H∞ fault detection and isolation for nonlinear systems with state delay in T–S form

Available online at www.sciencedirect.com

Journal of the Franklin Institute 353 (2016) 2030–2056 www.elsevier.com/locate/jfranklin

H
=H1 fault detection and isolation for nonlinear systems with state delay in T–S form Sung Chul Jeea, Ho Jae Leeb,n a

b

Korea Institute of Robot and Convergence, Pohang 790-834, South Korea Department of Electronic Engineering, Inha University, Incheon 402-751, Republic of Korea

Received 15 December 2014; received in revised form 4 September 2015; accepted 13 November 2015

Abstract An H
=H1 sensor fault detection and isolation problem is investigated for continuous- and discretetime Takagi–Sugeno fuzzy systems with state delay and immeasurable premise variables. A bank consisting of the sensor’s number of fuzzy observers is adopted. A fuzzy observer gain and a fuzzy residual gain in each observer are designed such that the residual is sensitive to a certain partial group of faults and robust against disturbance. Sufficient design conditions are derived in nonlinear matrix inequality format and a numerically tractable algorithm involving a convex optimization is presented based on the cone complementary linearization technique. A simulation is provided to verify the effectiveness of the proposed technique. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction For the last two decades, fault detection and isolation (FDI) problem has been investigated with considerable attentions. There are a lot of research works related to FDI, and among them, an interesting stream is the data-driven approach that is suitable for model-free processes [1]. Recently some schemes have been established utilizing the key-performance-indicator [2] and the particle filter framework [3]. Yet another effective technique for FDI of dynamical systems

Corresponding author. Tel.: þ82 32 860 7425; fax: þ82 32 868 3654. E-mail address: [email protected] (H.J. Lee).

n

http://dx.doi.org/10.1016/j.jfranklin.2015.11.008 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

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is a model-based approach. In this direction, an observer is used to generate a residual, and the FDI mission is accomplished by evaluating the residual through appropriate fault decision logic. To achieve a good fault detection (FD) performance, the observer should be designed so as to be sensitive to fault (e.g. in H
sense) and robust against disturbance (e.g. in H1 sense) simultaneously [4–9]. To attain an acceptable fault isolation performance, the map from the fault to the transfer-matrix sense), which is not an easy task even for a linear system, needless to say a nonlinear one [10]. Popular alternative strategy is to use a bank of as many observers as there are sensors. Then the fault can be isolated based on a suitable voting scheme [11,12]. A good basic FDI reference without performance measure for linear time-invariant systems is found in [11]. An FDI technique for linear parameter-varying systems is proposed in [12] with the consideration of the fault sensitivity in H1 sense. State-delay phenomenon is not considered therein, which is regarded as one of the obstacles against the desired FDI [13] although it often occurs in practical systems. A few efforts have been devoted to the nonlinear FDI problem by taking a differential geometric approach [10], a sliding mode control [14], an optimal linearization [15], and under sampled-data environment [16,17]. A discrete-time H
=H1 FD solution is developed in the coupled Hamilton–Jacobi–Bellman (HJB) inequalities form [18]. Optimal residual evaluation is studied using a post-filter and threshold in [19]. On the other hand, as the Takagi–Sugeno (T–S) fuzzy model has been recognized as an effective way to bridge the gap between nonlinear systems and well-matured linear system theories [20–22], outcomes in the context of T–S fuzzy model have recently emerged [9,13,23– 25], mainly focusing on FD. In [13,24], linear matrix inequality (LMI) design conditions for FD of discrete-time T–S fuzzy systems are reported without studying the H
performance in depth. Paper [25] suggests an H
=H1 FD observer design condition in terms of LMI for a continuous-time fuzzy system. However, the work does not introduce a residual gain. A residual gain may be considered in the observer dynamics to improve the FD performance [4,7,9]. In this setting, design conditions are formulated in terms of nonlinear matrix inequality, rather than LMI, which are generally difficult to solve. It is because the product of the residual gain and its transpose that arises in the H
performance imposition is not convexified [8]. This burden can simply be avoided by a matrix square root or the Cholesky factorization (with the restriction on the residual gain to be a symmetric positive definite or a triangular form), which is applicable only to a linear residual gain [5,9]. However when a fuzzy residual gain is adopted, such matrix manipulations cannot be utilized. In addition, sometimes premise variables in a fuzzy inference engine include immeasurable state variables [20,26], which makes the FDI observer design much more complicated. In this paper, we develop sensor FDI observer-bank techniques for continuous- and discretetime T–S fuzzy systems with state delay. Premise variables in the fuzzy observer are assumed to be immeasurable. The bank consists of the sensor’s number of observers. The observer bank is designed for each residual to be sensitive to a certain partial group of faults, but robust against disturbance as far as possible. Both the fuzzy observer gain and the fuzzy residual gain (that is not confined to be symmetric or triangular due to additional manipulation such as a matrix square root or the Cholesky factorization) are designed. Sufficient conditions to find the gains are derived in terms of nonlinear matrix inequality. An iterative algorithm involving a convex optimization is presented based on the cone complementary linearization technique. This paper is organized as follows. In Section 2 we review the T–S fuzzy system under consideration and formulate the sensor FDI observer-bank design problem. Section 3 provides the

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design condition for the continuous-time case and Section 4 parallelizes the main result to discretetime case. Section 5 provides some simulation results based on the inverted pendulum. Finally, concluding remarks are made in Section 6. We follow standard notations: A ¼ AT !0 is a negative definite matrix. kxk stands for a Euclidean norm while kxkL2 ðkxkl2 Þ means the L2 ðl2 Þ norm. A symbol n denotes a transposed element in a symmetric position. An ellipsis is adopted for long symmetric matrix expressions, e.g., " # " # T T He f S g n S þ S M T n≔K KT K: M QT n M QT Q For any vector yA Rm and matrix C A Rmn we define as follows: 3 3 2 2 y1 ⋯ C1n C 11 7 7 6 6 ⋮ 7 6 ⋮ 7 6 ⋮ 7 7 6 6 7 6y 6C ⋯ Cp
1;n 7 7 p 6 p
1 7 p 6 p
1;1 y ≔6 7; C ≔6 7: 6 ypþ1 7 6 Cpþ1;1 ⋯ C pþ1;n 7 7 7 6 6 6 ⋮ 7 6 ⋮ ⋮ 7 5 5 4 4 ym Cm1 ⋯ C mn 2. Preliminaries The ith rule of a continuous-time T–S fuzzy system is expressed by ( x_ ¼ Ai x þ Bxðt
dÞ Ri : IFz1 is Γi1 and ⋯ and zh is Γih THEN y ¼ C i x þ D wi w þ f

ð1Þ

where xA Rn is the state, y A Rm is the output, d A R½0;dM  , dM A R⩾0 is the known time-invariant state delay. wA Rl and f A Rm are the disturbance and the sensor fault, respectively, that all live in L2 . zj, jA IH ≔f1; 2; …; hg is the premise variable, and Γij, iA IR ≔f1; 2; …; rg is the fuzzy set for zj in Ri. Using a singleton fuzzifier, a product inference, and a center-average defuzzification, the global dynamics of Eq. (1) is inferred as 8 r X > > _ x ¼ θi ðAi x þ Bxðt
dÞÞ > > < i¼1 ð2Þ r X > > >y¼ θ ðC x þ D w þ f Þ i i i > : i¼1

where h

∏ Γij ðzj Þ

θi ðzÞ≔

j¼1 r P i¼1

h

∏

j¼1

! Γij ðzj Þ

is the firing strength of Ri and z≔ðz1 ; z2 ; …; zh ÞA Rh and Γij ðzj Þ is the membership value of zj in Γij.

