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Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models

Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Z ak, and Jian Zhang*

International Journal of Control, Automation and Systems 16(6) (2018) 2676-2687

http://dx.doi.org/10.1007/s12555-017-0582-4

International Journal of Control, Automation and Systems 16(6) (2018) 2676-2687http://dx.doi.org/10.1007/s12555-017-0582-4

ISSN:1598-6446 eISSN:2005-4092http://www.springer.com/12555

Fault Detection and Reconstruction for Discrete Nonlinear Systems viaTakagi-Sugeno Fuzzy ModelsShenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Zak, and Jian Zhang*

Abstract: Observer-based actuator fault detection and sensor fault reconstruction for a class of discrete-time non-linear systems with actuator and sensor faults are investigated in this paper. A descriptor Takagi-Sugeno (T-S) fuzzymodel is employed to construct observer-based systems for the purpose of fault detection and sensor fault recon-struction. Two methods for observer design are proposed. In the first method, the observer gains are computedoff-line. In the second method, the observer gains are computed on-line at each iteration. The observer designs areformulated using linear matrix inequalities. Sufficient conditions for the existence of the observer-based fault de-tection and sensor fault reconstruction systems are provided. Comparative simulation study to illustrate the validityof the proposed methods is performed.

Keywords: Discrete-time systems, fault diagnosis, singular systems, state estimation.

1. INTRODUCTION

The increasing demands of reliability and safety forlarge-scale complex control systems resulted in manyresearch centers investigating fault detection and isola-tion (FDI) methods, see, for example, [1–6] and refer-ences therein. Different approaches to fault estimationor fault reconstruction have been proposed. In particular,observer-based fault detection and isolation methods arereported in [7–14]. Overview papers, [7] and [8], sum-marized observer-based fault diagnosis techniques that in-clude H∞ theory, nonlinear unknown input observer the-ory, as well as adaptive observer theory. In [9] a state ob-server has been proposed that is used to cancel the systemprocess dynamics so that a residual vector signal sensitiveonly to faults, and disturbances can be constructed. Thedesign algorithm is formulated in terms of linear matrixinequalities using state-space techniques. An observer-based integrated robust fault estimation and accommoda-tion for a class of discrete-time uncertain nonlinear sys-tems was presented in [13]. Both full-order and reduced-order fault estimation observers in frequency domain fordiscrete-time systems are proposed in [14]. A robust slid-

Manuscript received September 24, 2017; revised April 7, 2018; accepted July 23, 2018. Recommended by Associate Editor Jun Yoneyamaunder the direction of Editor Euntai Kim. This work was supported by the National Nature Science Foundation of China (61703296,61573256).

Sheng-hui Guo is with the College of Electronics and Information Engineering, Suzhou University of Science and Technology, No. 1,Kerui Road, Suzhou 215009, China, with the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics,Nanjing 211106, China, and also with Seloon Elevator Co., Ltd., Suzhou 215213, China (e-mail: liwdedy@163.com). Fanglai Zhu iswith the College of Electronics and Information Engineering, Tongji University, No. 4800, Caoan Road, Shanghai 201804, China (e-mail:zhufanglai@tongji.edu.cn). Wei Zhang is with the Laboratory of Intelligent Control and Robotics, Shanghai University of EngineeringScience, No. 333, Longteng Road, Shanghai 201620, China (e-mail: wizzhang@foxmail.com). Stanisław H. Zak is with the School ofElectrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (e-mail: zak@purdue.edu). Jian Zhang is withthe School of Mechanical Engineering, Tongji University, No. 4800, Caoan Road, Shanghai 201804, China (e-mail: jianzh@tongji.edu.cn).* Corresponding author.

ing mode descriptor observer design method to estimatestate and disturbance simultaneously using singular sys-tem theory was proposed in [10]. An integrated designscheme for affine nonlinear systems using L2-stability the-ory based on the proof that an L∞ and an L2 observer-basedfault detection systems exist for output re-constructabilityand weak output re-constructability, respectively, werepresented in [12]. Although there are many significantresults that have been proposed, most of them are givenfor linear systems [1,9,14] and some for special classes ofnonlinear systems [3, 4, 10, 12].

