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Published in IET Control Theory and Applications

Received on 10th January 2011

Revised on 7th November 2011

doi: 10.1049/iet-cta.2011.0022

ISSN 1751-8644

Adaptive observer design for the uncertainTakagiSugeno fuzzy system with output disturbanceT. V. Dang1 W.-J. Wang1 L. Luoh2 C.-H. Sun3

1Department of Electrical Engineering, National Central University, JhongLi 32001,Taiwan2Department of Electrical Engineering, Chung Hua University, Hsin-Chu 10608,Taiwan3Department of Mechanical and Electro-Mechanical Engineering,Tamkang University,Tamsui,Taipei County 25137,

Taiwan

E-mail: [email protected]

Abstract: The study proposes an adaptive fuzzy observer for the uncertain TakagiSugeno ( TS) fuzzy system with outputdisturbance. First, an augmented fuzzy model is built by integrating the system state and the output disturbance together asnew variables. Then, the desired adaptive fuzzy observer is designed to estimate the unavailable system state and the unknownoutput disturbance simultaneously. Based on Lyapunov theory and linear matrix inequalities (LMIs) tools, two main conditionsare derived under which the fuzzy observer is designed. Finally, the procedure of the observer design is summarised and theeffectiveness of the designed observer is demonstrated with a numerical example.

1 Introduction

The design of observers for state estimation plays an

important role in the automatic control domain withvarious areas of application. Besides, several issues requiresystems state reconstruction, such as designing a statefeedback control law [1] or constructing a diagnosis systemunder supervisory [2] and reconfiguring the system withappearance of failures [3]. Diverse approaches for theobserver design have been proposed, such as transferfunction methods, geometric-observers, differential algebraicmodels and singular value decomposition [46].

In practical control problems, it is unavoidable to includeuncertain parameters and output disturbances becauseof modelling error, measurement error and linearisationapproximation. Hence, it is necessary to design a robustobserver which ensures the exactness of state estimate

between the states of dynamical systems and observer inthe presence of the uncertainties. Therefore the problem ofrobust state observer design for dynamical systems withuncertainty has received considerable attention of manyresearchers, and many approaches to design a robust stateobserver have been developed in the past decades.

Among the solutions to the problem of the stateestimation, the most famous estimator is the Luenbergersobserver [7]. The Luenbergers observer bases on thesynthesis of a static gain which is synthesised to stabilisethe state estimation error and to guarantee its asymptoticconvergence. However, the existence of disturbance oruncertain causes a bad reconstruction of the states system.Other solutions as the proportional integral (PI) observerdesign [8] or the proportional multiple-integral observerdesign [9] permits us to attenuate either measurementnoise or modelling error. Unfortunately, a conventional

PI observer cannot provide a satisfactory estimationperformance when the output disturbance occurs in the plant.

The same drawback of the conventional observer is pointedout in [8, 9], that is the output disturbance will be amplifiedby the conventional observer gain matrix.

Fuzzy technique has been widely used in modellingand controlling non-linear systems. The fuzzy control withIF-THEN rules can combine with the other control methodsto reach the some difficult control objective, because of itscapability of modelling complex non-linear systems [1015].Among various kinds of fuzzy models, TakagiSugeno (TS) fuzzy model [16] is one of the most popular classesof fuzzy model. It has recently gained much popularity

because of their special rule, consequent structure andsuccessful application in a functional approximation andsystem analysis. Therefore TS fuzzy model has been a

powerful tool in the analysis and synthesis to a non-linearcontrol system [1618].

Many papers have been studying the observer designfor fuzzy TS systems by using different techniques suchas the linear matrix inequalities (LMIs) approach, thesliding mode technique and the adaptive method to designthe fuzzy observer [1923]. However, the observer design

problems are more challenging in the presence of unknowntime-varying output disturbances [24]. Although there wasremarkable success in the observer design for linear crispsystem with measurement noise [25], the observer techniquefor the crisp system cannot be utilised directly in thefuzzy observer design because of the existence time-varyingweights in the fuzzy inference engine. In the presence of

bounded non-differentiable disturbances, adaptive observershave been developed in [2628] to guarantee bounded stateand parameter estimation errors.

