Robust fuzzy observer-based fault detection for nonlinear systems with disturbances

Author’s Accepted Manuscript Robust fuzzy observer-based fault detection for nonlinear systems with disturbances Linlin Li, Steven X. Ding, Ying Yang, Yong Zhang

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S0925-2312(15)01456-3 http://dx.doi.org/10.1016/j.neucom.2015.09.102 NEUCOM16175

To appear in: Neurocomputing Received date: 14 May 2015 Revised date: 10 September 2015 Accepted date: 28 September 2015 Cite this article as: Linlin Li, Steven X. Ding, Ying Yang and Yong Zhang, Robust fuzzy observer-based fault detection for nonlinear systems with disturbances, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.09.102 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust fuzzy observer-based fault detection for nonlinear systems with disturbances Linlin Lia , Steven X. Dinga , Ying Yangb,∗, Yong Zhangb,a a Institute

for Automatic Control and Complex Systems (AKS), Faculty of Engineering, University of Duisburg-Essen, 47057 Duisburg, Germany b State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, P.R. China

Abstract This paper studies the integrated design of fuzzy observer-based fault detection (FD) for nonlinear systems in the presence of external disturbances. To this end, we first approximate the nonlinear systems by a set of Takagi-Sugeno (T-S) fuzzy models. Then, the observer-based FD systems are studied with the aid of L2 stability theory. In the end, a numerical example is given to show the efficiency of the proposed method. Keywords: Nonlinear systems, fault detection (FD), observer-based residual generator, Takagi-Sugeno (T-S) fuzzy technique

1. Introduction Over the past decades, fault detection and diagnosis (FDD) have received considerably increasing attention [1, 2, 3, 4, 5, 6, 7], for the purpose of meeting high requirements on system performance and reliability from industrial processes. Since nonlinearity is considered to exist universally in real plants, a great deal of efforts have been paid to the observer-based fault detection (FD) and fault-tolerant control (FTC) for nonlinear systems [8, 9, 10, 11, 12]. More particularly, the integrated design of the nonlinear observer-based FD systems, which consists of an observer-based residual generator, a residual evaluator and a decision maker with an embedded threshold, have been extensively investigated in recent years [13, 14, 15, 16]. Takagi-Sugeno (T-S) Fuzzy technique, as an alternative approach to conventional control techniques, has been proved to be a powerful tool to analyze and approximate complex nonlinear systems [17, 18, 19, 20]. Encouraged by these studies, significant research efforts have been devoted to solve the controller [21] and filtering design problems [22, 23, 24, 25] for nonlinear systems. Besides, as reported in the literatures [26, 27, 28, 29], a great deal of attention has been paid to the construction of nonlinear residual generators based on the obtained universal T-S fuzzy model. Over the past few years, a trend of approximating general type of nonlinear systems to any degree of accuracy on any compact set by invoking control variables in the premise ∗ Corresponding

author. Tel: +86 10 6275 1815; fax: +86 10 6276 4044 Email addresses: [email protected] (Linlin Li), [email protected] (Steven X. Ding), [email protected] (Ying Yang), [email protected] (Yong Zhang) Preprint submitted to Neurocomputing October 6, 2015

