Unknown input observer design for fuzzy systems with uncertainties

Applied Mathematics and Computation 266 (2015) 108–118

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Unknown input observer design for fuzzy systems with uncertainties Xiao-Kun Du a,b, Hui Zhao a,c, Xiao-Heng Chang b,∗ a

School of Electrical Engineering and Automation, Tianjin University, Tianjin, China College of Engineering, Bohai University, Jinzhou, Liaoning, China c College of Engineering and Technology, Tianjin Agricultural University, Tianjin, China b

a r t i c l e

i n f o

Keywords: Unknown input observer (UIO) Uncertainties T–S fuzzy systems Lyapunov theory

a b s t r a c t This paper investigates the problem of unknown input observer design for both discrete and continuous-time T–S fuzzy systems with uncertainties. After doing appropriate processing to the model and reasonable analysis to the error expression of the system, the observer design conditions are proposed in LMI form based on Lyapunov theory. More important is the introduction of a new decoupling method which can further reduce the conservatism. The idea can eliminate the influence of the unknown inputs, and guarantee the error of the state estimation is bounded when the uncertainties are nonzero. Finally, an appropriate example is given to show the effectiveness of the algorithm, especially the excellent estimate ability of the observer in initial time. © 2015 Elsevier Inc. All rights reserved.

1. Introduction As everyone knows, most of the engineering, chemical and physical systems contain uncertainties and various disturbances [1–6]. The disturbances can mainly divide into two categories, external disturbances (space magnetic interference, pulse interference, temperature change, and so on) and internal disturbances (parameter variations, measurement errors, inaccurate modeling, etc). System state reconstruction will be seriously affected once these immeasurable or unpredictable signals, namely the unknown inputs (UI), are not processed correctly. Therefore, in recent decades considerable attention is focused on the study of the UI system in control and stability [7–12]. Tong et al. developed an adaptive fuzzy backstepping dynamic surface control approach for a class of MIMO nonlinear systems with immeasurable states. A fuzzy state observer is designed to estimate the immeasurable states [10,12]. Then Li et al. proposed an adaptive fuzzy backstepping output-feedback tracking control approach for a class of MIMO nonlinear systems. The MIMO systems are assumed to possess unstructured uncertainties, unknown dead-zones and unknown control directions [11]. And based on adaptive backstepping dynamic surface control technique and utilizing the prediction error between the system states observer model and the serial–parallel estimation model, a new fuzzy controller with the composite parameters adaptive laws are developed by Li et al. [30]. Jeong et al. [5] proposes a robust adaptive synchronization method for uncertain chaotic neural networks with timevarying delays and distributed delays. The uncertain factors including uncertainties and disturbances are estimated by the FDO without requiring any prior knowledge about the factors [4,6]. The so-called unknown input observer (UIO), as one of the important research fields is usually adopted as a solution to provide state estimations of a system [13–15]. After Basile and Marro started to concern the UIO from 1969 [16], considerable literatures ∗

Corresponding author: Tel.: 0416 3400908. E-mail addresses: [email protected] (X.-K. Du), [email protected] (H. Zhao), [email protected] (X.-H. Chang).

http://dx.doi.org/10.1016/j.amc.2015.05.046 0096-3003/© 2015 Elsevier Inc. All rights reserved.

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118

109

about UIO design emerged [17,18]. For linear time-invariant system models, there are several approaches for UIO design [19–21]. A minimal-order observer without any assumptions upon the unknown inputs was proposed for a linear time-invariant system by Wang et al. [22]. Then dozens of different UIO design methods are proposed after Wang, e.g., Chadli and his partners designed a multiple observer for multiple models with unknown inputs and put forward the interpolation principle [23]. Then Jamel et al. [28] discussed the UI multi model with uncertainties or unknown inputs; and Chadli and Karimi [24] reduced the conservative than the one given in [25] by introducing multiple matrices Xi > 0 instead of a single matrix X and nonsymmetrical slack variable V. In this paper, for unknown input observer design, the suggested technique consists in eliminating the unknown inputs from the dynamics of the state estimation error [23]. The asymptotic stability of estimation error relies on several constraints. Our contribution here lies in the robust observer design for state estimation. In order to be applicable to more general system, the uncertainty is also been considered. The contribution of this paper are: (i.) For a T–S fuzzy system with uncertainties, treat the uncertainties as the unknown input. This processing method can not only guarantee the gradual stability of the system, but also simplify the observer design greatly. (ii.) A new decoupling method is applied to avoid the nonsingular constraint to G so as to reduce the conversation. (iii.) After reasonable processing to the uncertainties, the authors propose the LMI design conditions of observer for both discrete and continuous-time systems based on Lyapunov stability theory and bound norm theory. 2. Problem statement Interference is ubiquitous in all practical systems. Some of them have direct influence on the system parameters or the input matrix [26]. The authors consider both of these two forms in the paper, and treat other disturbances as unknown inputs meanwhile. Consider the following T–S fuzzy system [27,28] (it is suitable for both discrete and continuous-time systems), the ith rule Ri is: if ζ 1 (t) is M1i and ζ 2 (t) is M2i and  and ζ p (t) is Mip (i = 1, 2, . . . , r),