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Assumption 1. System (2) is asymptotically stable and observable.

Assumption 2. There is an injective mapping from an immeasurable subset of x to z. Hence z is unavailable for synthesis. We adopt a bank of m observers shown in Fig. 1, whose pth observer takes the following fuzzy model-based form: 8 r X > p p > _ > ^ ¼ θ^ i ðAi x^ p þ B^x p ðt
dÞ
Lpi ðyp
y^ p ÞÞ x > > > > i¼1 > > > > > r < p X p p ^ ¼ y θ^ i C i x^ p ð3Þ OC : > i¼1 > > > > > r > > p X p > > r ¼ θ^ i H pi ðyp
y^ p Þ > > : i¼1 where x^ p A Rn is the estimated state, y^ p A Rm is the observer output, r p A Rm
1 is the residual, and h

p θ^ i ðz^ p Þ ≔

∏ Γij ð^z pj Þ

j¼1 r P i¼1

!

h

∏

j¼1

Γij ð^z pj Þ

where z^ p is mapped from a subset of x^ p . Matrices Lpi and Hpi , iA IR , are the observer gain and the residual gain to be designed, respectively. We note that yp
y^ p rather than y
y^ is fed back in Eq. (3). It means that r p is completely insensitive to the fault on the pth sensor. In what follows, unless otherwise indicated, the superscript p that indicates the pth observer will be omitted  for simplicity. Let η ≔ x
x x^ . Then the dynamics is augmented to 8 r X > p > ^
dÞ þ L^ j Dpg w þ L^ j f p Þ > θg θ^ h θi θ^ j ððA^ ij þ L^ j C^ gh Þη þ Bηðt η_ ¼ > > > < g;h;i;j ¼ 1 ð4Þ r > X > p > p p > θg θ^ h θ^ j ðH j C^ gh η þ H j Dg w þ H j f Þ >r ¼ > : g;h;j ¼ 1 where A^ ij ≔

"

Ai

0

Ai
Aj

Aj

# ;

B^ ≔



B

0

0

B

"

 ;

L^ i ≔

0 Li

# ;

h i p C^ ij ≔ C pi
C pj Cpj :

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S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

Fig. 1. Observer bank for FDI.

Definition 1. For a map from f p to r in Eq. (4), the H
performance is defined by [7,27] kO C k
≔

kr k  p L2 :  f A L2 ⧹f0g f  L2 inf

For a map from w to r in Eq. (4), the H1 performance is defined by kO C k1 ≔

kr kL2 : w A L2 ⧹f0g kwkL2 sup

Definition 2 (ðβ; γÞ-H
=H1 FDI). Let the fault sensitivity level β A R40 and the disturbance attenuation level γ A R40 be given. Under the initial condition xð0Þ ¼ x^ ð0Þ ¼ 0, ðβ; γÞ-H
=H1 FDI performance is guaranteed, if there exist gains Li and Hi, iA IR in Eq. (3) such that (C1) Eq. (4) is asymptotically stable when f¼ 0 and w ¼ 0; (C2) kOC k
4β when w ¼ 0; (C3) kOC k1 oγ when f¼ 0.

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Let the pth residual evaluation function and the threshold be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z t   1 2 rðτÞ dτ J pr ≔ T W t
TW and

n o J pth ≔ min J pth;w ; J pth;f

where J pth;w

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t  2 1   ≔ sup rðτÞ dτ; f ¼ 0 T W t
TW

J pth;f

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t  2 1   ≔ inf rðτÞ dτ w ¼ 0 TW t
T W

where T W A R40 is the constant time window [27]. Table 1 shows the FDI logic to be applied after residual generation. 3. Main results Before proceeding to our main results, the following lemmas will be needed throughout the proof. Lemma 1. Given any compatible nonzero matrices E1, F1, E2, and F2, we have " T# "  # " T# He E T1 E2 n E1 E  T
2T n: T n g FT E þ FT E F1 F2 1 2 2 1 He F 1 F 2 Proof. It is readily proved as follows: " T# "  He E T1 E 2 E1 T n
ð½E 1 F 1 
½E 2 F 2 Þ n ¼ F T1 E 2 þ F T2 E 1 F T1

#

"

E T2  T þ He F 1 F 2 F T2 n

ð5Þ

# n g 0 3ð5Þ:□

Lemma 2 (Park [28]). For given any vectors aðαÞ and bðαÞ, α A Ω, there exist compatible matrices X ¼ X T g 0 and N that satisfy. #" # Z " aðαÞ #T " X Z aðαÞ n T
2 b ðαÞaðαÞ dα ⩽ dα: N T X ðN T X þ IÞX
1 n bðαÞ Ω Ω bðαÞ Table 1 FDI logic. State (

J pr ⩽ J pth ; pAIM J qr 4J qth ; qAIM ⧹fpg

J pr ⩽ J pth , for all pAIM Otherwise

Decision The fault on the pth sensor is isolated Any fault is neither detected nor isolated A fault is not isolated but detected

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S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

Theorem 1 (ðβ; γÞ-H
=H1 FDI). For given β; γ; dM A R40 , Eq. (3) has ðβ; γÞ-H
=H1 FDI performance and guarantees asymptotic stability, if there exists P≔blockdiagfP1 ; P2 g ¼ PT g 0, Q ¼ QT g 0, X ¼ X T g 0, Z ¼ Z T g 0, H~ , Mi, Wi, and Y such that 1 20 n 3 p He PA^ ij þ M joC^ gh A 6@ 7 n n n n n 6 7 þ PB^ þ Y T B^ þ Q 6 7 6 7 6 7 T ^ 6 7
B Y
Q n n n n 6 7 6 7 p T T 2 ðDg Þ M j 0
γ I n n n ð6Þ 6 7!0 6 7 p 6 7 T 6 7 W j C^ gh 0 W j Dpg
H~ n n n 6 7 6 7 p p
1 ^ 6 7 ^ ^ PA ij þ M j C gh PB M j D g 0
PZ n n 4 5 Y þP

0

0

0

0


d M
1 X

3

20

n p 1 p He ðC^ gh ÞT W Tj C^ ii 6B C 6 B
PA^
M C^ p C 6@ ij j ogh A 6 6
PB^
Y T B^
Q 6 6 T 6 B^ Y 6 6 6 W T C^ p þ W C^ p
M T j gh 6 j ii j 6 p 6 6 H~ C^ ii 6 6 p 6 PA^ ij þ M j C^ gh 4 Y þP "


Z X B^

n

n

n

n

n

Q

n

n

n

n

0

HefW j g
β I

n

n

n

0

H~

I

n

n

PB^

Mj

0

PZ
1 n

n

0

dM
1 X

0

2

0

0

! 0;


dM
1 X

ði; j; g; hÞA IR  IR  IR  IR

where M i ≔ PLi , W i ≔ H~ H i . T

Proof. Introduce a Lyapunov function V≔ V 1 þ V 2 þ V 3 where V 1 ≔ ηT Pη Z 0 Z t T V2 ≔ η_ T ðαÞB^ X nη_ ðαÞ dα dβ Z
t d tþβ V3 ≔ ηT ðαÞQηðαÞ dα: t
d

Z η
ηðt
dÞ ¼

ð7Þ

#

n

Since

7 7 7 7 7 7 7 7 7 7g0 7 7 7 7 7 7 7 7 5

t t
d

η_ ðsÞ ds

ð8Þ

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

2037

is true, Eq. (4) can be rewritten by 8

 Z t r X > p p p > ^ ^ ^ ^ ^ ^ ^ ^ ^ > _ _ η ¼ C θ θ ð A þ L þ BÞη
B D w þ L f θ θ η ðαÞ dα þ L g h i j ij j gh j g j > > < t
d g;h;i;j ¼ 1 r X > p > > r ¼ θg θ^ h θ^ j ðH j C^ gh η þ H j Dpg w þ H j f p Þ: > > : g;h;j ¼ 1