variables have been observed [30]. In spite of the result mentioned in [30], the synthesis and construction of universal fuzzy controllers for general nonlinear systems have been investigated in [31, 32]. The main objective of this paper is to devoted to the integrated design of T-S fuzzy observerbased FD systems for general type of complex nonlinear systems involving external disturbances. To this end, the nonlinear plant is first interpreted in light of a class of generalized T-S dynamic fuzzy models. Then the integrated design procedure of FD systems is carried out by developing a fuzzy observer-based residual generator, an evaluator and a decision logic with the aid of L2 stability theory. The paper is organized as follows. The needed preliminaries and problem formulation are addressed in Section 2. In Section 3, robust fuzzy observer-based fault detection for nonlinear systems with external disturbances are presented. A numerical example is given in Section 4 to illustrate the effectiveness of the proposed method. Conclusions and future work are stated in Section 5. In addition, standard notations are adopted in this paper. R+ = [0, ∞). || · || denotes the Euclidean norm of a vector in some Euclidean space. u2 denotes L2 -norm of u(t) which is  ∞ 1/2 defined as u2 = 0 ||u(t)||2 dt . Sym{A} denotes A + AT . In a symmetric matrix,  represents the symmetric elements. 2. Problem formulation Generally speaking, a standard observer-based FD system consists of an observer-based residual generator, a residual evaluator and a decision maker with a threshold [5]. Along this line, numerous results have been proposed for the integrated design of observer-based FD systems for linear time-invariant (LTI) systems [5, 6]. Motivated by the integrated design schemes of observer-based fault detection systems for LTI systems, the major objective of this paper is to investigate the integrated design of robust fuzzy observer-based FD issues for general type of nonlinear systems with external disturbances, which is described by ⎧ ⎪ ⎪ ⎨ x˙ = f (x, u) + g(x, u)d Σ:⎪ (1) ⎪ ⎩y = h(x, u) + k(x, u)d where x(t) = [x1 (t) · · · xkx (t)]T ∈ Rkx , u = [u1 (t) · · · uku (t)]T ∈ Rku , y = [y1 (t) · · · yky (t)]T ∈ Rky denote the state, output and input vectors, respectively. f (x, u), h(x, u), g(x, u) and k(x, u) are continuously differentiable nonlinear functions with appropriate dimensions. d ∈ Rkd denotes the disturbances which is assumed to be L2 -bounded with d2 ≤ δd

(2)

Inspired by the Takagi-Sugeno (T-S) Fuzzy modelling technique [30, 31] for nonlinear systems, our first objective is to establish the T-S Fuzzy model for nonlinear systems (1). Then, the robust L2 fuzzy observer-based residual generator will be first investigated. As a result, an integrated FD system will be constructed, which consists of a residual generator, an evaluation function and a decision logic with a dynamic threshold. Our further task is dedicated to the robust L2 observer-based FD system in the presence of external disturbances for nonlinear systems (1). 2

3. Design of L2 robust fuzzy observer-based FD systems 3.1. Fuzzy dynamic modelling For our purpose, the following class of general T-S fuzzy models are employed to approximate nonlinear systems (1) first: Plant rule i : IF θ1 (t) is N1i and θ2 (t) is N2i and · · · and θ p (t) is N pi ⎧ ⎪ ⎪ ⎨ x˙(t) = Ai x(t) + Bi u(t) + Ei d(t) THEN ⎪ (3) ⎪ ⎩y(t) = Ci x(t) + Di u(t) + Fi d(t), i ∈ {1, 2, · · · , κ} i the ith fuzzy inference rule; κ denotes the number of inference rules; θ(t) =

where  represents θ1 (t) · · · θ p (t) denotes the premise variables assumed measurable; N ij ( j = 1, 2, · · · , p) indicates the fuzzy sets; Ai , Bi , Ci , Di , Ei and Fi are system matrices with appropriate dimensions; x(t), u(t) and y(t) denote the system state, input and output variables, respectively. d(t) represents the external disturbance variables. Let μi (θ(t)) be the normalized fuzzy membership function of the inferred fuzzy set N i := p i l=1 Nl , which is defined as p Nli μi (θ(t)) = κ l=1 . (4) p j j=1 l=1 Nl

Consequently, we have μi (θ(t)) ≥ 0, i = 1, 2, · · · , κ,

κ

μi (θ(t)) = 1.

(5)

i=1

Thus by using a center average defuzzifier, a singleton fuzzifier and product inference, the T-S fuzzy system in (3) can be inferred as follows: ⎧ ⎪ ⎪ ⎨ x˙(t) = f¯(x, u) (6) ⎪ ⎪ ¯ u) ⎩y(t) = h(x, where f¯(x, u) = ¯ u) = h(x,

κ

i=1 κ

μi (θ(t)) (Ai x(t) + Bi u(t) + Ei d(t)) μi (θ(t)) (Ci x(t) + Di u(t) + Fi d(t)) .

