Applied Mathematics and Computation 266 (2015) 108118

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Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amcUnknown input observer design for fuzzy systems with

uncertainties

Xiao-Kun Dua,b, Hui Zhaoa,c, Xiao-Heng Changb,

a School of Electrical Engineering and Automation, Tianjin University, Tianjin, Chinab College of Engineering, Bohai University, Jinzhou, Liaoning, Chinac College of Engineering and Technology, Tianjin Agricultural University, Tianjin, China

a r t i c l e i n f o

Keywords:

Unknown input observer (UIO)

Uncertainties

a b s t r a c t

This paper investigates the problem of unknown input observer design for both discrete and

continuous-time TS fuzzy systems with uncertainties. After doing appropriate processing toTS fuzzy systems

Lyapunov theory

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 in-

troduction of a new decoupling method which can further reduce the conservatism. The idea

can eliminate the inuence of the unknown inputs, and guarantee the error of the state es-

timation 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

[16]. The disturbances can mainly divide into two categories, external disturbances (space magnetic interference, pulse in-

terference, 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 [712]. 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 im-

measurable states [10,12]. Then Li et al. proposed an adaptive fuzzy backstepping output-feedback tracking control approach for

a class of MIMO nonlinear systems. TheMIMO 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 serialparallel estimationmodel, 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 elds is usually adopted as a solution to provide

state estimations of a system [1315]. After Basile and Marro started to concern the UIO from 1969 [16], considerable literatures

Corresponding author: Tel.: 0416 3400908.E-mail addresses: duxiaokun99@163.com (X.-K. Du), zhaohui3379@126.com (H. Zhao), changxiaoheng@sina.com (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) 108118 109about UIO design emerged [17,18]. For linear time-invariant systemmodels, there are several approaches for UIO design [1921].

A minimal-order observer without any assumptions upon the unknown inputs was proposed for a linear time-invariant system

byWang 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 conserva-

tive 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 TS 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 inuence on the system parameters or the in-

put matrix [26]. The authors consider both of these two forms in the paper, and treat other disturbances as unknown inputs

meanwhile.

Consider the following TS fuzzy system [27,28] (it is suitable for both discrete and continuous-time systems), the ith rule Ri

is: if 1(t) isMi1and 2(t) isM

i2and and p(t) isMip (i = 1,2, . . . , r),

then :

{x(t) = (Ai Ai)x(t) + (Bi Bi)u(t) + Qiu(t)y(t) = Cx(t) + Du(t) (1)

where (t) = [1(t), 2(t), . . . , p(t)]T Rp is the decision vector which is real time accessible variable; Mik (i = 1,2, . . . , r; k =1,2, . . . , p) is the fuzzy set; r is the number of fuzzy rules. x(t) represents the discrete-time system state x(t + 1) or continuous-time 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 andy(t) Rp represents the measured output. Ai Rnn, Bi Rnm and C Rpn dene the ith local model. Matrices Qi Rnq andD Rpq represent the inuence of the unknown inputs. Ai and Bi are the modeling uncertainties.

By using the TS fuzzy rules, the TS fuzzy control system model (1) can be rewritten as{x(t) =

ri=1

i(Aix(t) + Biu(t) + Qiu(t) + d(t))y(t) = Cx(t) + Du(t)

(2)

with i = i( (t))/r

j=1 j( (t)), i( (t)) =

pk=1

Mik(k(t)), M

ik(k(t)) represents the membership grade of k(t) towards M

ik,

and 0 i 1,r

i=1i = 1. Noting that d(t) = (Aix(t) Biu(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:{z(t) =

ri=1

i(iz(t) + iu(t) +Viy(t))x(t) = z(t) Ky(t)

(3)

The matrices i Rnn, i Rnm, Vi Rnp and K Rnp 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 uncer-

tainties d(t) are unmeasured. In order to assure the estimate state x(t) approach the true state x(t), we dene the estimation

error as

e(t) = x(t) x(t) (4)By the system model (2) and the observer expression (3), we have

e(t) = x(t) x(t) = x(t) z(t) + K(Cx(t) + Du(t))= (I + KC)x(t) z(t) + KDu(t) (5)

