑗≥0 are the transition probability rates from mode 𝑖 at time 𝑡 to mode 𝑗(𝑖≠𝑗) at time 𝑡+Δ𝑡, and ∑^𝑁_{𝑗=1,𝑖≠𝑗}𝜋_𝑖𝑗=−𝜋_𝑖𝑖.

Consider the following linear MJSs over the probability space (Ω,𝐹,𝜌): 𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥(𝑡−𝑑)+𝐵𝑟_𝑡𝑢(𝑡)+𝐵_𝑓𝑟_𝑡𝑓(𝑡),𝑦(𝑡)=𝐶𝑟_𝑡𝑥(𝑡),𝑥(𝑡)=𝜂(𝑡),𝑟(𝑡)=𝑟₀,𝑡∈−𝑑0,𝑖=1,2,…,𝑆, (2.2) where 𝑥(𝑡)∈ℜ^𝑛 is the state, 𝑦(𝑡)∈ℜ^𝑚 is the measured output, 𝑢(𝑡)∈ℜ^𝑙 is the controlled input, 𝑓(𝑡)∈ℜ^𝑝 is the unknown actuator fault, and we assume that its derivative is the norm bounded with ‖𝑓(̇𝑡)‖≤𝑓, wherein 0≤𝑓<∞. 𝑑>0 is the time delay constant, 𝜂(𝑡)∈ℜ^𝑛 is a continuous vector-valued initial function assumed to be continuously differentiable on [−𝑑0], and 𝑟₀ is the initial mode. 𝐴(𝑟_𝑡), 𝐴_𝑑(𝑟_𝑡), 𝐵(𝑟_𝑡), 𝐵_𝑓(𝑟_𝑡), and 𝐶(𝑟_𝑡) are known mode-dependent matrices with appropriate dimensions, 𝐵_𝑓(𝑟_𝑡) is of full column rank with rank[𝐵_𝑓(𝑟_𝑡)]=𝑝, and 𝑟_𝑡 represents a continuous-time discrete state Markov stochastic process with values in the finite set Λ.

For presentation convenience, we denote 𝑥(𝑡−𝑑), 𝐴(𝑟_𝑡), 𝐴_𝑑(𝑟_𝑡), 𝐵(𝑟_𝑡), 𝐵_𝑓(𝑟_𝑡), and 𝐶(𝑟_𝑡) as 𝑥_𝑑, 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐵_𝑓𝑖, 𝐶_𝑖 and, respectively.

Definition 2.1 (see Mao [23]). Let 𝑉(𝑥(𝑡),𝑟_𝑡,𝑡>0)=𝑉(𝑥(𝑡),𝑖) be the positive stochastic functional and define its weak infinitesimal operator as ℑ𝑉(𝑥(𝑡),𝑖)=lim_Δ𝑡→01𝐄𝑉𝑥()Δ𝑡𝑡+Δ𝑡,𝑟_𝑡+Δ𝑡,𝑡+Δ𝑡∣𝑥(𝑡),𝑟_𝑡=𝑖−𝑉(𝑥(𝑡),𝑖,𝑡). (2.3) Refer to observer design [2–4, 7] and consider the following systems: 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥_𝑑+𝐵_𝑓𝑖𝑓(𝑡)+𝐵_𝑖𝑢(𝑡)+𝐿_𝑖𝑦(𝑡)−𝑦(𝑡)+𝐿_𝑑𝑖𝑦_𝑑−𝑦_𝑑,𝑦(𝑡)=𝐶_𝑖𝑥(𝑡), (2.4) where 𝑥(𝑡)∈ℜ^𝑛 is the observer state vector, 𝑦(𝑡)∈ℜ^𝑚 is the observer output vector, and 𝑓(𝑡)∈ℜ^𝑝 is an estimate of actuator fault of 𝑓(𝑡). Denote 𝑒(𝑡)(𝑡)(𝑡),𝑧(𝑡)(𝑡)(𝑡),𝑒=𝑥−𝑥=𝑦−𝑦_𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑡). (2.5)

Then, we can present the error dynamics (2.4) as 𝑒̇(𝑡)=𝐴_𝑖𝑒(𝑡)+𝐴_𝑑𝑖𝑒_𝑑+𝐵_𝑓𝑖𝑒_𝑓(𝑡),𝑧(𝑡)=𝐶_𝑖𝑒(𝑡), (2.6) where 𝐴_𝑖=𝐴_𝑖−𝐿_𝑖𝐶_𝑖, 𝐴_𝑑𝑖=𝐴_𝑑𝑖−𝐿_𝑑𝑖𝐶_𝑖.