then :

δ x(t ) = (Ai ± Ai )x(t ) + (Bi ± Bi )u(t ) + Qi u(t ) y(t ) = Cx(t ) + Du(t )

(1)

where ζ (t ) = [ζ1 (t ), ζ2 (t ), . . . , ζ p (t )]T ∈ R p is the decision vector which is real time accessible variable; Mki (i = 1, 2, . . . , r; k = 1, 2, . . . , p) is the fuzzy set; r is the number of fuzzy rules. δ x(t) represents the discrete-time system state x(t + 1 ) or continuoustime system state x˙ (t ), x(t) ∈ Rn is the state vector, u(t) ∈ Rm represents the input vector, u(t ) ∈ Rq is the unknown input and y(t) ∈ Rp represents the measured output. Ai ∈ Rn×n , Bi ∈ Rn×m and C ∈ Rp×n define the ith local model. Matrices Qi ∈ Rn×q and D ∈ Rp×q represent the influence of the unknown inputs. Ai and Bi are the modeling uncertainties. By using the T–S fuzzy rules, the T–S fuzzy control system model (1) can be rewritten as



δ x(t ) =

r 

λi (Ai x(t ) + Bi u(t ) + Qi u(t ) + d (t ))

i=1

y(t ) = Cx(t ) + Du(t ) with λi = μi (ξ (t ))/ and 0 ≤ λi ≤ 1,

r  i=1

r  j=1

μ j (ξ (t )), μi (ξ (t )) =

p  k=1

(2)

Mki (ξk (t )), Mki (ξk (t )) represents the membership grade of ξ k (t) towards Mki ,

λi = 1. Noting that d (t ) = (±Ai x(t ) ± Bi u(t )), now we study the observer design problem towards the

derived system model (2). 3. UIO design and LMI synthesis conditions In order to estimate the system state, we built the following UIO:



δ z(t ) =

r  i=1

λi (i z(t ) + i u(t ) + Vi y(t ))

xˆ(t ) = z(t ) − Ky(t )

(3)

The matrices i ∈ Rn×n ,  i ∈ Rn×m , Vi ∈ Rn×p and K ∈ Rn×p are the observer gains which are remain to be determined. UIO gets the estimate state by using the known variables u(t) and y(t), meanwhile the unknown input u(t ) and the uncertainties d(t) are unmeasured. In order to assure the estimate state xˆ(t ) approach the true state x(t), we define the estimation error as

e(t ) = x(t ) − xˆ(t )

(4)

By the system model (2) and the observer expression (3), we have

e(t ) = x(t ) − xˆ(t ) = x(t ) − z(t ) + K (Cx(t ) + Du(t )) = (I + KC )x(t ) − z(t ) + KDu(t )

(5)

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Define L = (I + KC ), therefore

δ e(t ) = (I + KC )δ x(t ) − δ z(t ) + KDδ u(t ) = Lδ x(t ) + KDδ u(t ) −

r 

λi [i (x(t ) + Ky(t ) − e(t )) + i u(t ) + Vi y(t )]

i=1

let Ni = i K + Vi , we have

δ e(t ) =

r 

λi [i e(t ) + (LAi − i − NiC )x(t ) + Ld (t )

i=1

+(LBi − i )u(t ) + (LQi − Ni D )u(t )] + KDδ u(t )

(6)