IET Control Theory Appl., pp. 116 1

doi: 10.1049/iet-cta.2011.0022 The Institution of Engineering and Technology 2012

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Therefore how to design the fuzzy observer for TS fuzzysystems with output disturbances is worth being studied. The

paper [24] studied the observer design technique for theTS fuzzy system with output disturbance. However, itdid not deal with the TS fuzzy system with uncertain

parameters since uncertainties make the observer designmuch more complicated. There is a pioneer paperinvestigating the observer-controller design for TS fuzzy

uncertain system [29] in which the stability criterion androbustness area with respect to a single-grid-point in the

parameter space are derived. The paper [30] presents anadaptive approach for the synchronisation of TS chaoticsystems to estimate the uncertain parameters. Although theobjective can be completely achieved, the paper requiresa constraint (in Assumption 1) for the solution of LMI

problems. In the example, the authors have to use variablesin the state and input matrices which are found after thecommon positive definite matrix P. Although in the systemsthe state and input matrices are originally and normallyfixed.

The design of adaptive observer for uncertain non-linear systems containing disturbance has attracted great

attention recently. The paper [31] demonstrates a robustadaptive observer for multiple-in multiple-out non-linearsystem with uncertain non-linearities and disturbances. Theexisting problem in the paper is that the state estimationerror is uniformly bounded and does not converge to zeroasymptotically. The fuzzy observer-based fuzzy controlleris developed in [32]. The proposed methods efficientlyattenuate the peak of perturbation because of persistent

bounded disturbances. The question of bounded disturbanceis represented in [33] to deliver a method as using thestrictly positive real theory and Lyapunov stability theoryto design the state observer. All of the authors in [3133]use the popular lemma as MeyerKalmanYakubovich [34]

about assumption of common matrix P for all subsystems.This however seems to be a common practice in the relatedliterature.

In this paper, the problem to be handled is the estimationof state and disturbance subjected to the influence ofunknown output disturbances and uncertain parameters inthe TS fuzzy system. The work proposes a robust adaptivefuzzy observer to accurately estimate the system stateand the output disturbance at the same time under theexistence of uncertainties. First, this work generates anaugmented system for uncertain TS fuzzy model. Then,the main theorem for the adaptive observer design is

proposed and LMIs tool [35, 36] is used to find the solutionof parameters such that the observer system can achieve

the goals. Finally, the computer simulation is conductedto verify the effectiveness of the proposed method. Therest of paper is organised as follows. In Section 2, some

preliminaries for TS fuzzy model of uncertain system withoutput disturbance and problem formulation are given. InSection 3, the adaptive fuzzy state/disturbance observer isdesigned and the conditions ensuring global and asymptoticconvergence of estimation error are derived as a set of LMIterms. A numerical example is given in Section 4 to illustratethe design procedure and the effectiveness of the proposedapproach. Finally, a conclusion is given in Section 5.

2 System model and problem descriptionA TS fuzzy model described by IF-THEN rules isconsidered, and each rule represents the local linear

inputoutput relation from a non-linear uncertain system.The ith rule of the fuzzy model is of the following form:

Plant Rule iIF z1 is i1 and and zs is is, then

x(t) = (Ai + Ai)x(t) +Biu(t), i = 1, . . . , r

y(t) = Cix(t)(1)

where x(t) Rn is the unavailable state, u(t) Rm is thecontrol and y(t) Rp is the output. In general, zj Zj(j =1,2, . . .,s) are the antecedent variables which may be someof the states, the function of states or the function of output[36], but the states are unavailable, here, zj will be theoutput y or some function ofy. Moreover, ij (i = 1,2, . . ., r;

j = 1,2, . . .,s) are the fuzzy sets that are characterised bymembership functions, r is the number of IFTHEN rulesand s is the number of the premise variables. The matrices

Ai, Bi and Ci are with appropriate dimensions and Aidenotes the bounded uncertain constant matrix satisfyingAi i. The overall fuzzy model achieved from plant

rules in (1) is given by

x(t) =

ri=1

hi(z){(Ai + Ai)x(t) +Biu(t)}

y(t) =

ri=1

hi(z)Cix(t)

(2)

where

z(t) = [z1(t), . . . ,zs(t)], hi(z) =i(z)

r

i=1i(z)

,

i(z) =

sj=1

ji(z)

Here, hi(z) is regarded as the normalised weight of eachIF-THEN rule and hi(z) 0,

ri=1hi(z) = 1

In real systems, it is possible that the system containsuncertain parameters and output disturbance as follows

x(t) =

r

i=1 hi(z){(Ai + Ai)x(t) +Biu(t)}

y(t) =

ri=1

hi(z)Cix(t) + (t)

(3)

where (t) Rp stands for the unknown output disturbance.Since x(t) is unavailable and (t) is unknown, designing anobserver to estimate of the state and the disturbance for thesystem (2) is the main objective of this paper.

3 Observer design

First, let us consider the particular case which has equaloutput matrices in (3), that is, Ci = C for all i. Subsequently,this technique will be applied to the general case in whichthe system contains different output matrices.

