Adaptive Observer-Based Fault Estimation for Stochastic Markovian Jumping Systems

This paper studies the adaptive fault estimation problems for stochastic Markovian jump systems (cid:2) MJSs (cid:3) 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 su ﬃ cient 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. improving the rapidity of fault estimation. The aug-mented dynamic system is ﬁrstly constructed based on the adaptive fault estimation observer, and the observer parameters are designed on the system modes. Su ﬃ cient 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 (cid:2) LMIs (cid:3) (cid:5) 22 (cid:6) , 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 e ﬀ ectiveness of the developed techniques.


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 where Δt > 0 and lim Δt↓0 o 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 : where x t ∈ n is the state, y t ∈ m is the measured output, u t ∈ l is the controlled input, f t ∈ p is the unknown actuator fault, and we assume that its derivative is the norm bounded with ḟ t ≤ f, wherein 0 ≤ f < ∞. d > 0 is the time delay constant, η t ∈ n is a continuous vector-valued initial function assumed to be continuously differentiable on −d 0 , and r 0 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 Refer to observer design 2-4, 7 and consider the following systems: where x t ∈ n is the observer state vector, y t ∈ m is the observer output vector, and f t ∈ p is an estimate of actuator fault of f t . Denote e f t f t − f t .
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 termż t in estimation equation, that is, 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 t f z τ dτ, it is only an integral term in essence. It fails to deal with time-varying faults, that is,ḟ t / 0 though it assumes that the constant fault ḟ t 0 estimation is unbiased. In this paper, we add a derivative termż 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.

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: 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 Abstract and Applied Analysis 5 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 : in which P i ∈ n×n , Q ∈ 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: Given a symmetric positive definite matrix U, we can use the following relation:

3.7
By letting X i P i L i and X di P i L di , the derivative of V e t , e f t , i with respect to time follows that Thus, it concludes that V e t , e f t , i ≤ −λ ζ t 2 η, wherein λ min r∈Λ σ min −Π i . Obviously, we can get V 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,ḟ t 0, then the designed adaptive algorithm can achieve asymptotical convergence from 3.8 . Then we can get that V 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 n 2 0 ∞ . Then, lim t → ∞ e t → 0 holds by Barbalat's Lemma.

Extension to Multiple Time-Delayed MJSs
Consider the following multiple time-delayed MJSs over the probability space Ω, F, ρ : x t η t , r t r 0 , t ∈ − max d m 0 ,  Similar to Section 3, the following observer can be constructed: where x t ∈ n is the observer state vector, y t ∈ m is the observer output vector, and f t ∈ 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 Prior to the design of an adaptive diagnostic law, we can get the following results for multiple time-delayed MJSs 4.1 .
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: In order to make B T fi P i approximate to H i C 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.

Numeral Example
We consider the following time-delayed stochastic MJSs with parameters given by

5.1
The transition rate matrix that relates the two operation modes is given as Π

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 f 1 t and f 2 t in the simulation over the finite-time interval t ∈ 0 10 : 0.5 sin 5t , 4 < t ≤ 10,

5.3
Let r 0 2 and Γ 10, the jumping modes are shown in Figure 1. The estimated faults and estimation errors of f 1 t and f 2 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. Jumping modes t (s) Figure 1: The estimation of changing between modes during the simulation with initial mode 2.

Conclusions
In this paper, we have studied the design of adaptive fault estimation observer for timedelayed 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.