FAULT DETECTION FOR SINGULAR TS FUZZY SYSTEMS WITH TIME-DELAY

Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008

FAULT DETECTION FOR SINGULAR TS FUZZY SYSTEMS WITH TIME-DELAY ? Li Chen ∗,∗∗ Maiying Zhong ∗ ∗

School of Control Science and Engineering, Shandong University, Jinan 250061, P. R. China ( e-mail: [email protected]). ∗∗ School of Statistics and Mathematics, Shandong Economic University, Jinan 250014, P. R. China (e-mail: [email protected]) Abstract: The robust fault detection filter design problem for singular TS fuzzy systems with time-delay is studied. Using an observer-based fuzzy fault detection filter as the residual generator, the fault detection filter design is converted to an H∞ filtering problem such that the generated residual is the H∞ estimation of the fault. Sufficient conditions are given, which guarantee the robust H∞ fault detection filter exists. And by using the cone complementarity linearization iterative algorithm, the fault detection filter design is converted to solving a sequence of convex optimization problems subject to LMIs. The premise variables of the designed fuzzy filter are not demanded to be the same as the premise variables of the TS fuzzy model of the plant. 1. INTRODUCTION Over the past few decades, the problem of fault detection and isolation (FDI) in dynamic systems has attracted considerable attention of many researchers (Chen et al., 1999; Chen et al., 2000; Ding et al., 2000; Jiang et al., 2003; Zhong et al., 2005). Among the approaches for FDI, the model-based approach has been extensively studied. As is well known, the presence of time delays must be taken into account in a realistic FDI filter design. However, it seems that there are very few previous results on the FDI problem for time-delay nonlinear systems (B. Castillo-Toledo et al., 2005; Chen et al., 2006; E. Alcorta et al., 2003; Magdy et al., 2006; Sing, Peng and Steven, 2006; Sing, Ping and Steven, 2006). In (B. CastilloToledo et al., 2005; Chen et al., 2006; E. Alcorta et al., 2003; Magdy et al., 2006; Sing, Ping and Steven, 2006), fault detection problem for nonlinear systems was studied without considering time delays. In (Sing, Peng and Steven, 2006), fault estimation problem for timedelay nonlinear systems described by TS fuzzy models was studied and assumed that the premise variables of the residual generator were the same as the premise variables of the TS model of the plant. To the best of authors’ knowledge, however, there is a lack to the research on FDI for singular TS fuzzy systems with time-delay, which motivates the present study. In this paper, we will deal with the problem of fault detection for a class of singular time-delay nonlinear systems described by TS fuzzy models. Using an observer-based fuzzy fault detection filter as a residual generator, the design of fault detection filter (FDF) will be formulated as ? This work was supported by National Natural Science Foundation of P. R. China (60374021), Ph.D. Programs Foundation of Ministry of Education of P. R. China (20050422036), Ph.D. Foundation of Shandong Province (2005BS011007, 03BS091).

978-3-902661-00-5/08/$20.00 © 2008 IFAC

an H∞ -filtering problem firstly. Then sufficient conditions on the existence of a robust H∞ -FDF for singular timedelay fuzzy systems will be derived in terms of linear matrix inequalities (LMIs) by using the cone complementarity linearization iterative algorithm and a solution to the robust H∞ -FDF can be obtained.

2. PROBLEM FORMULATION Consider a class of singular time-delay system described by the following TS fuzzy model Rule i: IF z1 (t) is Θi1 and · · · and zq (t) is Θiq , THEN ( E x(t) ˙ = Ai x(t) + Aτ i x(t − τ ) + Bd d(t) + Bf f (t) y(t) = Ci x(t) + Dd d(t) + Df f (t) (1) x(θ) = φ(θ), ∀θ ∈ [−τ, 0] where i = 1, 2, · · · , r, Θij (j = 1, 2, · · · , q) are fuzzy sets; T z(t) = [ z1 (t) · · · zq (t) ] is the premise variable which may be a measurable variable or the state of the system; x(t) ∈ Rn , y(t) ∈ Rny , f (t) ∈ Rnf , d(t) ∈ Rnd are the state, measurement output, fault and unknown input, respectively; f and d are assumed to be L2 -norm bounded; rankE = p, 0 < p < n. E, Ai , Aτ i , Bd , Bf , Ci , Dd and Df are known matrices with appropriate dimensions; τ is an unknown constant delay satisfying 0 ≤ τm ≤ τ ≤ τM