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3.1 Particular case

If output matrices of all subsystems are equal, that is, Ci = Cfor all i, (3) becomes

x(t) =

r

i=1hi(z){(Ai + Ai)x(t) +Biu(t)}

y(t) = Cx(t) + (t)

(4)

Rewrite (4) as the following augmented form

Ex(t) =

ri=1

hi(z){( Ai + Ai)x(t) + Biu(t) + N(t)}

y(t) = Cx(t) = C0 x(t) + (t)

(5)

where

x(t) =

x(t)(t)

, E =

In OpOp Op

, Ai =

Ai OpOp Ip

,

Ai =

Ai OpOp Op

, Bi =

Bi

Opm

, N =

Onp

Ip

,

C = [C Ip] and C0 = [C Op] (6)

and Op denotes a p p zero square matrix and Onp is ann p zero matrix. Add Ly(t) to both sides of (5), it yields

( E+ LC) x(t) =

r

i=1hi(z){( Ai + Ai)x(t)

+ Biu(t) + N(t) + Ly(t)}

y(t) = Cx(t) = C0 x(t) + (t)

(7)

where L R(n+p)p will be determined later. Since (t) =y(t) C0 x(t), then (7) becomes

( E+ LC) x(t) =

ri=1

hi(z){ A0i x(t) + Ai x(t)

+ Biu(t) + N y(t) + Ly(t)} (8)

where A0i = Ai N C0. Suppose that, the pairs ( A0i, C) isobservable, the fuzzy observer is proposed as follows

( E+ LC)x(t) =

ri=1

hi(z){ A0i x(t) + Biu(t) + Hi(y(t)

y(t)) + Ai(t) x(t) + N y(t) + Ly(t)}

y(t) = Cx(t)(9)

where x(t) Rn+p is the estimate of x(t) and y(t) Rp is the

observer output, Ai is used to estimate Ai. Moreover,the gain Hi R

(n+p)p of the ith local observer will bedetermined later.

Let

e(t) = x(t) x(t) and

ey(t) = y(t) y(t) = Ce(t)(10)

From (9) to (10), the error dynamics is

( E+ LC) e(t) =

r

i=1

hi(z){( A0i HiC)e(t)

+ Ai x(t) Ai(t) x(t)} (11)

Furthermore

rank

E

C

= rank

In 00 0C Ip

= n +p

Thus, we can find a L R(n+p)p to satisfy rank[ E+ LC] =n +p such that ( E+ LC)1 exists. Rewrite (11) as the form

below

e(t) =

ri=1

hi(z){( E+ LC)1[( A0i HiC)e(t)

+ Ai x(t) Ai(t) x(t)]}

=

ri=1

hi(z){( A1i H1iC)e(t) + A1i x(t) A1i(t) x(t)}

=

ri=1

hi(z){ A2i e(t) + A1i x(t) A1i(t) x(t)} (12)

where

A1i = ( E+ LC)1 A0i, A1i = ( E+ LC)

1 Ai and

A1i = ( E+ LC)1 Ai (13)

H1i = ( E+ LC)1 Hi, A2i = A1i H1iC (14)

Now, suppose A1i and Ai can be represented as

A1i = Mi (15a)

and

A1i = Mi(t) (15b)

respectively, where M is a constant matrix with dimensions(n +p) p; i R

p(n+p) is uncertain constant matrices

satisfying i

iand

iis the estimate of

i. Furthermore,

since ||Ai|| i, then Ai i imply A1i 1i,where 1i = ( E+ LC)

1i. From (12) and (15), it yields

e(t) =

ri=1

hi(z){ A2i e(t) +Mi x(t) Mi(t) x(t)}

=

ri=1

hi(z){ A2i e(t) +Mi x(t) Mi x(t)

+Mi x(t) Mi(t) x(t)}

=

r

i=1 hi(z){A2i e(t) +Mie(t) +Mei(t) x(t)} (16)

where ei(t) = i i(t). Before deriving the main result, the

following lemma is required.



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Lemma 1 [37]: Let x and y denote two constant vectors,then xTy = tr(xyT), where tr(G) is the trace of the constantmatrix G.