Employing Lemma 2 with aðαÞ ≔ B^ η_ ðαÞ and bðαÞ ≔ Pη, V_ 1 satisfies the following relation:

 Z t r X p ^
B^ V_ 1 ¼ 2 θg θ^ h θi θ^ j ηT P ðA^ ij þ L^ j C^ gh þ BÞη η_ ðαÞ dα þ L^ j Dpg w þ L^ j f p t
d

g;h;i;j ¼ 1 r X

⩽

g;h;i;j ¼ 1

o n

p ^ þ dPðN T X þ IÞX
1 n η θg θ^ h θi θ^ j ηT He PðA^ ij þ L^ j C^ gh þ BÞ

Z T ^ þ2η P N X B

t

T

t
d

η_ ðαÞ dα þ L^ j Dpg w þ L^ j f p



Z þ

t t
d

T η_ T ðαÞB^ X nη_ ðαÞ dα

and V_ 2 and V_ 3 are calculated as follows: Z t T T V_ 2 ¼ d η_ T B^ X nη_
η_ T ðαÞB^ X nη_ ðαÞ dα t
d

V_ 3 ¼ ηT Qη
ηT ðt
dÞQηðt
dÞ: We substitute XNP for Y and assume that T Z g d M B^ X n 3ð8Þ

then the following relations hold: r o n

X p ^ þ dM ðY þ PÞT X
1 n η V_ ⩽ θg θ^ h θi θ^ j ηT He PðA^ ij þ L^ j C^ gh þ BÞ g;h;i;j ¼ 1

Z þ2ηT Y T B^

t

t
d

2

η

3T

6 7 6 ηðt
dÞ 7 7 ¼6 6 w 7 4 5 fp 0

η_ ðαÞ dα þ PL^ j Dpg w þ PL^ j f p

0

20

B 6B B X 6B B r 6B ^ h θi θ^ j 6 B B θ θ g B 6B Bg;h;i;j ¼ 1 6B @ 4@ 2

6 B 6 B X 6 B r ^ h θi θ^ j 6 θ þB θ g 6 B 6 Bg;h;i;j ¼ 1 4 @



 þ η_ T Z η_ þ ηT Qη
ηT ðt
dÞQηðt
dÞ

o

n p T^ ^ B þQ He PðA^ ij þ L^ j C^ gh þ BÞþY
B^ Y ðL^ j Dpg ÞT P T

n

n


Q

n

0

0

0

0

T L^ j P

p ðA^ ij þ L^ j C^ gh ÞT P Y T þ P T B^ P

0

ðL^ j Dpg ÞT P

0

T L^ j P

0

12

31 7C" 7C 7C PZ
1 P 7C 7C 0 7C 5A

n

dM
1 X

#
1 C C6

η

n

3

7 7 7 7 7 n7 5 0 n

3

7 ηðt
dÞ 7 C6 6 7: nC w 7 C6 4 5 A fp ð9Þ

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In order to prove that Eq. (6) guarantees (C3) in Definition 1, we consider the case of f¼ 0 and w a 0 in Eq. (9). The following relation holds: n ! 0 3 2 p ^ He PðA^ ij þoL^ j C^ gh þ BÞ 2 3T n n7 B 6 η r þ Y T B^ þ Q B X 7 6 B 6 7 7 6 V_ ⩽ 4 ηðt
dÞ 5 B θg θ^ h θi θ^ j 6 7 T Bg;h;i;j¼1 6 ^
B Y
Q n 7 @ 5 4 w ðL^ j Dpg ÞT P 0 0 0 2 31 p ðA^ ij þ L^ j C^ gh ÞT P Y T þ P r B X 6 7C B 6 7C ^ TP þB θg θ^ h θi θ^ j 6 7C B 0 @g;h;i;j¼1 4 5A p T ^ ðL j D g Þ P 0 2 3 "
1 #
1 1 η PZ n n 6 7  nA4 ηðt
dÞ 5: 0 dM
1 X w Let an H–J–B inequality be in the form of d kr k2
γ 2 kwk2 þ Vo0: dt Integrating Eq. (10) from zero to infinity yields Z 1 Z 1     rðτÞ2 dτ
γ 2 wðτÞ2 dτoVðeð0ÞÞ
Vðeð1ÞÞ ⩽ Vðeð0ÞÞ ¼ 0 0

0

which implies Eq. (10) ) (C3). LMI (6) is a sufficiency for Eq. (10) because 2 31 2 3T 00 2 3 p η ðH j C^ gh ÞT 0 n n X 6 7C 6 7 BB 7 ^ 6 B 7 Cn
γ 2 6 0 ð10Þ ( 4 ηðt
dÞ 5 B 40 0 n5 @@ g;h;j¼1 θg θ h θj 4 5A w ðH j Dpg ÞT 0 0 I 1 3 20 n p ^ He PðA^ ij þ L^ j C^ gh þ BÞ C 7 6B o n n7 A 6@ T^ r 7 6 X B þ Q þY 7 6 þ θg θ^ h θi θ^ j 6 7 7 6 g;h;i;j¼1 7 6 ^ TY
B
Q n 5 4 T p ðL^ j Dg Þ P 0 0 0 2 31 p ðA^ ij þ L^ j C^ gh ÞT P Y T þ P r B X 6 7C 6 B 7C ^ TP þB θg θ^ h θi θ^ j 6 7C 0 B @g;h;i;j¼1 4 5A p T ^ ðL j Dg Þ P 0 1 2 3 "
1 #T C η PZ n n C6 7 nC4 ηðt
dÞ 5o0 
1 C 0 dM X A w

ð10Þ

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2039

n 1 3 p ^ He PðA^ ij þ L^ j C^ gh þ BÞ C 7 6B o n n A 7 6@ T r 7 6 ^ X þY B þ Q 7 6 ^ ^ 3 θ g θ h θi θ j 6 7 7 6 T g;h;i;j¼1 7 6 ^
B Y
Q n 5 4 T p ðL^ j Dg Þ P 0
γ2I 0 2 31 p p ðH j C^ gh ÞT ðA^ ij þ L^ j C^ gh ÞT P Y T þ P r B X 6 7C B 6 7C T ^ þB θg θ^ h θi θ^ j 6 C 0 B P 0 7 @g;h;i;j¼1 4 5A T p T p ðH j Dg Þ ðL^ j Dg Þ P 0 2 3
1 I n n
1 6 7 n 4 0 PZ n 5 n!0 20

dM
1 X 2 n o p ^ þ Y T B^ þ Q He PðA^ ij þ L^ j C^ gh þ BÞ 6 6 T 6
B^ Y 6 6 r X 6 ðL^ j Dpg ÞT P 3 θg θ^ h θi θ^ j 6 6 p 6 g;h;i;j¼1 H j C^ gh 6 6 p 6 PðA^ ij þ L^ j C^ 4 0