In the Eq. (6), if the following constrains ➀–➃: ➀ LAi − NiC − i = 0 ➁ LBi − i = 0 ➂ LQi − Ni D = 0 ➃ KD = 0 can be fulfilled, then the expression of δ e(t) can be rewritten as

δ e(t ) =

r 

λi (i e(t ) + Ld (t ))

(7)

i=1

The robust state estimation problem is reduced to determine the observer gains matrices on condition that the estimate error e(t) asymptotic convergence towards zero if d (t ) = 0 and to ensure a bounded error in the case d(t) = 0, i.e.



lim e(t ) = 0 when d (t ) = 0 e(t )Qe ≤ γ d (t )Qd when d (t ) = 0 and e(0 ) = 0

t→∞

(8)

where γ > 0 is the attenuation level. To satisfy the constraints (8), it is sufficient to find a Lyapunov function such that

δV (t ) + eT (t )Qe e(t ) − γ 2 dT (t )Qd d (t ) < 0

(9)

where Qe and Qd are two positive definite matrices. Next, we will present new LMI conditions of the UI observer design for both discrete and continuous-time systems respectively.

3.1. The discrete-time system case For discrete-time T–S fuzzy systems with uncertainties, we try to find the sufficient constrain conditions to ensure the asymptotic stability of the observer in terms of LMIs. The following theorem can give a reference to other designers. Theorem 1. For a given scalar γ > 0, if there exist matrices P > 0, G, M, N, S, U, Ji, , and Ji , scalars α and β such that the following conditions hold:

⎡−P + Q

e

⎢ 0 ⎢ ⎢ 31 ⎢ ⎢ NVCAi ⎢ ⎢ 0 ⎢ ⎣ SWiC 0

∗

−γ Qd G + MVC NVC

33 0

ε

∗ ∗ ∗ ∗

0 0

0 0

G − MU 0

− αJi2 0

η

0

0

0

0

−G − MU

2

∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

⎤

⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ∗ ⎥ ⎥ ∗ ⎦

(10a)

J,

− βi2

UQi + VCQi = Wi D

(10b)

VD = 0

(10c)

where 31 = GAi + MVCAi , 33 = −G − GT + P + Ji + Ji, , ε = −α NU − αU T NT , and η = −β SU − β U T ST . Then the observer (3) converges asymptotically to the state of the discrete-time T-S fuzzy system model (1) is guaranteed, and the observer parameters can be

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118

111

determined as follows:

K = U −1V

(11a)

i = (I + U −1VC )Bi

(11b)

i = (I + U −1VC )Ai − U −1WiC

(11c)

Vi = U −1Wi − iU −1V

(11d)

Proof. For the discrete-time system, let us consider the normal Lyapunov function: V (t ) = eT (t )Pe(t ), then the differential expression of V(t) is

δV = V (t + 1 ) − V (t ) = eT (t + 1 )Pe(t + 1 ) − eT (t )Pe(t )

(12)

So the sufficient condition of (9) turns into

eT (t + 1 )Pe(t + 1 ) − eT (t )Pe(t ) + eT (t )Qe e(t ) − γ 2 dT (t )Qd d (t ) < 0 i.e. r 

 λi

i=1

 T





where  =



e(t ) d (t )

−P + Qe 0

      e(t ) i L T P i L +  <0 d (t )

(13)

(14)



0

−γ 2 Q

. So the core problem turns into meet the condition d

   i L T P i L +  < 0

(15)

By the Schur complement to (15), we can obtain that



−P + Qe 0

i

∗

−γ Qw L 2

∗ ∗ −P −1

 <0

(16)



I Now, pre- and post-multiplying (16) by 0 0



−P + Qe 0 Gi

∗

−γ Qw GL 2

∗ ∗ −GP −1 GT



0 0 I 0 and its transpose, respectively, (16) is rewritten as follows: 0 G

 <0

(17)

Note that −GP −1 GT ≤ −G − GT + P, P > 0, and substituting the definitions i = LAi − NiC and L = I + KC into the coupling term Gi and GL: Gi = GAi + GKCAi − GNiC, GL = G + GKC, then the (17) can be ensured by



−P + Qe 0 GAi + GKCAi − GNiC

∗

−γ 2 Qd G + GKC

∗ ∗ −G − GT + P



<0

(18)

It can be expressed as

 

 