Theorem 1: Consider the system (8) and the observer (9),suppose A1i satisfies (15) with A1i 1i andi i.Then, the estimation errors (16) of the state and the outputdisturbance converge to zero asymptotically if

i(t) = hi(z)iey(t) x

T(t) (17)

where i is any positive-definite matrix, and there exist acommon symmetric positive definite matrix P = PT and amatrix Hi such that

AT2iP+PA2i + 21iP < 0, i = 1,2, . . . , r (18)

and

MT = CP1 (19)

Proof: Let the Lyapunov function candidate be

V = eT(t)Pe(t) +

ri=1

tr(Tei(t)1i ei(t)) (20)

The time derivative of V(t) along function trajectories ofsystem (16) is shown as follows

V = eT(t)Pe(t) + eT(t)Pe(t) +

ri=1

tr( Tei (t)1i ei(t))

+

ri=1

tr(Tei (t)1i

ei(t))

As ei(t) = (i

i(t)) =

i(t) then

V =

ri=1

hi(z){eT(t)( AT2iP+PA2i)e(t)}

+ 2

ri=1

hi(z){eT(t)MiPe(t)}

+ 2

ri=1

hi(z){eT(t)PMei(t) x(t)}

2

ri=1

tr(

T

ei(t)

1

i

i(t)) (21)

By Lemma 1, (17) and (19), the last term of (21) is shownas follows

2

ri=1

tr(Tei (t)1i

i(t))

= 2

ri=1

hi(z){tr(Tei (t)

1i iey(t)

xT(t))}

= 2

r

i=1

hi(z){tr(Tei (t)ey(t))(

xT(t))}

= 2

ri=1

hi(z)eT

y (t)ei(t)x(t)

= 2

ri=1

hi(z){eT(t)CTei(t) x(t)}

= 2

ri=1

hi(z){eT(t)PMei(t) x(t)} (22)

According to (22), the sum of the last two terms of (21)equals to zero. Thus

2

ri=1

hi(z){eT(t)MiPe(t)}

2

ri=1

hi(z){eT(t)MiPe(t)}

= 2

ri=1

hi(z){eT(t) A1iPe(t)}

2

ri=1

hi(z){eT(t)1iPe(t)}

Then

V

ri=1

hi(z){eT(t)( AT2iP+PA2i)e(t)}

+ 2

ri=1

hi(z){eT(t)1iPe(t)}

If (18) holds

V

ri=1

hi(z){eT(t)[ AT2iP+PA2i + 21iP]e(t)} 0 (23)

The proof is completed.

Remark 1: Based on Theorem 1, the fuzzy observer (9)is obtained and the condition (19) is used for supportingthe solution of LMI (18) which is originally from MeyerKalmanYakubovich Lemma [34].

3.2 General case

The output matrices of all subsystems are not necessary tobe equal, that is, Ci = Cj, for i = j. Then, the TS fuzzysystem (3) can be rewritten as a new form

x(t) =

ri=1


y(t) = Cx(t) +

ri=1

hi(z)(Ci C)x(t) + (t)

(24)

where C is the matrix chosen from the set {C1, C2, . . ., Cr}.Let

0(t) =

ri=1

hi(z)(Ci C)x(t) + (t)

the (24) becomes

x(t) =

ri=1


y(t) = Cx(t) + 0(t)

(25)



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Similar to (4), (25) can be rewritten as

Ex0(t) =

ri=1

hi(z){( Ai + Ai)x0(t) + Biu(t) + N0(t)}

y(t) = Cx0(t) = C0 x0(t) + 0(t)

(26)

where xT0 = [x(t) 0(t)] and the state-space system

coefficients Ai, Bi, N, C, C0, E and Ai are defined as thesame as those in (6). Add Ly(t) to both sides of (26), ityields

( E+ LC) x0(t) =

ri=1

hi(z){( Ai + Ai)x0(t)

+ Biu(t) + N0(t) + Ly(t)}

y(t) = C0 x0(t) + 0(t) (27)

Substitute 0(t) = y(t) C0 x0(t) into (27)

( E+ LC) x0(t) =

ri=1

hi(z){ A0i x0(t) + Ai x0(t)

+ Biu(t) + N y(t) + Ly(t)} (28)

which is similar to (8). Therefore we can design the TSfuzzy state-space observer (30) to estimate x(t) and 0(t)of system (28) by using the approach given in Theorem 1.Since

x(t)(t)

=

In 0r

i=1hi(z)(Ci C) Ip

1

x0(t) (29)

where x0(t) is the estimate of x0(t) let the TS fuzzyobserver (30) be designed to estimate x(t) and 0(t)

( E+ LC)x0(t) =

ri=1

hi(z){ A0i x0(t) + Biu(t) + Hi(y(t)

y(t)) + Ai x0(t) + N y(t) + Ly(t)}

y(t) = Cx0(t) (30)

It is seen that (30) is similar to (9). Let e0(t) = x0(t) x0(t),and e0y(t) = y(t) y(t) = Ce0(t). Therefore by similar

derivations of (12) and (15)

e0(t) =

ri=1

hi(z){ A2i e0(t) +Mie0(t) +Mei(t) x0(t)} (31)

is obtained.

