Synthesis of Switching Controllers a Fuzzy Supervisor

8/13/2019 Synthesis of Switching Controllers a Fuzzy Supervisor

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Nonlinear Analysis 65 (2006) 16891704

www.elsevier.com/locate/na

Synthesis of switching controllers: A fuzzy supervisorapproach

Najib Essounbouli, Noureddine Manamanni, Abdelaziz Hamzaoui,Janan Zaytoon

Centre de Recherche en STIC, Faculte des Sciences, B.P.1039, 51687, Reims cedex 2, France

Abstract

In this paper the synthesis of multiple controllers for a class of time-varying nonlinear systems is

considered. The proposed hybrid control scheme is based on a fuzzy supervisor which manages the

combination of controllers of two types: with sliding mode control (SMC) and H control. A convex

formulation of the two controllers leads to a structure which benefits from the advantages of both controllersto (i) ensure a good tracking performance in both the transient state (SMC) and the steady state (H), (ii)

provide a fast dynamic response to enlarge the stability limits of the system, and (iii) efficiently reduce the

chattering phenomena induced by the SMC. The stability analysis uses the Lyapunov technique, inspired

from switching system theory, to prove that the system with the proposed controller remains globally stable

despite the configuration changing. Some simulation results and conclusions end the paper.

c 2006 Elsevier Ltd. All rights reserved.

Keywords:Sliding mode control; H control; Fuzzy supervisor; Riccati equation; Nonlinear systems

1. Introduction

Hybrid dynamical systems include continuous and discrete dynamics and a mechanics

(supervisor) managing the interaction between these dynamics. This paper is concerned with

a particular class of hybrid systems where the hybrid nature of the control scheme developed

consists of a fuzzy supervisor managing the combination between two controllers (SMC and

H). The switching action is gradual and is related to the system evolution between the

consecutive transient and steady state modes.

Corresponding author. Tel.: +33 326 91 83 86; fax: +33 326 91 31 06.E-mail address:[email protected](N. Essounbouli).

0362-546X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.12.039
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1690 N. Essounbouli et al. / Nonlinear Analysis 65 (2006) 16891704

An abrupt switch is not used in the proposed control scheme in order to attenuate the

controllability and the instability problems related to the induced jump phenomenon. So, in

this work, the supervisor determines the adequate mixing between the two controllers in each

mode. Furthermore, the proof of the global stability of a closed loop system using the proposed

method is related to the ones developed for switching systems theory based on multiple Lyapunovfunctions[5].

Combination of different techniques to obtain the best performances is widely used today.

Wong et al. [19] proposed a combination of three methods: SMC, fuzzy logic control (FLC), and

PI control. The resulting controller eliminates the chattering and the steady error introduced by

the FLC. Lin and Chen [12]used Genetic algorithms to optimize the mixing of SMC and FLC,

and hence to reduce chattering in the system. Barrero et al. [1] developed a FLC-based hybrid

controller to manage the switching between a SMC and a fuzzy PI controller. Nevertheless,

the above-mentioned works use a fixed combination or restrictive assumptions for the stability

analysis.

The aim of this paper is to propose a fuzzy supervisor for hybrid combination of SMC andHcontrollers to overcome their disadvantages, and to ensure the robustness and the stability

of the closed loop system.

According to Utkin [17], SMC is unique in its ability to achieve accurate, robust and

decoupled tracking for a class of nonlinear time-varying systems in the presence of disturbances

and parameter variations. It achieves these performances without precise calculations or

estimations of the system parameters, nonlinearities, and disturbances. SMC relies on the

presence of a high speed switching feedback control, and has its roots in relay and bangbang

control theory. The advent of faster switching circuitry, and the many advances in computer

technology, have made the implementation of SMC a reality and of increasing interest to control

system engineers[13,15,6]. Nevertheless, the major problem related to SMC is the chattering

phenomenon, which is quite undesirable in dynamic systems.