In this paper, by comparison with the conventional adaptive fault estimation algorithm [21], we consider the following fast adaptive fault estimation algorithm. In this algorithm, we add a derivative term 𝑧(̇𝑡) in estimation equation, that is, 𝑓̇(𝑡)=Γ𝐻_𝑖[𝑧̇(𝑡)+𝑧(𝑡)], (2.7) which can realize lim_𝑡→∞𝑧(𝑡)=0, where 𝐻_𝑖 is a given mode-dependent matrix and Γ=Γ^T>0 is a prespecified matrix which defines the learning rate for (2.6).

Remark 2.2. In the conventional adaptive fault estimation algorithm 𝑓(𝑡)=Γ𝐻∫^𝑡_𝑡𝑓𝑧(𝜏)𝑑𝜏, it is only an integral term in essence. It fails to deal with time-varying faults, that is, 𝑓(̇𝑡)≠0 though it assumes that the constant fault (𝑓(̇𝑡)=0) estimation is unbiased. In this paper, we add a derivative term 𝑧(̇𝑡) in estimation equation and improve the conventional adaptive fault estimation algorithm such that the time-varying faults can be considered. For the stochastic modes jumping case, we select the given matrix 𝐻_𝑖 as a mode-dependent one.

3. Adaptive Fault Estimation Observer Design

Theorem 3.1. If there exist a set of positive definite symmetric matrices 𝑃_𝑖, 𝑄, 𝑈 and mode-dependent matrices 𝐻_𝑖, 𝑋_𝑖, and 𝑋_𝑑𝑖, such that the following matrix equations hold for all 𝑖∈Λ: 𝐻_𝑖𝐶_𝑖=𝐵^T_𝑓𝑖𝑃_𝑖,Π_𝑖=Ξ_𝑖𝑃_𝑖𝐴_𝑑𝑖−𝑋_𝑑𝑖𝐶_𝑖−𝐴^T_𝑖𝑃_𝑖𝐵_𝑓𝑖+𝐶^T_𝑖𝑋^T_𝑖𝐵_𝑓𝑖∗−𝑄−𝐴^T_𝑑𝑖𝑃_𝑖𝐵_𝑓𝑖+𝐶^T_𝑖𝑋^T_𝑑𝑖𝐵_𝑓𝑖∗∗−2𝐵^T_𝑓𝑖𝑃_𝑖𝐵_𝑓𝑖+𝑈<0, (3.1)(3.2) where Ξ_𝑖=𝐴^T_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖−𝑋_𝑖𝐶_𝑖−𝐶^T_𝑖𝑋^T_𝑖+𝑄+∑^𝑁_𝑖=1𝜋_𝑖𝑗𝑃_𝑗. Then the fast adaptive fault estimation algorithm of (2.7) can be realized. And in the estimated time-interval, it can estimate errors with the uniformly boundedness of the states and faults. Moreover, the observer gains are respectively as 𝐿_𝑖=𝑃_𝑖⁻¹𝑋_𝑖,𝐿_𝑑𝑖=𝑃_𝑖⁻¹𝑋_𝑑𝑖. (3.3)