Theorem 2: Suppose A1i can be represented as the form

(15) with A1i 1i and i i, then the fuzzy

observer (30) can asymptotically estimate the state and theoutput disturbance of the fuzzy system (28), if the conditionsin (18) and (19) of Theorem 1 hold simultaneously and theadaptive law in (17) is updated by

i(t) = hi(z)ie0y(t) x

T0 (t) (32)

where i is any positive-definite matrix.

Proof: As the forms of the fuzzy system (28) and of thefuzzy observer (30) are similar to the forms of the fuzzysystem (8) and of the fuzzy observer (9), respectively, basedon Theorem 1, we obtain that

limt

x(t)0(t)

x(t)0(t)

= 0 (33)

where x(t) is the estimate of x(t). Since hi(t) is bounded forany i r we have

limt

In 0r

i=1

hi(z)(Ci C) Ip

1 x(t)0(t)

x(t)

0(t)

= 0

(34)

Using 0(t) =

r

i=1 hi(z)(Ci C)x(t) + (t) and the (29),it has (see (35))

From (34) and (35), it yields

limt

x(t)(t)

x(t)(t)

= 0 (36)

Therefore the proof is completed.

Remark 2: Let us define Qi = P H1i, based on (14), (18) canbe rewritten as the form

AT1iP+PA1i C

TQTi QiC + 21iP < 0 i r (37)

Here, P and Qi can be obtained by means of LMI methodand then by (14), Hi is obtained. Based on Theorems 1

x(t)(t)

x(t)(t)

=

x(t)(t)

In 0r

i=1

hi(z)(Ci C) Ip

1 x(t)

0(t)

=

In 0r

i=1

hi(z)(Ci C) Ip

1

In 0r

i=1

hi(t)(Ci C) Ip

x(t)(t)

x(t)

0(t)

=

In 0r

i=1

hi(z)(Ci C) Ip

1

x(t)

0(t)

x(t)

0(t)

(35)



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and 2, we can construct the procedure as follows to designthe fuzzy observer. Before the observer design, some known

parameters are stated in advance. For the system (3), Aiis the uncertainty satisfying (15), where Ai| i andi i with known bounds i andi, respectively. Finally,the procedure of the observer design is summarised asfollows.

Step 1: Find the matrix L such that ( E+ LC)1

exists, andthen 1i is also obtained.

Step 2: Give a positive value p to satisfy p (1i/Ci)and assume P1 p. The reason is as below. Withthe aids of (19), we have A1i P

1CTi P1Ci 1i. Therefore

P1 1i

Ci(38)

Step 3: Define Qi = P H1i by means of LMI method, P andH1i in (37) are obtained. Substitute P into (38) to check

whether P1

p holds? If not, go back to Step 2 andgive another p and re-run LMI again to solve (37) untilP1 p holds. Then, go to next step.

Fig.1 Flowchart of the fuzzy observer design

Step 4: Obtain M from (19).

Step 5: Choose the any positive symmetric matrix toobtain the adaptive law (32).

Finally, the observer design is completed. The aboveprocedure is presented as the flowchart in Fig. 1.

Remark 3: In Step 3, using LMI method to solve (37) andcheck P1 p are two iterative processes. We can givethe initial p to be equal to (1i/Ci), and decrease p to

p p until the suitable P is found, where p is a verysmall value.

4 Numerical example

Example 1: Consider an uncertain non-linear system des-cribed by the following TS fuzzy system.

Rule 1: If y21(t) is 1 (small), then

x(t) = (A1 + A1)x(t) +B1u(t)y(t) = C1x(t)

Rule 2: If y21(t) is 2 (large), thenx(t) = (A2 + A2)x(t) +B2u(t)

y(t) = C2x(t)

where y1 [01], and Ai, Bi and Ci are given as follows

x(t) = [x1(t) x2(t) x3(t)]T, y(t) = [y1(t) y2(t) y3(t)]

T

A1 = 1.25 2.05 02.025 1.025 0

1.05 0.05 3 , B1 =

10

0 ,

C1 =

1 0 0

1 1 00 0 1

, A2 =

2 1.05 0

0 0.5 1.0251.05 0 1

,

B2 =

010

, C2 =

1 0 01 0 10 1 0

Identical to the definition in the paper [36], let 1 = 1 y21(t) and 2 = y

21(t), then

h1(z) =i(z)

ri=1i(z)