For more general nonlinear systems, the most commontechnique is based on the Takagi-Sugeno (T-S) fuzzy mod-els [15]. The T-S fuzzy model, first proposed in [16], hasbecome an attractive approach to the control and obser-vation of dynamic systems and found a number of ap-plications in real-life systems [17–19]. In [19] a T-Smodel is employed to represent a truck-trailer system. T-Sfuzzy models are also used to describe networked controlsystems with different network-induced delays, the pre-treatment of wastewater, and space vehicle attitude dy-namics, see, for example, [17, 18, 20]. Methods for ap-proximating a nonlinear system using T-S fuzzy models

c⃝ICROS, KIEE and Springer 2018

Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models 2677

can be found in [16] and in [21, 22]. T-S fuzzy mod-els, have been analyzed from many different angles, suchas stability analysis [23, 24], observer design [15, 25–27],fault detection and isolation [28–32], fault-tolerant con-trol [33,34]. Only a handful of papers address the issues offault estimation for T-S fuzzy models, and most of the pro-posed methods consider sensor faults and actuator faultsseparately. Proportional integral (PI) observers have beenused to deal with systems with both actuator and sensorfaults [35–37], but most of PI observers have their ownconservatism, for they are usually good at coping withslow varying parameters or time-invariant unknown inputsor faults [35, 36]. Chadli et al. [29] made a significantcontribution to the simultaneous state and fault estimationusing multiple Lyapunov functions with slack variables toreduce conservatism in the observer design. Most of theproposed design methods are based on off-line comput-ing [15, 25, 27, 29], little work of on-line observer gaincomputation can be found in the literature.

In this paper, we discuss the problem of simultaneousactuator fault detection and sensor fault reconstruction fornonlinear systems with bounded disturbances using a de-scriptor T-S framework. The main contributions of thispaper are: (i) new sufficiency conditions are given for ac-tuator fault detection and sensor fault reconstruction interms of linear matrix inequalities (LMIs); (ii) the pro-posed schemes are shown to be robust to additive distur-bances; (iii) novel on-line approach to simultaneous sen-sor and actuator fault detection is proposed.

The paper is organized as follows: The problem state-ment and some preliminary results are given in Section 2.In Sections 3 and 4, two methods for constructing a sys-tem for fault detection and sensor fault reconstruction arepresented in the form of linear matrix inequalities. Next,we discuss the method for actuator fault detection by con-structing an appropriate residual. In Section 5, two simu-lation examples are presented to demonstrate the validityof the proposed method. Conclusions are drawn in Sec-tion 6.

2. PROBLEM STATEMENT

We consider a class of discrete-time, nonlinear systemswith actuator fault and sensor fault subject to bounded un-certainties modeled as,

x(k+1) = ϕ(x(k),u(k), fa(k),η(k)), (1a)

y(k) =Cx(k)+F fs(k), (1b)

where x(k)∈Rn is the state, u(k)∈Rm is the control input,y(k) ∈ Rp is the measured output, fa(k) ∈ Rq models theactuator fault and fs(k) ∈ Rs models the sensor fault. Thevector η(k) ∈ Rd models the uncertainty/disturbance inthe system. We make the following assumption.

Assumption 1: The matrix C ∈ Rp×n has full row rankand F ∈ Rp×s has full column rank.

Remark 1: Assumption 1 is a common condition forobserver designing, and similar assumptions can be foundin [1, 3, 4, 10, 33]. Note that the assumption that F ∈ Rp×s

has full column rank implies that p ≥ s.

We refer to the model (1) as the truth model. For thedesign purpose, we construct a design model using the T-S fuzzy model,

x(k+1) =r

∑i=1

µi(ξ (k)) [Aix(k)+Biu(k)

+Di fa(k)+Giη(k)] , (2a)

y(k) =Cx(k)+F fs(k), (2b)

where ξ (k) is a vector of measured variables, for ex-ample, we can have ξ (k) = y(k). The activation func-tions µi(·) (i = 1 . . .r) are known functions which satisfy

µi(ξ (k)) ≥ 0 andr∑

i=1µi(ξ (k)) = 1, where r ∈ N is the

number of local models comprising the T-S model. Fora method of generating these local models, see, for exam-ple, [21] and [22].

A consequence of Assumption 1 is that we can con-struct a state variable transformation such that the outputmatrix C of the plant model (1) in the new coordinates willhave the following form,

C =[Ip 0

]. (3)

Therefore, without loss of generality, we consider C to beof the form (3).

To proceed, we represent the above model as a descrip-tor T-S fuzzy system. We first partition the state as,

x1 =Cx, and x2 =[0 In−p

]x.

To this end, we define an augmented state,

x(k) =

x1(k)x2(k)fs(k)

.

Let,

E =

[In 00 0s

]=

Ip 0 00 In−p 00 0 0s

,

Ai =

[Ai 00 0s

]=

A11,i A12,i 0A21,i A22,i 0

0 0 0s

,

Bi =

[Bi

0s×m

]=

B1,i

B2,i

0s×m

,

Di =

[Di

0s×q

]=

D1,i

D2,i

0s×q

,

2678 Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Żak, and Jian Zhang

Gi =

[Gi

0s×d

]=

G1,i

G2,i

0s×d

,

and,

C =[C F

]=[Ip 0 F

].

Here, x1(k) ∈ Rp, A11,i ∈ Rp×p, B1,i ∈ Rp×m, D1,i ∈ Rp×q,G1,i ∈ Rp×d . In the following, we omit the arguments inthe membership functions so that no confusion arises.