Proof. Let the mode at time 𝑡 be 𝑖, that is, 𝑟_𝑡=𝑖∈Λ. Take the stochastic Lyapunov-Krasovskii function 𝑉(𝑒(𝑡),𝑒_𝑓(𝑡),𝑟_𝑡,𝑡>0)∶ℜ^𝑛×ℜ^𝑛×Λ×ℜ₊→ℜ₊ as 𝑉𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=𝑉₁𝑒(𝑡),𝑒_𝑓(𝑡),𝑖+𝑉₂𝑒(𝑡),𝑒_𝑓(𝑡),𝑖+𝑉₃𝑒(𝑡),𝑒_𝑓(𝑡),𝑖, (3.4) where 𝑉₁(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)=𝑒^T(𝑡)𝑃_𝑖𝑒(𝑡), 𝑉₂(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)=∫^𝑡_𝑡−𝑑𝑒^T(𝑡)𝑄𝑒(𝑡)𝑑𝜏, 𝑉₃(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)=𝑒^T_𝑓(𝑡)Γ⁻¹𝑒_𝑓(𝑡), in which 𝑃_𝑖∈ℜ^𝑛×𝑛, 𝑄∈ℜ^𝑛×𝑛 are the given mode-dependent symmetric positive-definite matrix for each modes 𝑖∈Λ.
According to Definition 2.1 and along the trajectories of the error dynamics MJSs (2.6), we can derive the following: ℑ𝑉₁𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=2𝑒^T(𝑡)𝑃_𝑖𝑒̇(𝑡)+𝑒^T(𝑡)^𝑁_𝑖=1𝜋_𝑖𝑗𝑃_𝑗𝑒(𝑡)=2𝑒^T(𝑡)𝑃_𝑖𝐴_𝑖𝑒(𝑡)+𝐴_𝑑𝑖(𝑟)𝑒_𝑑+𝐵_𝑓𝑖𝑒_𝑓(𝑡)+𝑒^T(𝑡)^𝑁_𝑖=1𝜋_𝑖𝑗𝑃_𝑗𝑒(𝑡)=𝑒^T(𝑡)𝐴^T_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_𝑖=1𝜋_𝑖𝑗𝑃_𝑗𝑒(𝑡)+2𝑒^T(𝑡)𝑃_𝑖𝐴_𝑑𝑖𝑒_𝑑+2𝑒^T(𝑡)𝑃_𝑖𝐵_𝑓𝑖𝑒_𝑓(𝑡),ℑ𝑉₂𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=𝑒^T(𝑡)𝑄𝑒(𝑡)−𝑒^T_𝑑𝑄𝑒_𝑑+^𝑁_𝑖=1𝜋_𝑖𝑗^𝑡_𝑡−𝑑𝑒^T(𝑡)𝑄𝑒(𝑡)𝑑𝜏=𝑒^T(𝑡)𝑄𝑒(𝑡)−𝑒^T_𝑑𝑄𝑒_𝑑+^𝑁_𝑖=1𝜋_𝑖𝑗^𝑡_𝑡−𝑑𝑒^T(𝑡)𝑄𝑒(𝑡)𝑑𝜏=𝑒^T(𝑡)𝑄𝑒(𝑡)−𝑒^T_𝑑𝑄𝑒_𝑑,ℑ𝑉₃𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=2𝑒^T_𝑓(𝑡)Γ⁻¹𝑒_𝑓̇(𝑡)=2𝑒^T_𝑓(𝑡)Γ⁻¹𝑓̇(𝑡)−𝑓̇(𝑡)=−2𝑒^T_𝑓(𝑡)𝐻_𝑖[𝑧̇(𝑡)+𝑧(𝑡)]+2𝑒^T_𝑓(𝑡)Γ⁻¹𝑓̇(𝑡)=−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝑒̇(𝑡)−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝑒(𝑡)+2𝑒^T_𝑓(𝑡)Γ⁻¹𝑓̇(𝑡)=−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐴_𝑖𝑒(𝑡)−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐴_𝑑𝑖𝑒_𝑑−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐵_𝑓𝑖𝑒_𝑓(𝑡)−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝑒(𝑡)+2𝑒^T_𝑓(𝑡)Γ⁻¹𝑓̇(𝑡). (3.5)