= 1 y21(t)

h2(z) =i(z)

ri=1i(z)= y21 (t)

(39)

In practical, the fuzzy system may be disturbed by someoutput disturbance as follows

x(t) =

2i=1

hi()(Ai + Ai)x(t) +Biu(t)

y(t) =

2i=1

hi( )Cix(t) + (t)

(40)

where (t) represents the unknown disturbance output.Suppose the constant uncertainty Ai, i = 1, 2 are boundedby 1 = 0.0608 and 2 = 0.1216, respectively. Moreover,



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(15) holds and i, i = 1, 2 are bounded by 1 = 0, 01 and2 = 0, 02, respectively. Assume Ai is formulated as below

A1 =

0.0608 0 0

0 0.06 00 0 0.03

,

A1 = 0.1216 0 0

0 0.1 0

0 0 0.03

(41)

where is any constant belonging to the interval [1, 1].Now, according to the procedure in Section 3, let us start

the fuzzy observer design.

Step 1: Choose

LT =

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

and C = C1 such that ( E+ LC)1 exists in which

( E+ LC)1 =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

1 0 0 1 0 01 1 0 0 1 00 0 1 0 0 1

Step 2: Based on Remark 1, give an initial p to be equal to(1i/Ci) = 6.6990

Step 3: Utilise LMI-Toolbox of MATLAB to obtain thefeasible solution P and H1i of (37), then based onRemark 3, the matrix P satisfies P1 6.6987 = p, and(see equations at the bottom of the page)

Step 4: Obtain M from (19)

M =

0.7358 0.7358 0.27000.0000 0.7358 0.38120.0000 0.0000 1.62480.7358 0.0000 0.27000.0000 0.7358 0.11130.0000 0.0000 0.2149

Step 5: Let 1 = 2 = diag[5,1,1], then the adaptive law(32) is established as follows

1(t) = h1(z)1e0y(t) x

T0 (t)

2(t) = h2(z)2e0y(t) x

T0 (t)

Suppose u(t) = sin(t) and (t) = [1(t) 2(t) 3(t)]T,

where

1(t) =

0.2 sin2(3(t 2) ), t 2

0 else

2(t) =

0.3 cos2(4(t 3) ), t 3

0 else

and 3(t) = 0.4(sin2 2 t + cos3 t), t 0.

Let the initial conditions be selected as x(0) =(0.5, 0.7, 0.5)T and x(0) = (0,0,0,0,0,0)T. The results ofsimulation are shown in Figs. 28. It can be observed from

Figs. 24 that the state estimate of x(t) the proposed adaptivefuzzy observer (30) indeed converges asymptotically to theoriginal state x(t) of the uncertain fuzzy system (3). Onthe other hand, it can be seen from the Figs. 57, theestimated output (t) indeed converges asymptotically tothe original disturbance (t) of the uncertain fuzzy system(3). Specifically, Fig. 8 shows the curves of the state errors.One can see that the estimation performance is desired inthe disturbance and perturbed environment.

Next, we consider about the estimate of state and sensorfault signal in Example 2.

Example 2: Consider the following fuzzy system with sensor

fault [24]

x(t) =

2i=1

hi()(Ai + Ai)x(t) +Biu(t)

y(t) =

2i=1

hi()Cix(t) +Dsfs

(42)

P =

1.0840 0.0017 0.1464 0.2751 0.2769 0.05560.0017 1.2450 0.2543 0.0017 0.1158 0.23060.1464 0.2543 0.6544 0.1464 0.1079 0.5807

0.2751 0.0017 0.1464 1.0840 0.2769 0.05560.2769 0.1158 0.1079 0.2769 0.9664 0.17500.0556 0.2306 0.5807 0.0556 0.1750 0.9025

Q1 =

99.5520 99.3547 24.020999.1200 0.7214 36.6378

12.0081 36.1252 1.22770.2736 100.2149 11.7282

100.6824 0.2151 36.277111.8713 36.4382 0.0028

; Q2 =

62.5805 61.9457 102.221462.3312 0.2077 74.451926.2671 74.9067 0.86580.1426 62.8342 25.907862.2580 0.0966 75.377526.0239 74.9049 0.1093

and

H1 =

69.5174 62.9193 16.511377.2680 14.4091 27.4455

19.9466 58.8165 5.744673.0467 146.8385 9.0447

220.2577 73.7918 71.323021.6974 67.0093 2.2743

; H2 =

53.3387 24.9351 76.384535.8656 28.7691 54.325743.4246 122.3873 4.9702

46.1502 91.8100 94.2744

137.7150 45.6598 185.452948.2914 138.1335 1.9888



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Fig.2 State x1(t) and the estimated state x1(t)





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Fig.5 Output disturbance 1(t) and the estimated output disturbance 1(t)





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Fig.8 State estimated errors

a x1(t) x1(t)

b x2(t) x2(t)

c x3(t) x3(t)

where fs Rk is the sensor fault, Ds Rpk is full column,and all other matrices are defined as before. Let

(t) = Dsfs(t) (43)

The (42) has the similar form to (24). Then, wecan apply the fuzzy observer (30) for estimating thesensor fault fs(t). Suppose x(t) = [x1(t) x2(t) x3(t)]

T, y(t) =[y1(t) y2(t) y3(t)]

T and AiBi and Ci are same in [24].

