THE Takagi Sugeno (T-S) fuzzy model proposed in [1]

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1 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 9 Fault Detection for -S Fuzzy ime-delay Systems: Delta Operator and Input-Output Methods Hongyi Li, Yabin Gao, Ligang Wu, Senior Member, IEEE, and H. K. Lam, Senior Member, IEEE Abstract his paper focuses on the problem of fault detection for akagi Sugeno fuzzy systems with time-varying delays via delta operator approach. By designing a filter to generate a residual signal, the fault detection problem addressed in this paper can be converted into a filtering problem. he time-varying delay is approximated by the two-term approximation method. Fuzzy augmented fault detection system is constructed in δ-domain, and a threshold function is given. By applying the scaled small gain theorem and choosing a Lyapunov Krasovskii functional in δ-domain, a sufficient condition of asymptotic stability with a prescribed H disturbance attenuation level is derived for the proposed fault detection system. hen, a solvability condition for the designed fault detection filter is established, with which the desired filter can be obtained by solving a convex optimization problem. Finally, an example is given to demonstrate the feasibility and effectiveness of the proposeethod. Index erms Delta operator, fault detection, filter design, -S fuzzy system, time-varying delay. I. INRODUCION HE akagi Sugeno -S fuzzy model proposed in has been recognized as an effective and popular approach to control nonlinear systems 7. By using -S fuzzy rules, a set of complex nonlinear systems can be possibly represented as a weighted sum of some simple linear subsystems. hus, it is effective to analyze and synthesize nonlinear systems 8 such as in chemical processes, automotive systems, robotics systems, anany manufacturing processes. During the past few decades, the results on stability analysis and controller synthesis based on -S fuzzy model approach Manuscript received November 7, 3; revisearch 5, 4; accepted April 3, 4. Date of publication June 5, 4; date of current version January 3, 5. his work was supported in part by the National Natural Science Foundation of China under Grant 6333, Grant 63, Grant 6746, Grant 63, and Grant 663, in part by the Fok Ying ung Education Foundation under Grant 459, in part by the Program for New Century Excellent alents in University under Grant NCE-3-696, in part by the NKSP under Grant BAF9G, in part by the Key Laboratory of Integrated Automation for the Process Industry, Northeast University, and in part by the Fundamental Research Funds for the Central Universities under Grant HI.BREIV.33. his paper was recommended by Associate Editor P. Shi. H. Li is with the College of Engineering, Bohai University, Jinzhou 3, China lihongyi9@gmail.com. Y. Gao is with the College of Information Science and echnology, Bohai University, Jinzhou 3, China gaoyb@gmail.com. L. Wu is with the Research Institute of Intelligent Control and Systems, Harbin Institute of echnology, Harbin 5, China ligangwu@hit.edu.cn. H. K. Lam is with the Department of Informatics, King s College London, London WCR LC, U.K. hak-keung.lam@kcl.ac.uk. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier.9/CYB have been developed in and 9. In, some novel delay-dependent conditions for the stability and H control problems of discrete-time fuzzy systems were obtained by employing a fuzzy-basis-dependent Lyapunov functional. Liu et al. 4 considered the problem of fuzzy-model-based fault estimations and fault-tolerant control in the simultaneous presences of sensor and actuator faults, and proposed an interesting descriptor discontinuous observer approach to solve this problem. For discrete-time -S fuzzy systems with stochastic perturbation and time-varying delay, the problems of dissipativity analysis and synthesis were concerned in. In fault detection systems, various faults are likely to be encountered, especially faults from actuators and sensors. In the past few years, the investigation of fault detection has been developed, see 4. Yin et al. 5 served as pioneers to firstly offer a comprehensive reference for basic data-driven fault diagnosis and process monitoring schemes for modern complex processes. More recently, fault detection systems have been studied via -S fuzzy model approach, and the filter design method was considered to deal with the fault detection for -S fuzzy systems, 6, 7. he fault detection problem for -S fuzzy Itô stochastic systems was studied in 7. By designing a fuzzy fault detection filter, the problem of fault detection for -S fuzzy systems with intermittent measurements was studied in. Moreover, based on the delta operator approach 8, the robust fault-detection was investigated for uncertain -S fuzzy systems, and the fuzzy fault detection filter was designed in 9. However, it should be mentioned that the effect of the time delay was not considered in 9. It is very challenging to solve the fault detection problem of -S fuzzy systems with sensor faults and timevarying delays via delta operator approach, which motivates this paper. In this paper, the problem of fault detection is investigated for -S fuzzy systems with time-varying delays via delta operator approach. Firstly, by designing a filter to generate a residual signal, the fault detection problem is considered to be converted into a filtering problem. Based on delta operator approach, in -S fuzzy system, a new model transformation is processed by the input-output IO 3, 3 approach. Secondly, fuzzy augmented fault detection system is constructed in δ-domain, and a threshold function is given. hen, by applying the scaled small gain SSG theorem and choosing a Lyapunov Krasovskii functional LKF in δ-domain, a sufficient condition of asymptotic stability with a prescribed H disturbance attenuation level is obtained for the proposed fault detection system. A new fuzzy filter is designed to guarantee c 4 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 3 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 that the proposed system is asymptotically stable with the prescribed H performance. he existence condition of the filter can be solved by convex optimization techniques. Finally, a practical example is provided to illustrate the feasibility and effectiveness of the proposeethod. he key contributions of this paper are summarized as follows. A reference model based on delta operator is selected to handle the fuzzy filter, and new augmented fuzzy system is obtained based on delta operator approach. he time-varying delay is considered in the fault detection system and new method handling the delay is proposed for the fuzzy system. 