0

gh

n

n

n

n

n

n

n

n


γ2I PL^ g Dpg

n

n

0


PZ

0

0

0

Y þP 3

n


1

n

n


dM
1 X

n


Q 0 0 PB^ 0

7 7 7 7 7!0 7 7 5

3ð6Þ where we have used the Schur complement and a congruence transform, and replaced PL^ i and H~ H i by Mi and Wi, respectively. This also implies (C1). Now, we prove that Eq. (7) is a sufficiency for (C2) in Definition 1. In the case of f a 0 and w ¼ 0 in Eq. (9), the following relation is still true: 20 n 3 0 1 p ^ He PðA^ ij þ L^ j C^ gh þ BÞ 6B 7 C 2 3T B o 6@ B n n7 A η 6 7 B X T r þY B^ þ Q 6 7 6 ηðt
dÞ 7 B 6 7 B _ V ⩽4 5 B 6 7 T 6 B p g;h;i;j¼1 f
B^ Y
Q n 7 6 7 B 4 5 @ T L^ j P 0 0

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0

2

BXr 6 B 6 þB g;h;i;j¼1 θg θ^ h θi θ^ j 6 @ 4 " 

PZ


1

0

n

n

dM
1 X

ðA^ ij þ L^ j C^ gh ÞT P

YT þ P

T B^ P T L^ j P

0 0

p

2 3 #
1 1 η 6 7 nA4 ηðt
dÞ 5:

31 7C 7C 7C 5A

fp

Similar to Eq. (10), the implication  2 d ðC2Þ ( kr k2
β2 f p 
V40 dt holds as long as Eq. (7) is true because 2 3 2 3T 0 p ðH j C^ gh ÞT 0 r X 6 7 6 6 ηðt
dÞ 7 B 2 7n
β 4 0 0 θg θ^ h θ^ j 6 4 5 B 4 5 @ p g;h;j ¼ 1 T f Hj 0 2

η

2


6 6 θg θ^ h θi θ^ j 6 4 g;h;i;j ¼ 1 r X



PZ
1 n 0


B^ Y T L^ j P T

r B X 6 6 B
B θg θ^ h θi θ^ j 6 @g;h;i;j ¼ 1 4

"

n

0

n5

0

I

7

n o p ^ þ Y T B^ þ Q He PðA^ ij þ L^ j C^ gh þ BÞ

2

0

3

n

ðA^ ij þ L^ j C^ gh ÞT P p

YT þ P

T B^ P T L^ j P

2 3 #
1 1 η n 6 7 nA4 ηðt
dÞ 540: dM X fp

0 0

31

n

n


Q

n

0

0

3 7 7 7 5

7C 7C 7C 5A

ð11Þ

P P p By P employing Lemma 1 with E 1 ≔ rg;h;j ¼ 1 θg θ^ h θ^ j H j C^ gh , F 1 ≔ rh ¼ 1 θ^ j H j , p r E2 ≔ i ¼ 1 θi H~ C^ ii , F 2 ≔ H~ , the following relations are established: 2 n 3 o p p He ðH j C^ gh ÞT H~ C^ ii n n r r 6 7 X X 6 7 0 0 n θg θ^ h θi θ^ j 6 ð11Þ ( n o7 4 5 p p T g;h;j¼1 i¼1 H Tj H~ C^ ii þ H~ H j C^ gh 0 HE H Tj H~ 2 2 31 p 0 ðH~ C^ ii ÞT r X B 6 7C 26
@ θi 4 n
β 40 5A 0 i¼1 T 0 H~ 0

3

n

n

0

n5

0

I

7

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

2

6 6 ^ ^
θg θ h θi θ j 6 6 4 g;h;i;j¼1 r X

0



2

PZ
1 n

n

0

dM
1 X


B^ Y T L^ j P T

r B X 6 B 6
B θg θ^ h θi θ^ j 6 @g;h;i;j¼1 4

"

 n o p T^ ^ ^ ^ ^ He PðA ij þ L j C gh þ BÞ þ Y B þ Q

ðA^ ij þ L^ j C^ gh ÞT P p

T B^ P T L^ j P

#
1

YT þ P 0 0

2041

3 n


Q 31

0

7C 7C 7C 5A

n

3 n p p 1 He ðH j C^ gh ÞT H~ C^ ii 7 6B C 7 6B C 7 6 B
PðA^ þ L^ C^ p þ BÞ ^ C n n ij j gh 7 6B C 7 6@ o A 7 6 T^ r B
Q
Y 7 6 X 7 6 ^ ^ ¼ θg θ h θi θ j 6 7 7 6 T g;h;i;j¼1 7 6 B^ Y Q n 7 60 n o 1 0 1 p p 7 6 T ~ 7 6 H Tj H~ C^ ii þ H~ T H j C^ gh H He H j 7 6@ A @ A 0 5 4 T 2
L^ j P
β I 0 2 T 31 ~ j H~ C^ pii ÞT ðA^ ij þ L^ j C^ pgh ÞT P Y T þ P ð H r B X 6 7C C
B θg θ^ h θi θ^ j 6 0 B^ T P 0 7 @ 4 5A T g;h;i;j¼1 0 L^ j P H~ T 2 3
1 I n n
1 6 7 n 4 0 PZ n 5 ng0
1 0 0 dM X 1 20 n p p He ðH j C^ gh ÞT H~ C^ ii
PðA^ ij C 6B o A n 6@ T 6 þL^ j C^ p þ BÞ ^
Y B^
Q gh 6 6 6 T 6 r B^ Y Q X 6 ^ ^ 6 3 θ g θ h θi θ j 6 p p T T T ~ ^ ^ ~ ^ 0 6 H j H C ii þ H H j C gh
L j P g;h;i;j¼1 6 p 6 H~ C^ ii 0 6 6 p 6 PðA^ ij þ L^ j C^ gh PB^ 4 20

Y þP

0

n

7 7 7 n7 5 0

2042

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056 n

n

n

n

n no He H Tj H~
β2 I

n

n

n

n

n

n

I

n

n

H~ PL^ j 0


1

0 PZ 0 0

n

n

dM
1 X

3 7 7 7 7 7 7g0 7 7 7 5

3ð7Þ T where we have used the Schur complement, replacements H~ H j and PL^ j by Wj and Mj, respectively.□

When there is an injective mapping from a measurable subset of x to z, the problem is reduced to a standard fuzzy FDI problem and we have the following corollary. Corollary 1 (Measurable case). If z is measurable, for given β; γ; dM A R40 , the bank consisting of m observers in the form of Eq. (3) has ðβ; γÞ-H
=H1 FDI performance and guarantees asymptotic stability, if there exists P ≔ blockdiagfP1 ; P2 g ¼ PT g 0, Q ¼ QT g 0, X ¼ X T g 0, Z ¼ Z T g 0, H~ , Mi, Wi, and Y such that 3 20 n 1 p He PA^ ii þ M i C^ jj 7 6B C o 7 6@ n n n n n 7 6 þPB^ þ Y T B^ þ Q A 7 6 7 6 7 6 T 7 6 ^
B Y
Q n n n n 7 6 7 6 p T T 2 7!0 6 ðDj Þ M i 0
γ I n n n 7 6 7 6 p T p 7 6 ^ ~ W i C jj 0 W i Dj
H n n n 7 6 7 6 p 7 6 p 7 6 PA^ ii þ M i C^ jj PB^ M i Dj 0
PZ
1 n n 5 4 Y þP 0 0 0 0
d M
1 X n p p 1 He ðC^ hh ÞT W Ti C^ jj 6B C 6B C 6 B
PA^ ii
M i C^ p C jj C 6B o 6@ A 6
PB^
Y T B^
Q 6 6 6 T 6 B^ Y 6 6 6 W T C^ p þ W C^ p
M T i jj 6 jj i i 6 p 6 ^ ~ 6 H C jj 6 6 p 6 PA^ ii þ M i C^ jj 4

3

20

Y þP

n

n

n

n

n

Q

n

n

n

n

0

HefW i g
β I

n

n

n

0

H~

I

n

n

PB^

Mi

0

PZ
1 n

n

0

dM
1 X

0

2

0

0

7 7 7 7 7 7 7 7 7 7 7g0 7 7 7 7 7 7 7 7 7 5

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

"


Z X B^

2043

#

n


dM
1 X

! 0;

ði; j; hÞA IR  IR  IR

T where M i ≔ PLi , W i ≔ H~ H i , A^ ii ≔ blockdiagfAi ; Ai g, and C^ ii ≔ blockdiagfC i ; C i g.