(2)

where τm and τM are known constants; φ(·) is a continuous vector valued initial function; r is the number of IFTHEN rules. In this paper, it is supposed that system (1) with f (t) = 0 and d(t) = 0 is asymptotically stable and E = diag(I, 0). The resulting fuzzy system model is inferred as the weighted average of the local models of the form

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10.3182/20080706-5-KR-1001.1703

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

 r X    E x(t) ˙ = µi (z(t))[Ai x(t) + Aτ i x(t − τ )    i=1   +Bd d(t) + Bf f (t)] r X    µi (z(t))[Ci x(t) + Dd d(t) + Df f (t)] y(t) =     i=1  x(θ) = φ(θ), ∀θ ∈ [−τ, 0] q Q j=1

≥ 0, Θij (zj (t))

¸ φ(θ) , ∀θ ∈ [−τ, 0], Ds = [ Dd Df ], 0 = [ 0 CW ].

(3)

Csf

Consider the following fuzzy FDF of observer-type Rule i: IF zˆ1 (t) is Θi1 and · · · and zˆq (t) is Θiq , THEN  Es x ˆ˙ s (t) = Asi x ˆs (t) + Aτ si x ˆs (t − τav )     +H (y(t) − yˆ(t)) i

Θij (zj (t))

where µi (z(t)) = P q r Q

·

φs (θ) =

r P i=1

yˆ(t) = Csi x ˆs (t)   r(t) = C x ˆ  ri s (t)  x ˆs (θ) = φs (θ), ∀θ ∈ [−τ, 0]

µi (z(t)) = 1.

(8)

i=1 j=1

Θij (zj (t)) is the grade of membership of zj (t) in Θij . Note that the premise variable z(t) may be a measurable variable or the state of the system, so system (3) is a nonlinear system. For the convenience of notations, we let µi = µi (z(t)). Defining τav = 12 (τM + τm ), using x(t − τ ) − x(t − τav ) = R t−τ x(θ)dθ, ˙ we can rewrite system (3) as t−τav  r X R t−τ    E x(t) ˙ = ˙ µi [Ai x(t) + Aτ i t−τav x(θ)dθ    i=1   +Aτ i x(t − τav ) + Bd d(t) + Bf f (t)] (4) r X    µ [C x(t) + D d(t) + D f (t)] y(t) =  i i d f    i=1  x(θ) = φ(θ), ∀θ ∈ [−τ, 0] The main objective of this paper is to design an asymptotically stable FDF such that the generated residual signal r satisfies the H∞ performance ||r − Wf (s)f ||2 ≤ γ||w||2

(5)

for a prescribed γ > 0, where Wf (s) is a given stable £ ¤T weighting matrix and w(t) = dT (t) f T (t) . Remark 1. The introducing of a suitable weighting matrix Wf (s) was used to limit the frequency interval, in which the fault should be identified, and the system performance could be improved. Without loss of generality, we suppose that one minimal realization of Wf (s) is ½ x˙ f (t) = AW xf (t) + BW f (t), xf (0) = 0 (6) rf (t) = CW xf (t) where xf (t) ∈ RnW , rf (t) ∈ Rnf , AW , BW and CW are known constant matrices. Augmenting (4) and (6) yields  r X    Es x˙ s (t) = µi [Asi xs (t) + Aτ si xs (t − τav )     i=1  R t−τ    +Aτ si t−τav x˙ s (θ)dθ + Bs w(t)] r (7) X   y(t) = µi [Csi xs (t) + Ds w(t)]     i=1    rf (t) = Csf xs (t)   xs (θ) = φs (θ), ∀θ ∈ [−τ, 0] where · ¸ · ¸ · ¸ x(t) E 0 Ai 0 xs (t) = ,Es = , Asi = , x (t) 0 I 0 AW · f ¸ · ¸ Aτ i 0 Bd Bf Aτ si = , Bs = ,Csi = [ Ci 0 ], 0 0 0 BW