Next, we represent model (2) as a descriptor T-S fuzzysystem as follows,

Ex(k+1) =r

∑i=1

µi(ξ (k))[Aix(k)+ Biu(k)

+Di fa(k)+ Giη(k)], (4a)

y(k) = Cx(k). (4b)

Note that if,

M =

Ip −F 00 0 In−p

0 Is 0

∈ R(n+s)×(n+s),

then

CM = [ Ip 0 ].

Let θ(k) = M−1x(k) ∈ R(n+s)×(n+s). Then the model (4)can be represented as,

Eθ(k+1) =r

∑i=1

µi(ξ (k))[Aiθ(k)+ Biu(k)

+Di fa(k)+ Giη(k)], (5a)

y(k) = Cθ(k). (5b)

where E = EM, Ai = AiM, and C = CM.In the next two sections, we propose LMI conditions

for the design of robust observers with constant and time-varying gains. These observers are capable of rejectingdisturbances while, at the same time, being sensitive toactuator faults fa(k) under a different set of sufficient con-ditions. In addition, these observers can be used to recon-struct the sensor faults fs(k).

3. OBSERVER WITH CONSTANT GAINS

3.1. Technical resultTo proceed, we require the following lemma.

Lemma 1: There is a nonsingular matrix N ∈R(n+s)×(n+s) such that,

NE =

[E1 0E2 In+s−p

], (6)

where E1 ∈ Rp×p and E2 ∈ R(n+s−p)×p.

Proof: Since F ∈ Rp×s is full column rank it followsfrom Assumption 1 that there exists a nonsingular matrixS ∈ Rp×p such that

SF =

[0(p−s)×s

Is

].

We note that

E = EM =

Ip −F 00 0 In−p

0 0s 0

.

Hence, the nonsingular matrix,

N =

−[

Ip−s 00 0s

]S 0

[0Is

]−[

0 Is]

S 0 00 In−p 0

,

satisfies (6), which completes the proof.

3.2. Design of the observer with constant gainsWe begin with the following state transformation that

relates z(k) to θ(k),[θ1(k)θ2(k)

]=

r

∑j=1

µ j(ξ (k−1))[

Ip 0−E2 −L jE1 In+s−p

][z1(k)z2(k)

].

Therefore,

z1(k) = θ1(k), (7a)

z2(k) =r

∑j=1

µ j(ξ (k−1))(L jE1 +E2)θ1(k)+θ2(k)

=r

∑j=1

µ j(ξ (k−1))[L j I

]NEθ(k), (7b)

where, θ1(k),z1(k) ∈ Rp, L j ∈ R(n+s−p)×p and the selec-tion of gains L j will be discussed later in this subsection.

Remark 2: The above state transformation is used inthe observer design to eliminate the effects of uncertain-ties/disturbances on the plant state estimate.

Note that to transform the system model into the z-coordinates we need an expression for NEθ(k+1). SinceN is nonsingular, we can pre-multiply by N both sidesof (5a) to obtain

NEθ(k+1) =r

∑i=1

µi(ξ (k))[NAiθ(k)+NBiu(k)

+NDi fa(k)+NGiη(k)], (8a)

y(k) = Cθ(k) =[Ip 0

][θ1(k)θ2(k)

]= θ1(k), (8b)

where θ1(k) ∈ Rp.

Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models 2679

Next, we partition the matrices in (8a) as follows:

Ai = NAi =

[A1,i A2,i

A3,i A4,i

], Bi = NBi =

[B1,i

B2,i

],

Di = NDi =

[D1,i

D2,i

], Gi = NGi =

[G1,i

G2,i

],

where all the partitioned matrices are of appropriate di-mensions. We are now ready to use the transformationgiven by (7) to obtain

z1(k) = y(k), (9a)

z2(k+1) =r

∑i=1

r

∑j=1

µi(ξ (k))µ j(ξ (k))

× [(L jA2,i +A4,i)z2(k)

+r

∑l=1

µl(ξ (k−1))((L jA1,i +A3,i)

−(L jA2,i +A4,i)(LlE1 +E2))y(k)

+(L jD1,i +D2,i) fa(k)

+(L jG1,i +G2,i)η(k)

+(L jB1,i +B2,i)u(k)] . (9b)

Using (9), we propose the observer with constant gains ofthe form,

z1(k) = y(k), (10a)

z2(k+1) =r

∑i=1

r

∑j=1

µi(ξ (k))µ j(ξ (k))

× [(L jA2,i +A4,i) z2(k)

+r

∑l=1

µl(ξ (k−1))((L jA1,i +A3,i)

−(L jA2,i +A4,i)(LlE1 +E2))y(k)

+(L jB1,i +B2,i)u(k)] . (10b)

Theorem 1: Let

Zi j =12(PA4,i +Q jA2,i +PA4, j +QiA2, j),

and let fa(k) = 0 in (9). If there exist matrices P = P⊤ ∈R(n+s−p)×(n+s−p) and Qi ∈R(n+s−p)×p such that the follow-ing linear matrix inequalities

minδ , (11a)[−δ I

[Q j P

]Gi

([

Q j P]Gi)

⊤ −δ I

]≺ 0, (11b)[

−P Zi j

Z⊤i j −P

]≺ 0, (11c)

P ≻ 0, (11d)

i < j. (11e)

are feasible, then the system (10) characterized by gainsL j = P−1Q j produces error dynamics that are asymptoti-cally stable.