Proof. Since z is measured, it is obvious that θi ¼ θ^ i . So we have " # r r X X Ai n ^ ^ θi θ j A ij ¼ θi : 0 Ai i;j ¼ 1 i¼1 Similarly, r X i;j ¼ 1

θi θ^ j C^ ij ¼

r X

θi C^ ii :

i¼1

By substituting these relations to Theorem 1, the claim is directly proved.□ Remark 1. The matrix inequality condition in [9] that considers the linear residual gain (denoted by W therein) possesses the nonlinear term W T W. It is solved by pre-choosing or defining W T W as a variable. Then W can be found in a confined (positive definite or triangular) form by a matrix square root or the Cholesky factorization. However, such a scheme is not applicable to the fuzzy residual gain in Eq. (3). Whereas the developed technique does not constrain the structure of the fuzzy residual gain matrices by introducing Lemma 1. In addition, our result takes account of the state delay that the result in [9] does not.

Remark 2. The matrix inequalities (6) and (7) in Theorem 1 are not linear due to the quadratic terms T H~ n and
PZ
1 n. They make Eqs. (6) and (7) difficult to solve because it is categorized to a nonconvex feasibility problem. To alleviate the difficulty, one may fix H~ and P to I and Z thereby can convexify the nonlinear feasibility problem. However, it could yield a conservative result. We take the cone complementary linearization technique [29] to solve Theorem 1. Introduce V and S such that " # 
1  n S
1 V n I ≽ 0; ≽ 0: ð12Þ
1 P
1 Z
1 H~

Fig. 2. Iterative algorithm.

2044

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

Define Eqs. (6)0 and (7)0 be Eqs. (6) and (7) where V and S are substituted for H~ n and PZ
1 n,
1 implying Eqs. (6)0 , (7)0 , (8) ) (6)–(8). Letting fN; K; U; J; Rg ≔ fV
1 ; H~ ; S
1 ; P
1 ; Z
1 g, Eq. (12) is represented as     U n N n ≽ 0: ð13Þ ≽ 0; J R K I Then, Theorem 1 is converted to the following nonlinear minimization one involving LMIs: MP 1 :

Minimize nP;Q;X;H~ ;V o trðPJ þ ZR þ SU Þ þ trðH~ K þ VNÞ

S≔

S;W i ;M i ;Y;U J;R;Z;N;K

subject to Eqs: ð6Þ0 ; ð7Þ0 ; ð8Þ; ð13Þ; and "       S n P n Z n H~ ≽ 0; ≽ 0; ≽ 0; I J I U I R I

# n

K

 ≽ 0;

V

n

I

N

 ≽ 0:

ð14Þ

An iterative LMI approach to MP 1 appeared in Fig. 2. Remark 3 (Threshold). The role of TW is to average the integration in J pr thus eliminate the unwanted chattering in J pr caused from w. When an appropriate TW is not chosen, it could happen that J pr ≯J pth alarming no fault detection, although a fault occurs under disturbance (i.e., f a 0 and w a 0), because it is possible that the effects of the disturbance on a residual and that of the fault cancel each other. On the other hand, when f ¼ 0 and w a 0 (f a 0, w ¼ 0), J pr oJ pth;w (J pr 4J pth;f ) always holds by definition. Remark 4. In general, introducing new decision variables in the derivation of the numerical design condition reduces the conservatism or sharpens the optimality. The price to be paid for this benefit is the increased computational intractability. The complexity of an LMI optimization problem is known to be proportional to N 3d N l , where Nd is the number of scalar decision variables and Nl is the total row size of the LMIs. Theorem 1 has N l ¼ ð16n þ 3m þ l
1Þr 4 þ 4n and N d ¼ ð2n þ mÞmr þ 4nð3n þ 1Þ þ mðm
1Þ. The main aim of this paper is the observerbank design, rather than the relaxation of the conservatism. As discussed in [30], how to further reduce the computational hardness would be another critical issue to be investigated. Remark 5. The present development focuses on the system without uncertainties. One knows that the error dynamics in this case can be made independent of the control input. Therefore for discussional simplicity, we do not introduce the control input to Eq. (1). When the system is subjected to uncertainties, this is not the case and the control input needs to be included. 4. Parallelizing to discrete-time case In this section, we present the result for discrete-time case. Consider the following discretetime T–S fuzzy system: 8 r X > > x ¼ θ ðA x þ Bi xk
dk Þ > > < kþ1 i ¼ 1 i i k ð15Þ r X > > > yk ¼ θ ðC x þ D w þ f Þ i i k i k > k : i¼1

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

2045

where wk , f k A l2 , and dk A ½0; dM   Z⩾0 , dM A Z40 . We adopt a bank of m discrete-time observers whose pth observer is expressed by 8 r X > > > x^ kþ1 ¼ θ^ i ðAi x^ k þ Bi x^ k
dk
Li ðypk
Cpi x^ k ÞÞ > > > > i¼1 > > > r X < θ^ i C i x^ k ð16Þ OD : y^ k ¼ > i ¼ 1 > > > r > X > > > θ^ i H i ðypk
y^ pk Þ: > rk ¼ > : i¼1 The augmented system is 8 r X > p > > η ¼ θg θ^ h θi θ^ j ððA^ ij þ L^ j C^ gh Þηk þ B^ ij ηk
d þ L^ j Dpg wk þ L^ j f pk Þ > kþ1 > < g;h;i;j ¼ 1 r X > p > > ¼ θg θ^ h θ^ j ðH j C^ gh ηk þ H j Dpg wk þ H j f pk Þ r k > > : g;h;j ¼ 1

where ηk ≔



xk xk
x^ k

ð17Þ

 .

Lemma 3 (Moon et al. [29]). For any compatible matrices X ¼ X T g 0, Z ¼ Z T g 0, Y, N, the following inequality holds:  T    a X n a
2aT Nb ⩽ b YT
NT Z b where   X n ≽ 0: ð18Þ YT Z

Theorem 2 (ðβ; γÞ-H
=H1 FDI). For given β, γ, d M A R40 , Eq. (16) has ðβ; γÞ-H
=H1 FDI performance in l2-norm sense and is asymptotically stable, if there exist P ≔ blockdiagfP1 ; P2 g ¼ PT g 0, Q ¼ QT g 0, X ¼ X T g 0, Z ¼ Z T g 0, H~ , Mi, Wi, and Y such that 1 3 20
P þ dM X C 7 6B n n n n n 7 6 @ þHefY g A 7 6 7 6 þðd M þ 1ÞQ 7 6 7 6 T
Y
Q n n n n 7 6 7 6 2 7!0 6 ð19Þ 0 0
γ I n n n 7 6 7 6 p T p 7 6 W j C^ gh 0 W j Dg
H~ n n n 7 6 7 6 p 7 6 PA^ þ M C^ p PB^ ij M j Dg 0
P n ij j gh 7 6 5 4 p PA^ ij þ M j C^ gh
P PB^ ij M j Dpg 0 0
dM
1 PZ
1 n

2046

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

20

n p o1 p He ðC^ gh ÞT W Tj C^ ii
Y A 6@ n 6 þP
d X
ðd þ 1ÞQ 6 M M 6 6 YT Q 6 6 p p 6 W Tj C^ ii þ W j C^ gh 0 6 6 6 H~ C^ ii 0 6 6 p 6 ^ PA ij þ M j C^ gh PB^ ij 6 4 p PA^ ij þ M j C^ gh
P PB^ ij