where i = 1, 2, · · · , r, zˆi (t) is the estimation of zi (t), x ˆs (t) ∈ Rn+nW is the estimation of xs (t) and r(t) ∈ Rnf is the residual of FDF. Hi and Cri are the matrices to be designed. The final fuzzy FDF is inferred as the weighted average of the local models of the following form  r X   ˙ s (t) =  E x ˆ µ ˆi [Asi x ˆs (t) + Aτ si x ˆs (t − τav ) s     i=1    +Hi (y(t) − yˆ(t))]   r  X yˆ(t) = µ ˆi Csi x ˆs (t) (9)   i=1   r  X    r(t) = µ ˆi Cri x ˆs (t)     i=1  x ˆs (θ) = φs (θ), ∀θ ∈ [−τ, 0] Remark 2. In (B. Castillo-Toledo et al., 2005; E. Alcorta et al., 2003; Magdy et al., 2006; Sing, Peng and Steven, 2006; Sing, Ping and Steven, 2006), the premise variables of the residual generator are assumed to be the same as the premise variables of the fuzzy systems model. This actually means that the premise variables of the fuzzy systems model are assumed to be measurable. However, in general, it is extremely difficult to derive an accurate fuzzy systems model by imposing that all the premise variables are measurable. In this paper, we do not impose that condition, we choose the premise variables of the residual generator to be the estimated premise variables of the plant. Defining e(t) = xs (t) − x ˆs (t), re (t) = r(t) − rf (t), we have  r X r X   E e(t) ˙ = µ ˆi µ ˆj (Asi − Hi Csj )e(t)  s    i=1 j=1   r  X    + µ ˆi Aτ si e(t − τav )     i=1   r  X    + (µi − µ ˆi )Asi xs (t)   i=1 (10a) r X r X    − µ ˆ (µ − µ ˆ )H C x (t)  i j j i sj s    i=1 j=1   r  X    + (µi − µ ˆi )Aτ si xs (t − τav )     i=1   r  X R t−τ    + (µi − µ ˆi )Aτ si t−τav x˙ s (θ)dθ  i=1

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17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

           

+

r X

µ ˆi Aτ si

·

R t−τ t−τav

+

 µ ˆi (Bs − Hi Ds )w(t)

i=1   r r  X X    r (t) = µ ˆ (C − C )x (t) − µ ˆi Cri e(t)  e i ri sf s    i=1 i=1  e(θ) = 0, ∀θ ∈ [−τ, 0]

(10b)

£ ¤T Defining x ˜(t) = xTs (t) eT (t) , using x ˜(t − τ ) − x ˜(t − R t−τ ˙ τav ) = t−τav x ˜(θ)dθ, one obtains  r X   ˜ ˜ ix ˜x(t) ˙  E x ˜ (t) = G + µ ˆ i ∆H ˜(t) + ∆A˜  ij    i=1   +∆A˜τ x ˜(t − τ ) (11) r X   r (t) = ˜  µ ˆi Cri x ˜(t) e     i=1   ˜ x ˜(θ) = φ(θ), ∀θ ∈ [−τ, 0] where Gij =

r X r X

˜wi w(t) µ ˆi µ ˆj [(A˜ij + A˜τ i,2 )˜ x(t) + B

i=1 j=1

Rt −A˜τ i,2 t−τav x ˜˙ (θ)dθ + A˜τ i,1 x ˜(t − τ )], · ¸ · ¸ 0 ˜ = Es 0 , A˜ij = Asi E , 0 Es 0 Asi − Hi Csj · ¸ · ¸ 0 0 Aτ si 0 A˜τ i,2 = , A˜τ i,1 = , −Aτ si Aτ si Aτ si 0 · ¸ · ¸ 0 0 0 0 ∆A˜ = , ∆A˜τ = , ∆A 0 ∆Aτ 0 · ¸ · ¸ Bs 0 0 ˜wi = ˜i = B , ∆H , B − Hi Ds −Hi ∆C 0 · s ¸ φs (θ) ˜ φ(θ) = , ∀θ ∈ [−τ, 0], 0 r P C˜ri = [ Cri − Csf −Cri ], ∆A = (µi − µ ˆi )Asi , ∆Aτ =

r P i=1

(µi − µ ˆi )Aτ si , ∆C =

ω11 P T A˜τ i,1  * −Q  *  *  *  *  * * * *

(13)