Proof: Note that (11a) and (11b) implies that

[Q j P

][ G1,i

G2,i

]= 0 ⇔

[L j I

][ G1,i

G2,i

]= 0

⇔ (L jG1,i +G2,i) = 0.

Let z2(k) = z2(k)− z2(k), and subtract (10b) from (9b), weobtain the error dynamic system,

z2(k+1)

=r

∑i=1

r

∑j=1

µi(ξ (k))µ j(ξ (k))[(L jA2,i +A4,i)z2(k)

+(L jD1,i +D2,i) fa(k)]. (12)

For fa(k) = 0, the above takes the form,

z2(k+1)

=r

∑i=1

r

∑j=1

µi(ξ (k))µ j(ξ (k))(L jA2,i +A4,i)z2(k)

=r

∑i=1

r

∑j=1

µi(ξ (k))µ j(ξ (k))Si j z2(k), (13)

where Si j = L jA2,i +A4,i.Now, consider the Lyapunov function candidate V (k) =

z⊤2 (k)Pz2(k). We evaluate the first forward time differ-ence of V (k) on the trajectories of the error dynamic sys-tem (13) to obtain (14).

∆V = z⊤2 (k+1)Pz2(k+1)− z⊤2 (k)Pz2(k)

=r

∑i=1

r

∑j=1

r

∑l=1

r

∑t=1

µiµ jµl µt z⊤2 (k)[S⊤

i j PSi j −P]

z2(k)

=14

r

∑i=1

r

∑j=1

r

∑l=1

r

∑t=1

µiµ jµl µt z⊤2 (k)

×[(Si j +S ji)

⊤P(Skl +Slk)−4P]

z2(k)

≤ 14

r

∑i=1

r

∑j=1

µiµ j z⊤2 (k)

×[(Si j +S ji)

⊤P(Si j +S ji)−4P]

z2(k)

=r

∑i=1

r

∑j=1

µiµ j z⊤2 (k)

×

[(Si j +S ji

2

)⊤P(

Si j +S ji

2

)−P

]z2(k).

(14)

Therefore, ∆V < 0 if(Si j +S ji

2

)⊤P(

Si j +S ji

2

)−P ≺ 0.

Then (11c) is obtained using the Schur complement andchoosing Q j = PL j. This completes the proof.

2680 Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Żak, and Jian Zhang

Following [38], we further evaluate (14) to obtain

∆V

≤r

∑i=1

µ2i (ξ (k))z⊤2 (k)

[S⊤

ii PSii −P]

z2(k)

+2r

∑i< j

µi(ξ (k))µ j(ξ (k))z⊤2 Qz2(k)

≤r

∑i=1

µ2i (ξ (k))z⊤2 (k)

[S⊤

ii PSii −P]

z2(k)

+(γ −1)r

∑i=1

µ2i (ξ (k))z⊤2 Qz2(k)

=r

∑i=1

µ2i (ξ (k))z⊤2 (k)

[S⊤

ii PSii −P+(γ −1)Q]

z2(k),

(15)

where 0< γ < r, i< j, j = 1,2, ...,r. Using (15), we obtainthe following relaxed sufficient conditions,

minδ , (16a)[−δ I

[Q j P

]Gi

([

Q j P]Gi)

⊤ −δ I

]≺ 0, (16b)[

−P PA4,i +QiA2,i

A⊤4,iP+A⊤

2,iQi −P+(γ −1)Q

]≺ 0, (16c)[

−P Zi j

Z⊤i j −P

]≺ 0, (16d)

P ≻ 0, (16e)

Q ≻ 0. (16f)

The estimated states in the original coordinates are ob-tained by applying the inverse transformation to obtain,

ˆx(k) = Mr

∑j=1

µ j(ξ (k−1))[

I 0−E2 −L jE1 I

]z(k).

Remark 3: Although the above LMIs form relaxedconditions for constructing constant gains for the pro-posed observer, the infeasibility may still be an issue, asdemonstrated in Example 2. To remedy this problem, wedevelop a new methodology for the observer design that ispresented in Section 4.

3.3. Actuator fault detectionWe note that we stated and proved Theorem 1 under the

assumption that no actuator fault occurs. When the actu-ator fault occurs, that is, fa(k) = 0, the observer error dy-namic is governed by (12). For the actuator fault detectionpurpose, the following result is crucial.