Given a symmetric positive definite matrix 𝑈, we can use the following relation: 2𝑒^T_𝑓(𝑡)Γ⁻¹𝑓̇(𝑡)≤𝑒^T_𝑓(𝑡)𝑈𝑒_𝑓(𝑡)+𝑓^Ṫ(𝑡)Γ⁻¹𝑈⁻¹Γ⁻¹𝑓̇(𝑡)≤𝑒^T_𝑓(𝑡)𝑈𝑒_𝑓(𝑡)+𝑓²̇𝜎_maxΓ⁻¹𝑈⁻¹Γ⁻¹. (3.6)
Then, we can get ℑ𝑉𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=ℑ𝑉₁𝑒(𝑡),𝑒_𝑓(𝑡),𝑖+ℑ𝑉₂𝑒(𝑡),𝑒_𝑓(𝑡),𝑖+ℑ𝑉₃𝑒(𝑡),𝑒_𝑓(𝑡),𝑖=𝑒^T(𝑡)𝐴^T_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_𝑖=1𝜋_𝑖𝑗𝑃_𝑗+𝑄𝑒(𝑡)+2𝑒^T(𝑡)𝑃_𝑖𝐴_𝑑𝑖𝑒_𝑑−𝑒^T_𝑑𝑄𝑒_𝑑−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐴_𝑖𝑒(𝑡)−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐴_𝑑𝑖𝑒_𝑑−2𝑒^T_𝑓(𝑡)𝐵^T_𝑓𝑖𝑃_𝑖𝐵_𝑓𝑖𝑒_𝑓(𝑡)+𝑒^T_𝑓(𝑡)𝑈𝑒_𝑓(𝑡)+𝑓²̇𝜎_maxΓ⁻¹𝑈⁻¹Γ⁻¹. (3.7)
By letting 𝑋_𝑖=𝑃_𝑖𝐿_𝑖 and 𝑋_𝑑𝑖=𝑃_𝑖𝐿_𝑑𝑖, the derivative of ℑ𝑉(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖) with respect to time follows that ℑ𝑉𝑒(𝑡),𝑒_𝑓(𝑡),𝑖≤𝜁^T(𝑡)Π_𝑖𝜁(𝑡)+𝜂, (3.8) where 𝜁(𝑡)=^{𝑒𝑒(𝑡)𝑑}_{𝑒𝑓(𝑡)}, 𝜂=𝑓²̇𝜎_max(Γ⁻¹𝑈⁻¹Γ⁻¹).
Thus, it concludes that ℑ𝑉(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)≤−𝜆‖𝜁(𝑡)‖²+𝜂, wherein 𝜆=min_𝑟∈Λ𝜎_min(−Π_𝑖). Obviously, we can get ℑ𝑉(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)<0 if 𝜂<𝜆‖𝜁(𝑡)‖². According to stochastic Lyapunov-Krasovskii stability theory, the trajectory of 𝜁(𝑡) will converge to the small set Φ={𝜁(𝑡)∣‖𝜁(𝑡)‖²≤𝜂/𝜆}, though it is outside set 𝑆. Therefore, 𝜁(𝑡) is ultimately bounded. This completes the proof.