 ˜wi ω15 ω16 ω13 P T B 0 0 ω25 ω26   −S 0 0 ω36  <0 * −γ 2 I 0 ω46  * * −I 0  * * * ω66

(14)

where ω11 = P T A˜ij + A˜Tij P + τav X + Y + Y T + Q + S + ε21 I, £ ¤ ω13 = P T A˜τ i,2 − Y , ω16 = τav A˜Tij Z 0 0 0 , · ¸ √ √ 2 T 2 T ˜T ω15 = ε2 P T ε3 P T P Ωi Φ Cri 0 , ε2 · · ¸ √ ε1 ¸ 0 0 2 T ω25 = 0 0 0 0 0 , Ω = , Ψ i 0 Hi Cˆ ε3 £ ¤ ω26 = τav A˜Tτi,1 Z τav A˜Tτi,1 Z 0 0 , £ ¤ ω36 = τav A˜Tτi,2 Z τav A˜Tτi,2 Z τav A˜Tτi,2 Z 0 , £ ¤ T T T T ˜wi ˜wi ˜wi ˜wi ω46 = τav B Z τav B Z τav B Z τav B Z , ω66 =· diag(−τ ¸ av Z,·−τav Z, ¸ −τav Z, −τav Z), ˆ ˆ Φ = A 0 , Ψ = Aτ 0 , i = 1, 2, · · · , r, 0 0 0 0 r r r P P P Aˆ = Asi , Aˆτ = Aτ si , Cˆ = Csj . i=1

i=1

j=1

Proof. Choose a Lyapunov function candidate as Rt ˜x V (t) = x ˜T (t)P T E ˜(t) + t−τ x ˜T (θ)Q˜ x(θ)dθ Rt T + t−τav x ˜ (θ)S x ˜(θ)dθ R0 Rt T ˜ E ˜x + ds x ˜˙ (θ)EZ ˜˙ (θ)dθ −τav

t+s

Taking the derivative of V (t) with respect to t along the trajectory of (11) yields r X ˜ ix ˜x(t) µ ˆ i ∆H ˜(t) + ∆A˜ V˙ (t) = 2˜ xT (t)P T [Gij + i=1

+∆A˜τ x ˜(t − τ )] − x ˜T (t − τ )Q˜ x(t − τ ) T T +˜ x (t)Q˜ x(t) − x ˜ (t − τav )S x ˜(t − τav ) T ˜ E ˜x +˜ xT (t)S x ˜(t) + τav x ˜˙ (t)EZ ˜˙ (t) Rt T ˜ E ˜x − t−τav x ˜˙ (s)EZ ˜˙ (s)ds

i=1 r P

j=1

¸ ≥0

x˙ s (θ)dθ

i=1

r X

X Y ˜ E ˜ * EZ

(µj − µ ˆj )Csj .

Based on the above discussion, the FDF problem to be addressed is stated as follows. The FDF Problem: Design an FDF of observer-type in form (8) such that it is a robust H∞ -FDF of system (1) if (i) system (11) with w(t) = 0 is asymptotically stable; (ii) the H∞ performance ||re ||2 < γ||w||2 is guaranteed for all nonzero w(t) ∈ L2 [0, ∞) and a prescribed γ > 0 under the condition φ(θ) = 0, ∀θ ∈ [−τ, 0].

Using (13), we have

Rt −2˜ xT (t)P T A˜τ i,2 t−τav x ˜˙ (θ)dθ · ¸T · ¸· ¸ Rt x ˜(t) x ˜(t) 0 −P T A˜τ i,2 = t−τav ˙ dθ x ˜(θ) x ˜˙ (θ) * 0 · ¸T · ¸ · ¸ Rt (15) x ˜(t) x ˜(t) X Y − P T A˜τ i,2 ≤ t−τav ˙ dθ ˜ E ˜ x ˜(θ) x ˜˙ (θ) * EZ = τav x ˜T (t)X x ˜(t) + 2˜ xT (t)(Y − P T A˜τ i,2 )(˜ x(t) Rt T ˜ E ˜x −˜ x(t − τav )) + x ˜˙ (θ)EZ ˜˙ (θ)dθ

3. MAIN RESULTS

t−τav

The following theorem is essential for solving the FDF problem formulated in the previous section.