Assumption 2: Recall that,

[Di Gi

]=

[D1,i G1,i

D2,i G2,i

],

where Di ∈Rn×q, D1,i ∈Rp×q, Gi ∈Rn×d , and G1,i ∈Rp×d .We assume that,

rank[

Di Gi]= q+d,

and,

rank[

D1,i G1,i]< q+d.

Lemma 2: Suppose Assumption 2 holds. If L j satisfyL jG1,i +G2,i = 0, then L j satisfy L jD1,i +D2,i = 0.

Proof: We prove the lemma by contradiction. Supposethat L jG1,i+G2,i = 0 and L jD1,i+D2,i = 0 hold simultane-ously, that is, we have G2,i =−L jG1,i, and D2,i =−L jD1,i,then

rank[Di Gi

]= rank

[D1,i G1,i

D2,i G2,i

]= rank

[D1,i G1,i

−L jD1,i −L jG1,i

]= rank

[D1,i G1,i

].

On the other hand,

rank[Di Gi

]= rank

[NDi NGi

]= rank

[Di Gi

]= rank

[Di Gi

].

Hence, by Assumption 2,

rank[D1,i G1,i

]= rank

[Di Gi

]= q+d.

By the definition of[Di Gi

], we have[

D1,i G1,i

D2,i G2,i

]=[Di Gi

]= N

[Di Gi

]= N

D1,i G1,i

D2,i G2,i

0 0

,

that is,[D1,i G1,i

D2,i G2,i

]

=

−[

Ip−s 00 0s

]S 0

[0Is

]−[0 Is

]S 0 0

0 In−p 0

D1,i G1,i

D2,i G2,i

0 0

,

and therefore,[D1,i G1,i

]=

−[

Ip−s 00 0s

]SD1,i −

[Ip−s 0

0 0s

]SG1,i

−[0 Is

]SD1,i −

[0 Is

]SG1,i

Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models 2681

=

−Ip−s 0 00 0s Is

0 −Is 0

[S 00 I

][D1,i G1,i

0 0

].

Thus,

rank[D1,i G1,i

]= rank

[D1,i G1,i

]= q+d,

which contradicts Assumption 2 and the proof is com-plete.

Remark 4: It should be noted that Assumption 2 is notnecessary for the observer design, but rather for the actu-ator fault detection.

The observation error equation for the case when a faultoccurred, that is, fa(k) = 0, is (12). It then follows fromLemma 2 that the proposed observer is robust with respectto the uncertainty η(k) but sensitive to actuator faultsfa(k). We proceed by constructing a residual as the outputerror

ey(k) = y(k)− y(k),

where, y(k) =Cx(k). Then, by checking the residual ey(k)to determine if a fault has occurred or not, we can use thefollowing scheme,

∥ey(k)∥ ≤ λ , No fault,

∥ey(k)∥> λ , Fault alarm!,

where λ > 0 is the user-defined alarm threshold.

4. OBSERVER WITH VARIABLE GAINS

In the case when a given system does not satisfy con-ditions for the observer existence given in Theorem 1,we provide an alternative. Herein we present a methodfor constructing an observer-based system for the actuatorfault detection and sensor fault reconstruction where theobserver has time-varying gains that are computed on-lineat each iteration.

4.1. Preliminary transformationWe begin our discussion with an equivalent state trans-

formation,[θ1(k)θ2(k)

]=

[Ip 0

−E2 −LkE1 In+s−p

][z1(k)z2(k)

],

where the matrices Lk ∈ R(n+s−p)×p are computed on-lineand a method to compute them will presented later in thissubsection. The inverse of the above transformation hasthe form,

z1(k) = θ1(k),

z2(k) = (LkE1 +E2)θ1(k)+θ2(k)

=[

Lk I]

NEθ(k),

where, θ1(k),z1(k) ∈ Rp. Then the system (5a) is trans-formed into the form,

z1(k) = θ1(k), (17a)

z2(k+1) =[

Lk I]

Nr

∑i=1

µi(ξ (k))[Aiθ(k)+ Biu(k)

+Di fa(k)+ Giη(k)], (17b)

let

Ak = Nr

∑i=1

µi(ξ (k))Ai =

[Ak1 Ak2Ak3 Ak4

],

Bk = Nr

∑i=1

µi(ξ (k))Bi =

[Bk1Bk2

],

Dk = Nr

∑i=1

µi(ξ (k))Di =

[Dk1Dk2

],

Gk = Nr

∑i=1

µi(ξ (k))Gi =

[Gk1Gk2

],

where all the partitioned matrices are of appropriate di-mensions, then (17) can be rewritten as

z1(k) = θ1(k),

z2(k+1) =[

Lk I][Akθ(k)+ Bku(k)

+Dk fa(k)+ Gkη(k)],

or

z1(k) = θ1(k), (18a)

z2(k+1) = (LkAk2 + Ak4)z2(k)

+((LkAk1 + Ak3)

−(LkAk2 + Ak4)(LkE1 +E2))y(k)