H techniques guarantee the robustness of the disturbed systems. Many combinations of

fuzzy logic intelligence and H technique efficiency have been proposed in the literature[4,3,7,

11]. In these works, the plant is approximated by two adaptive fuzzy systems; an H supervisor

computed from a Riccati-like equation attenuates the effects of both the external disturbances and

the approximation errors. Using the linear matrix inequality (LMI) technique, Tseng et al. [16]

proposed a FLC based on the TakagiSugeno system. The robustness and the tracking problem

are treated using LMI. However, there is a trade-off between the attenuation level and the system

time response on one hand, and the behaviour of the applied control signal on the other hand.The contribution of the work presented in this paper is combining SMC and Hcontrollers

using a supervisor, which manages the gradual transition from one controller to another. This

method is applied to a class of uncertain nonlinear systems subject to external disturbances, to

overcome the drawbacks of each controller. The control signal is obtained via a weighting sum of

the two signals given by the SMC and the H controllers. This weighting sum is managed thanks

to a fuzzy supervisor, which is adapted to obtain the desired closed loop system performances

by benefiting from the robustness of the SMC in the approaching phase and the ability of the

Hcontrol to eliminate the chattering and to guarantee the system robustness near the sliding

surface. So, the SMC mainly acts in the transient state providing a fast dynamic response and

enlarging the stability limits of the system, while the H control acts mainly in the steady state toreduce chattering and maintain the tracking performances. This method is particularly attractive

for nonlinear systems since it can result in many cases in invariant control systems, i.e. systems

completely insensitive to parametric uncertainties and external disturbances. Furthermore, the


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N. Essounbouli et al. / Nonlinear Analysis 65 (2006) 16891704 1691

global stability of the system even if the system switches from one configuration to another

(transient to steady state and vice versa) is guaranteed.

Section2presents the system definition, and the two controllers used. In Section3,the fuzzy

supervisor, and the proposed control law and its stability analysis are described. An example is

given in Section4to illustrate the efficiency of the proposed method.

2. Problem statement

2.1. System definition

Most of electromechanical SISO systems can be described by the following differential

equations which represent an nth-order nonlinear dynamic Single-Input Single-Output (SISO)

system in the canonical form:

x(n) = f(x,x, . . . ,x(n1)) + g(x,x, . . . ,x(n1))u+ d

y = x (1)

where f and g are two continuous uncertain and bounded functions; u R and y R are

respectively the input and output of the system, and ddenotes the external disturbances (due

to system load, external noise, etc) which are assumed to be unknown but bounded. It should

be noted that more general classes of nonlinear control problem could be transformed into this

structure[15,4]. Let X =(x, x, . . . ,x(n1))T Rn be the system state vector, which is assumed

to be available for measurement. For the system(1)to be controllable, the condition g (X) = 0

must be satisfied in a given controllability region, Uc Rn.

In the case where the system model is unknown and/or no information from the human

expert describing the system dynamic behaviour is available, the model can be approximated

by two adaptive fuzzy systems, which are used to synthesize the H and the sliding mode

controllers. Hence, the corresponding adaptation laws are deduced from the stability analysis [8].

Nevertheless, the control design requires a large computation time which makes it difficult to

implement. To overcome this problem, we can benefit from the ability of the system to be

described by a collection of local nominal models [9]. Thus, the functions f andg can be written

as sums of a known nominal part and an uncertain but bounded part:

f(x)= f0(x) + f(x)

g(x)= g0(x) + g(x).

Hence, the system described in(1)becomes

x(n) = f0(x) + g0(x)u+ w+ d

y = x(2)

wherew =f(x) + g(x)u.

The control objective is to force y to follow a given bounded reference trajectory, yr, under

the constraint that the stability and the robustness of the closed loop system are guaranteed.

2.2. Sliding mode control

In order to determine the SMC signal, let us first define the sliding surface, given by

s(x, t)= k1e k2e kn1e(n2) e(n1) (3)


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where e = yr y denotes the tracking error, and the factors ki are calculated such that the

Hurwitzian stability criterion is satisfied.

The process of sliding control can be divided into two phases: the approaching phase where

s(x, t)=0, and the sliding phase where s (x, t)= 0. A sufficient condition for guaranteeing the

transition of the tracking error trajectory from the approaching phase to the sliding one is [15]:

1

2

d

dts2(x, t)= s (x, t)s(x, t) |s(x, t)|; >0 (4)

with

s(x, t)= k1e k2e kn1e(n1) e(n). (5)

The following control law [15] can be used to ensure the stability of the closed loop system and

to satisfy the transition condition(4):

uSMC = g10 (x)

f0(x) +

n1i=1

ki e(i) +y (n)r Dsign(s)

(6)

where Dis a positive constant chosen such that D |w| + |d| + .