Remark 3.2. It is necessary to point out that if the presented faults are constant, that is, 𝑓(̇𝑡)=0, then the designed adaptive algorithm can achieve asymptotical convergence from (3.8). Then we can get that ℑ𝑉(𝑒(𝑡),𝑒_𝑓(𝑡),𝑖)≤−𝜆‖𝜁(𝑡)‖²≤0, which proves the stability of the origin 𝑒(𝑡)=0, 𝑒_𝑓(𝑡)=0 and the uniformly boundedness of 𝑒(𝑡) and 𝑒_𝑓(𝑡) with 𝑒(𝑡)∈𝐿^𝑛₂(0∞). Then, lim_𝑡→∞𝑒(𝑡)→0 holds by Barbalat’s Lemma.

4. Extension to Multiple Time-Delayed MJSs

Consider the following multiple time-delayed MJSs over the probability space (Ω,𝐹,𝜌): 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+^𝑀_𝑚=1𝐴_𝑑𝑚𝑖𝑥𝑡−𝑑_𝑚+𝐵_𝑖𝑢(𝑡)+𝐵_𝑓𝑖𝑓(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡),𝑥(𝑡)=𝜂(𝑡),𝑟(𝑡)=𝑟₀,𝑡∈−max𝑑_𝑚0, (4.1) where 𝑑_𝑚, 𝑚=1,2,…,𝑀 are multiple time delays with 0≤𝑑_𝑚<∞, and other notations are the same as in Section 2.

Similar to Section 3, the following observer can be constructed: 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+^𝑀_𝑚=1𝐴_𝑑𝑚𝑖𝑥𝑡−𝑑_𝑚+𝐵_𝑓𝑖𝑓(𝑡)+𝐵_𝑖𝑢(𝑡)+𝐿_𝑖𝑦(𝑡)−𝑦(𝑡)+^𝑀_𝑚=1𝐿_𝑑𝑚𝑖𝑦𝑡−𝑑_𝑚−𝑦𝑡−𝑑_𝑚,𝑦(𝑡)=𝐶_𝑖𝑥(𝑡) (4.2) where 𝑥(𝑡)∈ℜ^𝑛 is the observer state vector, 𝑦(𝑡)∈ℜ^𝑚 is the observer output vector, and 𝑓(𝑡)∈ℜ^𝑝 is an estimate of actuator fault of 𝑓(𝑡).

Then, we can obtain the error dynamics by using the same notations of 𝑒(𝑡), 𝑒_𝑓(𝑡), and 𝑧(𝑡) as follows: 𝑒̇(𝑡)=𝐴_𝑖𝑒(𝑡)+^𝑀_𝑚=1𝐴_𝑑𝑚𝑖𝑒_𝑑𝑡−𝑑_𝑚+𝐵_𝑓𝑖𝑒_𝑓(𝑡),𝑧(𝑡)=𝐶_𝑖𝑒(𝑡) (4.3) where 𝐴_𝑖=𝐴_𝑖−𝐿_𝑖𝐶_𝑖, 𝐴_𝑑𝑚𝑖=𝐴_𝑑𝑚𝑖−𝐿_𝑑𝑚𝑖𝐶_𝑖.

Prior to the design of an adaptive diagnostic law, we can get the following results for multiple time-delayed MJSs (4.1).

Theorem 4.1. If there exist a set of positive definite symmetric matrices 𝑃_𝑖, 𝑄, 𝑈 and mode-dependent matrices 𝐻_𝑖, 𝑋_𝑖, and 𝑋_𝑑𝑚𝑖, 𝑚=1,2,…,𝑀, such that the following matrix equations hold for all 𝑖∈Λ: 𝐻_𝑖𝐶_𝑖=𝐵^T_𝑓𝑖𝑃_𝑖,Ψ_𝑖=Ξ_𝑖𝑃_𝑖𝐴_𝑑𝑖−∑^𝑀_𝑚=1𝑋_𝑑𝑚𝑖𝐶_𝑖−𝐴^T_𝑖𝑃_𝑖𝐵_𝑓𝑖+𝐶^T_𝑖𝑋^T_𝑖𝐵_𝑓𝑖∗−𝑄−𝐴^T_𝑑𝑖𝑃_𝑖𝐵_𝑓𝑖+𝐶∑^T_𝑖^𝑀_𝑚=1𝑋^T_𝑑𝑚𝑖𝐵_𝑓𝑖∗∗−2𝐵^T_𝑓𝑖𝑃_𝑖𝐵_𝑓𝑖+𝑈<0. (4.4)(4.5) Then, the fast adaptive fault estimation algorithm of (2.7) can be realized. And in the estimated time interval, it can estimate errors with the uniformly boundedness of the states and faults. Moreover, the observer gains are respectively as follows: 𝐿_𝑖=𝑃_𝑖⁻¹𝑋_𝑖,𝐿_𝑑𝑚𝑖=𝑃_𝑖⁻¹𝑋_𝑑𝑚𝑖,𝑚=1,2,…,𝑀. (4.6)

