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1861-5252/ c© 2012 TSSD Transactions onSystems, Signals & Devices

Vol. 7, No. 1, pp.1-14

Discrete Fuzzy High Gain Observer

for a Class of Nonlinear Systems

M. Oueder, M. Mihoub, R. Ben Abdennour and M. Farza

1Research Unit CONPRI, ENIG, University of GabesOmar Ibn EL Khattab Street, 6029 Gabes, Tunisia.

2GREYC UMR 6072 CNRS, University of Caen, ENSICAEN6 Marechal Juin Street, 14050 Caen Cedex, France

Abstract In order to improve the robustness of the discrete high gain ob-server with respect to noise measurements and enhance the con-vergence dynamics, we propose, in this paper, a fuzzy supervisorfor the observation of a class of nonlinear systems. A suitable online tuning of the observer’s gain is made by a fuzzy supervisor.Simulation results are given in order to show the performancesamelioration relatively to the case where the classical discretehigh gain observer is applied.

Keywords: Nonlinear systems, Discrete high gain observer, Fuzzy supervi-sion, Robustness, Noise measurements, Accuracy of convergence.

1. Introduction

In the last decade many works were interested in the observer synthesisfor continuous time nonlinear systems [8, 10, 14–18, 22]. Nevertheless,results related to discrete-time nonlinear systems are still quite limited(see e.g [2, 9, 11, 13]). For systems that are uniformly observable forany input u(t), a high gain observer has been suggested in [24]. Oneof the advantages of this observer is its excellent robustness properties.By choosing the observer gain large enough (therefore the name “highgain”) the observer error can be made arbitrary small. The difficulty inpractical applications is, however, the determination of an appropriatevalue for the observer gain. For relatively low values, the desired bounds

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on the observer error cannot be achieved. For values unnecessarily high,the sensitivity to noise increases, thus limiting the practical applications[7]. Many works were focused on this problem. Some authors suggestedthe determination of the observer gain by the LMI approach [23], byquadratic optimisation [3]. Others suggested an on line tuning of theobserver gain : adaptive gain [7], switched gain [1].

This paper proposes a fuzzy supervision for the single design parameterthat tunes the nonlinear discrete-time observer proposed in [13].

This paper is organized as follows. Firstly, we present the design ob-server’s procedure. Then, we illustrate the limits of this observer dueto the compromise in choosing the gain by an academic example. Inthe fourth section, a solution for this problem by exploiting a fuzzy su-pervisor for the online tuning of the design parameter is developed. Asimulation example illustrating the efficiency of the proposed observer isthen given.

2. Observer synthesis

Let’s consider the following continuous-time class of nonlinear systems:

x1 (t) = F 1[

u (t) , x1 (t)]

x2 (t) + b1[

u (t) , x1 (t)]

x2 (t) = b2[

u (t) , x1 (t) , x2 (t)]

y (t) = C x = x1

(1)

where the state x(t) =

(

x1

x2

)

∈ Rn, xi ∈ R

ni , i = 1, 2 with p =

n1 ≥ n2 and n1 + n2 = n. F 1 is a n1 × n2 rectangular matrix; the

input u ∈ Rm, the output y ∈ R

p, B(u, x) =

(

b1(

u, x1)

b2 (u, x)

)

∈ Rn with

bi(u, x) ∈ Rni , i = 1, 2; finally:

C = [In1, 0n1 ×n2

] (2)

where In1is a n1 × n1 identity matrix and 0n1×n2

is the n1 × n2 nullmatrix.

A simple forward Euler discretization of system (1), up to output injec-tion, gives:

x1k+1

= x1k + Ts

[

F 1 (uk, yk) x2k + b1 (uk, yk)

]

x2k+1 = x2

k + Ts

[

b2(

uk, yk, x2k

)]

yk = C xk = x1k

(3)

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where Ts is the sampling time and the notation ξk stands for the valueof any variable ξ at the sampling time k.