3 A fuzzy filter is designed to solve the fault detection problem for the fuzzy systems with sensor faults. he rest of this paper is organized as follows. he fault detection system is represented and some referenced definitions and lemmas are introduced in Section II. he designed threshold is given in Section III, while the new transformed model and H performance index is presented in Section IV. In Section V, stability analysis and fuzzy H filter design are proposed, and some simulation results are provided to show the effectiveness of the proposeethod in Section VI. Section VII concludes this paper. A. Notation he notation used throughout the paper is fairly standard. L, denotes the space of square-integrable vector functions over,. G G represents the series connection of mapping G and G. he notation X > respectively, X, for X R n n means that the matrix X is real symmetric positive definite respectively, positive semi-definite. he symbol * in a matrix A R n n stands for the transposed elements in the symmetric positions. he superscripts and denote the matrix transpose and inverse, respectively. he shorthand diag {M, M,...,M r } denotes a block diagonal matrix with diagonal blocks being the matrices M, M,...,M r. Identity matrices of appropriate dimensions are denoted by I. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. II. PROBLEM FORMULAION AND PRELIMINARIES A. Delta-Domain Modeling Firstly, we give a linear state-space model of continuoustime form as follows: { ẋ t = As x t + B s u t y t = C s x t + D s u t where x t R n denotes the state vector, u t R v stands for the control input, y t R m is the output. A s, B s, C s, and D s are system matrices with appropriate dimensions. According to the definition of discrete-time system, let q be the forwardshift operator, when u t is processed via a zero-order-hold, the corresponding discrete-time model from the system can be written in the following standard forward shift form: { qx k = Aq x k + B q u k y k = C q x k + D q u k where qx k = x k +, x k = x k, u k = u k, y k = y k, A q = e A s, B q = e τa sdτb s, is the sample period, C q = C s and D q = D s. When, it can be seen from system that A q I n and B q. Obviously, the forward shift representation of a discrete-time model becomes extremely sensitive to roundoff errors under fast sampling. Recently, a delta-domain δ-domain discrete-time system for a given continuous-time system was proposed in 3, and the work in 33 obtained a reliable algorithm for deriving these δ-domain discrete-time model from continuous-time model in a finite precision arithmetic. Based on the discussion in 3, the delta operator is defined by { ddt x t, = δx t = xt+ xt, =. Using the alternative discrete-time representation of delta operator, a discrete-time form of the model in is written as { δx tk = Ax t k + Bu t k 3 y t k = Cx t k + Du t k where A = A q I n / = Ϝ As, A s, B = B q / = Ϝ A s, B s, C = C q, D = D q, and Ϝ A s, = I + A s! + A 3 s 3 3! +. denotes the sample period. Let, it follows that A A s and B A s. his implies that the representation of δ-domain converges to the continuous-time form while the sampling period increases. herefore, the delta operator not only achieves better numerical conditions of the algorithms, but also guarantees that the discrete-time result converges to the continuous result as. hese features are useful to unify continuous-time and discrete-time control theory and applications. B. Delta-Operator-Based -S Fuzzy Systems Consider the following nonlinear system: ẋ t = f x t, u t 4 where f x t, u t is a known nonlinear continuous-time function satisfying f, =. Under the concept of sector nonlinearity, the nonlinear system in 4 can be represented by the following r-rule -S fuzzy system. Plant Rule i: If θ t is N i, and..., and θ j t is N ij, and..., and θ p t is N ip, then { ẋ t = Asi x t + B si u t y t = C si x t + D si u t where A si, B si, C si, and D si i =,,...,r are system matrices with appropriate dimensions. θ j t and N ij are the premise variable and the fuzzy set, respectively. Remark : In practice, the effect of time delay and disturbance should be taken into account in the modeled system. In addition, a fault signal is considered for solving the fault detection problem. In this paper, we establish the deltaoperator-based -S fuzzy model to represent a nonlinear system with time delay, disturbance input and sensor fault in the frame of the delta operator approach.