Noticing that

Theorem 1. For given scalars γ > 0 and τ satisfying (2), system (8) is a robust H∞ -FDF of system (1), if there exist scalars ε1 > 0, ε2 > 0, ε3 > 0, matrices Q > 0, S > 0, X ≥ 0, Z > 0 and P , Y , such that the following matrix inequalities hold for i, j = 1, 2, · · · , r ˜ = PTE ˜≥0 EP (12)

10132

r r X X ˜ ix ˜ iT )P x x ˜T (t)P T ( µ ˆ i ∆H ˜(t)) + ( µ ˆi x ˜T (t)∆H ˜(t) i=1

i=1

r 1 X 2 T ˜ i) µ ˆi x ˜T (t)P T ∆H ≤ ε1 x ˜ (t)˜ x(t) + 2 ( ε1 i=1 r X ˜ iT P x ×( µ ˆ i ∆H ˜(t)) i=1

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

r 1 X ˜ i ∆H ˜ iT P x µ ˆi x ˜T (t)P T ∆H ˜(t) 2 ε1 i=1 (16) r 2 X 2 T T T T ≤ ε1 x ˜ (t)˜ x(t) + 2 µ ˆi x ˜ (t)P Ωi Ωi P x ˜(t) ε1 i=1

≤ ε21 x ˜T (t)˜ x(t) +

Similarly, we have ˜x(t) + x x ˜T (t)P T ∆A˜ ˜T (t)∆A˜T P x ˜(t) 2 T (17) 2 T T T ≤ ε2 x ˜ (t)P P x ˜(t) + 2 x ˜ (t)Φ Φ˜ x(t) ε2 x ˜T (t)P T ∆A˜τ x ˜(t − τ ) + x ˜T (t − τ )∆A˜Tτ P x ˜(t) 2 (18) T T 2 T T ˜ (t − τ )Ψ Ψ˜ x(t − τ ) ≤ ε3 x ˜ (t)P P x ˜(t) + 2 x ε3 £ T ¤T Denote ξ1 (t) = x ˜ (t) x ˜T (t − τ ) x ˜T (t − τav ) . In the case of w(t) = 0, from (15)-(18) we obtain r X r X V˙ (t) ≤ µ ˆi µ ˆj ξ1T (t)Υ1 ξ1 (t) (19) i=1 j=1

where

 ω ¯ 11 ω ¯ 12 ω ¯ 13 ¯ 22 τav A˜Tτi,1 Z A˜τ i,2 , Υ1 =  * ω * * τav A˜Tτi,2 Z A˜τ i,2 − S T ˜ ω ¯ 11 = P Aij + A˜Tij P + τav X + Y + Y T + Q +S + ε21 I + τav A˜Tij Z A˜ij + (ε22 + ε23 )P T P 2 2 + 2 P T Ωi ΩTi P + 2 ΦT Φ, ε1 ε2 ω ¯ 12 = P T A˜τ i,1 + τav A˜Tij Z A˜τ i,1 , ω ¯ 13 = P T A˜τ i,2 − Y + τav A˜Tij Z A˜τ i,2 , ω ¯ 22 = ε22 ΨT Ψ − Q + τav A˜Tτi,1 Z A˜τ i,1 . 3 Noticing that (14) implies Υ1 < 0, there exists a scalar δ > 0 such that Υ1 + diag(δI, 0, 0) < 0 (20) From (19)-(20), we have V˙ (t) < −δ x ˜T (t)˜ x(t) which means that the system (11) with w(t) = 0 is asymptotically stable. Define R +∞ J = 0 (reT (t)re (t) − γ 2 wT (t)w(t))dt under zero initial condition, it can be shown that for any nonzero w(t) ∈ L2 [0, ∞) and t > 0 R +∞ T J≤ (re (t)re (t) − γ 2 wT (t)w(t) + V˙ (t))dt 0

Observing that r r X X T ( µ ˆi x ˜T (t)C˜ri )( µ ˆi C˜ri x ˜(t)) i=1

≤

r X

i=1 T

µ ˆi x ˜

T ˜ (t)C˜ri Cri x ˜(t)

i=1

we have reT (t)re (t) ≤

r X

T ˜ µ ˆi x ˜T (t)C˜ri Cri x ˜(t)