0  0     + 0 K CAi C 0 + CAi C 0 T K T 0 T G G     0 0     + 0 Ni C 0 0 + C 0 0 T NiT 0 T < 0 −G −G





(19)

−P + Qe ∗ ∗ with  = 0 −γ 2 Qd ∗ and  = −G − GT + P. Defining UK = V, UNi = Wi and considering the matrices M, N, and S, G  GAi with U, N, and S are nonsingular matrices, it follows that

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X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118

the left of inequality (19)

 

0  0 U −1 N−1 NV CAi G

=+

 T









×N N

−T



+ CAi

U

C



−T

0 0 G − MU

0 V

× C

T



0 0 + C

 

0  + 0 Wi C M

+



× NV CAi







0  0 V CAi M

C

0











0 0

T

∗ ∗

T

0 0 M







0 0 + U −1 N−1 G − MU 

−γ Qd G + MVC 2



T

C

0 0 −G − MU

 T WiT

∗

T

T

0 V N N



0  0 + U −1 S−1 SWi C −G − MU



−T



U

0 0 G − MU

−T

T

T

0 0 WiT ST S−T U −T

0 0 + C



0 0 −G − MU

Based on XY + Y T X T ≤ XJX T + Y T J−1Y, J > 0, for positive matrices Ji and Ji, , one gives





0  0 U −1 N−1 NV CAi G − MU



× NT N−T U −T



× C



0 0 G − MU



0 0 + C

0



C

T

0



C







WiT ST S−T U −T

 

0  0 (G − MU )U −1 N−1 NV CAi I

=

 T

T

×N N

−T



× Wi C



≤ CAi

C

U

−T

(G − MU )





0 0 + C



T

0 0 I

0 0



0 TV T

0 + CAi

0 0 + U −1 S−1 SWi −G − MU

T

T

T



0 WiT ST S−T U −T



0 WiT ST S−T U −T

0 0 −G

0 TV T

C

0

0 + CAi

C



0 + CAi

C



T

0

T

−P + Qe 0 GAi + MVCAi + MWiC





0 0 + U −1 S−1 SWi −G − MU



=



0 0 M

0 + C

0

0 T V T NT N−T

 

 T

T



T



C

0 0 + C



0  0 =+ U −1 N−1 NV CAi G − MU T



0  + 0 U −1 S−1 SWi C −G

0 0 G

× U −T



0 + CAi

C

C





0 0 −G − MU

0 + CAi

C

T



0 TV T

 

0 + 0 (−G − MU )U −1 S−1 S I

 T

WiT ST S−T U −T

0

(−G − MU ) 0 I

0 T V T NT N−T U −T (G − MU )T Ji−1 (G − MU )U −1 N−1

T <0

(20)

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118



× NV CAi ×

Ji,−1

C





0 + C

113

T

0 WiT ST S−T U −T (−G − MU )T

0



   T



0

0

I

I

(−G − MU )U S SWi C 0 0 + 0 Ji 0 −1 −1

   T

0 0 + 0 Ji, 0 I I

So, once the following condition is satisfied, the inequality (20) holds



−P + Qe 0 GAi + MVCAi + MWC

   T

0 0 + 0 Ji, 0 I I

∗

−γ Qd G + MVC 2



+ CAi

C

∗ ∗



   T

0 0 + 0 Ji 0 I I 



0 T V T NT N−T U −T (G − MU )T Ji−1



× (G − MU )U −1 N−1 NV CAi

C





0 + C



0 0 T WiT ST S−T



× U −T (−G − MU )T Ji,−1 (−G − MU )U −1 S−1 SWi C



0 0 <0

(21)

By applying the Schur complement to (21), it can transform into

⎡ −P + Q ∗ ∗ e 0 −γ 2 Qd ∗ ⎢ ⎢GAi + MVCAi G + MVC  + Ji + J, i ⎣ NVC 0

NVCAi SWiC

1 = 2 =

−T

−T

(

0 0

)

(

∗ ∗ ∗

∗ ∗ ∗ ∗

1

⎤ ⎥ ⎥<0 ⎦

(22)

2

0

)

( − MU )T Ji−1 (G − MU )]−1U T NT ,

−[N U G − MU T Ji−1 G − MU U −1 N−1 ]−1 = −NU[ G −[S−T U −T −G − MU T Ji,−1 −G − MU U −1 S−1 ]−1 = −SU[

(

)

(

(−G − MU )T Ji,−1 (−G − MU )]−1U T ST .

