Robust fuzzy-observer-design-for-nonlinear-systems

Robust Fuzzy Observer Design for Nonlinear Systems

Michal Polanský, and Cemal Ardil

Abstract—This paper shows a new method for design of fuzzyobservers for Takagi-Sugeno systems. The method is based on Linearmatrix inequalities (LMIs) and it allows to insert H constraint into the design procedure. The speed of estimation can tuned be specification of a decay rate of the observer closed loop system. Wediscuss here also the influence of parametric uncertainties at theoutput control system stability.

Keywords— H norm, Linear Matrix Inequalities, Observers,Takagi-Sugeno Systems, Parallel Distributed Compensation

I. INTRODUCTION

N the past some design techniques were developed for modeling and control of nonlinear uncertain systems. Very

interesting approach were done in the fuzzy modeling andcontrol, especially with Takagi-Sugeno (T-S) fuzzy modeling[4] and related Parallel Distributed Compensation (PDC)control algorithm [1][2]. This method uses local linear modelsin the consequent of fuzzy rules. Decision variables are designed to divide the state space of the system into areas, where the linear local models describe precisely the nonlinearsystem. In the overlapped parts of these areas are local modelsinterpolated according to fuzzy membership functions. Takagi-Sugeno model based control of nonlinear systems isquite popular now for its simple and effective design based usually on Linear Matrix Inequalities (LMIs)[5].

The design of an observer in Takagi-Sugeno fuzzy systemsis very important part of a control system design. Estimationof unmeasured states must be fast and should depend on thenoise and system uncertainties as less as possible. Since we can look at the parametric uncertainties as at an inputdisturbance signal, which should be attenuated, H norm of the closed loop system should be minimized. It could be alsouseful, if we can implement the parametric uncertainties intothe design method. This problem is discussed here, but it isquite difficult and exceeds the scope of this paper.

Although many design methods for a fuzzy controller(Parallel Distributed Compensator – PDC) are developed[1][5][6], there can be found only a few methods for a T-S fuzzy observer design in a literature.

Michal Polanský is with the Department of Control and Instrumentation,Faculty of Electrical Engineering and Communication,Brno University of Technology, Kolejní 4, 612 00 Brno, Czech Republic (tel.: +420 5 41141128, fax : +420 5 41141123)

The observer and controller has some similar propertiesfrom a mathematical point of view [2][3][7], that can simplifythe development of observer design methods. We can say, thatthey are dual problems. Presented research is inspired by thepaper published in [1], which deals with a robust H PDC controller design. On its base is derived a method for an observer design.

II. TAKAGI-SUGENO SYSTEM WITH OBSERVER AND CONTROLLER

A. T-S model

The standard Takagi-Sugeno fuzzy model consists of theset of fuzzy rules with linear consequent, that describe systemin a local areas i. For our purpose we suppose following form:

Rule i:IF is and is and … and is)(1 tz iM1 )(2 tz iM 2 )(tzn

inM

THEN)()()()()()()( 2211 tttt iiiiii uBBwBBxAAx ,

)()( tt ixCy (1)

where )(),...,()( 1 tztzt nTz are some premise variables,

)(),...,()( 1 tytyt lTy is an output vector,

)(),...,()( 1 txtxt nTx is a state vector,

)(),...,()( 1 tutut mTu is a control input vector and

)(),...,()( 1 twtwt pTw is a disturbance vector.

i = 1,2,…,r denotes the area’s number. r is the number of areas and thus also of fuzzy rules. is a fuzzy set

( is the grade of membership of premise variable z

ijM

))(( tM ij z j(t)

in the area number i). m is the number of inputs and l is the number of outputs of T-S fuzzy system. Matrixes ,

, , describe the system in thearea number i.

nni RA

mni R1B mn

i R2B nli RC

Matrixes ii 1, BA a i2B represent uncertainties in the system. We will suppose, that uncertainties in the outputmatrix Ci can be transformed into matrices i1B a i2B . Toincorporate the uncertainties into a PDC controller designmethods is commonly used the following assumption:

I

World Academy of Science, Engineering and TechnologyInternational Journal of Mathematical, Computational Science and Engineering Vol:2 No:2, 2008

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IFF

EEEDFBBA

)()(

],,)[(),,( 32121

tt

t

iTi

iiiiiii (2)

where D, are known, real, constant matrixes with appropriate dimensions. I is identity matrix and areunknown matrix functions with Lebesgue-measurableelements.

iii 321 ,, EEE)(tiF

The defuzzified output can be represented as follows:

(3)

)())(()(

)}(][)(][

)(]{[))(()(

1

2211

1

ttht

tt

ttht

i

r

ii

iiii

ii

r

ii

xCzy

uBBwBB

xAAzx

where r

i

n

jj

ij

n

jj

ij

i

tzM

tzMth

1 1

1

))((

))(())((z (4)

If z(t) is in the specified range, then 1))((1

r

ii th z

This representation can be easily implemented into a Matlabmodel.