(21)

i=1

Denote £ T ¤T ξ2 (t) = x ˜ (t) x ˜T (t − τ ) x ˜T (t − τav ) wT (t) , From (15)-(18) and (21), we have r X r X R +∞ J≤ µ ˆi µ ˆj 0 ξ2T (t)Υ2 ξ2 (t)dt (22) i=1 j=1

where

˜wi + τav A˜Tij Z B ˜wi  ω ˜ 11 ω ˜ 12 ω ˜ 13 P T B ˜wi  * ω  ˜ 22 ω ˜ 23 τav A˜Tτi,1 Z B , Υ2 =  T  * * ω  ˜wi ˜ 33 τav A˜τ i,2 Z B 2 T ˜ ˜ * * * −γ I + τav Bwi Z Bwi ω ˜ 11 = P T A˜ij + A˜Tij P + τav X + Y + Y T + Q +S + ε21 I + τav A˜Tij Z A˜ij + (ε22 + ε23 )P T P 2 2 T ˜ + 2 P T Ωi ΩTi P + 2 ΦT Φ + C˜ri Cri , ε1 ε2 ω ˜ 12 = P T A˜τ i,1 + τav A˜Tij Z A˜τ i,1 , ω ˜ 23 = τav A˜Tτi,1 Z A˜τ i,2 ,˜ ω33 = τav A˜Tτi,2 Z A˜τ i,2 − S, ω ˜ 13 = P T A˜τ i,2 − Y + τav A˜Tij Z A˜τ i,2 , ω ˜ 22 = ε22 ΨT Ψ − Q + τav A˜Tτi,1 Z A˜τ i,1 . 3 Applying the Schur complement formula to (14), we obtain Υ2 < 0. Thus J < 0, i.e. ||re ||2 < γ||w||2 , which implies system (8) is a robust H∞ -FDF of system (1). This completes the proof. Now, we are in the position to solve the robust H∞ -FDF problem. Theorem 2. For given scalars γ > 0 and τ satisfying (2), system (8) is a robust H∞ -FDF of system (1), if there exist scalars ε1 > 0, ε2 > 0, ε3 > 0, matrices P1 , P2 , Li , Cri , Xk , Qk , Sk , Zk , Uk and Yl , i = 1, 2, · · · , r, 1 ≤ k ≤ 3, 1 ≤ l ≤ 4, where X1 ≥ 0, X3 ≥ 0, Q1 > 0, Q3 > 0, S1 > 0, S3 > 0, Z1 > 0, Z3 > 0, U1 > 0, U3 > 0, such that the following matrix inequalities hold for i, j = 1, 2, · · · , r Es P1 ≥ 0, Es (P2 − P1 ) ≥ 0 (23)   T Λ11 X1 + X2 Λ13 Y1 + Y3  * X1 Y1 + Y2 Y1  ≥0 (24)  * * Λ33 Λ34  * * * Λ44 · ¸ T ˆ ˆ ˆ ˆT Ξ11 Z1 M U2 M + Z2 M U3 M (25) ˆ U2 M ˆ T + Z3 M ˆ U3 M ˆT = I Ξ21 Z2T M · ¸ Γ11 τav X1 + τav X2T + Y1 + Y1T + Y2 + Y3 ≥ 0 (26) * τav X1 + Y1 + Y1T   η11 η12 η13 η14 η15 η16 η17 η18 η19  * η22 η23 η24 η25 η26 η27 η28 η29   * * η 0 0 0 η38 η39  33 η34    * * * −Q1 0 0 0 η48 η49     * * * * η55 η56 0 0 0   < 0 (27)   * * * * * −S1 0 0 η69     * * * * * * −γ 2 I 0 η79    * * * * * * * −I 0 * * * * * * * * η99 where Λ11 = X1 + X2 + X2T + X3 ,Λ13 = Y1 + Y2 + Y3 + Y4 , ˆ U1 M ˆ T P T Es , Λ33 = Es P1 M 1 ˆ U1 M ˆ T P T E s + E s P1 M ˆ U2 M ˆ T (P T − P T )Es , Λ34 = Es P1 M 2 1 2 ˆ UT M ˆ T P T Es + Es P2 M ˆ U2 Λ44 = Es (P1 − P2 )M 2 2 ˆ T (P1T − P2T )Es + Es P2 M ˆ U1 M ˆ T P2T Es ×M T T T ˆ ˆ +Es (P1 − P2 )M U3 M (P1 − P2 )Es , £ ¤ ˆ = M ˆ 11 , M ˆ 23 and M ˆ 32 are identity matrices ˆ ij M ,M 3×3 with appropriate dimensions, other block matrices are zero matrices. Γ11 = τav X1 + τav X2 + τav X2T + τav X3 + Y1 +Y1T + Y2 + Y2T + Y3 + Y3T + Y4 + Y4T ,