110 X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118Dene L = (I + KC), therefore

e(t) = (I + KC)x(t) z(t) + KDu(t)

= Lx(t) + KDu(t) r

i=1i[i(x(t) + Ky(t) e(t)) + iu(t) +Viy(t)]

let Ni = iK +Vi, we have

e(t) =r

i=1i[ie(t) + (LAi i NiC)x(t) + Ld(t)

+(LBi i)u(t) + (LQi NiD)u(t)] + KDu(t) (6)

In the Eq. (6), if the following constrains : LAi NiC i = 0 LBi i = 0 LQi NiD = 0 KD = 0

can be fullled, then the expression of e(t) can be rewritten as

e(t) =r

i=1i(ie(t) + Ld(t)) (7)

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.{limt

e(t) = 0 when d(t) = 0e(t)Qe d(t)Qd when d(t) = 0 and e(0) = 0

(8)

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

V (t) + eT (t)Qee(t) 2dT (t)Qdd(t) < 0 (9)

where Qe and Qd are two positive denite 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 TS fuzzy systemswith uncertainties, we try to nd the sucient constrain conditions to ensure the asymp-

totic stability of the observer in terms of LMIs. The following theorem can give a reference to other designers.

Theorem 1. For a given scalar > 0, if there exist matrices P > 0, G, M, N, S, U, J,i, and Ji, scalars and such that the following

conditions hold:

P + Qe 0 2Qd

31 G + MVC 33 NVCAi NVC 0

0 0 0 G MU Ji2

SWiC 0 0 0 0 0 0 0 0 0 G MU J

,

i

2

< 0 (10a)

UQi +VCQi = WiD (10b)

VD = 0 (10c)

where 31 = GAi + MVCAi,33 = G GT + P + Ji + J,i , = NU UTNT , and = SU UT ST . Then the observer (3) con-

verges 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) 108118 111determined as follows:

K = U1V (11a)

i = (I +U1VC)Bi (11b)

i = (I +U1VC)Ai U1WiC (11c)

Vi = U1Wi iU1V (11d)

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

V = V (t + 1) V (t) = eT (t + 1)Pe(t + 1) eT (t)Pe(t) (12)So the sucient condition of (9) turns into

eT (t + 1)Pe(t + 1) eT (t)Pe(t) + eT (t)Qee(t) 2dT (t)Qdd(t) < 0 (13)i.e.

ri=1

i

([e(t)d(t)

]T([

i L]T P[i L

]+

)[e(t)d(t)

])< 0 (14)

where =[P + Qe 0

0 2Qd

]. So the core problem turns into meet the condition

[i L

]T P[i L

]+ < 0 (15)

By the Schur complement to (15), we can obtain that[P + Qe 0 2Qw i L P1

]< 0 (16)

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

[I 0 0

0 I 0

0 0 G

]and its transpose, respectively, (16) is rewritten as follows:

[P + Qe 0 2Qw

Gi GL GP1GT

]< 0 (17)

Note that GP1GT G GT + P, P > 0, and substituting the denitions i = LAi NiC and L = I + KC into the coupling termGi and GL: Gi = GAi + GKCAi GNiC, GL = G + GKC, then the (17) can be ensured by[ P + Qe

0 2Qd GAi + GKCAi GNiC G + GKC G GT + P

]< 0 (18)

It can be expressed as

+[00G

]K[CAi C 0

]+[CAi C 0

]TKT

[00G

]T

+[

00

G

]Ni[C 0 0

]+[C 0 0

]TNT

i

[00

G

]T < 0

(19)

with =[P + Qe

0 2Qd GAi G

]and = G GT + P. Dening UK = V, UNi = Wi and considering the matrices M, N, and S,with U, N, and S are nonsingular matrices, it follows that

112 X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118the left of inequality (19)

= +[00G

]U1N1NV

[CAi C 0

]+[CAi C 0

]TV TNTNT

UT[00G

]T+[

00

G

]U1S1SWi

[C 0 0

]+[C 0 0

]TWTi S

T STUT

[00

G

]T

= +[

00

G MU

]U1N1NV

[CAi C 0

]+[CAi C 0

]TV T

NTNTUT[

00

G MU

]T+[00M

]V[CAi C 0

]