+(LkBk1 + Bk2)u(k)+(LkDk1 + Dk2) fa(k)

+(LkGk1 + Gk2)η(k) . (18b)

4.2. The variable-gain observer designThe proposed robust observer for system (18) has the

form,

z1(k) = θ1(k) = y(k), (19a)

z2(k+1) = (LkAk2 + Ak4)z2(k)

+((LkAk1 + Ak3)

−(LkAk2 + Ak4)(LkE1 +E2))y(k)

+(LkBk1 + Bk2)u(k) . (19b)

Theorem 2: Let fa(k) = 0 in (18), then (19) is a robustobserver for system (18) if there exist symmetric positive

2682 Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Żak, and Jian Zhang

definite matrices Pk ∈ R(n+s−p)×(n+s−p) and matrices Qk ∈R(n+s−p)×p, such that the following LMIs,

minδ , (20a)[−δ I

[Qk Pk+1

]Gk

([

Qk Pk+1]Gk)

⊤ −δ I

]≺ 0,

(20b)[−Pk+1 Pk+1Ak4 +QkAk2

Ak⊤4 Pk+1 + Ak

⊤2 Q⊤

k −Pk

]≺ 0,

(20c)

Pk ≻ 0, (20d)

are feasible, and the observer gain matrices are given byLk = P−1

k+1Qk.

Proof: Note that (20a) and (20b) implies that,[Qk Pk+1

][ Gk1Gk2

]= 0 ⇔

[Lk I

][ Gk1Gk2

]= 0

⇔ LkGk1 + Gk2 = 0.

Subtracting (19b) from (18b) yields the error dynamic sys-tem,

z2(k+1) = (LkAk2 + Ak4)z2(k)

+(LkDk1 + Dk2) fa(k), (21)

where z2(k) = z2(k)− z2(k). When fa(k) = 0, the abovetakes the form,

z2(k+1) = (LkAk2 + Ak4)z2(k). (22)

Now, consider the Lyapunov function candidate V (k) =z⊤2 (k)Pk z2(k). The first forward time difference of V (k)along the trajectories of the error dynamic system (22) is

∆V = z⊤2 (k+1)Pk+1z2(k+1)− z⊤2 (k)Pk z2(k)

= z⊤2 (k)(Ω⊤

k Pk+1Ωk −Pk)

z2(k),

where Ωk = LkAk2 + Ak4.We obtain ∆V < 0 if

Ω⊤k Pk+1Ωk −Pk < 0.

We satisfy (20c) using the Schur complement selectingQk = Pk+1Lk. This completes the proof.

An asymptotic state estimation of system (4) is obtainedas,

ˆx(k) = M[

Ip 0−E2 −LkE1 In+s−p

]z(k).

Remark 5: The proposed observer rejects the uncer-tainty when its gains are obtained by solving the LMIs(20). This in turn makes it possible to deal with actua-tor fault detection problem as discussed in the followingsubsection.

Remark 6: The existing design methods use a constantcommon quadratic Lyapunov function or multiple con-stant Lyapunov functions However, a common symmet-ric positive definite P may not always exist, as shown inLMIs (16). On the other hand, time-varying symmetricpositive definite gain matrices Pk determined LMIs (20)can be easily computed on-line. The advantage of on-linecomputation of the observer gains is shown in Example 2.The proposed observer is of reduced-order, so it can makethe structure of the overall system simpler, especially forhigh-dimensional plants.

4.3. Actuator fault detectionTo proceed, we need the following lemma.

Lemma 3: Suppose Assumption 2 holds, then if amatrix Lk satisfies LkGk1 + Gk2 = 0, then it must satisfyLkDk1 + Dk2 = 0.

Proof: The proof is similar to that of Lemma 2.

The error observation equation in the presence of anactuator fault, that is, when fa(k) = 0, is given by (21).Applying Lemma 3 to (21) implies that the proposed ob-server is robust with respect to the uncertainty η(k) butsensitive to the actuator fault fa(k). We then can proceedas in the previous case to construct a residual output errorey(k) = y(k)− y(k) and then use it to decide if an actuatorfault occurred or not.

5. EXAMPLES

In this section, we test our proposed observer-basedfault detection and sensor fault reconstruction systems ontwo numerical examples. In the first example, we startwith a nonlinear model. We then construct a T-S fuzzydesign model and use the fuzzy model to compute the ob-server gains. Then we perform simulations on the nonlin-ear model to demonstrate the validity of the method. Wecompare the performance of two proposed methods.