The condensed form of system (3) can be written as follows:

{

xk+1 = (In + Ts Fk (uk, yk)) xk + TsBk

(

uk, yk, x2k

)

yk = C xk = x1k

(4)

where Fk(uk, yk) =

[

0n1×n1F 1k (uk, yk)

0n2×n10n2×n2

]

is a (n × n) square matrix;

Bk(uk, yk, x2k) =

[

b1k(uk, yk)

b2k(uk, yk, x2k)

]

.

Predictive form of the discrete high gain observer

The observer synthesis necessitates the following assumption:

Assumption: [13]

The function b2(uk, yk, x2k) is global lipschitz with respect to x2

k locallyuniformly in u and y.

An observer for system (4) can be written in the following predictiveform:

xk+1 = xk+1 |k − Lk C(

xk+1 |k − xk+1

)

xk+1|k = (In + Ts Fk) xk + B(

uk, yk, x2k

)

Lk = θ Ts Λ+

k (In − Ts A) ∆− 1

θ S− 1

1 CT

(5)

where:

• x =

[

x1

x2

]

∈ Rn with xi ∈ Rni , i = 1, 2;

uk and yk are the input and the output of system (4), respectively.

• A =

[

0n1In1

0n10n1

]

.

• S1 is the unique solution of the algebraic Lyapunov equation:

S1 + AT S1 + S1 A − CT C = 0 (6)

where C is given by equation (2). It can be shown that the explicitsolution of (6) is given by:

S1 =

[

In1− In1

− In12 In1

]

(7)

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• ∆θ = diag(

In1, 1

θIn1

)

.

• Λ(uk, yk) = diag[In1, F 1(uk, yk)] and Λ+(uk, yk) its left inverse.

• θ is a parameter design chosen such that 0 < θTs < 1.

Indeed, we state the following:

Theorem 1 [13]

Assuming that system (4) satisfies the assumption (2), then, system (5)is an exponential convergent observer for system (4).

Notice that the term xk+1|k. allows to predict the value of xk+1 betweenthe samples k and k+1. When the value Cxk+1 becomes available, it isused to update the predicted value xk+1|k. in order to provide the onlineestimation.

3. Limits of the classical discrete-time high gainobserver

The discrete high gain observer provides fast and accurate state esti-mations for discrete nonlinear systems thanks to the large value of itsgain. However, this large gain increases the observer’s sensitivity tonoise measurements which are frequent in the industrial environment. Areduction of the observer’s gain slows down the convergence dynamics.So the design of the observer is confronted to a compromise: rapidityand accuracy versus sensitivity to noise measurements. This problem isillustrated by the following simulation example.

Let’s consider the following nonlinear dynamical system:

x1(k + 1) = x1(k) + 1

5Ts

[(

[x2(k)]2 + 1)

x2(k) + u(k)]

x2(k + 1) = x2(k)− Ts

[

x2(k) + 2x1(k)1 + 3[x2(k)]2

]

y(k) = x1(k)

(8)

where x =

(

x1

x2

)

∈ Rn1+n2 , with n1 = n2 = 1. u(k) = cos(2πkTs) and

y(k) respectively denote the measured inputs and outputs. The retainedsampling time Ts is equal to 0.01s.

It is easy to see that system (8) is under form (4). In particular, we have

F 1k = 1

5

(

[x2(k)]2 + 1)

and b2k =x2(k) + 2x1(k)1 + 3[x2(k)]2

.

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So, as a result, an observer of the form (5) can be synthesized.

In order to simulate practical situations, the output of system (8) iscorrupted by a noise measurements with zero mean value and a standarddeviation equals to

√0.5 (see Fig. 1).

0 500 1000 1500 2000

−20

−15

−10

−5

0

5

10

15

20

25

k

Fig. 1. Noisy output measurements.

We choose, in first time, to estimate the state variable x2 by the classicalhigh gain observer using two constant values of the design parameter,(small and large : θ = 1 and θ = 10).

Figures 2 and 3 show respectively the estimation of the state variablesx2(k) provided by the discrete high gain observer for the two retaineddifferent values of θ. We can clearly conclude that, on one hand, witha small value of θ, the result is more insensitive to noise measurementsrelatively to the case where a high gain is used. However, on the otherhand, using the small value of θ, estimated variables have a slow conver-gence to the real ones.