(7)

i=1

In what follows, the approximation capability of the T-S fuzzy models in (3) will be addressed based on the results given in [30, 31]. To this end, the following lemma is introduced first. Lemma 1. [30] If vector value function s(z) = [s1 (z1 , · · · , zN ) · · · sn (z1 , · · · , zN )]T is m(≥ 1) time continuously differentiable on Z with s(0) = 0, then for i = 1, · · · , N, the vector value function Qi (z) = qi (z1 , · · · , zN ) ⎧ s(0,··· ,0,z ,z ,··· ,z )−s(0,··· ,0,z ,··· ,z ) i i+1 N i+1 N ⎪ ⎪ , ⎨ zi =⎪ ∂s(0,··· ,0,z ,··· ,z ) ⎪ i+1 N ⎩ , ∂zi

3

(8) zi  0 zi  0

is m − 1 continuously differentiable on Z and s(z) =

N

Qi (z)zi =

i=1

N

qi (z1 , · · · , zN )zi

(9)

i=1

Theorem 1. Consider nonlinear systems given in (1), where f (x, u) and g(x, u) are continuously differentiable on the compact set X × U and f (0, 0) = 0, g(0, 0) = 0. Then, for any positive  i , i = 1, · · · , 4 and any (x, u) ∈ X × U, there exist T-S fuzzy models (7) such that f (x, u) + g(x, u)d = f¯(x, u) + ΔA (x, u)x + ΔB (x, u)u + ΔE (x, u)d ¯ u) + ΔC (x, u)x + ΔD (x, u)u + ΔF (x, u)d h(x, u) + k(x, u)d = h(x,

(10)

with ||ΔA (x, u)|| ≤  1 , ||ΔB (x, u)|| ≤  2 ||ΔC (x, u)|| ≤  3 , ||ΔD (x, u)|| ≤  4 ||ΔE (x, u)|| ≤  5 , ||ΔF (x, u)|| ≤  6 .

(11)

Proof. The proof follows directly from [31] and thus is omitted here. As a result, the nonlinear systems (1) can be also rewritten as the following T-S fuzzy models with the norm bounded uncertainties: Plant rule Ri : IF θ1 (t) is N1i and θ2 (t) is N2i and · · · and θ p (t) is N pi ⎧ ⎪ ⎪ x˙(t) = Ai x(t) + Bi u(t) + Ei d(t) + ΔA (x, u)x(t) + ΔB (x, u)u(t) + ΔE (x, u)d(t) ⎪ ⎪ ⎪ ⎨ THEN ⎪ y(t) = Ci x(t) + Di u(t) + Fi d(t) + ΔC (x, u)x(t) + ΔD (x, u)u(t) + ΔF (x, u)d(t), ⎪ ⎪ ⎪ ⎪ ⎩ i ∈ {1, 2, · · · , κ}

(12)

Thus the final state of the fuzzy system can be inferred as follows: x˙(t) = y(t) =

κ

i=1 κ

μi (θ(t)) (Ai x(t) + Bi u(t) + Ei d(t)) + ΔA (x, u)x(t) + ΔB (x, u)u(t) + ΔE (x, u)d(t) μi (θ(t)) (Ci x(t) + Di u(t) + Fi d(t)) + ΔC (x, u)x(t) + ΔD (x, u)u(t) + ΔF (x, u)d(t)

(13)

i=1

3.2. L2 Robust fuzzy observer-based residual generator By adopting the following fuzzy observer-based residual generator: Plant rule Ri : IF θ1 (t) is N1i and θ2 (t) is N2i and · · · and θ p (t) is N pi ⎧ ⎪ ⎪ x˙ˆ(t) = Ai xˆ(t) + Bi u(t) + Li (y(t) − yˆ (t)) ⎪ ⎪ ⎪ ⎨ THEN ⎪ yˆ (t) = Ci xˆ(t) + Di u(t) ⎪ ⎪ ⎪ ⎪ ⎩r(t) = y(t) − yˆ (t), i ∈ {1, 2, · · · , κ}

4

(14)

As a consequent, the overall T-S fuzzy residual generator can be further represented as the following form x˙ˆ(t) = yˆ (t) =

r

i=1 r

μi (θ(t)) (Ai xˆ(t) + Bi u(t) + Li (y(t) − yˆ (t))) μi (θ(t)) (Ci xˆ(t) + Di u(t))

i=1

r(t) =y(t) − yˆ (t)

(15)

Denote

T

T e(t) = x(t) − xˆ(t), x¯(t) = eT (t) xT (t) , z(t) = uT (t) dT (t)    