3 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 3 Based on the delta operator approach, we consider the following -S fuzzy system with time-varying delays, and formulate the i-th rule of -S model as follows. Plant Rule i: If θ t k is N i, and..., and θ j t k is N ij, and..., and θ p t k is N ip, then δx t k = A i x t k + A di x t k n k + B i w t k + B ei f t k, y t k = C i x t k + C di x t k n k + D i w t k + D ei f t k, x t k = φ t k, k = n M, n M +,..., 5 where x t k R n denotes the state variable, y t k R m stands for the measured output, w t k R l denotes the unknown disturbance input which belongs to L,, f t k R q is the fault signal to be detected and belongs to L,, and φ t k is a continuous vector-valued initial function. he parametric variable d k = n k is the bounded time-varying delay in the whole dynamic process and satisfies < d k with = n m and = n M n m and n M are the known positive integers. For conveniences we let t k = k. he matrices A i, A di, B i, B ei, C i, C di, D i, and D ei i =,,...,r are system matrices with appropriate dimensions. hen the defuzzifieodel of system 5 is inferred as follows: δx t k = r h i θ t k A i x t k + A di x t k n k + B i w t k + B ei f t k y t k = r 6 h i θ t k C i x t k + C di x t k n k + D i w t k + D ei f t k x t k = φ t k, k = n M, n M +,..., where h i θt k = μ i θt k / r μ i θt k, μ i θt k = j= p N ij θ j t k and N ij θ j t k is the degree of the membership of θ j t k in fuzzy set N ij. It can be assumed that: μ i θt k for i =,,...,r and r μ i θt k for all t k. herefore, h i θt k fori =,,...,r and r h i θt k =. In this paper, the premise variable θt k of the fuzzy model is assumed to be available. Besides, suppose that the premise variable of filter is the same as that of the original plant. Based on the parallel distributed compensation, the fuzzy filter is constructed as follows. 3 Filter Rule i: If θ t k is N i, and..., and θ j t k is N ij, and..., and θ p t k is N ip, then δx f t k = A fi x f t k + B fi y t k r f t k = C fi x f t k + D fi y t k 7 x f =, k = n M, n M +,..., where x f t k R n is the state vector, and r f t k R v is the residual signal; A i, B i, C i, and D i are filter parameters to be determined for i =,,...,r. hen the defuzzifieodel of filter 7 is inferred as follows: δx f t k = r h i Afi x f t k + B fi y t k r f t k = r h i Cfi x f t k + D fi y t k 8 where we denote h i = h i θ t k for simplicity. o improve the performance of the fault detection system, we append a weighting matrix function to the fault f s. hus, f w s = W s f s, where f s and f w s denote the Laplace transforms of f w t k and f t k, respectively. In this paper, a state-space realization of f w s = W s f s is given as follows: δx w t k = A w x w t k + B w f t k f w t k = C w x w t k + D w f t k x w =, k = n M, n M +,..., where x w t k R s is the state vector of reference model 9, and f w t k R v is reference signal, A w, B w, C w, and D w are constant matrices. Defining ξ t k = 9 x t k x f t k x w t k and e tk = r f t k f w t k, and then from 6, 8, and 9, the fault detection system can be inferred as δξ t k = r r h i h j Āij ξ t k + Ā dij ξ t k n k j= + B ij ζ t k e t k = r r h i h j C ij ξ t k + C dij ξ t k n k j= + D ij ζ t k where ζ t k = w t k f t k and A i Ā ij = B fj C i A fj A w A di Ā dij = B fj C di B i B ei B ij = B fj D i B fj D ei C dij = D fj C di C ij = D fj C i C fj C w D ij = D fj D i D fj D ei D w. o develop the main results, some definitions and lemmas are introduced as follows. Definition 34: he conditions for the asymptotic stability of a delta operator system hold: a Vxt k, with equality if and only if xt k =, b δvxt k = Vxt k + Vxt k / <, where Vxt k is a Lyapunov functional in δ-domain. For Lyapunov functional both in s-domain and z-domain, the condition a Vxt k in Definition is given for the asymptotic stability. On the other hand for the condition b, when, there exists

4 3 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 lim δv x t V x t k+ V x t k k = lim = dv x t k < dt k and when =, there exists δvxt k = Vxt k + Vxt k / = Vxt k + Vxt k <. Obviously, the Lyapunov functional in δ-domain can be reduced to the traditional Lyapunov functional in s-domain or z-domain when the time period tends to or is. Definition : he closed-loop system is said to be asymptotically stable if under ζ t k = lim ξ t k =. t k Correspondingly, the control output e = {e t k } L, when ζ = {ζ t k } L, for the asymptotic stability of closed-loop system. Definition 3: Given a scalar γ >, the closed-loop system is said to be asymptotically stable with an H performance index γ if it is asymptotically stable and e <γ ζ, = ζ L, where e = e t k e t k. Define ζ t k = γζ t k, and then the fault detection system in is rewritten as δξ t k = r r h i h j Āij ξ t k + Ā dij ξ t k n k j= + γ B ij ζ t k e t k = r r h i h j C ij ξ t k + C dij ξ t k n k j= + γ D ij ζ t k ξ =, k = n M, n M +,...,. hus, the H performance defined in is equivalent to e < ζ, = ζ L,. Remark : he auxiliary system is introduced to obtain model transformation and apply the IO approach in the next section. It can be seen that the H performance index γ of the auxiliary system is transferred into system parameters, and is equivalent to the H performance in forthe original fault detection system. Our purpose in this paper is to determine filter parameters in filter system 7 such that the fault detection system is asymptotically stable with an H performance. Lemma 34: For any of the time functions x t k and y t k δ x t k y t k = δ x t k y t k + x t k δ y t k +δ x t k δ y t k where is a time period. Lemma : For any of two positive integers r and r satisfying r r holds r r r x i M x i r r + x i Mx i r=r r=r r=r where M is a constant positive semi-definite symmetric matrix. Lemma 3 35: Given any matrices X, Y and Z with appropriate dimensions such that Y >. hen, the following inequality holds: X Z + Z X X YX + Z Y Z. Lemma 4 36: Consider an interconnection system consisting of two subsystems : ς t k = Gσ t k 3 : σ t k = ς t k where the forward subsystem is a known linear timeinvariant system with operator G mapping σ t k to ς t k,the feedback subsystem is an unknown linear time-varying one with operator I D which has a block-diagonal structure, a mapping from ς t k to σ t k. he symbol I D denotes a compact set of appropriately dimensioned time-varying matrices with a diagonal structure specified by I D = diag {μ t k I n,μ t k I n,...,μ s t k I n } where μ i t k R, μ i t k, i =,,...,s i is the position of repeated scalar. As a direct result of the small gain theorem 36 38, a sufficient condition regarding asymptotic stability of the interconnection in 3 is given as follows. Assumed that