5.1. Example 1We consider a truck-trailer system model from [19, 27]

subjected to disturbance, actuator and sensor faults de-scribed asx1(k+1)

x2(k+1)x3(k+1)

=

(1− vtsL )x1(k)

( vtsL )x1(k)+ x2(k)

x3(k)+ vts sin(x2(k)+ vts2L x1(k))

+

vtsl00

u(k)+

0.1−0.05−0.05

fa(k)

+

0.1−0.05

0.1

η(k),

Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models 2683[y1(k)y2(k)

]=

[x1(k)+ fs(k)

x2(k)+1.2 fs(k)

],

where x1(k) is the angle difference between truck andtrailer, x2(k) is the angle of trailer, x3(k) is the vertical po-sition of the rear end of the trailer, u(k) is the steering an-gle, η(k) is the uncertainty of the system, fa(k) and fs(k)are actuator and sensor faults, respectively. The length ofthe trailer is L = 5.5 m, l is the length of the truck wherel = 2.8 m, ts = 2 sec is the sampling time, and v = 1 m/secis the constant speed of backing up.

The fuzzy rules are,Rule 1: IF x2(k)+ vts

2L x1(k) is 0, THEN,x(k+1) = A1x(k)+B1u(k)+D1 fa(k)+G1η(k),

y(k) =Cx(k)+F fs(k).

Rule 2: IF x2(k)+ vts2L x1(k) is ±π , THEN,

x(k+1) = A2x(k)+B2u(k)+D2 fa(k)+G2η(k),y(k) =Cx(k)+F fs(k).

We obtain a T-S fuzzy model as in [19], where the activa-tion functions are selected to be

µ1(y(k)) =sin(y2(k)+

vtsy1(k)2L )

y2(k)+vtsy1(k)

2L

,

and

µ2(y(k)) = 1−µ1(y(k)).

The local models’ matrices are

A1 =

1− vtsL 0 0

vtsL 1 0

(vts)2

2L vts 1

, A2 =

1− vtsL 0 0

vtsL 1 00 0 1

,

B1 = B2 =

vtsl00

, D1 = D2 =

0.1−0.05−0.05

,

G1 = G2 =

0.1−0.05

0.1

, C =

[1 0 00 1 0

],

F =

[1

1.2

].

The control input u(k), the actuator fault fa(k), the sensorfault fs(k), and the uncertainty of the system η(k) are

u(k) = 0.3cos(0.2k+2.5),

fs(k) =

−0.8 0 ≤ k < 402.5sin(0.2k) 40 ≤ k < 80,

fa(k) =

0 0 ≤ k < 402sin(0.2) 40 ≤ k < 550 55 ≤ k < 80,

η(k) = 0.1cos(0.1k+2).

We compute the observer-based system constant gain ma-trices by solving the LMIs (16) to obtain

L1 =

[−0.0298 04.5045 0

], L2 =

[0.3334 00.2625 0

].

We also compute time-varying gain matrices of the time-varying observer based system by solving the LMIs (20).Plots of time history of the gains of Lk are shown in Fig. 1.We compare the performance of the system with constantand time-varying gains in Figs. 2-4. In Fig. 2 we showthe state x3(k) and its estimate for the constant gain andthe time-varying gain observers, respectively. Sensor faultand its reconstruction are shown in Fig. 3. Fig. 4 showsthe actuator fault detection. It follows from our analysisthat the actuator fault detection plots are the same for thetwo methods.

Fig. 1. Time-varying gains of the gain matrices Lk.

Fig. 2. States x3(k) (black solid) and its estimates for con-stant gain method (red dotted), for time-varyinggains (blue dot dash).

2684 Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Żak, and Jian Zhang

Fig. 3. Sensor fault (black solid) and its reconstructionsfor constant gain method (red dotted), for time-varying gains (blue dot dash).

Fig. 4. Actuator fault detection for constant gain method(red dotted), for time-varying gains (blue dotdash).

5.2. Example 2In this Example, we consider a discrete-time T-S model

that is a modification of the example system from [25],

x(k+1) =2

∑i=1

µi(ξ (k))(Aix(k)+Biu(k)

+Di fa(k)+Giη(k)),

y(k) =Cx(k)+F fs(k),

where

A1 =

−0.4 0.2 0.30.3 −0.6 0.30.4 0.2 0.6

,

A2 =

−0.45 0.375 0.3750.15 −0.45 00.75 0.75 −0.45

,

B1 =

1−0.5−0.5

, B2 =

−0.51

−0.5

,

D1 =

1−0.5−0.5

, D2 =

−0.51

0.5

,

G1 =

−0.40.2−0.4

, G2 =

0.2−0.4−0.1

,

C =

[1 0 00 1 0

], F =

[1

1.2

].

The control input u(k), the actuator fault fa(k), the sensorfault fs(k), and the uncertainty of the system η(k) are

u(k) =

2, 0 ≤ k < 20,2sin(0.2k+1), 20 ≤ k < 40,5, 40 ≤ k < 80,

fa(k) =

0, 0 ≤ k < 20,2, 20 ≤ k < 40,0, 40 ≤ k < 80,

fs(k) =

0, 0 ≤ k < 20,2sin(0.5k+5), 20 ≤ k < 40,2, 40 ≤ k < 80,

η(k) =

2cos(0.1k+2), 0 ≤ k < 20,2, 20 ≤ k < 80,

with µ1(y(k)) = 1− 0.02y22(k) and µ2(y(k)) = 0.02y2

2(k).We obtained the following gain matrices using the LMItoolbox of MATLAB to solve the LMIs (16),

L1 =

[0.4638 0−0.2556 0

],L2 =

[0.4545 0−0.2260 0

].