Ai − LiC j 0 0 Ei Ai, j = , Bi = , Ci = C i 0 , D i = 0 F i Bi Ei 0 Ai  

0 ΔA (x, u) − Li ΔC (x, u) , ΔC = 0 ΔC (x, u) ΔAi = 0 ΔA (x, u)  

ΔB (x, u) − Li ΔD (x, u) ΔE (x, u) − Li ΔF (x, u) , ΔD = ΔD (x, u) ΔF (x, u) ΔBi = ΔE (x, u) ΔB (x, u) we obtain x˙¯(t) =

κ

i=1

r(t) =

κ

μi (θ(t))

κ

  μ j (θ(t)) (Ai, j + ΔAi ) x¯(t) +(Bi + ΔBi )z(t)

j=1

μi (θ(t)) ((Ci + ΔC ) x¯(t) + (Di + ΔD )z(t))

(16)

i=1

The following theorem provides a design scheme for the determination of gain matrices Li , i = 1, · · · , κ, which can be applied to solve the following (robust) FD problem. Theorem 2. Given nonlinear systems (1) and the fuzzy residual generator (15). Suppose that there exists a positive matrix P and two constants α > 0, ξ > 0, such that the following inequalities hold ⎤ ⎡ PBi CTi PEi ⎥⎥ ⎢⎢⎢ PAi,i + ATi,i P + G ⎥⎥ ⎢⎢⎢ BTi P −αI + λI 0 0 ⎥⎥⎥⎥ ⎢⎢⎢ ⎥ < 0, 1 ≤ i ≤ κ ⎢⎢⎢⎢ Di −I F ⎥⎥⎥⎥ Ci ⎦ ⎣ 0 F T −ξI ET P ⎤ ⎡ PBi, j CTi, j PEi, j ⎥⎥ ⎢⎢⎢ Ξi, j ⎥⎥ ⎢⎢⎢ T 0 0 ⎥⎥⎥⎥ ⎢⎢⎢ Bi, j P −2αI + 2λI ⎥ < 0, 1 ≤ i < j ≤ κ (17) ⎢⎢⎢⎢ Ci, j Di, j −2I 2F ⎥⎥⎥⎥⎥ ⎢⎣ T ⎦ T 0 2F −2ξI Ei, j P

5

where

   T Ξi, j = P Ai, j + A j,i + Ai, j + A j,i P + 2G



Bi, j = Bi + B j , Ci, j = Ci + C j , Di, j = Di + D j , Ei =       0 0 I −L j I −Li G= , Ei, j = + I 0 I 0 0 λI

2 2 2 2 2 F = 0 I , λ =  1 +  2 +  3 +  4 +  5 +  26

I I

−Li 0



Then, it holds r22 < α z22 + V( x¯(0))

(18)

Proof. Consider the following Lyapunov candidate V( x¯(t)) = x¯T (t)P x¯(t)

(19)

where P is a positive definite matrix. Suppose ˙ x¯(t)) + rT (t)r(t) − αzT (t)z(t) < 0 V( it yields



∞

V( x¯(∞)) − V( x¯(0)) <



0

Hence, we have



∞

rT (t)r(t)dt

(21)

0



∞

rT (t)r(t)dt < α

0

∞

αz (t)z(t)dt − T

(20)

zT (t)z(t)dt + V( x¯(0))

(22)

0

In what follows, we are devoted to seek the solvability of (20). To this end, we first take the time derivative of V( x¯(t)) along the trajectory of system (16) ˙ x¯(t)) = V(

κ

μi (θ(t))

i=1

κ

 μ j (θ(t)) x¯T (t)(P(Ai, j + ΔAi ) + (Ai, j + ΔAi )T P) x¯(t)

j=1



+ x¯ (t)P(Bi + ΔBi )z(t) +uT (t)(Bi + ΔBi )T P x¯(t) T

(23)

Note that

⎛ κ ⎞T κ ⎜⎜⎜ ⎟⎟ r (t)r(t) = ⎜⎜⎝ μl (θ(t)) ((Ci + ΔC ) x¯(t) + (Di + ΔD )z(t))⎟⎟⎟⎠ μi (θ(t)) ((Ci + ΔC ) x¯(t) + (Di + ΔD )z(t)) T

i=1

≤

κ

i=1

μi (θ(t)) ((Ci + ΔC ) x¯(t) + (Di + ΔD )z(t))T ((Ci + ΔC ) x¯(t) + (Di + ΔD )z(t))