+[CAi C 0

]TV T

[00M

]T+[

00

G MU

]U1S1SWi

[C 0 0

]+[C 0 0

]TWTi S

T STUT

[00

G MU

]T

+[00M

]Wi[C 0 0

]+[C 0 0

]TWTi

[00M

]T

=[ P + Qe

0 2Qd GAi + MVCAi + MWiC G + MVC

]+[

00

G MU

]U1N1

NV[CAi C 0

]+[CAi C 0

]TV TNTNTUT

[00

G MU

]T

+[

00

G MU

]U1S1SWi

[C 0 0

]+[C 0 0

]TWTi S

T STUT

[00

G MU

]T< 0 (20)

Based on XY + YTXT XJXT + YT J1Y, J > 0, for positive matrices Ji and J,i , one gives[00

G MU

]U1N1NV

[CAi C 0

]+[CAi C 0

]TV T

NTNTUT[

00

G MU

]T+[

00

G MU

]U1S1SWi

[C 0 0

]+[C 0 0

]TWTi S

T STUT

[00

G MU

]T

=[00I

](G MU)U1N1NV

[CAi C 0

]+[CAi C 0

]TV T

NTNTUT (G MU)T[00I

]T+[00I

](G MU)U1S1S

Wi[C 0 0

]+[C 0 0

]TWTi S

T STUT (G MU)[00I

]T

[ ]T T T T T T 1 1 1 CAi C 0 V N N U (G MU) Ji (G MU)U N

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118 113

U NV[CAi C 0

]+[C 0 0

]TWTi S

T STUT (G MU)T

J,1i

(G MU)U1S1SWi[C 0 0

]+[00I

]Ji

[00I

]T

+[00I

]J,i

[00I

]T

So, once the following condition is satised, the inequality (20) holds[ P + Qe 0 2Qd

GAi + MVCAi + MWC G + MVC

]+[00I

]Ji

[00I

]T

+[00I

]J,i

[00I

]T+[CAi C 0

]TV TNTNTUT (G MU)T J1

i

(G MU)U1N1NV[CAi C 0

]+[C 0 0

]TWTi S

T ST

UT (G MU)T J,1i

(G MU)U1S1SWi[C 0 0

]< 0 (21)

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

P + Qe 0 2Qd

GAi + MVCAi G + MVC + Ji + J,i NVCAi NVC 0 1 SWiC 0 0 0 2

< 0 (22)

1 = [NTUT (G MU)T J1i (G MU)U1N1]1 = NU[(G MU)T J1i (G MU)]1UTNT ,2 = [STUT (G MU)T J,1i (G MU)U1S1]1 = SU[(G MU)T J,1i (G MU)]1UTST .

For a scalar , note that (L Q )Q1(L Q )T 0, Q > 0 implies that LQ1LT L LT + 2Q . Therefore, one has

1 = NU[(G MU)T J1i (G MU)]1UTNT NU UTNT + 2(G MU)T J1i (G MU) = ,1,2 = SU(G MU)T J,1i (G MU)UTST SU UTST + 2(G MU)T J,1i (G MU) = ,2.

So the (22) can be rewritten as

P + Qe 0 2Qd

GAi + MVCAi G + MVC + Ji + J,i NVCAi NVC 0

,1

SWiC 0 0 0

,2

< 0 (23)

Using the Schur complement to the (23), with = NU UTNT and = SU UT ST , the (10a) can be obtained easily.The restricted condition of can be expressed as: (I + KC)Qi = NiD, multiply the both sides of the equation by U: UQi +

KCQi = UNiD. Then the turns intoUQi +VCQi = WiD based on the above denitions. Meanwhile, the convert to VD = 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 dene E = GK and F = GNi, so get the following LMI condition:[ P + Qe 0 2Qd

GAi + ECAi FC G + EC G GT + P

]< 0

The constraint conditions and change to GQi + ECQi = FD, ED = 0 correspond. Notice that the denitions E = GK, F = GNican be applied based on the premise of Gmust 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 denitions of UNi = Wi, Ji, and J,i holdthe 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.

114 X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118Fig. 1. States and their estimates with d(t) = +Aix(t) + Biu(t).