A1 =

1 2 0

2 1 01 0 3

, B1 =

100

, C1 =

1 0 0

1 1 00 0 1

A2 =

2 1 0

0 0.5 11 0 1

, B2 =

010

, C2 =

1 0 01 0 10 1 0

The constant uncertainties Ai, i = 1,2 are bounded by1 = 0.1199 and 2 = 0.2512, respectively. Moreover, (15)holds and i, i = 1, 2 are bounded by 1 = 0.01 and 2 =

0.02, respectively. Assume Ai is formulated as below

A1(t)

0.1199 0 00 0.1 0

0 0 0.009

,

A2(t)

0.2512 0 00 0.135 0

0 0 0.03

(44)

where is any constant belonging to the interval [1, 1].Choose fs(t) = [fs1(t) fs2(t) fs3(t)]

T, where

fs1(t) =

0.2sin(3(t 2) , t 2

0 else

fs2(t) =

0.2sin(4(t 3)), t 3

0 else(45)

fs3(t) = 0.4(sin 2 t+ cos3 t), t 0

and fs3(t) = 0.4(sin 2 t + cos3 t), t 0.Now, continue to establish the fuzzy observer design. We

will obtain as follows (see equations at bottom of the page)

P =

1.0985 0.0050 0.1473 0.2634 0.2684 0.05520.0050 1.2579 0.2537 0.0050 0.1090 0.23110.1473 0.2537 0.6561 0.1473 0.1064 0.5819

0.2634 0.0050 0.1473 1.0985 0.2684 0.05520.2684 0.1090 0.1064 0.2684 0.9845 0.17580.0552 0.2311 0.5819 0.0552 0.1758 0.9040

Q1 =

65.4924 65.2214 11.370565.0034 0.7170 15.21404.4217 15.7164 1.22800.3128 66.1576 4.7099

66.5728 0.2609 15.55404.5701 15.3885 0.0041

; Q2 =

157.2621 157.9240 35.0218157.5289 0.2073 52.118288.6387 52.5717 0.87650.1320 157.0138 88.9878

157.6156 0.0911 53.044588.8707 52.5709 0.1129

and

H1 =

48.8341 51.5411 9.440045.8567 5.2319 10.70116.7262 25.1635 5.674647.8579 96.4646 4.8905

144.6969 48.6067 30.94008.5007 28.1772 2.2373

; H2 =

138.3019 130.0065 24.623282.6335 19.7876 37.8285

141.8137 85.2032 4.9397115.3722 231.2420 91.0537

346.8630 115.8698 51.5007162.0611 96.4272 1.9691



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Let 1 = 2diag[10,1,1] and the initial conditionsbe selected as x(0) = (0.2, 0.3, 0.5)T and x(0) = (0,0,0,0,0,0)T. The simulation results are shown inFigs. 915. Figs. 911 show the curves of state estimation.

Figs. 1214 express the estimated sensor fault fsi(t) withthe different form of sensor fault fsi(t) respectively.Fig. 15 shows the curves of the state estimationerrors.

The above two examples show that the estimationperformance of the designed fuzzy observer is achievedunder the existence of uncertain parameters and outputdisturbance.

To reinforce the application value of the proposed fuzzyobserver, the following Example 3 shows the approachto be applied into a real example adopted from the

paper [8].

Example 3: Consider the dynamics of single link robot armwith a revolute elastic joint in a vertical plane in [8]

J1q1 + k(q1 q2) + mglsin q1 = 0

J2q2 k(q1 q2) = u(46)

All parameters are the same as those in the example of [8].