4. A fuzzy supervision for the high gainobserver

The Fuzzy Logic (FL) is a problem-solving methodology that provides asimple way to arrive at a definite conclusion based upon vague, ambigu-ous, imprecise, noisy, or missing input information [26]. FL incorporatesa rule approach to solve an identification, a control or an observationproblems [5, 6, 19, 20].

Many combinations of FL to many types of observers have been proposed[12, 21, 25]. In this paper, we propose to exploit this approach in orderto supervise the high gain observer.

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0 500 1000 1500 2000−4

−3

−2

−1

0

1

2

3

4

5

6

kSIMULATED

ESTIMATED

x2

Fig. 2. Estimation of x2(k) (Case

of θ, relatively small).

0 500 1000 1500 2000−6

−4

−2

0

2

4

6

k

ESTIMATED

SIMULATED

x2

Fig. 3. Estimation of x2(k) (Caseof θ, relatively large).

To do so, a fuzzy supervisor is then considered [4–6, 19, 20]. The su-pervisor inputs are the indicators of the noise and the convergence. Itsoutput is the supervised high gain noted θFn

(k) (Fig. 4).

Fuzzy supervisor

Inoisen(k)

Iconvn(k)

θFn(k)

Fig. 4. The fuzzy supervisor.

The supervision procedure involves three stages : the fuzzifucation, thefuzzy inference and the defuzzifucation.

4.1 Fuzzifucation

The fuzzifucation consists in converting the normalized real inputs (Inoise(k)and Iconv(k)) to fuzzy inputs by associating, for example, a set triangu-lar membership functions centred on the numbers Ncei to the consideredlinguistic variables (see Fig. 5.a).

Having the number ne of fuzzy sets associated to each input, the nor-malized variables Inoisen(k) and Iconvn(k) (normalized in the interval [0,1]) and the central numbers Ncei can be written:

Inoisen (k) =Inoise (k) − Inoise min

Inoise max− Inoise min

(9)

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Discrete Fuzzy High Gain Observer for Nonlinear Systems 7

Iconv n(k) =

Iconv (k) − Iconv min

Iconv max− Iconv min

, Ncei =i − 1

ne − 1(10)

with: 1 ≤ i ≤ ne.

In the present case, three fuzzy sets have been reserved for each super-vision input (ne = 3).

4.2 Fuzzy inference

The synthesis of the fuzzy supervisor requires an experimentation phasepermitting the development of a logical and methodological rule basis.The stage of inference consists in applying the linguistic rules providedby the rule basis to the fuzzy inputs descended of the fuzzifucation toevaluate the supervisor output (θFn

(k)). The fuzzy inference rules thatmanage the on line supervision of the observer gain are summarized inTab. 1, with VS: Very Small, S: Small, M: Medium, L: Large, and VL:Very Large.

Table 1. Fuzzy inference table relative to the supervision output θFn (k).

θFn(k) Iconv n

(k)

S M L

S VS VS S

Inoise n(k) M VS S M

L M L VL

The ith rule can be formulated as shown by the following example:

Ri: If [Iconvn(k)] is S and [Inoisen (k)] is M Then [θFn(k)] is VS.

The inference table associates ns fuzzy sets to the supervisor outputvariable, which are described also by triangular memberships (see Fig.5.b).

These functions are defined in the interval [0, 1] and centered on thenumber Ncsl :

Ncsl =l − 1

ns − 1(11)

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with: 1 ≤ l ≤ ns.

The MIN-MAX inference method is retained for the evaluation of thefuzzy rules’ contribution. This method assigns to the logical operation“AND” a minimum and to the logical operation “OR” a maximum.Therefore, the supervisor output is determined by the calculation of thecoefficients Dl which are given by the following formula:

Dl = Maxrule l

{

Mini→ l

[µi (Iconvn(k) , Inoisen (k))]

}

(12)

with 1 ≤ i ≤ ne et 1 ≤ l ≤ ns.