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η11 = P1T Asi + ATsi P1 + 2ε21 I + Q1 + Q2 + QT2 +Q3 + τav X1 + τav X2 + τav X2T + τav X3 +S1 + S2 + S2T + S3 + Y1 + Y2 + Y3 + Y4 +Y1T + Y2T + Y3T + Y4T , T η12 = P1T Asi + ATsi P1 + Csj Li + ε21 I + Q1 + QT2 T +τav X1 + τav X2 + S1 + S2T + Y1 + Y1T +Y2T + Y3 , η13 = η14 = η23 = η24 = P1T Aτ si ,η16 = −Y1 − Y3 , η15 = −Y η17 = P1T Bs , ¸ · 1 − Y2 − Y3 − Y4 , √ 2 ˆT T η18 = ε2 P1T 0 ε3 P1T 0 0 A −Csf 0 , ε 2 £ T η19 = τav ATsi P1 τav (ATsi P1 + Csj Li ) 0 0 0 0 0 0], η22 = P2T Asi + ATsi P2 + ε21 I + Q1 + S1 + τav X1 +Y1 + Y1T , η25 = −Y1 − Y2 , η26 = P2T Aτ si − P1T Aτ si − Y1 , T T T η27 = P s ,η33 = −Q1 − Q2 − Q2 − Q3 , £ 1 BsT + Li D T T T T η28 = ε2 P2 ε2 (P1 − P2 ) ε3 P2 ε3 (P1 − P2T ) ¸ √ √ 2 T ˆ 2 ˆT T T − L C A Cri − Csf 0 , ε1 i ε2 £ ¤ η29 = τav ATsi P1 τav ATsi P2 0 0 0 0 0 0 , · ¸ √ 2 ˆT η38 = η48 = 0 0 0 0 0 0 0 , A ε3 τ £ T T η39 = η49 = τav Aτ si P1 τav Aτ si P1 τav ATτsi P1 ¤ τav ATτsi P1 0 0 0 0 , η55 = £−S1 − S2 − S2T − S3 , η56 = −S1 − S2T , η69 = 0 τav ATτsi (P2 − P1 ) 0 τav ATτsi (P2 − P1 ) ¤ 0 τav ATτsi (P2 − P1 ) 0 0 , £ η79 = τav BsT P1 τav T1 τav BsT P1 τav T1 ¤ τav BsT P1 τav T1 τav BsT P1 τav T1 , T1 = BsT P1 + DsT Li , η34 = −Q1 − QT2 , η99 =diag(T2 , T2 , T2 , T2 ),  −τav (Z1 + Z2 T −τ (Z + Z ) av 1 2 . T2 =  +Z2T + Z3 ) * −τav Z1 In this case, a desired robust H∞ -FDF is given in the form of (8) with parameters as follows Hi = (P2 − P1 )−T LTi , i = 1, 2, · · · , r

(28)

Proof. Suppose (23)-(27) hold and the coefficient matrices of the filter (8) are designed in the form of (28). Now we will prove that there exist matrices Q > 0, S > 0, X ≥ 0, £ Z ¤> 0 and P , Y satisfying (12)-(14). Denote P1 = P˜ij 3×3 . From (23) we have P˜12 = 0, P˜32 = 0 T and P˜31 = P˜13 . Then we obtain P1T Es = Es P1 ≥ 0, T P2 Es = Es P2 ≥ 0. From (26)-(27), it is easy to prove that P1 is nonsingular. Without loss of generality, it is assumed that P1 − P2 is −1 nonsingular (Shengyuan Denote · −1 ¸ et al., 2003). · ¸ P = Π2 Π1 , P1 I I P2 where Π1 = , Π2 = . Then using 0 P1 − P2 P1−1 0 theorem 1 in (Shengyuan et al., 2003), we obtain P = · ¸ P2 P1 − P2 is nonsingular and satisfies (12). * −(P1 − P2 ) · ¸ · ¸ · ¸ Q1 Q2 S1 S2 X1 X2 Denote Q = , S = , X = , * Q3 * S3 * X3