)

For a scalar α , note that −(L − α Q )Q −1 (L − α Q )T ≤ 0, Q > 0 implies that −LQ −1 LT ≤ −α L − α LT + α 2 Q. Therefore, one has

1 = −NU[(G − MU )T Ji−1 (G − MU )]−1U T NT ≤ −α NU − αU T NT + α 2 (G − MU )T Ji−1 (G − MU ) = 1, , 2 = −SU (−G − MU )T Ji,−1 (−G − MU )U T ST ≤ −β SU − βU T ST + β 2 (−G − MU )T Ji,−1 (−G − MU ) = 2, . So the (22) can be rewritten as

⎡ −P + Q ∗ ∗ e 0 −γ 2 Qd ∗ ⎢ ⎢GAi + MVCAi G + MVC  + Ji + J, i ⎣ NVCAi SWiC

NVC 0

0 0

∗ ∗ ∗

1, 0

∗ ∗ ∗ ∗

⎤ ⎥ ⎥<0 ⎦

(23)

2,

Using the Schur complement to the (23), with ε = −α NU − αU T NT and η = −β SU − β U T ST , the (10a) can be obtained easily. The restricted condition of ➂ can be expressed as: (I + KC )Qi = Ni D, multiply the both sides of the equation by U: UQi + UKCQi = UNi D. Then the ➂ turns into UQi + VCQi = Wi D based on the above definitions. Meanwhile, the ➃ convert to V D = 0. The proof is completed.  By the LMI decoupling approach in [29], the appearance of crossing terms between G and K, G and Ni have been avoided in (10a), it enables us to obtain more strict LMI conditions for designing UIO. Meanwhile, another easier decoupling method to (18) is to define E = GK and F = GNi , so get the following LMI condition:



−P + Qe 0 GAi + ECAi − FC

∗

−γ 2 Qd G + EC

∗ ∗ −G − GT + P



<0

The constraint conditions ➂ and ➃ change to GQi + ECQi = F D, ED = 0 correspond. Notice that the definitions E = GK, F = GNi can be applied based on the premise of G must be a nonsingular matrix. These conditions (10a)–(10c) are less conservative than the easier one because the paper introduce G without nonsingular constraint, and the definitions of UNi = Wi , Ji , and Ji, hold the multiple property of the system. Another important advantage is the more accurate estimate ability in the initial time, meanwhile, the estimation error is asymptotic convergence towards zero in other time.

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Fig. 1. States and their estimates with d (t ) = +Ai x(t ) + Bi u(t ).

3.2. The continuous-time system case For continuous-time T–S fuzzy systems with uncertainties, we have the following theorem. Theorem 2. For a given scalar γ > 0, if there exist matrices P > 0, X, and Y such that the following conditions hold





P + CT X T

∗

−γ 2 Qd



<0

(24a)

PQi + XCQi = Y D

(24b)

XD = 0

(24c)

where

 = ATi P + ATi C T X T − C T Y T + PAi + XCAi − YC + Qe . Then the observer (3) converges asymptotically to the state of the continuous-time T–S model (1) is guaranteed, and the observer parameters can be obtained as follows:

K = P −1 X

(25a)

i = (I + P−1 XC )Bi

(25b)

i = (I + P−1 XC )Ai − P−1YC

(25c)

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118

115

Fig. 2. States and their estimates with d (t ) = +Ai x(t ) − Bi u(t ).

Vi = P −1Y − i K

(25d)

Proof. For a continuous-time system (1), consider the Lyapunov function V (t ) =

eT (t )Pe(t ),

V˙ = e˙ T (t )Pe(t ) + eT (t )P e˙ (t ) + eT (t )Qe e(t ) − γ 2 dT (t )Qe d (t ) =

r 



λi

i=1

  T

e(t ) d (t )

        Q i L T P I 0 + I 0 T P i L + e 0

So if the following inequality:

 i.e.