B. Fuzzy State Observer

Here we can suppose, that the disturbance signal is notmeasured. Also the observer don’t have information aboutparametric uncertainties. Fuzzy state observer is described bythe following fuzzy rules:

Observer rule i: IF is and is and … and is)(1 tz iM1 )(2 tz iM 2 (t)zn

inM

THEN ,)](ˆ)([)()(ˆ)(ˆ 2 ttttt iii yyGuBxAx),(ˆ)(ˆ tt ixCy (5)

where Gi is an observer gain in the area number i.)(ˆ),...,(ˆ)(ˆ 1 txtxt n

Tx is estimated state vector and

)(ˆ),...,(ˆ)(ˆ 1 tytyt lTy is a vector of estimated outputs.

After defuzzyfication we get the following equations:

r

iiiii tttttht

12 )](ˆ)([)()(ˆ))(()(ˆ yyGuBxAzx ,

r

iii ttht

1)(ˆ))(()(ˆ xCzy . (6)

The aim of the observer is to eliminate estimation error asfast as possible. Estimation error is defined as

)(ˆ)()( ttt xxeIt will be useful to convert the equation (6) in the form with

estimation error instead of . The computation of timederivative of estimation error is then:

)(te )(ˆ tx

r

iiiii

r

i iii

iiii

ttttth

tttttt

th

ttt

12

1 21

21

)](ˆ)([)()(ˆ))((

)()()()()()(

)).((

)(ˆ)()(

yyGuBxAz

uBwBxAuBwBxA

z

xxe

r

i iii

iiii ttt

tttttth

1 21

1

)()()()()](ˆ)([)](ˆ)([

))((uBwBxA

wByyGxxAz

r

i iii

ijiir

jji

r

i

iii

i

r

jjji

i

i

ttttt

thth

ttt

tttth

tt

th

1 21

1

1

1

21

11

)()()(

)()()())(())((

)()()(

)()](ˆ)([))((

)](ˆ)([

))((

uBwBxAwBeCGA

zz

uBwBxA

wBxxCzG

xxA

z

(7)

In the case that the matrixes ,0iA ,01iB 02iBwe get

r

iii

i

r

ijii

r

jji

ttht

ttththt

1

11 1

)())(()(

)]()()[())(())(()(

eCz

wBeCGAzze , (8)

where )(ˆ)()( ttt yy is an output estimation error. If we define jiiij CGAL , then we get

r

iiij

r

jji ttththt

11

1)]()([))(())(()( wBeLzze . (9)

C. PDC Fuzzy Control Algorithm

The control signal of PDC controller is computed as

)())(()(1

ttht i

r

ii xKzu (10)

where is a constant feedback gain.nmi RK

If some state variables have to be estimated by a fuzzy observer, then the control output is computed in the followingway:

)(ˆ))(()(1

ttht i

r

ii xKzu (11)

The closed loop system we get by combination of (10) and(3). Then


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)]()()())([(

))(())(()(

1122

1 1

ttt

ththt

iijiiii

r

ij

r

ji

wBwBxKBBAA

zzx (12)

In the case that the matrixes ,0iA ,01iB 02iBwe get

)]()()[())(())(()( 11 1

ttththt ijii

r

ij

r

ji wBxKBAzzx (13)

If we define , then we get the system equationjiiij KBAF

)]()([))(())(()( 11 1

ttththt iij

r

ij

r

ji wBxFzzx , (14)

which is formally identical to (9). The important result is thatfuzzy observer and PDC controller are dual problems and wemay transform the PDC controller design methods to be used for design of an observer.

III. ROBUST CONTROLLER DESIGN

The method have been published by Lee, Jeung and Park in[1]. The objective is to minimize H norm, that determine a disturbance attenuation and is frequently used also as a robustness measure. H norm is defined as

.)()(

sup2

2

02tt

wy

Tw

yw (15)

where is transfer matrix of a system (13) and supremumis taken from all trajectories from the initial conditions

.

ywT

0)0(xWe suppose, that

22)()( tt wy (16)

The method minimizes , that is an upper bound of ywT .

Decay rate is specified by a condition

))((2))(( tVtV xx , (17)

where is a Lyapunov function, that must be alwayspositive with a negative time derivative.

))(( tV x

The controller design method is summarised in Theorem 1.