in 3 is internally stable, the closedloop system of interconnection system described by 3 is asymptotically stable for all I D if there exists R G R <, where R G = G, R =, and = diag {,,..., s } >. Remark 3: he IO approach actually is based on the application of Lemma 4, that is the SSG theorem. o apply Lemma 4 in the time-varying delay systems, it needs to derive the time delay uncertainty and put it into a feedback subsystem, which converts the original system into a feedback interconnection formulating. herefore, Lemma 4 is given to handle the H performance for the interconnection system proposed in the next section. III. HRESHOLD DESIGN Generally, a fault detection system consists of a residual generator and a residual evaluation stage including an evaluation function and a prescribed threshold 7. here are many methods to define evaluation functions and determine thresholds. Zhao et al. proposed a novel fault detection method for -S fuzzy systems with intermittent measurements. Similar to the fault detection method proposed in, in this section, the result of threshold design is given for the -S fuzzy systems based on delta operator approach. In order to detect the fault, an appropriate threshold J th can be chosen as J th = sup J r rf, k e 4 =w L, f = where J r is a residual evaluation function, and selected as

5 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 33 J r rf, k e = k=k +k e k e k=k r f t k r f t k 5 where k denotes the initial evaluation time instant, and k e denotes the evaluation time steps. hus, the occurrence of faults can be detected by comparing the following logic rule: J r f, k e > Jth Fault Alarm J r f, k e Jth No Fault. IV. MODEL RANSFORMAION In this section, we firstly handle the time-varying delay in for transforming model system. By applying the IO approach, the time-varying term ξ t k d k in system is approximated by the two-term approximation method 3 which results in a approximation error bound. In this paper, the two-term approximation ξ t k + ξ t k results in a small estimation error σ d t k = { ξ t k d k } d ξ t k + ξ t k = n k δξ t k + i n m δξ t k + i d i= n M i= n k n m = τ i ς d t k + i 6 n M n m i= n M where d =, ς d t k = δξ t k and {, i nk τ i =, i > n k. hus, by applying the IO approach, the following auxiliary system is introduced to replace system : δξ t k = Ā t k ξ t k + Ād t k ξ t k + ξ t k + dσ d t k + γ B t k ζ t k e t k = C t k ξ t k + C d t k ξ t k + ξ t k + dσ d t k + γ D t k ζ t k 7 where Ā t k = B t k = C d t k = r j= r j= r j= r h i h j Ā ij, Ā d t k = r h i h j B ij, C t k = r h i h j C dij, D t k = r j= r j= r r h i h j Ā dij r h i h j C ij j= r h i h j D ij. Based on 7, the interconnection frame is reformulated as δξ t k Ɣ ξ t k d : ς d t k = G γ σ d t k 8 e t k ζ t k d : σ d t k = d t k ς d t k, ξ = where Ɣ ξ t k = ξ t k ξ t k ξ t k and Ɣ t k = Ā t k Ād t k Ād t k Ɣ t k = C t k C d t k C d t k Ɣ t k d Ād t k γ B t k G γ Ɣ t k d Ād t k γ B t k. Ɣ t k d C d t k γ D t k t k may be considered as n m d t k : ς d t k σ d t k = τ i ς d t k + i n M n m i= n M 9 which denotes the mapping from ς d t k to σ d t k. hen Lemma 5 can be obtained as to provide the upper bound of L norm of d t k. Li and Gao 3 proposed a new model transformation of uncertain linear discrete-time systems with time-varying delays. Similar to the Lemma in 3, we give the following lemma for the -S fuzzy systems with time-varying delays via delta operator method. Lemma 5: Operator d t k in 9 guarantees the property that d t k. Proof: On the basis of equality in 6, the following inequality about σ d t k can be obtained under the zero initial condition: σ d t k n m n = n τ j ςd t m k + i τ j ς d t k + i i= n M i= n M n m n n M n m τ i ςd t k + i ς d t k + i i= n M n m n = i= n M ςd k ς d k ςd t k ς d t k = ς d t k where n = / n M n m, which means σ d t k = sup d ς d. ς d = he proof is completed. Remark 4: Based on Lemma, obviously, the L norm of d in 8 from input to output is bounded by. herefore, in the next section, the analysis of the SSG of d for the interconnection frame 8 is developed based on Lemma 4. Lemma 6: Assumed that the d is internally stable in 8, the interconnection system described by 8 is asymptotically stable and has an H performance index γ for d t k if there exists matrix X = diag { X, I } > such that X G γ X <. Proof: Considering 8 and, we have: ς d t k + e t k < σ d t k + ζ t k. hus, the above inequality combining Lemma 5 yields. Based on Lemma 4 and Lemma 5, the proof is completed.

6 34 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 Remark 5: Along the interconnection frame 8, the sufficient condition in Lemma 6 can be converted to the following condition. Assumed that d in 8 is internally table, the closed-loop system of interconnection system described by 8 is asymptotically stable with an H performance index γ for d t k if there exists matrix X = X X such that J = ς d t k Xς d t k σd t k Xσ d t k + e t k e t k ζ t k ζ t k <. herefore, on the basis of inequality, the stability analysis and filter synthesis with H performance are developed for the fault detection system in the next section. V. MAIN RESULS In this section, the stability analysis and filter design for the proposed system are investigated on the basis of new model transformation. Based on linear matrix inequality LMI approach, a sufficient condition for the solvability of the proposed H filter design problem is given for the fault detection system in 8. Firstly, the H performance stability analysis criterion is derived for fault detection system 8 inthe following theorem. heorem : Considering the fault detection system in 8, for given positive integers n m, n M < n m < n M and the time period >, system in 8 is asymptotically stable and has an H performance γ>ifthere exist symmetric matrices P >, Q >, R >, R >, S >, S > and X > with appropriate dimensions, such that for i, j =,,...,r ii < r ii + ij + ji <, j = i 3 where ij X ij =, = ij I d XĀij ij = XĀ dij XĀ dij XĀ dij X B ij d C ij C dij C dij C dij D ij PĀ ij PĀ dij ij = ij PĀ dij + d m S 4 Q R d m S PĀ d dij PĀ dij P B ij ij = PĀ dij + d d M S PĀ dij P B ij 4 Q d 4 Q 4 Q R d M S d 4 Q 3ij = d 4 Q X γ I ij ij = ij, = P + S + S 3ij ij = PĀ ij + Ā ij P + n M n m + Q +R + R S S. Proof: Choose an LKF Vt k for system asfollows: Vt k = 4 V s t k s= where V t k = ξ t k Pξ t k, and V t k = V 3 t k = n M V 4 t k = + i=n m j= n m i ξ t k j Qξ tk j ξ t k i R ξ t k i + n m i ςd j= n M i j= ς d n M tk j S ς d tk j tk j S ς d tk j ξ t k i R ξ t k i where symmetric matrices P >, Q >, R >, R >, S >, S >, and ς d t k = δξ t k. Applying Lemma to V t k and along the trajectory of the system 7, it can be obtained that δv t k = δ ξ t k Pξ t k +δ ξ t k Pδ ξ t k. 4 Similarly, applying the delta operator manipulation of V t k and V 3 t k, it can be obtained as follows: δv t k = n M n M i=n m j= n M i ξ t k j Qξ tk j i=n m j= i ξ t k j Qξ tk j n M = ξ t k Qξ t k ξ t k i Qξ t k i i=n m i=n m n M n m + ξ t k Qξ t k ξ t k nk Qξ tk nk = n M n m + ξ t k Qξ t k ξ t k + ξ t k + d σ d t k Q ξ t k + ξ t k + d σ d t k 5 δv 3 t k = nm ξ t k i R ξ t k i + n M n m ξ t k i R ξ t k i ξ t k i R ξ t k i n M ξ t k i R ξ t k i = ξ t k R + R ξ t k ξ t k R ξ t k ξ t k R ξ t k. 6

7 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 35 Along the trajectory of system 7 and applying Lemma 3 to V 4 t k, it can be obtained that δv 4 t k = n m j= n m j= i ς t k j S ς t k j i ς t k j S ς n M t k j + S ς t k j n M j= nm = ς t k S t k ς t k + n M ς t k S t k ς t k j= i ς t k j i ς t k j S ς t k j n m n M ς t k i S ς t k i ς t k i S ς t k i ς t k n m S + n M S ς t k nm ς t k i S n m nm ς t k i nm nm ς t k i S ς t k i n M = δ ξ t k S + S δ ξ t k ξ t k ξ t k S ξ t k ξ t k ξ t k ξ t k S ξ t k ξ t k. 7 For matrix P >, the following equation holds: = δ ξ t k P δξ t k Ā ij ξ t k γ B ij ζ t k ξ t k + ξ t k + dσ d t k. 8 Ādij Considering the H performance in Lemma 3, the H performance index γ of system can be established as follows: J = ς d t k Xς d t k σd t k Xσ d t k + e t k e t k ζ t k ζ t k <. hen putting δv t k involving 4 7, and 8, under zero initial condition, J can be expressed as J ς d t k Xς d t k σd t k Xσ d t k + e t k e t k ζ t k ζ t k + δv t k = η t k r r j= where η d t k = ξ t k h i h j ij ij ij η t k < ξ t k and η t k = δ ξ t k ξ t k η d t k σ d t k ζ t k. Applying the Schur s complement, ij ij ij < is equivalent to ij <. From the method in, we know that the conditions in and 3 can guarantee ij <. hus, for all nonzero ζ = ζ t k L,θ, J <, which means e < ζ. Moreover, under the conditions of the zero inputs σ d t k ζ t k =, δv t k can be expressed as δv t k = 4 δv s t k = η t k s= r j= r h i h j ij η t k where η t k = δ ξ t k ξ t k ηd t k, and PĀ ij dij ij = 3 3 PĀ dij + d M S 4 Q. 4 Q R d M S Obviously, for X > and Q >, inequalities ij < can be obtained from ij <. hus, δv t k <, which means the system in 8 is asymptotically stable. herefore, the LMIs conditions in heorem guarantee the asymptotic stability of the -S fuzzy delta operator fault detection system with H performance in 8. he proof is completed. Remark 6: Under the premise of the known system parameters, heorem provides the sufficient condition of asymptotic stability with H performance for the fault detection system 8. On the basis of heorem, the fault detection filter in 8 will be given in next step, thus, the relevant residual signal r f t k can be generated for further fault detection. In the following theorem, the H fuzzy filter design for fault detection system is investigated based on heorem. heorem : Considering the fault detection system in for given positive integers n m, n M < n m < n M and the time period >, system is asymptotically stable with H performance γ > if there exist symmetric matrices P >, W >, Q >, R >, R >, S >, S >, X >, Q >, R >, R >, S >, S >, X > and >, where Q Q Q = X X, X Q = 3 X 3 R R R = R R, R = S = R 3 S S, S S = 3 R 3 S S S 3 and Ā j, B j, C j, D j with appropriate dimensions, such that the following LMIs hold for i, j =,,...,r: ii < 9 r ii + ij + ji <, j = i 3 where ij = ij ij, ij = ij ij ij = ij, ij = ij ij ij 3ij

8 36 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 X ϒ = X P W,ϒ = W I ϒ ij = Bij ϒBij 7 γ I ϒBeij ϒBeij 7 γ I ϒ Aij ϒ Adij ϒ Adij ij = A w ϒ Cij C w ϒ Cdij ϒ Cdij d ϒ Adij ϒ Bij ϒ Beij ij = d ϒ Cdij D j D i D j D ei D w ϒ Aij ϒ Adij A w ij = ij 6ij S 3 3 ϒ d Adij ϒ Adij d 7ij ij = ϒ Adij S 4 Q d 4 Q 4 Q d 4 Q 4 d 4 Q 3ij = 4 d 4 Q 5 5 PA ϒ Aij = i + B j C i Ā j W A i + B j C i Ā j PA ϒ Adij = di + B j C di W A di + B j C di PB ϒ Bij = i + B j D i W B i + B j D i,ϒ Cdij = D j C di PB ei + B j D ei W,ϒ B ei + B j D Cij = D j C i C j ei ϒ Beij = = ϒ + S + S = + S + S ij = ϒ Aij + ϒAij + n M n m + Q + R + R S S = A w + A w + n M n m + Q +R + R S S 3 = 4 Q R S 3 = 4 Q R S 4 = 4 Q R S, 4 = 4 Q R S 5 = d 4 Q X, 5 = d 4 Q X 6ij = ϒ Adij + S, 7ij = ϒ Adij + S. Moreover, if the above inequalities have a feasible solution, then the matrices for the desired H filter in the form of 7 for system can be designed as Afj B fj W = C fj D fj I Āj B j. 3 C j D j Proof: Firstly, similar to heorem, from 9 3, we have ij <. Let = 3 where >, >, and is invertible otherwise, can be made invertible through slight perturbation. Let I W =. 3 Define the following variables: P =, W = 3 Q = W Q W, X = W X W and R = W R W, R = W R W S = W S W, S = W S W Ā j = A fj 3, B j = B fj 3 C j = C fj 3, D j = D fj. 33 By premultiplying and postmultiplying { } diag W, I,I, W, I, W, I, W, I, W, I, W, I,I,I and its transpose, respectively, to ij <, one can have ˆ ij < where ˆ ˆ ˆ ij = ij ˆ ij with X ˆ = X I ˆ ij = ˆ ij ˆ ij, ˆ ij = ˆ ij ˆ ij ˆ ˆ ij = ij ˆ ij ˆ 3ij d Ā dij B ij B eij ˆ ij = d C dij D fj D i D eij Ā d dij Ā dij d 7ij Ā dij ˆ ij = S 4 Q d 4 Q 4 Q d 4 Q