Actually, the termsDl correspond to the heights of the trapezes obtainedby the levelling of the triangular membership functions corresponding tothe supervisor output.

4.3 Defuzzifucation

The defuzzifucation operation that consists in calculating the numericalvalue of the supervision output from its fuzzy value, constitutes the laststage of supervision. The supervisor output θFn

(k) in our case, can becalculated using the formula of gravity center:

θFn(k) =

ns∑

l=1

Dl Ncsl

ns∑

l=1

Dl

(13)

At the end of each supervision cycle, the activation of the associatedrules’ set gives the necessary value of the design parameter.

The fuzzy supervisor structure of the design parameter of the high gainobserver is given by Fig. 6.

5. Simulation Results

For comparaison purposes with the classical high gain observer, the dis-crete fuzzy high gain observer is applied on the system (8). The equationof the observer can be written in the original coordinates as follows:

xk+1 = (In + TsFk)xk +B(uk, yk, x2k)

−LkC((In + TsFk)xk +B(uk, yk, x2k)− xk+1)

Lk = θFn(k)TsΛ

+

k (In − Ts A)∆−1

θfuzzyS−1

1 CT

(14)

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Discrete Fuzzy High Gain Observer for Nonlinear Systems 9

µ(Iconvn) and µ(Inoisen

)

S M L

0 0.5 1

Iconvn(k)

Inoisen(k)

µ(θFn)

S M L

θFn(k)

0 0.5 1

VLVS

0.750.25

(a)

(b)

Fig. 5. Membership functionsrelative to the supervisor inputsInoisen(k) and Iconv n(k). and thesupervisor output θFn(k)

Discrete HighGain Observer

Process

Noise

Inputs u(k) Output y(k)

x(k)

θ(k)

Performanceindices

FuzzySupervisor

Fig. 6. The fuzzy supervisor struc-ture of the discrete high gain ob-server.

In order to highlight the performances of the discrete fuzzy high gainobserver in terms of accuracy estimations and sensitivity to noise mea-surements, we corrupted the output of system (8) by a noise measure-ments with zero mean value and a standard deviation

√1.5 (we can see

that with comparaison to the classical discrete high gain observer simu-lations, we have increased the noise measurements) and at k = 1500, wechose to return at the initial conditions to provide a kind of parametervariations. The noisy output is given in Fig. 7.

The estimation of the state variable x2(k) provided by the discrete fuzzyhigh gain observer is given in Fig. 8 while the estimation with twodifferent constant design parameters (θ = 1 and θ = 10) is illustratedby Fig. 9. An obvious improvement of the performances in the caseof the fuzzy supervision of the discrete high gain observer relatively tothe classical one is observed. In fact, Fig. 8 shows a fast and accurateestimation with insensitivity to noise measurements. So, we can concludethat the fuzzy discrete high gain observer solves the compromise betweenthe convergence rapidity and the insensitivity to noise measurements.

The evolution of the fuzzy design parameter is illustrated in Fig. 10.We can notice that the design parameter continuously takes a low valueto ensure the robustness of the observer to noise measurements and in-creases only in the presence of a convergence problem.

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0 500 1000 1500 2000 2500 3000−25

−20

−15

−10

−5

0

5

10

15

20

25

k

Fig. 7. Noisy output measure-ments.

0 500 1000 1500 2000 2500 3000−4

−3

−2

−1

0

1

2

3

4

5

6x2(k), x2(k)

k

Fig. 8. Estimation of x2(k) (On lineFuzzy Supervision).

0 500 1000 1500 2000 2500 3000

−4

−2

0

2

4

6

0 500 1000 1500 2000 2500 3000

−4

−2

0

2

4

6

k

k

ESTIMATED

ESTIMATED

SIMULATED

x2

x2

Large gain

Small gain

SIMULATED

Fig. 9. Estimation of x2(k) (Classi-cal Discrete High Gain Observer).

0 500 1000 1500 2000 2500 3000

0

1

2

3

4

5

6

7

8

9

10

k

θ(k)

Fig. 10. Evolution of the fuzzy su-pervisor output.