(24)

i=1

By the substitution of (23) and (24) into (20), we get ˙ x¯(t)) + rT (t)r(t) − αzT (t)z(t) V(  T      κ

x¯(t) (Ci + ΔC )T 0 0 ≤ μl (θ(t)) () − (Di + ΔD )T z(t) 0 αI i=1 !  T  κ κ

x¯(t) Sym P(Ai, j + ΔAi ) + μi (θ(t)) μ j (θ(t)) z(t) (Bi + ΔBi )T P i=1 j=1 6

 x¯(t) z(t)   x¯(t)  <0 z(t) 0

(25)

As a result, it is easy to see that (20) is fulfilled providing that (26) holds for l = 1, · · · , r. 

 " # $ Sym P Ai,i + ΔAi P(Bi + ΔBi ) μi (θ(t)) μ j (θ(t)) −αI (Bi + ΔBi )T P i=1 j=1   

(Ci + ΔC )T Ci + ΔC (Di + ΔD ) <0 + (Di + ΔD )T

κ

κ

By applying Schur complement, (26) can be transformed to ⎡ ⎤ $ " # ⎢⎢⎢ Sym P Ai,i + ΔAi κ κ   ⎥⎥⎥

⎢⎢⎢ ⎥ −αI  ⎥⎥⎥⎥ (Bi + ΔBi )T P μi (θ(t))μ j (θ(t)) ⎢⎢ ⎣ ⎦ i=1 j=1 Di + ΔD −I Ci + Δ C ⎛⎡ ⎤ ⎡  ⎥⎥⎥ ⎢⎢⎢ PΔAi + ΔTAi P  ⎜⎜⎜⎢⎢⎢ PAi, j + ATi, j P  κ κ

⎜⎜⎜⎢⎢⎢ ⎥ ⎢ 0 ΔTBi P μi (θ(t))μ j (θ(t)) ⎜⎜⎢⎢ = −αI  ⎥⎥⎥⎥ + ⎢⎢⎢⎢ BTi P ⎝⎣ ⎦ ⎣ i=1 j=1 Di −I Ci ΔC ΔD !  ⎤ ⎛⎡ ⎜⎜⎜⎢⎢⎢ Sym P Ai, j + A j,i κ κ−1   ⎥⎥⎥

⎥⎥ ⎜⎢ μi (θ(t))μ j (θ(t)) ⎜⎜⎜⎜⎜⎢⎢⎢⎢⎢ = −2αI  ⎥⎥⎥⎥ 2BTi P ⎝ ⎦ ⎣ i=1 j=i+1 2Di −2I 2Ci $ " ⎤⎞ ⎡   ⎥⎥⎥⎟⎟⎟ ⎢⎢⎢ Sym PΔAi ⎢⎢⎢ 0  ⎥⎥⎥⎥⎥⎟⎟⎟⎟⎟ ΔTBi P +2 ⎢⎢ ⎦⎠ ⎣ ΔC ΔD 0 $ " ⎛⎡ ⎤⎞ ⎤ ⎡ $ "   ⎥⎥⎥⎟⎟⎟ ⎜⎜⎜⎢⎢⎢ Sym PAi,i κ   ⎥⎥⎥ ⎢⎢⎢ Sym PΔAi

⎜⎢ ⎥ ⎢ 0  ⎥⎥⎥⎥⎥⎟⎟⎟⎟⎟ < 0 ΔTBi P −αI  ⎥⎥⎥⎥ + ⎢⎢⎢⎢ BTi P + μ2i (θ(t)) ⎜⎜⎜⎜⎢⎢⎢⎢ ⎝⎣ ⎦⎠ ⎦ ⎣ i=1 Di −I Ci ΔC ΔD 0 Note that $ " ⎡ ⎢⎢⎢⎢ Sym PΔAi ⎢⎢⎢ ΔTBi P ⎣⎢ ΔC  0 ΔA (x, u) ≤ξ 0 ΔC (x, u)