When there are difficulties in solving (3.1) or (4.4), we can transform them into the following SDP problems via disciplined convex programming [24]: min𝛿s.t.𝛿𝐼𝐵^T_𝑓𝑖𝑃_𝑖−𝐻_𝑖𝐶_𝑖𝑃_𝑖𝐵_𝑓𝑖−𝐶^T_𝑖𝐻^T_𝑖𝛿𝐼>0. (4.7) In order to make 𝐵^T_𝑓𝑖𝑃_𝑖 approximate to 𝐻_𝑖𝐶_𝑖 with a satisfactory precision, we can firstly select a sufficiently small scalar 𝛿>0 to meet (4.7).

Remark 4.2. The solutions of Theorems 3.1 and 4.1 can be obtained by solving an optimization problem with (4.7). By using the Matlab LMI Toolbox, it is straightforward to check the feasibility of LMIs. In order to illustrate the effectiveness of the developed techniques, we will give several numerical examples about fuzzy jump system with time delays in Section 5.

5. Numeral Example

We consider the following time-delayed stochastic MJSs with parameters given by 𝐴₁=−0.5−0.3−0.1−1.0,𝐴₂=0.5−0.4−0.1−1.06,𝐴_𝑑1=0.05−0.100.05,𝐴_𝑑2=,𝐵0.07−0.10−0.05₁=𝐵₂=1.00.5,𝐵_𝑓1=𝐵_𝑓2=00.1,𝐶₁=0.2−0.1,𝐶₂=−0.20.4. (5.1) The transition rate matrix that relates the two operation modes is given as Π=^−0.50.5_0.3−0.3. By solving the LMIs in (3.1), (3.2), and (4.7) with 𝛿=8.9252×10⁻⁴, we can get the following solutions: 𝐿₁=4.58532.2033,𝐿₂=,𝐿−8.37971.6944_𝑑1=0.11100.0691,𝐿_𝑑2=,𝐻−0.2448−0.1343₁=−0.0061,𝐻₂=0.0019. (5.2)

To show the effectiveness of the designed methods, the time-delay 𝑑 is assumed to be 0.2 s, and we consider two kinds of actuator faults 𝑓₁(𝑡) and 𝑓₂(𝑡) in the simulation over the finite-time interval 𝑡∈[010]:𝑓₁(𝑡)=()𝑓0,0≤𝑡≤4,0.5sin5𝑡,4<𝑡≤10,₂(𝑡)=0.5,2𝑘−1<𝑡≤2𝑘,𝑘∈𝑁⁺,1≤𝑘≤5,0,others. (5.3)

Let 𝑟₀=2 and Γ=10, the jumping modes are shown in Figure 1. The estimated faults and estimation errors of 𝑓₁(𝑡) and 𝑓₂(𝑡) are shown in Figures 2 and 3, respectively. From the simulation results and design algorithm, it can be concluded that the adaptive fault diagnosis observer can enhance the performance of fault estimation for slow and fast time-varying faults.

Figure 1 

The estimation of changing between modes during the simulation with initial mode 2.

Figure 2 

Fault 𝑓1(𝑡) and the estimated 𝑓1(𝑡).

Figure 3 

Fault 𝑓2(𝑡) and the estimated 𝑓2(𝑡).

6. Conclusions

In this paper, we have studied the design of adaptive fault estimation observer for time-delayed MJSs. It ensures the rapidity and accuracy performance of fault estimation of the designed observer. By selecting the appropriate Lyapunov-Krasovskii function and applying matrix transformation and variable substitution, the main results are provided in terms of LMIs form and then extended to multiple time-delayed MJSs case. Simulation example demonstrates the effectiveness of the developed techniques.

Acknowledgments

This work was supported in part by the State Key Program of National Natural Science Foundation of China (Grant no. 61134007), the Key Program of Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2012A014), and Doctor Research Project of Anhui University (Grant no. 32030017).