3 n

n

n

n

n

n

n

He W j
β I

n

n

n

H~

I

n

n

Mj

0 P

n

Mj

0

n





2

d M
1 PZ
1 n

0

7 7 7 7 7 7 7 7 7g0 7 7 7 7 7 7 5

ð20Þ

and Eq. (18). Proof. Define the Lyapunov function V k ≔ V 1k þ V 2k þ V 3k þ V 4k , where V 1k ≔ ηTk Pηk k
1 X V 2k ≔ ηTi Qηi V 3k ≔

i ¼ k
dk 1 X

k
1 X

ηTi Qηi

j ¼
dM þ2 i ¼ kþj
1

V 4k ≔

k
1 X


1 X

i ¼
dM l ¼ kþi

ΔηTl ZΔηl

P
1 where Δηl ≔ ηlþ1
ηl . Since ηk
ηk
dk ¼ kl ¼ k
dk Δηl , we can reexpress Eq. (17) as ! ! 8 r k
1 X X > p p > p > η ¼ θg θ^ h θi θ^ j ðA^ ij þ L^ j C^ gh þ B^ ij Þηk
B^ ij Δηl þ L^ j Dg wk þ L^ j f k > > < kþ1 g;h;i;j ¼ 1 l ¼ k
d k

r X > p > > r ¼ θg θ^ h θ^ j ðH j C^ gh ηk þ H j Dpg wk þ H j f pk Þ: > k > : g;h;j ¼ 1

Let ΔV k ≔ V kþ1
V k . By assigning a ≔ ηk and b ≔ Δηl , and ! ! r r X X p T ^ ^ ^ ^ N≔ θg θ h θi θ j ðA ij þ L^ j C^ þ B^ ij Þ P θi θ j B^ ij gh

i;j ¼ 1

g;h;i;j ¼ 1

in Lemma 3, one gets r X

ΔV 1k ¼ ηTk

! ! p T ^ ^ θg θ h θi θ j ðA^ ij þ L^ j C^ gh þ B^ ij Þ Pn
P ηk

g;h;i;j ¼ 1 k
1 X

þ

!T

Δηl

l ¼ k
dk

þwTk

r X r X g¼1h¼1

r X i;j ¼ 1

! T θi θ^ j B^ ij

θg θ^ j ðL^ j Dpg ÞT

!

! Pn

! Pn wk þ

k
1 X

! Δηl

l ¼ k
dk

ðf pk ÞT

r X h¼1

! T θ^ j L^ j

! Pn f pk

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

θg θ^ h θi θ^ j ðA^ ij þ

g;h;i;j ¼ 1

r X



!

r X

þ2ηTk

g;h;j ¼ 1


2

l ¼ k
dk

!T

r X

Δηl

i;j ¼ 1

l ¼ k
dk r X

þ2wTk

2

i;j ¼ 1

!

θi θ^ j ðL^ j Dpg ÞT P

6η 7 6 k
dk 7 6 7 ⩽6 7 4 wk 5 f pk 0

r X

þ B^ ij ÞT P !

!

!

r X r X

P

! θ^ j L^ j f pk


P þ d M X þ HefY g B6
YT B6 B6 @4 0

!! þ L^ j Dpg wk þ L^ j f pk

Δηl

T θi θ^ j B^ ij

h¼1

3 T 02

ηk

p L^ j C^ gh

k
1 X

θg θ^ h θ^ j
B^ ij

k
1 X

2047

g¼1h¼1

n

n

n

0

n

n

0

0

θg θ^ j ðL^ j Dpg wk þ

L^ j f pk Þ

3

7 7 7 n5

0 0 0 0 31 1 2

T p 3 2 A^ ij þ L^ j C^ gh P B 7C C 6 ηk C B 7 C 6 T B X 7 7C C6 6 k
1 B^ ij P X B r 7C
1 C6 ηk
dk 7 6 ^ ^ B 7þ 7 CP n C6 6 þB θg θ h θi θ j 6 ΔηTl ZΔηl :

T 7C C6 wk 7 ^L j Dpg P 7C Bg;h;i;j ¼ 1 5 l ¼ k
dM C4 6 B 7C C 6 f pk @ 5A A 4 T L^ j P

Other terms in ΔV k are calculated as ΔV 2k ¼

k X

k
1 X

ηTi Qηi


i ¼ kþ1
d kþ1

i ¼ k
dk

⩽ ηTk Qηk
ηTk
dk Qηk
dk þ

ΔV 3k ¼

1 X j ¼
dM þ2

ηTi Qηi k X

i ¼ kþ1
d M

ηTi Qηi

ηTk Qηk
ηTkþj
1 Qηkþj
1 ¼ d M ηTk Qηk


ΔV 4k ¼ dM ΔηTk ZΔηk


k
1 X l ¼ k
dM

k X i ¼ kþ1
dM

ηTi Qηi

ΔηTl ZΔηj :

From the foregoing relations, ΔV k is majorized by 2

ηk

3 T 02

6 7 6 ηk
d k 7 7 ΔV k ⩽ 6 6 wk 7 4 5 f pk

B6 B6 B6 @4


P þ d M X þ HefY g þ ðdM þ 1ÞQ

n

n


YT


Q

n

0 0

0 0

0 0

n

3

7 7 7 n5 n

0

2048

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

0

2

B 6 B X 6 B r 6 ^ ^ 6 þB θ θ θ θ g h i j6 B Bg;h;i;j ¼ 1 6 @ 4 " 

P 0

ðA^ ij þ L^ j C^ gh ÞT P

ðA^ ij þ L^ j C^ gh
IÞT P

T B^ ij P

T B^ ij P

ðL^ j Dpg ÞT P

ðL^ j Dpg ÞT P

p

p

T L^ j P

T L^ j P 3 2 ηk 1 #
1 6 7 n 6 ηk
d k 7 7 6 A n 6 wk 7: d M
1 PZ
1 n 5 4 f pk

31 7C 7C 7C 7C 7C 7C 5A

ð21Þ

In case of f k ¼ 0 and wk a 0, Eq. (21) is written as 2

ηk

3T 02

6 7 B6 ΔV k ⩽ 4 ηk
dk 5 @4 wk 0


P þ dM X þ HefY g þ ðd M þ 1ÞQ

n

n


YT 0


Q 0

n

2

r B X 6 B 6 þB θg θ^ h θi θ^ j 6 @g;h;i;j ¼ 1 4

" 

P 0

7 5

0

p ðA^ ij þ L^ j C^ gh ÞT P

p ðA^ ij þ L^ j C^ gh
IÞT P

T B^ ij P

T B^ ij P

ðL^ j Dpg ÞT P 2 3 #
1 1 ηk n 6 7 nA4 ηk
d k 5: dM
1 PZ
1 n wk

3

ðL^ j Dpg ÞT P

31 7C 7C 7C 5A

Summing up the discrete-time H–J–B inequality kr k k2
γ 2 kwk k2 þ ΔV k o0

ð22Þ

along Eq. (17) from 0 to 1 yields 1 X

kr k k2
γ 2

k¼0

K X

kwk k2 oV ðe0 Þ
V ðe1 Þ ⩽ V ðe0 Þ ¼ 03

k¼0

kr k kl2 oγ: wk A l2 ⧹f0g kwk kl2 sup

Inequality (19) guarantees this γ-H1 performance in l2-norm sense, because one can derive 2 3 2 3T 0 p ðH j C^ gh ÞT 0 r X 6 7 6η 7 B 26 ^ ^ 6 7 B 0 ð22Þ3 4 k
dk 5 @ θg θ h θ j 4 5n
γ 4 0 T g;h;j ¼ 1 p wk 0 ðH j Dg Þ 2