3.2. The continuous-time system case

For continuous-time TS fuzzy systems with uncertainties, we have the following theorem.

Theorem 2. For a given scalar > 0, if there exist matrices P > 0, X, and Y such that the following conditions hold[

P +CTXT 2Qd

]< 0 (24a)

PQi + XCQi = YD (24b)

XD = 0 (24c)where

= ATi P + ATi CTXT CTYT + PAi + XCAi YC + Qe.Then the observer (3) converges asymptotically to the state of the continuous-time TS model (1) is guaranteed, and the observer

parameters can be obtained as follows:

K = P1X (25a)

i = (I + P1XC)Bi (25b)

i = (I + P1XC)Ai P1YC (25c)

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118 115Fig. 2. States and their estimates with d(t) = +Aix(t) Biu(t).

Vi = P1Y iK (25d)Proof. For a continuous-time system (1), consider the Lyapunov function V (t) = eT (t)Pe(t), so we have

V = eT (t)Pe(t) + eT (t)Pe(t) + eT (t)Qee(t) 2dT (t)Qed(t)

=r

i=1i

[e(t)d(t)

]T

([i L

]T P[I 0

]+[I 0

]T P[i L

]+[Qe 0

0 2Qd

])[e(t)d(t)

]

So if the following inequality:[i L

]T P[I 0

]+[I 0

]T P[i L

]+[Qe 0

0 2Qd

]< 0

i.e. [T

iP + Pi + Qe

LTP 2Qd

]< 0 (26)

holds, then the (9) will be fullled.

For there has coupling terms inTiP, Pi, L

TP and PL, substitutei = (I + KC)Ai NiC and L = (I + KC) into last inequality anddene X = PK, Y = PNi, so the (24a) can be obtained. Meanwhile, the restricted condition of can be derived as: (I + KC)Qi =NiD, multiply the both sides of the equation by P: PQi + PKCQi = PNiD. Then the turns into PQi + XCQi = YD by using thedenitions X = PK and Y = PNi. Meanwhile, the convert to XD = 0.

The proof is completed.

116 X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118Fig. 3. States and their estimates with d(t) = Aix(t) + Biu(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]{x(t) =

2i=1

i(Aix(t) + Biu(t) + Qiu(t) + d(t))y(t) = Cx(t) + Du(t)

with x(t) = [x1(t) x2(t)]T , A1 =[

0 1

g/(4l/3 aml) 0

], A2 =

[0 1

2g/(4l/3 aml2) 0

], B1 =

[0

a/(4l/3 aml)

],

B2 =[

0

a/(4l/3 aml2)

], C =

[1 0

], and the Q1, Q2, D are null matrices, a = 1/(M + m). The carts quality M = 1,

the pendulums quality m = 0.1, the pendulums length 2l = 1, g = 9.8m/s2 and = cos(88). Using the Theorem 1,the observer (3) gains are obtained: K =

[10

], 1 =

[0 0

3.18 0

], 2 =

[0 0

3.18 0

], 1 =

[0

1.44

], 2 =

[0

0.04

],

V1 =[

0

15.68

], V2 =

[0

15.68

]. Taking Ai = 0.1 Ai and Bi = 0.1 Bi, and considering the four conditions unknown in-

put as d(t) = +Aix(t) + Biu(t), d(t) = +Aix(t) Biu(t), d(t) = Aix(t) + Biu(t) and d(t) = Aix(t) Biu(t),respectively. By doing this, the effective performance of this method can be displayed comprehensively, and the simulation

results are given in Figs. 14. They show the comparison between the actual state and the estimate state.

From the Figs. 14, 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 signicant.

X.-K. Du et al. / Applied Mathematics and Computation 266 (2015) 108118 117Fig. 4. States and their estimates with d(t) = Aix(t) Biu(t).

At the initial moment, the perfect tting results fully show the effectiveness of the algorithm. So, when the system described by

TS 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.

Sucient 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).

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Unknown input observer design for fuzzy systems with uncertainties1 Introduction2 Problem statement3 UIO design and LMI synthesis conditions3.1 The discrete-time system case3.2 The continuous-time system case

4 Simulation example5 Conclusion Acknowledgment References