The time-varying gain matrices can be obtained by solv-ing the LMIs (20) and the time histories of the elements ofLk are shown in Fig. 5. The performance of the constantgain and time-varying gain systems are shown in Figs. 6–8. We show plots of the state x3(k) and its estimates inFig. 6. Sensor fault and its reconstruction are given inFig. 7. Fig. 8 illustrates the actuator fault detection. Inthis example, the time-varying gain system clearly out-performs the constant gain system. From the two exam-ples, we find that the performances of the two methods aredifferent in different situations. In Example 1, the perfor-mance of the constant gain system is somewhat better thanthat of the time-varying gain system. But it is exactly op-posite in Example 2. The cause of differences might be thesolutions of the LMIs for the two systems. In Example 2,

Fault Detection and Reconstruction for Discrete Nonlinear Systems via Takagi-Sugeno Fuzzy Models 2685

Fig. 5. The variation of gain Lk for the time-varying gain.

Fig. 6. State x3(k) (black solid) and its estimates for con-stant gain method (red dotted), for time-varyinggains (blue dot dash).

the solutions of the LMIs (16) are close to 0. On the otherhand, the solutions for the observer with the time-varyinggains in this example are better conditioned.

Remark 7: On the one hand, the constant gain is ob-tained under the assumption that the LMIs (11) have acommon positive definite matrix solution P which is ac-tually a very strong constraint condition, so the total per-formance of the constant gain method is inferior to that ofthe time-varying gain method, which can be seen in Figs.6-7. On the other hand, however, it is frequent varying ofthe gain in the time-varying gain method that leads to un-smoothed curves of the estimations, so it has sometimesan unsatisfied performance as shown in Figs. 2-3.

Fig. 7. Sensor fault (black solid) and its reconstructionsfor constant gain method (red dotted), for time-varying gains (blue dot dash).

Fig. 8. Actuator fault detection for constant gain method(red dotted), for time-varying gains (blue dotdash).

6. CONCLUSIONS

In this paper, observer-based actuator fault detectionand sensor fault reconstruction systems are proposed fora class of discrete-time nonlinear systems using Takagi-Sugeno (T-S) models. In the design, the given plant modelis first transformed into a descriptor model, and then novelrobust observer design methods are developed by deter-mining observers’ gain matrices using linear matrix in-equalities (LMIs). The proposed observer-based architec-tures can be used as sensor fault estimators because theycan estimate the system states and sensor faults simulta-neously. On the other hand, they can also be used as actu-ator fault detectors because the proposed observer-based

2686 Shenghui Guo, Fanglai Zhu, Wei Zhang, Stanisław H. Żak, and Jian Zhang

systems are robust with respect to the uncertainty but sen-sitive to actuator faults. Numerical examples are given toshow the validity of the proposed architectures.

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Sheng-hui Guo received his Ph.D. de-gree in Control Theory and Control Engi-neering from Tongji University, China, in2016. He is currently an associate profes-sor with the College of Electronics and In-formation Engineering, Suzhou Universityof Science and Technology, China, andalso a post-doctor with the College of Au-tomation Engineering, Nanjing University

of Aeronautics and Astronautics, China. His research interestsinclude observer design, model-based fault detection, and fault-tolerant control.

Fang-lai Zhu received the Ph.D. degreein Control Theory and Control Engineer-ing from Shanghai Jiao Tong Universityin 2001. Now he is a professor of TongjiUniversity, China. His research interestsinclude nonlinear observer design, chaoticsynchronization based on observer, systemidentification, and model-based fault de-tection and isolation.

Wei Zhang received his Ph.D. degreein Control Theory and Control Engineer-ing from Shanghai Jiao Tong University,Shanghai, China, in 2010. He is currentlyan associate professor with the Labora-tory of Intelligent Control and Robotics,Shanghai University of Engineering Sci-ence, Shanghai, China. His current re-search interests lie in the areas of nonlinear

observer design, robust control and networked control systems.

Stanisław H. Zak received the Ph.D. de-gree from Warsaw University of Technol-ogy in 1977. Now he is a professor of Pur-due University, U.S.A. His research inter-ests include control, optimization, nonlin-ear systems, neural networks, fuzzy logiccontrol.

Jian Zhang received his Ph.D. degree inSchool of Mechatronics Engineering fromHarbin Institute of Technology in 2005.Now he is a vice professor of Tongji Uni-versity, China. His research interests in-clude robot control theory, system identifi-cation and computer vision.