9 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 37 4 d 4 Q ˆ 3ij = 4 d 4 Q 5 5 B ˆ ij ij = B ij 7 γ I B eij B eij 7 γ I Ai Ā ij =, B fj C i A fj Adi Ā dij = B fj C di Bi B ij =, B fj D i Bei B eij = B fj D ei C ij = D fj C i C fj, C di = D fj C di = + S + S ij = Ā ij + Ā ij + nm n m + Q +R + R S S 3 = 4 Q R S 4 = 4 Q R S D eij = D fj D ei D w, 5 = d 4 Q X 6ij = Ā dij + S 7ij = Ā dij + S Ā ij Ā dij Ā dij ˆ ij = A w C ij C w C dij C dij Ā ij Ā dij A w ˆ ij = ij 6ij S. 3 3 Next, let P = diag { n n, s s } >, and Q = diag { } Q n n, Q s s > X = diag { } X n n, X s s > R = diag { } R n n, R s s > R = diag { } R n n, R s s > S = diag { } S n n, S s s > S = diag { } S n n, S s s >. hus, matrix ˆ ij can be rewritten as ˇ ˇ ˇ ij = ij ij where X P ˇ = I d PĀij ˇ ij = PĀ dij PĀ dij PĀ dij P B ij d C ij C dij C dij C dij D ij which means ˇ ij <. For X >, the inequality X P X X P holds, then PX P X P. 34 By utilizing inequality 34, and performing a congruence transformation by diag { XP, I, I, I, I, I, I, I }, it can meet the condition ij < in heorem, which can guarantee the asymptotic stability for the fault detection system with H performance in. In addition, it can be seen from 3 and 33 that Afj B fj = Āj B j C fj D fj I C j D 3 j I = 3 W I I Āj B j C j D j I It can be found that 35 is a similarity transformation from the filter in 7. Let 3 = I, which yields the filter parameters in 3. herefore, all conditions in heorem are satisfied. he proof is completed. Remark 7: heorem provides a sufficient condition for the solvability of H filter design for a class of -S fuzzy systems with time-varying delay and disturbance input. Definitely, a desired filter in 3 can be determined by solving the following convex optimization problem: min ρ subject to 9 and 3 with ρ = γ. 36 Remark 8: From the results of heorem, the residual signal r f t k can be obtained, then a fault detection measure can be set up, and accordingly, the residual evaluation function J r f, k e and threshold Jth can be selected as the form of 5 and 4, respectively. Remark 9: Gao et al. 39 first time proposed a novel basis dependent Lyapunov function method for the fuzzy control problem of nonlinear systems under unreliable communication links. By this approach, the results proposed in this paper can be significantly improved, which is our future work. VI. NUMERICAL EXAMPLE In this section, a practical system 4 is used to demonstrate the effectiveness of the proposed fault detection design method. Fig. depicts the tunnel diode circuit, which is characterized by i D t k =.v D t k +.v 3 D t k. Let x t k = v C t k and x t k = i L t k be the state variables, the state equations is derived as follows: { Cẋ t k =.x t k.x 3 t k + x t k Lẋ t k = x t k Rx t k + w t k

10 38 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 Fig.. unnel diode circuit. Fig.. MSVs of the fault detection system. where w t k is the disturbance input. And with the parameters C = mf, L = mh, and L = in 4, the nonlinear system is rewritten as { ẋ t k =.x t k.5x t k x t k + 5x t k, 37 ẋ t k = x t k x t k + w t k. A two-rule fuzzy system is utilized to approximate the nonlinear system 37. In order to use the nonlinear system to test the effectiveness of the proposed design method, we consider the time delay and the sensor fault. When time period is chosen as =. s, the -S fuzzy systems in δ-domain can be represented as follows. A. Plant Rule If x t k 3, then { δx tk = A x t k + A d x t k n k + B w t k + B e f t k y t k = C x t k + C d x t k n k + D w t k + D e f t k. B. Plant Rule If x t k > 3, then { δx tk = A x t k + A d x t k n k + B w t k + B e f t k y t k = C x t k + C d x t k n k + D w t k + D e f t k where rr A =, B = A =, B =.95 and, the membership functions are assumed as x t k + 3 /3, 3 x t k h θ t k = 3 x t k /3, x t k 3, else h θ t k = h θ t k. In addition, the other system parameters are given as follows:...8 A d =, C.3.4 =.4 C d =.4., D = A d =, C.4.3 =. C d =.3., D =.. In order to design a fuzzy filter in 8 to verify the effectiveness of the proposeethod, we assume capacitor voltage is abnormal, and the abnormal matrices are given as B e = B e =.5., De = D e =.. he system matrices of reference model 9 are chosen as A w =., B w =., C w =., D w =.4 and the time delay bounds may be chosen as =. s and =.4 s. hus, under the conditions in heorem, by using the MALAB Control oolbox to solve the optimization problem in 36, it can be obtained the H performance level γ min =.4. Fig. depicts the maximum singular values of the fault detection system in the sense of the disturbance attenuation performance level. Moreover, the matrix parameters of the fuzzy filter in 8 are given as follows: A f = B f = C f = 3.. A f = B f = C f = , D f = , D f = Next, under the zero initial condition x t k =,adisturbance input w t k is assumed as random disturbance shown in Fig. 3, where {.randn,.3 tk.8 w t k =, else. In order to analyze the dynamic features of the fault detection system, a fault signal f t k is assumed as {.,.4 tk.7 f t k =, else. Fig. 4 plots the response of the weighting fault signal reference signal. Fig. 5 plots the responses of the residual signal without fault case and with fault case, respectively. he different error values e t k = r f t k f w t k are depicted in Figs. 6 and 7. When the residual signal is generated, the next step is to