Theorem 1: Equilibrium of a T-S fuzzy system with a PDC control (10) is asymptotically stable in large with a decay rate

0 and ywT , if there exists a common positive

definite matrix Z > 0, > 0 and matrices Mi, such that the following LMIs conditions hold.

ri ,...,2,1

00

021

1

IZCIB

ZCB

i

Ti

Tiiii

, (18),,...,2,1 ri

02002 2

11

11

IZCZCIBB

ZCZCBB

ji

Tj

Ti

Tj

Tijijiij

, ,0 rji (19)

where ,ZBMMBZAZA 222T

iTjji

Tiiij

The gain of the robust controller the will be:

(20)1ZMK ii

The optimization objective is to minimize .

Proof: We choose the following substitution: r

i

r

jjiiji thth

1 12 ],))[(())(( KBAzzA

r

iiiw th

11 ,))(( BzB

r

iiiy th

1,))(( CzC

Then the closed loop system (13) will be transformed to

)()()()()(

ttttt

y

w

xCywBxAx

(21)

Lyapunov function we choose , where . If we suppose zero initial conditions, then from (17) we

get

)()())(( tttV T Pxxx0P

0)()2)(( tt TT xPAPPAx (22)

The system then will be stable with a decay rate > 0. Itholds if there exists matrix that fulfill the matrixinequality

0P

02 PAPPAT (23)

To implement the H norm we will now suppose that

0)()()()())(( 2 tttttV TT wwyyx . (24)

Then H norm of the system (21) is less than [5]. The condition (24) is equivalent to

0)()()()(

)()()())((2 tttt

ttttTT

wT

wT

yTy

TT

wwPxBw

wBPxxCCAPPAx (25)

This is equivalent to


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02IPBBPCCAPPA

Tw

wyTy

T

(26)

To fulfill both conditions (22) and (25) must hold:

022IPB

BPPCCAPPATw

wyTy

T

(27)

This inequality is not linear, but we can transform it into a LMI. At first we introduce a matrix and we will

multiply (27) by . Using S-procedure we obtain

1PZ

IZ0

0

00

02

2

IZCIB

CZBZZAAZ

y

Tw

Tyw

T

, (28)

that is equivalent to

00

02

))(())(( 21

1

1 1 IZCIB

ZCBZZFZFzz

i

Ti

Tiiij

Tijr

ij

r

ji thth (29)

where .jiiij KBAF 2

This LMI can be also written as

02002

))(())((0

0))((

211

11

1

21

1

1

2

IZCZCIBB

ZCZCBB

zzIZC

IBZCB

z

ji

Tj

Ti

Tj

Tijijiij

r

ij

r

jii

i

Ti

Tiiiir

ii ththth

(30)

where ZBMMBZAZA 222T

iTjji

Tiiij

and .ZKM ii

This is equivalent to the LMI conditions in the Theorem 1.

IV. FUZZY OBSERVER DESIGN

Now we will derive the LMI method for an observer design. The objective is again to minimize H norm, that determinenow a disturbance attenuation on the output estimation error.H norm ewT is then defined as

.)()(

sup2

2

02tt

wT

ww (31)

where is transfer matrix of a system (8).wT

We suppose, that

22)()( tt e w (32)

The method minimizes e, that is an upper bound ofwT .

The decay rate e is specified by a condition

))((2))(( tVtV eee xx , (33)

where is a Lyapunov function, that must be alwayspositive with a negative time derivative.

))(( tVe x

We will start from the equation (27), where remain matricesr

iiiw th

11))(( BzB and

r

iiiy th

1))(( CzC and newly defined is

r

i

r

jjiijie thth

1 1]))[(())(( CGAzzA that replaces matrix

A . Matrix P is replaced by Y. The closed loop system (8) isthen transformed to

)()()()()(

ttttt

y

we

eCwBeAe

(34)

Lyapunov function we choose

)()())(( tttV Te Yeee , (35)

where Y > 0.For H norm must hold that

0)()()()())(( 2 tttttV Te

Te wwe . (36)

Then the H norm of a system with observer (31) is lessthan e [5]. The condition (27) is transformed to

022IYB

BYYCCAYYA

eTw

wyTy

T

(37)

Inequality (37) consists all requirements, whose are thestability, the H norm determined by its upper bound e and the decay rate . This inequality we can write also as

02))(())(( 21

1

1 1 IYBYBYCCYLYLzz

eTi

iiTiij

Tij

r

ij

r

ji thth (38)

or

02

))(())((

))((

12

11

11

21

1

1

2

r

i eT

jTi

jijiijj

r

jii

eTi

iiir

ii

thth

th

IYBYBYBYB

zz

IYBYBz

(39)

where .YCCCJYAJCYA 2iTijii

Ti

Tj

Tiij


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We are now able to formulate the LMI conditions for the observer design method.