6. Conclusion

In this work, the limits of the classical discrete high gain observer wasdiscussed. The sensitivity to noise measurements when the observergain is large and the convergence slowness when the observer gain issmall were illustrated by a simulation example on a nonlinear system.The proposed fuzzy supervised discrete high gain observer was, then,introduced and applied on that same nonlinear system. Comparisonbetween the results obtained by the classical discrete high gain observerand the fuzzy one showed a notable enhancement offered by the fuzzysupervision, in terms of convergence rapidity and estimation accuracy.

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References

[1] J. H. Ahrens and H. K. Khalil. High-Gain Observers in the Presence ofMeasurement Noise: A Switched-Gain Approach. 17th World CongressThe Int. Federation of Automatic Control, Seoul, Korea 2008.

[2] V. C. Aitken and H. M. Schwartz. On the exponential stability of discrete-time systems with applications in observer design. IEEE Trans. on Auto-matic Control, 39(9):1959-1962, 1994.

[3] R. Aloui and N. B. Braıek. On the determination of an optimal stateobserver gain for multivariable systems: Application to induction motors.J. Automation and Systems Engineering, 2(3):2008.

[4] R. Ben Abdennour, P. Borne, M. Ksouri and F. M’sahli. Identification etcommande numerique des procedes industriels. Editions Technip, Paris,2001.

[5] R. Ben Abdennour, G. Favier and M. Ksouri. Fuzzy trace identification al-gorithms for non-stationnary systems. J. of Intelligent and Fuzzy Systems,6(4):403–417, 1998.

[6] R. Ben Abdennour, M. Ltaief and M. Ksouri. Un coefficientd’apprentissage flou pour les reseaux de neurones artificiels. J. Europeendes Systemes Automatises, JESA, 35(9):1089–1103, 2001.

[7] E. Bullinger and F. Allgower. An adaptive high-gain observer for nonlinearsystems. 36th Conf. on Descion and Control, San Diego, California, USA,1997.

[8] K. Busawon, M. Farza and H. Hammouri. Observer design for a specialclass of nonlinear systems. Int. J. of Control, 71(3):405–418, 1998.

[9] K. Busawon, M. Saif and M. Farza. A discrete-time observer for a class ofnonlinear systems. 36th Conf. on Descion and Control, San Diego, USA,1997.

[10] F. Deza, E. Busvelle and J.P. Gauthier. Exponentially converging ob-servers for distillation columns and internal stability for the dynamic out-put feedback. Chemical Engineering Science, 47(15-16):3935–3941, 1992.

[11] A. Chaari, S. Hajji, M. Farza and M. M’Saad. Discrete-time estimationof the rotor fluxes and the load torque in an induction motor. 3rd Int.Conf. on Signals, Systems Decision and Information Technology, SSD’05,Sousse, Tunisia, 2005.

[12] M. Dong, B. Jiang and J. L. Wang. Fuzzy observer based fault accomoda-tion for nonlinear systems. IEEE Int. Conf. on Control and Automation,:1649–1652, 2007.

[13] M. Farza and M. M’Saad. Discrete-time observer synthesis for a classof nonlinear systems. 2nd Int. Conf. on Signals Systems Decision andInformation Technology, SSD’03, Sousse, Tunisia, 2003.

PRO

OF

12 M. Oueder et al.

[14] M. Farza, M. M’Saad and L. Rossignol. Observer design based on trian-gular form generated by injective map. Automatica, 40(1):135–143, 2004.

[15] J.P. Gauthier and I.A.K. Kupka. Observability and observers for nonlin-ear systems. SIAM J. on Control & Optimization, 32:975–994, 1994.

[16] M. Guay. Observer linearization by output-dependent time-scale trans-formations. IEEE Trans. on Automatic Control, 47(10):1730–1735, 2002.

[17] M. Hou and A. C. Pugh. Observer with linear error dynamics for nonlinearmulti-output systems. Systems & Control Letters, 37(1):1–9, 1999.

[18] A. J. Krener and A. Isidori. Linearization by output injection and non-linear observers. Systems & Control Letters, 3(1):47–52, 1983.