11 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 39 Fig. 3. Unknown disturbance input w t k. Fig. 6. Error value e t k without fault. Fig. 4. Weighting fault signal f w t k. Fig. 7. Error value e t k with fault. Fig. 5. Residual signal r f t k. Fig. 8. State response of the fault detection filter. process the fault detection measure. According to the threshold in 4, the threshold of the fault detection system is J th = sup J r rf, = =w L,f = = r f t k r f t k From the result that J r rf, = J r rf, = k= r f t k r f t k = r f t k r f t k = obviously, J r rf, < J th < J r rf,, then we know that the appeared fault can be detected after.4 s. Fig. 9 illustrates that the fault detection filter can detect the fault immediately and effectively when fault occurs under the disturbance input. Finally, Fig. 8 plots the states responses of the filter. hese simulation results show the effectiveness of the proposed fault detection method. Fig. 9. Response of evaluation function J r. VII. CONCLUSION In this paper, the problem of fault detection has been investigated for the fault detection system with sensor faults and time-varying delays via delta operator approach. he fault detection problem has been expressed as a filter design problem by designing a filter to generate a residual signal. Firstly, the time-varying delay has been approximated by using the two-term approximation method. Secondly, the fuzzy augmented fault detection system has been constructed in δ-domain. By applying the SSG theorem, sufficient conditions of asymptotic stability with a prescribed H disturbance

12 4 IEEE RANSACIONS ON CYBERNEICS, VOL. 45, NO., FEBRUARY 5 attenuation level have been obtained for the resulting augmented fault detection system. A fuzzy filter has been designed to guarantee that the fault detection system is asymptotically stable with H performance. he existence condition of the filter can be formulated as a convex optimization problem. Finally, a practical example has been given to show the effectiveness of the proposeethod. In future work, the actuator faults and the stochastic disturbance will be considered in the fault detection problem. In addition, the fuzzy weighting-dependent approach to fault detection will be investigated for fuzzy systems to overcome the conservativeness. REFERENCES. akagi an. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE rans. Syst., Man, Cybern., vol. 5, no., pp. 6 3, Feb H. Gao, X. Liu, and J. Lam, Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay, IEEE rans. Syst., Man, Cybern. B, Cybern., vol. 39, no., pp , Apr H. Gao and. Chen, Stabilization of nonlinear systems under variable sampling: A fuzzy control approach, IEEE rans. Fuzzy Syst., vol. 5, no. 5, pp , Oct H.-N. Wu and H.-X. Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with timevarying delay, IEEE rans. Fuzzy Syst., vol. 5, no. 3, pp , Jun S. ong, Y. Li, Y. Li, and Y.-J. Liu, Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems, IEEE rans. Syst., Man, Cybern. B, Cybern., vol. 4, no. 6, pp , Dec.. 6 S. ong, B. Huo, and Y. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE rans. Fuzzy Syst., vol., no., pp. 5, Feb Y.-J. Liu, S. ong, and C. P. Chen, Adaptive fuzzy control via observer design for uncertain nonlinear systems with unmodeled dynamics, IEEE rans. Fuzzy Syst., vol., no., pp , Apr C.-H. Lee and W.-Y. Lai, Nonlinear control of benchmark problems using SK-type fuzzy neural network, Neural Comput. Appl., vol. 3, no., pp , 3. 9 C.-H. Lee, F.-Y. Chang, and C.-M. Lin, An efficient interval type- fuzzy CMAC for chaos time-series prediction and synchronization, IEEE rans. Cybern., vol. 43, no. 3, pp , Mar. 3. H. Li, J. Yu, C. Hilton, and H. Liu, Adaptive sliding mode control for nonlinear active suspension vehicle systems using -S fuzzy approach, IEEE rans. Ind. Electron., vol. 6, no. 8, pp , Aug. 3. J. Qiu, G. Feng, and J. Yang, Improved delay-dependent H filtering design for discrete-time polytopic linear delay systems, IEEE rans. Circuits Syst. II, Exp. Briefs, vol. 55, no., pp. 78 8, Feb. 8. B. Zhang and S. Xu, Delay-dependent robust H control for uncertain discrete-time fuzzy systems with time-varying delays, IEEE rans. Fuzzy Syst., vol. 7, no. 4, pp , Aug J. Qiu, G. Feng, and J. Yang, A new design of delay-dependent robust H filtering for discrete-time -S fuzzy systems with timevarying delay, IEEE rans. Fuzzy Syst., vol. 7, no. 5, pp , Oct M. Liu, X. Cao, and P. Shi, Fault estimation and tolerant control for fuzzy stochastic systems, IEEE rans. Fuzzy Syst., vol., no., pp. 9, Apr X. Su, P. Shi, L. Wu, and Y.-D. Song, A novel control design on discrete-time akagi Sugeno fuzzy systems with time-varying delays, IEEE rans. Fuzzy Syst., vol., no. 4, pp , Aug M. R. Soltanpour, B. Zolfaghari, M. Soltani, an. H. Khooban, Fuzzy sliding mode control design for a class of nonlinear systems with structured and unstructured uncertainties, Int. J. Innov. Comput. Inf. Contr., vol. 9, no. 7, pp , 3. 7 H. Lin, Y. Xu, and Y. Zhao, Frequency analysis of -S fuzzy control systems, Int. J. Innov. Comput. Inf. Contr., vol. 9, no., pp , 3. 8 X. Su, P. Shi, L. Wu, and Y.-D. Song, A novel approach to filter design for -S fuzzy discrete-time systems with time-varying delay, IEEE rans. Fuzzy Syst., vol., no. 6, pp. 4 9, Dec.. 9 M. Liu, X. Cao, and P. Shi, Fuzzy-model-based fault-tolerant design for nonlinear stochastic systems against simultaneous sensor and actuator faults, IEEE rans. Fuzzy Syst., vol., no. 5, pp , Oct. 3. L. Wu, X. Yang, and H. K. Lam, Dissipativity analysis and synthesis for discrete-time -S fuzzy stochastic systems with time-varying delay, IEEE rans. Fuzzy Syst., vol., no., pp , Apr. 3. Y. Zhao, J. Lam, and H. Gao, Fault detection for fuzzy systems with intermittent measurements, IEEE rans. Fuzzy Syst., vol. 7, no., pp , Apr. 9. M. Zhong, S. Li, and Y. Zhao, Robust H fault detection for uncertain LDV systems using Krein space approach, Int. J. Innov. Comput. Inf. Contr., vol. 9, no. 4, pp , 3. 3 Z. Gao, X. Wang, and Z. Kai, A study of negative selection algorithmbased motor fault detection and diagnosis, Int. J. Innov. Comput. Inf. Contr., vol. 9, no., pp , 3. 4 S. Yin, H. Luo, and S. Ding, Real-time implementation of faulttolerant control systems with performance optimization, IEEE rans. Ind. Electron., vol. 64, no. 5, pp. 4 4, May 4. 5 S. Yin, S. X. Ding, A. Haghani, H. Hao, and P. Zhang, A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark ennessee Eastman process, J. Process Contr., vol., no. 9, pp ,. 