Theorem 2: State fuzzy observer (13) maintains asymptoticalstability of an estimation error , H0)(te norm ewT

and decay rate , if there will exist matrix , positivedefinite matrix and scalars

iJ0P 0e and 0 , that the

following LMI conditions hold

021

1

IYBYB

eTi

iii (40),,...,2,1 ri

02 2

11

11

IYBYBYBYB

eT

jTi

jijiij (41),,...,2,1, rji

where ,YCCCJYAJCYA 2iTijii

Ti

Tj

Tiij

and .ii YGJ

The observer gain is then computed as .ii JYG 1

Proof: It implies from the above results.

V. OBSERVER BASED FUZZY CONTROL DESIGN

Observer is a very important part of the control system. Thequality of the observer may dramatically change the speed andparameters of a controller, e.g. the H norm. The observer should attenuate disturbances, be as fast, as possible andshould not depend much on parametric uncertainties.Hopefully Ma et Sun in [3] found so called separationproperty that if both controller and observer are stable, thenthe overall observer based control system is stable too. Onlythe control quality parameters will change. We are thusallowed to design these parts separately.

By combination of the T-S fuzzy system (3) withoutparametric uncertainties, observer (6) and PDC control (11) we get the overall state equation, from which is the separationproperty well observable:

)()()(

0)()(

1

1 ttt

tt

i

i

jii

jijii wBB

ex

CGAKBKBA

ex (42)

If we incorporate parametric uncertainties into the system,then the design problem is more complicated.

)()()()(

))(())(()()(

2

2222

1 1

tt

ththtt

jiijii

jiijiiii

r

ij

r

ji

ex

CGAKBAKBBKBBAA

zzex

(43)

)(~)()(~))(())(()(

1 111

11 tttththt ii

r

ij

r

ji

ii

ii wBexAzzwBB

BB

We can see, that the stability can be changed since the zero

element disappeared. The separation property will not hold.The problem is difficult to solve since the design method mustassure the Lyapunov stability of the overall system. In thesense of quadratic Lyapunov stability, that is commonly used for T-S fuzzy systems it means that the there must exist a common matrix P~ that

0~~~~i

Ti APPA (43)

holds for ri ,...,2,1 .This condition have to be incorporated into a design

method. The parametric uncertainties can be included in theform (2). In further research we will concern on this problem.

V. CONCLUSIONS

The paper deals with a fuzzy observer design problem. The method is based on a convex numerical optimization of a setof LMIs. The LMI conditions are derived here and allow us toincrease robustness against input disturbances and parametricuncertainties by minimizing of the upper bound of the Hnorm. The fast state estimation is ensured by specifying ofminimal decay rate parameter.

The duality of the controller and observer is shown here.The observer design method is then derived on a base ofresults in [1], where a method for a PDC controller design ispublished.

The paper also discuss the problem of design of theobserver based control fuzzy system, where can be both partsdesigned separately due to a separation principle. Thedifferent situation comes if we would like to incorporate theparametric uncertainties into the design process. It was shown,that there are some differences between the observer and PDCcontroller in this case and that it is possible to find a method,that could cope with this problem, only if we will work withan overall system consisting both state variables an estimationerror variables.

REFERENCES

[1] K. R. Lee, E. T. Jeung, H. B. Park, “Robust fuzzy H control foruncertain nonlinear systems via state feedback: an LMI approach”. Fuzzy Sets and Systems. 2001, vol. 120, p. 123–134

[2] M. Polansky, “Nová metoda ARPDC pro zvýšení kvality robustníhoízení nelineárních systém ” Dissertation thesis, Faculty of electrical

engineering and Communication at Brno University of Technology,2005, in Czech

[3] X. J. Ma, Z. Q. Sun, “Analysis and design of fuzzy controller and fuzzyobserver” IEEE Trans. Fuzzy Systems. 1998, vol. 6, no. 1, p. 41–51.

[4] T. Takagi, M. Sugeno, “Fuzzy identification of systems and itsapplications to modeling and control” IEEE Transactions on Systems,Man and Cybernetics. 1985, vol. 1, no. 1, p. 116–132

[5] S. Boyd, L. El Ghaoui, E. Ieron, V. Balakrishnan, “Studies in AppliedMathematics : Linear Matrix Inequalities in System and ControlTheory”. SIAM, Philadelphia, PA, 1994

[6] S.G.Cao, N.W.Rees, G. Feng, H control of uncertain fuzzy continuous-time systems. Fuzzy Sets and Systems. 2000, vol. 115, p. 171-190

[7] V. Ku era, Analysis and design of discrete linear control systems.Prentice Hall, 1991


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Robust fuzzy-observer-design-for-nonlinear-systems

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