[19] A. Messaoud, M. Lataief and R. Ben Abdennour. Fuzzy supervision fora multimodel generalized predictive control based on performances index.Int. J. on Sciences and Techniques of Automatic control and computerengineering, 1(1):181–195, 2007.

[20] M. Mihoub, A. Messaoud, M. Lataief, A. S. Nouri and R. Ben Abdennour.Fuzzy Discontinuous Term for a Second Order DSMC: An ExperimentalValidation on a Chemical Reactor. Asian Journal of Control, 13, 2011,(under press).

[21] J. H. Park and G.-T. Park. Adaptive fuzzy observer with minimal dy-namic order for uncertain nonlinear systems. IEE Proc. control theoryapplicatons,150(2):189–197, 2003.

[22] R. Rajamani. Observers for Lipschitz Nonlinear Systems. IEEE Trans.on Automatic Control, 43(3):397–401, 1998.

[23] M. Rodrigues, H. Hammouri, C. Mechmeche and N. Benhadj Braıek. Ahigh gain observer based LMI approach. 17th IFAC World Congress, Seoul,Korea, 2008.

[24] A. Tronambe. High-gain observers for nonlinear systems. Int. J. of Sys-tems Science, 23(9):1475–1489, 1992.

[25] C. S. Tseng, B. S. Chen and Y. F. Li. Robust fuzzy observer-based fuzzycontrol design for nonlinear systems with persistent bounded disturbances:A novel decoupled approach. Fuzzy sets and systems, 160(19):2824–2843,2009.

[26] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.

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Biographies

Monia Oueder was born in Gabes, Tunisia in 1983.

She received her National Engineering Diploma in Elec-

tric and Automatic Engineering in 2006 and her Master

degree in Automatic and Intelligent Techniques in 2007

from the National School of Engineers of Gabes-Tunisia.

Since 2008, she registered in Ph.D, in the National School

of engineers of Gabes. Her specific research interests deals

with nonlinear observation, high gain observers and non-

linear control. She is currently teaching electrical and automatic control in

the High Institute of Industrial Systems of Gabes, Tunisia. She is a member

of the Research Unit of Numerical Control of Industrial Processes.

Mohamed Mihoub received his National Engineering

Diploma in Electric and Automatic Engineering in 2005,

his Master degree in Automatic and Intelligent Techniques

in 2006 and his PhD in 2010 from the National School of

Engineers of Gabes-Tunisia. His research deals with slid-

ing mode control, state estimation, fuzzy supervision and

multimodel approach. He is currently teaching automatic

control in the High Institute of Application Sciences and

Technology of Sousse, Tunisia. He is a member of the Research Unit of Nu-

merical Control of Industrial Processes.

Ridha Ben Abdennour received the Doctorat de spe-

cialite degree from the Ecole Normale Superieure de l’Ens-

eignement Technique in 1987, and the Doctorat d’Etat

degree from the Ecole Nationale d’Ingenieurs de Tunis in

1996. He is Professor in Automatic Control at the Na-

tional School of Engineering of Gabes-Tunisia. He was

chairman of the Electrical Engineering Department and

the Director of the High Institute of Technological Stud-

ies of Gabes. He is the Head of the Research Unit of Numerical Control

of Industrial Processes and is the President of the Tunisian Association of

Automatic and Numerisation. His research is on Identification, Multimodel

& Multicontrol approaches, Numerical Control and Supervision of Industrial

Processes. He is the co-author of a book on Identification and Numerical Con-

trol of Industrial Processes and he is the author of more than 300 publications.

He participated in the organization of many Conferences and he was member

of some scientific committees of congresses.

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Mondher Farza received his degrees of engineer and

M.Sc. in computer sciences and applied Mathematics

from ENSEEIHT Toulouse in 1988, his PhD in control

sciences from INPG Grenoble in 1992. Actually, he is Pro-

fessor at the University of Caen Basse-Normandie where

he is the head of the Control Group at the GREYC. His

research interests are in nonlinear control and systems and

applications.