T 





T 

i L P I 0 + I 0 P i



Ti P + Pi + Qe LT P

∗

−γ 2 Qd





Qe L + 0

0

−γ 2 Qd

0

−γ 2 Qd



so we have



e(t ) d (t )

 <0

 <0

(26)

holds, then the (9) will be fulfilled. For there has coupling terms in Ti P, Pi , LT P and PL, substitute i = (I + KC )Ai − NiC and L = (I + KC ) into last inequality and define X = PK, Y = PNi , so the (24a) can be obtained. Meanwhile, the restricted condition of ➂ can be derived as: (I + KC )Qi = Ni D, multiply the both sides of the equation by P: PQi + PKCQi = PNi D. Then the ➂ turns into PQi + XCQi = Y D by using the definitions X = PK and Y = PNi . Meanwhile, the ➃ convert to XD = 0. The proof is completed. 

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Fig. 3. States and their estimates with d (t ) = −Ai x(t ) + Bi u(t ).

4. Simulation example In order to illustrate the proposed approach, this section will consider an inverted pendulum on a cart. After reasonable hypothesis and approximation, a 2-rules inverted pendulum system model can be expressed as [31]



δ x(t ) =

2  i=1

λi (Ai x(t ) + Bi u(t ) + Qi u(t ) + d (t ))

y(t ) = Cx(t ) + Du(t ) with



x(t ) = [x1 (t )

x2

(t )]T ,





0 A1 = g/(4l/3 − aml )

 0 , C= 1 B2 = −aβ /(4l/3 − aml β 2 )





1 , 0



0 A2 = 2g/π (4l/3 − aml β 2 )



1 , 0





0 B1 = , −a/(4l/3 − aml )

0 , and the Q1 , Q2 , D are null matrices, a = 1/(M + m ). The cart’s quality M = 1,

◦ the pendulum’s quality m = 0.1, the pendulum’s 2l = 1, g= 9.8 m/s2 and β =   length   cos(88 ). Using the Theorem  1, −1 0 0 0 0 0 0 the observer (3) gains are obtained: K = , 1 = , 2 = , 1 = , 2 = , 0 3.18 0 −3.18 0 −1.44 −0.04









0 0 , V2 = . Taking Ai = 0.1 ∗ Ai and Bi = 0.1 ∗ Bi , and considering the four conditions unknown in15.68 15.68 put as d (t ) = +Ai x(t ) + Bi u(t ), d (t ) = +Ai x(t ) − Bi u(t ), d (t ) = −Ai x(t ) + Bi u(t ) and d (t ) = −Ai x(t ) − Bi u(t ), respectively. By doing this, the effective performance of this method can be displayed comprehensively, and the simulation results are given in Figs. 1–4. They show the comparison between the actual state and the estimate state. From the Figs. 1–4, the estimated state and the actual state are superimposed especially in the vicinity of the origin, it is better than [9] which cannot coincide in the origin nearby, while the estimate ability in the original time is usually more significant.

V1 =

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108–118

117

Fig. 4. States and their estimates with d (t ) = −Ai x(t ) − Bi u(t ).

At the initial moment, the perfect fitting results fully show the effectiveness of the algorithm. So, when the system described by T–S model with input uncertainties, the designed UIO can estimate the system state successfully. It can be concluded that the proposed method can estimate the system state very well even under the circumstances that input uncertain. 5. Conclusion In this paper, the UIO design problem for both discrete and continuous-time uncertain fuzzy systems have been studied. Sufficient constrain conditions for both discrete and continuous-time uncertain fuzzy systems are presented in terms of LMIs, especially new decoupling approach was applied in discrete-time system UIO design. It is shown, when the uncertainties are seen as unknown inputs, how to design an robust UIO by using the principle of Lyapunov stability theory and bound norm theory. Moreover, the case for some of the system matrices are uncertain has been considered. The modeling and outputs uncertainties will be considered when dealing with the extension of the proposed method in the future works. Acknowledgment The work was supported in part by the National Natural Science Foundation of China (grant no. 61104071), by the Program for Liaoning Excellent Talents in University, China (grant no. LJQ2012095), by Tianjin science and technology support program, China (grant no. 13ZCZDGX03800), by the Open Program of the Key Laboratory of Manufacturing Industrial Integrated Automation, Shenyang University, China (grant no. 1120211415). References [1] X. Chang, Robust nonfragile h∞ filtering of fuzzy systems with linear fractional patamentric uncertainties, IEEE Trans. Fuzzy Syst. 20 (6) (2012) 1001–1011.

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