We obtain the state equations of (47)

x1 = x2

x2 = x3 10 sinx1 10x1x3 = x4

x4 = 50x1 5x3 + 5u

y = x1 + d

(47)


Fig. 10 State x2(t) and the estimated state x2(t)



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Fig. 11 State x3(t) and the estimated state x3(t)

Fig. 12 Fault f s1(t) and the fault estimate fs1(t)




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Fig. 15 State estimated errors

a x1(t) x1(t)

b x2(t) x2(t)

c x3(t) x3(t)

where x1 (/2, /2), Let sin(x1(t)) be represented asfollows

(t) = sin(x1(t)) =

2

i=1

hi( (t))bi

x1(t) (48)

Let the membership functions be the form as (49a) and (49b)which satisfy the property: h1( (t)) + h2( (t)) = 1, where

h1( (t)) =

(t) (2/ ) sin1( (t))

(1 2/) sin1( (t)), (t) = 0

1 otherwise

(49a)

and

h2( (t)) =

sin1( (t)) (t)

(1 2/) sin1( (t)), (t) = 0

0 otherwise

(49b)

which are shown in Fig. 16. Then, we arrive at the followingTS fuzzy model

Rule 1: If (t) is zero, then

x(t) = A1x(t) +B1u(t)

y(t) = C1x(t) + d(t)

Rule 2: If (t) is not zero, then

x(t) = A2x(t) +B2u(t)

y(t) = C2x(t) + d(t)

where, d(t) is the same form as the disturbance in [8], thatis d(t) is a measurement noise of 5% of the maximum value



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Fig. 16 Membership functions of two-rule model

of the output with zero mean and standard deviation 1

A1 =

0 1 0 0

20 0 1 0

0 0 0 1

50 0 5 0

, A2 =

0 1 0 0

20

10 0 1 0

0 0 0 1

50 0 5 0

,

B1 = B2 =

0

0

0

5

, CT1 = CT2 =

1

0

0

0

Construct the augmented system (50) as below

Ex(t) =

ri=1

hi(){ Ai x(t) + Biu(t) + N(t)}

y(t) = C0 x(t) + d(t) = Cx(t)

(50)

where

A1 =

0 1 0 0 020 0 1 0 0

0 0 0 1 050 0 5 0 00 0 0 0 1

;

A2 =

0 1 0 0 020/ 10 0 1 0 0

0 0 0 1 050 0 5 0 0

0 0 0 0 1

;

B1 = B2 =

00050

; N =

00001

and CT =

10001

Now, let us establish the fuzzy observer design as follows:Choose LT = [0 0 0 0 1] to guarantee ( E+ LC)1

existing, such that

( E+ LC)1 =

1 0 0 0 0

0 1 0 0 00 0 1 0 00 0 0 1 0

1 0 0 0 1

Utilise LMI-Toolbox of MATLAB to obtain the feasiblesolution P and Hi as follows

P =

2.2592 0.2010 0.5134 0.0768 2.7653

0.2010 1.0801 0.0789 0.1328 0.2981

0.5134 0.0789 0.1644 0.0119 0.1504

0.0768 0.1328 0.0119 0.0583 0.0878

2.7653 0.2981 0.1504 0.0878 16.3296

QT1 = [1.1295 14.9640 1.1771 0.0706 31.2045]

QT2 = [1.4065 13.0016 1.0337 0.1706 30.7511]

HT1 = [27.8133 8.7405 89.7378 65.7083 22.5315]

and

HT2 = [29.2736 6.4259 94.4479 67.1429 23.7811]

The simulation was carried out to compare with theperformance among the proposed observer, the proportionalhigh gain observer (PO) and the PI high gain observer (PIO)in of [8]. Fig. 17 shows the estimation of x3(t) using POand Fig. 18 shows the estimation of x3(t) applying PIO.Furthermore, Fig. 19 shows the curve of the estimation state

x3(t) using the proposed adaptive fuzzy observer. It is seenthat there exists chattering phenomenon in Fig. 17 and theestimation performance is not perfect in Fig. 18. Althoughin Fig. 19, the estimation performance is much better thanthe above two figures.

Fig. 17 State x3(t) and state estimation x3(t) using PO [8]

Fig. 18 State x3(t) and state estimation x3(t) using high gain PI

observer [8]



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Fig. 19 State x3(t) and state estimation x3(t) using the proposed adaptive fuzzy observer

5 Conclusion

In this paper, an adaptive fuzzy observer design for thefuzzy system with unknown output disturbance and boundedconstant parameter uncertainty has been presented. Thefuzzy system is with equal output matrices or with differentoutput matrices, the estimation of the state and disturbanceis achieved. Based on the proposed main theorems, thefuzzy observer is designed under the existence of uncertain

parameters and output disturbance. The procedure of the

observer design is also summarised. Simulation results havebeen shown that the observer design is successful. Finally,the proposed fuzzy observer has been applied into a realexample and the estimation performance comparison isshown in the last example.

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