6 S. K. Nguang, P. Shi, and S. Ding, Fault detection for uncertain fuzzy systems: An LMI approach, IEEE rans. Fuzzy Syst., vol. 5, no. 6, pp. 5 6, Dec L. Wu and D. Ho, Fuzzy filter design for Itô stochastic systems with application to sensor fault detection, IEEE rans. Fuzzy Syst., vol. 7, no., pp. 33 4, Feb C. P. Neuman, ransformations between delta and forward shift operator transfer function models, IEEE rans. Syst., Man, Cybern., Syst., vol. 3, no., pp , Jan./Feb H. Yang, X. Li, Z. Liu, and C. Hua, Fault detection for uncertain fuzzy systems based on the delta operator approach, Circuits Syst. Signal Process., vol. 33, no. 3, pp , Mar K. Gu, Y. Zhang, and S. Xu, Small gain problem in coupled differentialdifference equations, time-varying delays, and direct Lyapunov method, Int. J. Robust Nonlinear Contr., vol., no. 4, pp ,. 3 X. Li and H. Gao, A new model transformation of discrete-time systems with time-varying delay and its application to stability analysis, IEEE rans. Autom. Control, vol. 56, no. 9, pp. 7 78, Sep.. 3 R. H. Middleton and G. C. Goodwin, Digital Control and Estimation: A Unified Approach, vol. 56. Englewood Cliffs, NJ, USA: Prentice Hall, C. P. Neuman, Properties of the delta operator model of dynamic physical systems, IEEE rans. Syst., Man, Cybern., Syst., vol. 3, no., pp. 96 3, Jan./Feb J. Qiu, Y. Xia, H. Yang, and J. Zhang, Robust stabilisation for a class of discrete-time systems with time-varying delays via delta operators, IE Contr. heory Appl., vol., no., pp , L. Xie and E. de Souza Carlos, Robust H control for linear systems with norm-bounded time-varying uncertainty, IEEE rans. Autom. Control, vol. 37, no. 8, pp. 88 9, Aug K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, vol. 4. Upper Saddle River, NJ, USA: Prentice Hall, C. A. Desoer an. Vidyasagar, Feedback Systems: Input-Output Properties, vol. 55. Philadelphia, PA, USA: SIAM, G. Zames, On the input-output stability of time-varying nonlinear feedback systems part one: Conditions derived using concepts of loop gain, conicity, and positivity, IEEE rans. Autom. Control, vol., no., pp. 8 38, Apr H. Gao, Y. Zhao, and. Chen, H fuzzy control of nonlinear systems under unreliable communication links, IEEE rans. Fuzzy Syst., vol. 7, no., pp , Apr S. K. Nguang and W. Assawinchaichote, H filtering for fuzzy dynamical systems with D stability constraints, IEEE rans. Circuits Syst. I, Fundam. heory Appl., vol. 5, no., pp , Nov. 3.

13 LI et al.: FAUL DEECION FOR -S FUZZY IME-DELAY SYSEMS: DELA OPERAOR AND INPU-OUPU MEHODS 4 Hongyi Li received the B.S. an.s. degrees in mathematics from Bohai University, Jinzhou, China, in 6 and 9, respectively, and the Ph.D. degree in intelligent control from the University of Portsmouth, Portsmouth, U.K., in. He is currently a Professor with the College of Engineering, Bohai University. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong and Hong Kong Polytechnic University, Hong Kong, from June to September and from September to December, respectively. His current research interests include fuzzy control, robust control, and their applications. Prof. Li is also an Associate Editor/Editorial Boarember for several international journals including Neurocomputing, Circuits, Systems, and Signal Processing, andshock and Vibration. Yabin Gao received the B.S. degree in management from Bohai University, Jinzhou, China, in, and is pursuing the M.S. degree in control theory from the Automation Research Institute and the College of Information Science and echnology, Bohai University. His current research interests include fuzzy control. H. K. Lam M 98 SM received the B.Eng. Hons. and Ph.D. degrees from the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, in 995 and, respectively. From to 5, he was a Post-Doctoral Fellow and a Research Fellow with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, respectively. In 5, he joined King s College London, London, U.K., as a Lecturer and currently is a Senior Lecturer. His current research interests include intelligent control systems and computational intelligence. Dr. Lam has served as a Program Committee Member and International Advisory Boarember for various international conferences and a Reviewer for various books, international journals, and international conferences. He is an Associate Editor for IEEE RANSACIONS ON FUZZY SYSEMS, International Journal of Fuzzy Systems, andjournal of Intelligent Learning Systems and Applications; and a Guest Editor for a number of international journals. He is on the editorial board of a number of journals including IE Control heory and Applications. He is the co-editor for two edited volumes Control of Chaotic Nonlinear Circuits World Scientific, 9 and Computational Intelligence and Its Applications World Scientific,, and the co-author of the book Stability Analysis of Fuzzy-Model-Based Control Systems Springer,. Ligang Wu SM received the B.S. degree in automation from the Harbin University of Science and echnology, Harbin, China, in, the M.E. degree in navigation guidance and control from the Harbin Institute of echnology, Harbin, in 3, and where he received the Ph.D. degree in control theory and control engineering, in 6. From 6 to 7, he was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong. From 7 to 8, he was a Senior Research Associate with the Department of Mathematics, City University of Hong Kong, Hong Kong. From to 3, he was a Research Associate with the Department of Electrical and Electronic Engineering, Imperial College London, London, U.K. In 8, he joined the Harbin Institute of echnology as an Associate Professor and was then promoted to a Professor, in. His current research interests include complex hybrid dynamical systems, sliding mode control, optimal filtering, anodel reduction. Dr. Wu currently serves as an Associate Editor for a number of journals including IEEE RANSACIONS ON AUOMAIC CONROL, Information Sciences, Signal Processing, andie Control heory and Applications. Heis also an Associate Editor for the Conference Editorial Board, IEEE Control Systems Society. He is the author of the book Sliding Mode Control of Uncertain Parameter-Switching Hybrid Systems Wiley, 4.