·

¸ · ¸ Y1 Y2 Z1 Z2 ¯ −1 . Pre, Z¯ = , Z −1 = P −T ZP Y3 Y4 * Z3 and post-multiplying (27) by diag(P1−T , I, P1−T , I, P1−T , I, I, I, P1−T , I, P1−T , I, P1−T , I, P1−T , I) and its transpose, it can be shown that the LMIs in (27) are equivalent to the LMIs in (14). Similarly, it can be proved that the matrix inequalities in (24)-(25) is equivalent to the matrix inequality in (13). From theorem 1, we know that system (8) is a robust H∞ -FDF of system (1). This completes the proof. Y =

Remark 3. The introducing of the parameters ε1 , ε2 , ε3 and the matrices X, Y , Z is used to make the matrix inequalities in theorem 1 less conservative. Remark 4. It is clear that the nonlinear terms in (24)(25) make that (24)-(25) are not conformable to LMIs. However, by using the cone complementarity linearization iterative algorithm proposed in (Ghaoui et al., 1997) by minor modification, we can convert (24)-(25) to solving a sequence of convex optimization problems subject to LMIs. 4. CONCLUSION In this paper, fault estimation is adopted as the residual. The robust residual generator design problem for singular time-delay nonlinear systems is converted to a H∞ filtering problem. Sufficient conditions are given, which guarantee the robust H∞ fault detection filter exists. And by using the cone complementarity linearization iterative algorithm, the fault detection filter design is converted to solving a sequence of convex optimization problems subject to LMIs. The premise variables of the residual generator are chosen to be the estimated premise variables of the plant so that the premise variables of the residual generator are not demanded to be measurable. REFERENCES B. Castillo-Toledo and J. Anzurez-Marin (2005). Modelbased fault diagnosis using sliding mode observers to takagi-sugeno fuzzy model. Proceedings of the 2005 IEEE International Symposium on Mediterrean Conference on Control and Automation pp. 652–657. Chen, J. and R. J. Patton (1999). Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Boston. Chen, J. and R. J. Patton (2000). Standard H∞ filtering formulation of robust fault detection. Proceedings of the IFAC Symposium SAFEP-ROCESS 2000 pp. 256–261. Chen, W. and Saif, M (2006). An iterative learning observer for fault detection and accommodation in nonlinear time-delay systems. International Journal of Robust and Nonlinear Control, 16, 1–19. Ding, S. X., Jeinsch. T., Frank, P. M. and E. L. Ding (2000). A unified approach to the optimization of fault detection systems. International Journal of Adaptive Control and Signal Processing, 14, 725–745. E. Alcorta and S. Sauceda (2003). Fault detection and isolation based on takagi-sugeno modeling. Proc. of IEEE Int. Symp. on Intelligent Control pp. 673–678. Ghaoui, L. E., Oustry, F. and Rami, M. A.(1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42, 1171–1176.

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Jiang, B., Staroswiecki, M. and Cocquempot, V. (2003). H∞ fault detection filter design for linear discretetime systems with multiple time delays. International Journal of Systems Science, 34, 365–373. Magdy, G. E., Steven, X. D. and Zhiwei, G. (2006). Robust fuzzy fault detection for continuous-time nonlinear dynamic systems. Proceedings of the IFAC SAFEPROCESS’06 pp. 271–276. Shengyuan, X., James, L. and Yun, Z. (2003). H∞ filtering for singular systems. IEEE Transa-ctions on Automatic Control, 48, 2217–2222. Sing, K. N., Peng, S. and Steven, D. (2006). Delay dependent fault estimation for uncertain time delay nonlinear systems: an LMI approach. Proceedings of the 2006 American Control Conference pp. 5135–5140. Sing, K. N., Ping, Z. and Steven, D. (2006). Parity based fault estimation for nonlinear systems: an LMI approach. Proceedings of the American Control Conference pp. 5141–5146. Zhong, M., Ye, H., Shi, P. and Wang, G. (2005). Fault detection for markovian jump systems. IEE Proceedings Control Theory and Applications, 152, 97–402.

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