⎤ ⎤ ⎧⎡ ⎪ ⎥⎥⎥ ⎢ PEi ⎥⎥⎥  ⎪ ⎪ 0 ΔA (x, u) ⎨⎢⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ 0 ⎥⎥⎥⎥⎥ ⎥⎥⎦ = Sym ⎪ ⎪ ⎪ 0 ΔC (x, u) ⎦ ⎣ ⎩ F ⎡ T ⎢ PEi 1 ⎢⎢⎢⎢ ΔB (x, u) ΔE (x, u) 0 () + ⎢⎢⎢ 0 ΔD (x, u) ΔF (x, u) 0 ξ⎣ F

 0 ΔD

  0

ΔB (x, u) ΔD (x, u)

ΔE (x, u) ΔF (x, u)

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦ ()

holds for any constant ξ > 0 and i = 1, · · · , κ. It can be easily proved that for λ =  21 +  22 +  23 +  24 +  25 +  26 , it holds T    ΔA (x, u) ΔB (x, u) ΔE (x, u) ΔA (x, u) ΔB (x, u) ΔE (x, u) ≤ λI. ΔC (x, u) ΔD (x, u) ΔF (x, u) ΔC (x, u) ΔD (x, u) ΔF (x, u)

(26)

  0

⎤⎞ ⎥⎥⎥⎟⎟⎟ ⎥⎥⎥⎟⎟⎟ ⎥⎥⎦⎟⎟⎠

(27)

0 0

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (28)

(29)

By applying Schur Complement to (27), we obtain (17). As a result, (20) is met. The proof is thus completed. 3.3. Fuzzy observer-based fault detection scheme Theorem 1 provides us with an algorithm for the design of a robust L2 fuzzy observer-based residual generator. Moreover, by (i) defining the residual evaluation function as  ∞ J= rT (t)r(t)dt (30) 0

7

(ii) determining the threshold as 

∞

Jth = α

zT (t)z(t)dt + max x¯T (0)P x¯(0) x(0), xˆ(0)

0

(31)

and (iii) setting the decision logic, a robust L2 fuzzy observer-based FD system can be realized. Then, we are able to apply the following scheme to the realization of a FD system: • Run the residual generator (15) • Set

  Jth = α u22 + δ2d + sup V( x¯(0))

(32)

x(0), xˆ(0)

• Set the detection logic

⎧ 2 ⎪ ⎪ ⎨ J = r2 > Jth =⇒ faulty ⎪ ⎪ ⎩ J = r22 ≤ Jth =⇒ fault-free

(33)

4. A numerical example Consider the following fuzzy dynamic system: Plant rule i : IF y1 (t) is N1i ⎧ ⎪ ⎪ ⎨ x˙(t) = Ai x(t) + Bi u(t) + Ei d(t) + ΔA (x, u)x + ΔB (x, u)u + ΔE (x, u)d THEN ⎪ ⎪ ⎩y(t) = Ci x(t) + ΔC (x)x, i ∈ {1, 2, 3} where ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ ⎢⎢⎢ −0.4 0.01 ⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢ 0.2 ⎥⎥⎥ 0 ⎥⎥⎥

⎢ ⎢ ⎥ ⎢ ⎥ ⎥ −0.2 0 ⎥⎥⎥⎥ , B1 = ⎢⎢⎢⎢ 2 ⎥⎥⎥⎥ , E1 = ⎢⎢⎢⎢ 0.2 ⎥⎥⎥⎥ , C1 = 2 1 −1 A1 = ⎢⎢⎢⎢ 0 ⎣ ⎣ ⎦ ⎣ ⎦ ⎦ 0 0 −0.3 2 0.3 ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ ⎢⎢⎢ −0.3 0.01 ⎢⎢⎢ 2 ⎥⎥⎥ ⎢⎢⎢ 0.1 ⎥⎥⎥ 0 ⎥⎥⎥

⎢ ⎢ ⎥ ⎢ ⎥ ⎥ −0.3 0 ⎥⎥⎥⎥ , B2 = ⎢⎢⎢⎢ 2 ⎥⎥⎥⎥ , E2 = ⎢⎢⎢⎢ 0.1 ⎥⎥⎥⎥ , C2 = 2 −1 1 A2 = ⎢⎢⎢⎢ 0 ⎣ ⎣ ⎦ ⎣ ⎦ ⎦ 0 0 −0.3 1 0.3 ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ ⎢⎢⎢ −0.1 0.01 ⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢ 0.2 ⎥⎥⎥ 0 ⎥⎥⎥