ηk

2


P þ dM X þ HefY g þ ðd M þ 1ÞQ 6
YT þ4 0

n

n


Q

n

0

0

3

n

n

0 0

n5

3 7 5

I

7

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

0

2

r B X 6 B 6 þB θg θ^ h θi θ^ j 6 @g;h;i;j ¼ 1 4

ðA^ ij þ L^ j C^ gh ÞT P

ðA^ ij þ L^ j C^ gh
IÞT P

T B^ ij P

T B^ ij P

p

p

2049

31 7C 7C 7C 5A

ðL^ j Dpg ÞT P ðL^ j Dpg ÞT P 2 3 " #
1 1 ηk P n 6 7  nA4 ηk
dk 5o0 0 dM
1 PZ
1 n wk 2 3 n 0
P þ d M X þ HefY g þ ðd M þ 1ÞQ 6
YT
Q 0 7 34 5 0 0
γ2 I 0 2 31 p p p ðH j C^ gh ÞT ðA^ ij þ L^ j C^ gh ÞT P ðA^ ij þ L^ j C^ gh
IÞT P r B X 6 7C T T B 6 7C θg θ^ h θi θ^ j 6 þB 0 B^ ij P B^ ij P 7C @g;h;i;j ¼ 1 4 5A p T p T p T ^ ^ ðH j Dg Þ ðL j Dg Þ P ðL j D g Þ P 2 3
1 n I n 6 7 n 4 0 P 5 n!0 0 0 dM
1 PZ
1 n r X 3 θg θ^ h θi θ^ j g;h;i;j ¼ 1

2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 4


P þ d M X þ HefY g

!

3

þðdM þ 1ÞQ

n

n


YT


Q

n

0

0

p H j C^ gh p PðA^ ij þ L^ j C^ gh Þ p PðA^ ij þ L^ j C^ gh Þ
P

n

n

n

n

n

n


γ I

n

n

n

0

H j Dpg


I

n

n

PB^ ij

PL^ j Dpg

0


P

n

PB^ ij

PL^ j Dpg

0


d M
1 PZ
1 n

2

0

7 7 7 7 7 7 7 7!0 7 7 7 7 7 7 5

3ð19Þ T where we have used the Schur complement, a congruence transform, replacements H~ H i and PL^ i by Wi and Mi, respectively. This also implies the asymptotic stability of Eq. (16). Next, we prove that Eq. (20) implies the β-H
performance in the l2-norm sense with Eq. (21) under f k a 0 and wk ¼ 0 by inducing the followings:

inf f k A l2 ⧹f0g

‖r k ‖l2 ‖f pk ‖l2

4β

( ‖r k ‖2
β2 ‖f pk ‖2
ΔV k 40

2050

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

2

3T 0 0

2

31

2 0 n 6 7 BBXr 6 7C 6 2 η ^ ^6 7 BB 7Cn
β 4 0 0 0 (6 4 k
dk 5 @@ g;h;j¼1 θg θ h θ j 4 5A p T fk Hj 0 0 2 3
P þ dM X þ HefYg þ ðdM þ 1ÞQ n n 6 7 T
Y
Q n 5
4 ηk

0

0

ðH j C^ gh ÞT p

0

2

p ðA^ ij þ L^ j C^ gh ÞT P BX r 6 T B 6 B^ ij P
B g;h;i;j¼1 θg θ^ h θi θ^ j 6 @ 4 T L^ j P

n

3 7

n5

I

0

ðA^ ij þ L^ j C^ gh
IÞT P p

T B^ ij P T L^ j P

31 7C 7C 7C 5A

2 3 ηk #
1 1 P n 6 7  nA6 ηk
dk 740
1
1 4 5 0 dM PZ n f pk 2 n 3 o p p He ðH j C^ gh ÞT H~ C^ ii n n r 6 7 X 6 7 0 0 n ( θg θ^ h θi θ^ j 6 7 n o 4 5 p p T g;h;i;j¼1 H Tj H~ C^ ii þ H~ j H C^ gh 0 He H Tj H~ 2 3
P þ dM X þ PHefYg þ ðd M þ 1ÞQ n n 6
YT
Q n 7
4 5 "

0

2

0

B B r B X 6 B 6 θg θ^ h θi θ^ j 6
B Bg;h;i;j¼1 6 B 6 @ 4

0

6 ðH~ C^ p ÞT ii 6

2

I n 60 P 4 0 0 3

Xr g;h;i;j¼1

n n

dM
1 PZ
1 n θg θ^ h θi θ^ j

ðA^ ij þ

0 T H~ 3
1

7 5

ng0

p L^ j C^ gh ÞT P

β2 I 0 @

p ðA^ ij þ L^ j C^ gh


IÞT P

T B^ ij P

T B^ ij P

T L^ j P

T L^ j P

1 31 A 7C 7C 7C 7C 7C 7C 7C 5A

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

3

20

n p p1 He ðH j C^ gh ÞT H~ C^ ii 6B C o 6B 6 B
Y þ P
d X C C M 6@ A 6 6
ðd þ 1ÞQ M 6 6 6 YT 6 6 6 n o p 6 He H Tj H~ C^ ii 6 6 6 6 0 1 6 T ~ ^p 6 H C H ii j 6 @ A 6 p T 6 þH~ j H C^ gh 6 6 6 p 6 PðA^ ij þ L^ j C^ gh ÞÞ 6 4 p PðA^ ij þ L^ j C^ Þ
P gh

2051

n

n

Q

0 @

0

n

n

He H Tj H~

o1 A


β2 I

n

n

n

n

n

n

n

n

n

0

H~

I

n

n

PB^ ij

PL^ j

0

P

n

PB^ ij

PL^ j

0

0

dM
1 PZ
1 n

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7g0 7 7 7 7 7 7 7 7 7 7 7 7 5

3ð20Þ where we have used the Schur complement, a congruence transform, Lemma 1 with the assignments E1 ≔

r X g;h;j ¼ 1

θg θ^ h θ^ j H j C^ gh ; p

F1 ≔

r X

θ^ j H j ;

E2 ≔

r X i¼1

h¼1

θi H~ C^ ii ; p

F 2 ≔ H~

T and H~ H i ¼ : W i , PL^ i ¼ : M i .□

Design condition in Theorem 2 is casted in nonlinear form. It can be efficiently solved by Algorithm 2 through the following conversion:   MP 2 : Minimize nP;Q;X;H~ ;V o trðPJ þ ZR þ SU Þ þ tr H~ K þ VN S≔

S;W i ;M i ;Y;U J;R;Z;N;K

subject to Eqs: ð13Þ; ð14Þ; ð18Þ; ð19Þ0 ; ð20Þ0 where Eqs. ðð19Þ0 ; ð20Þ0 Þ ≔ ðð19Þ; ð20ÞÞjH~ T H~

≔ V;PZ
1 P ≔ S

.

Fig. 3. A flexible-joint robot-arm.