⎢ ⎢ ⎥ ⎢ ⎥ ⎥ −0.2 0 ⎥⎥⎥⎥ , B3 = ⎢⎢⎢⎢ 2 ⎥⎥⎥⎥ , E3 = ⎢⎢⎢⎢ 0.1 ⎥⎥⎥⎥ , C3 = 2 −1 1 . A3 = ⎢⎢⎢⎢ 0 ⎣ ⎣ ⎦ ⎣ ⎦ ⎦ 1 0 0 −0.3 0.3 The membership functions for premise variable y1 are depicted in Fig. 1. It is assumed that the model uncertainties ΔA (x, u), ΔB (x, u), ΔC (x) and ΔE (x, u) have the upper bound as  1 = 0.05,  2 = 0.01,  3 = 0.01 and  4 = 0.001, respectively. According to Theorem 2, the gain matrices for each local residual generator is given by

L1 = 5.2909 9.3079 3.7690

L2 = 5.2904 9.3059 3.7681

L3 = 5.2904 9.3060 3.7681 8

with α = 0.68. To demonstrate our method, disturbance signal is simulated as shown in Fig. 2. In addition, the system is in a steady operation, as shown in Fig. 3. It can be observed from the simulation results given in Fig. 4 that the residual signal r(t) of the proposed fuzzy residual generator is L2 bounded in the fault-free case. In order to illustrate the fault detection performance, a 0.4 offset of the measurable variable y is simulated at 80s, which causes the deviation of residual signal, as depicted in Fig. 5. By adopting the residual evaluation and threshold computation method provided above, it is evident that the fault can be detected in both cases, as shown in Fig. 6. 1.2

1

μ(y1 )

0.8

0.6

0.4

rule 1 rule 2 rule 3

0.2

0

0

1.8

3.4

y1

5.7

8

Figure 1: Membership functions for fuzzy modelling

5. Conclusion and future works This paper is devoted to the integrated design of robust fuzzy observer-based FD for general nonlinear systems in the presence of external disturbances. To this end, the nonlinear systems have been first approximated by a class of generalized T-S fuzzy models by taking into account of the input signal in the premise variables. Then the robust fuzzy observer-based residual generator has been carried out with the aid of L2 stability theory. Together with a evaluation window and a decision making logic with an embedded threshold, the integrated design of observer-based FD systems have been accomplished. The interesting topics for future works include fuzzy observer-based fault-tolerant control (FTC) for so-called affine nonlinear systems along the lines of the FTC configuration proposed in [33]. Acknowledgement: The authors are grateful to the Associate Editor and anonymous reviewers for their constructive comments based on which the presentation of this paper has been greatly improved. In addition, this work has been supported by the National Natural Science Foundation of China under grants 61433001, 61174052 and 61473004.

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0.4

0.3

d(t)

0.2

0.1

0

−0.1

−0.2

0

10

20

30

40

50

40

50

40

50

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Figure 2: The disturbance signal Input signal of the process

u

0.25 0.2 0.15 0.1

0

10

20

30

Output signal of the process 5.3

y

5.25 5.2 5.15 5.1

0

10

20

30 Time(s)

Figure 3: Input and output signals of the process in fault-free case

References [1] J. Chen, R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems, Kluwer Academic Publishers, 1999. [2] E. L. Russell, L. Chiang, R. D. Braatz, Data-driven Techniques for Fault Detection and Diagnosis in Chemical Processes, Springer-Verlag, London, 2000. [3] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecke, Diagnosis and Fault-Tolerant Control, Springer, 2003. [4] S. X. Ding, Y. Yang, Y. Zhang, L. Li, Data-driven realization of kernel and image representations and their application to fault detection and control system design, Automatica 50 (10) (2014) 2615–2623. [5] S. X. Ding, Model-Based Fault Diagnosis Techniques - Design Schemes, Algorithms and Tools, 2nd Edition, Springer-Verlag, London, 2013.

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0.2 0.15 0.1

r

0.05 0 −0.05 −0.1 −0.15 −0.2

0

10

20

30

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Time(s)

Figure 4: Residual signal r(t) in fault-free case

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r1

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50

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70 Time(s)

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Figure 5: Residual signal r(t) in faulty case

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1 threshold evaluation function

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Figure 6: Fault detection of a sensor fault

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