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S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

5. Example Consider an one-joint robot-arm actuated by the DC motor. In the arm model, the joint is assumed to have a flexibility modeled as a linear spring of stiffness k. Fig. 3 shows the schematic diagram. The dynamics is described by ( J m θ€ m þ Bθ_ m þ kðθm
θl Þ ¼ K τ τ J l θ€ l þ MgL sin ðθl Þ þ kðθl
θm Þ ¼ 0 where θ the angular rotation and J the inertia, where the subscripts m and l denote the motor and the link, respectively; τ the motor torque input; B the viscous damping coefficient; M the total mass of the arm; L the distance to the joint from the mass-center of the axis of rotation; g the gravity constant; and K τ the input coupling coefficient. Let ðx1 ; x2 ; x3 ; x4 Þ ≔ ðθm ; θ_ m ; θl ; θ_ l Þ and u ≔ τ. We assume that x3 is immeasurable but lives in ½
0:79π; 0:79π. It is required to transform this nonlinear system into the T–S fuzzy model. Consider the following convex combination: 2 X sin ðx3 Þ ¼ θi ðx3 Þai : x3 i¼1

ð23Þ

Solving Eq. (23) results in 8 sin ðx3 Þ > >
a2 < x 3 ; θ i ð x3 Þ ¼ a1
a2 > > : 1
θ1 ðx3 Þ; Here, from the fact that a1 ⩾

sup

x A ½
0:79π;0:79π

for i ¼ 1

ð24Þ

otherwise:

9 sin ðx3 Þ > > > x3 =

) θi ðx3 ÞA R½0;1 sin ðx3 Þ > > > ; x A ½
0:79π;0:79π x3

ð0:79πÞ we determine ða1 ; a2 Þ ¼ 1; sin0:79π . This builds the two-rule T–S fuzzy model of the flexible-joint robot-arm parameterized by 2 3 2 3 0 1 0 0 0 6
k
B 7 k Kτ 7 6 0 6 Jm 7 Jm Jm J 7 7; Bui ¼ 6 Ai ¼ 6 6 m 7: 6 0 7 0 0 1 405 4 5 k 0
Jkl
MgL 0 0 Jm J l ai a2 ⩽

inf

Their parameter setting is borrowed from [18]. Before designing the observer bank, we close the loop by means of the controller u ¼ ½0:5636
0:1906
0:9161
0:0094x

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

to fulfill Assumption 1. Then the system matrices alter to 2 3 2 0 1 0 0 0 6
48:6
1:25 7 6 48:6 07 6 6
48:6 A1 ¼ 6 7; A2 ¼ 6 4 0 5 4 0 0 0 1 19:5 0
22:83 0 19:5 We further suppose the following parameters: 2 3
0:1 0 0 0  6 0 1 0 0 0 0 07 6 7 Ad ¼ 6 7; C 1 ¼ 4 0 5 0 0:9 0 0 0 0 0  D1 ¼

0:5 0:1

0

0

 D2 ¼

0

1

0


1:25

48:6



0 0

 ;

C2 ¼

0

3

07 7 7: 15

0
20:3224 0

1

0

0 0

0

1:1

0 0



0 

;

0

2053


0:4



0:2

to apply the developed method. Since y A R2 , we are required to design a two-observer bank. Let β1 ¼ 2, β2 ¼ 1:4, γ 1 ¼ γ 2 ¼ 1, dM ¼ 2, ε1 ¼ ε2 ¼ 0:1. The observer and the residual gains are found through Algorithm 2 as 2 3 2 3 0:0037
0:0057 6
0:0775 7 6
0:0711 7 6 7 6 7 L11 ¼ 6 7; L12 ¼ 6 7; H 11 ¼ ½2:7913; H 12 ¼ ½2:7710 4 0:0026 5 4
0:0058 5 0:0360 and

2


0:0190

0:0728 3

6 0:3703 7 6 7 L21 ¼ 6 7; 4
0:0273 5 0:0754

2


0:0459

3

6
0:4140 7 6 7 L22 ¼ 6 7; 4
0:0275 5

H 21 ¼ ½1:6296;

H 22 ¼ ½1:7881

0:2720

in 12 and 47 iterations run, respectively.

Fig. 4. Residual evaluation when f¼ 0 but wa 0: J r (solid) and J th (dashed).

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S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

We apply a delay d¼ 2 and a disturbance w AL2 randomly varying within ð
1:2; 1:2Þ, a fault f A L2 with the time profile 8 > < ð0:8; 0Þ; t A ½5; 10Þ f ¼ ð0; 0:8Þ; t A ½15; 20Þ > : ð0; 0Þ; otherwise: Set T W ¼ 3. Then one can estimate J pth;w ¼ 1:2γ p and J pth;f ¼ 0:8βp . Thus the thresholds are determined by ( 1 J th ¼ minf1:2; 1:6g ¼ 1:2 J 2th ¼ minf1:2; 1:12g ¼ 1:12: Set xð0Þ ¼ x^ 1 ð0Þ ¼ x^ 2 ð0Þ ¼ 0. To illustrate the design, three cases are simulated: (i) First, faultfree-disturbance-activated case is considered. As shown in Fig. 4, even though irregular disturbance is injected to the system, it is clear that J pr oJ pth for all time horizon. According to Table 1 one does not alarm  any fault, which is true enough. The evaluations  O1  ¼ 0:279793oγ 1 ¼ 1 and O2  ¼ 0:816150oγ 2 ¼ 1 confirm that (C3) (and (C1) C 1 C 1 automatically) in Definition 2 is satisfied. (ii) Next, time simulation is performed for fault-

Fig. 5. Residual evaluation when w ¼0 but f a0: J r (solid) and J th (dashed).

Fig. 6. Residual evaluation when f a 0 and w a0: J r (solid) and J th (dashed).

S.C. Jee, H.J. Lee / Journal of the Franklin Institute 353 (2016) 2030–2056

2055

Fig. 7. Residual evaluation by [7] when f a0 and wa0: J r (solid) and J th (dashed).

activated-disturbance-free case to verify H
performance. Time  1 responses of the evaluation O  ¼ 2:7795754β1 ¼ 2 and functions are depicted in Fig. 5. H -performance indices are
C
 2 O  ¼ 1:6292844β2 ¼ 1:4, which satisfies (C2) in Definition 2. By the FDI decision logic in C
Table 1, we canannounce that the faults on y1 and y2 (i.e. f1 and f2) are isolated for t A ½7:1; 11Þ and t A ½15:8; 22:2Þ, respectively with a retard less than 2 s. (iii) The fault-disturbance-activated case is concerned, whose time responses are drawn in Fig. 6. f1 and f2 are isolated for t A½6:7; 11:3Þ and t A ½15:8; 22:2Þ, respectively. This exhibits that the observer bank designed by the proposed method successfully accomplishes FDI task in spite of the disturbance, the state delay, and the unknown premise variables. Finally, the previous approach based on linear time-invariant model without taking account of the state delay [7] is simulated for comparison purpose. For design, the statespace parameters for R1 are utilized. Fig. 7 illustrates that the compared method fails in detecting and isolating the fault on y1, which reveals the effectiveness of the proposed method. 6. Conclusions In this paper, we presented the H
=H1 FDI observer design problem for the T–S fuzzy systems with state delay and immeasurable premise variables. The design conditions were derived in the format of nonlinear matrix inequalities. We suggested an iterative approach to solve the condition using the cone complementary linearization technique. Simulation results clearly visualized the effectiveness of the developed methodologies. Acknowledgments This research was supported by INHA UNIVERSITY Research Grant. References [1] S. Yin, S.X. Ding, X. Xie, H. Luo, A review on basic data-driven approaches for industrial process monitoring, IEEE Trans. Ind. Electron. 61 (11) (2014) 6418–6428. [2] S. Yin, X. Zhu, O. Kaynak, Improved PLS focused on key-performance-indicator-related fault diagnosis, IEEE Trans. Ind. Electron. 62 (3) (2015) 1651–1658. [3] S. Yin, X. Zhu, Intelligent particle filter and its application on fault detection of nonlinear system, IEEE Trans. Ind. Electron. (2015) 3852–3861.

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