Let us consider the following Lyapunov function for the stability and the robustness analysis

of the closed loop system:

V =1

2s2(x, t). (7)

Using Eqs.(2),(5)and(6),the time derivative of(7)becomes

V = s(x, t)

n1i=1

ki ei f0(x) g0(x)uSMC

+

n1i=1

ki ei Dsign(st) + f0(x) +g0(x)uSMC+ w+ d

.

Then,

V =s (x, t)(w+ d) D|s(x, t)| 0.

Hence, the global stability of the closed loop system is guaranteed, as well as the property ofattractivity of the sliding surface. Nevertheless, one needs to attenuate the chattering phenomena

induced by the sign function. The main idea of this work is to use an H control, which will act

mainly in the steady state to reduce chattering without degrading the tracking performances and

the systems stability.

2.3. H control

The synthesized H control ensures the robustness of the system subjected to external

perturbations. Consider the following control law [7]:

u = g10 (x)[f0(x) +y(n)r + TE uh ] (8)

where E = [e,e, . . . , en1]T is the error vector, and = [n, n1, . . . , 1]T is the dynamic

error coefficient vector chosen such that the Hurwitzian stability criterion is satisfied.uh is the


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H supervisor which attenuates the effect of external disturbances on the tracking error and the

structural uncertainties [4,7].

Using(8)the tracking error dynamic can be written as

E= A E+ B[uh w d] (9)

where

A =

0 1 0 0 . . . 0 0

0 0 1 0 . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

0 0 0 0 . . . 0 1

1 2 . . . . . . . . . . . . n

, B = [0 . . . 0 1]T.

Since Ais a stable matrix, it can be associated with the following algebraic Riccati equation:

A P T + P A + Q 2P B

1r

122

B T P =0 (10)

where Q is a positive definite matrix given by the designer,ris a weighting factor, and is the

attenuation level. This equation has a unique positive definite solution, P = P T, if and only if

2

r

1

2 0.

To determine uhand to prove the global stability of the closed loop system subject to this control,

the following Lyapunov function is considered:

V = 12

ET P E. (11)

Using Eqs.(9)and(10),the time derivative of(11)can be given by

V =1

2ET Q E

1

2ET P B B T P E+ET P B

1

rB T P E+ uh w d

. (12)

On choosing

uh =1

rET P B. (13)

Eq.(12)implies that

V 1

2ET Q E+

2(w+ d)2

2 .

By integrating the above inequality from t = 0 to the response time T, the following H

criterion is obtained: T0

ET Q E.dt ET(0)P E(0) +

T0

2(w+ d)2.dt. (14)

Hence, the stability and the robustness of the closed loop system are guaranteed.

On the other hand, this last inequality(14)can be rewritten as T0

ETE.dt1

QET(0)P E(0) +

1

Q

T0

2(w+ d)2.dt


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whereQ is the minimal eigenvalue ofQ and the right side of this inequality is bounded. Thus,

from Barbalats lemma, we can conclude that the tracking error converges towards zero in a finite

time [18,7].

Now, the following remarks can be stated concerning the implementation of the control laws

(6)and(8):

Remarks. (1) Real processes often use a saturation condition on the control signal to preserve

the systems actuators or for security reasons. In this case, it is advisable to take into account

the variations of the control signal u h in the Hcriterion. Hence, the Riccati equation and

the corresponding criterion respectively become[4]

AT P + P A P B

1

r

1

2

B T P =0 (15)

T0

[ET Q E+ r u2

h].dt ET(0)P E(0) + T

0

2(w+ d)2.dt. (16)

(2) The control laws(6)and(8)are functions of the nominal model. Thus in the case where only

partial information (around some functioning points) is available, we have to establish the

local models and then, by using a fuzzy system, deduce an averaged model which will be

considered as nominal. For a functioning point j , this can be done as follows:

IF x1 is a neighbourhood ofxj1 AND x2 is a neighbourhood ofx

j2 AND ANDxn is a

neighbourhood ofxjn THEN x= A

jfx+ b

jf u.

This can be given by

IFx1

is Hj

1 ANDx

2is H

j

2 AND . . . ANDx

nis H

j

n THENx(n)

=

ni=1

ajf ixi + b

jf n u (17)

where Ajf = [a

jf i k]1i,kn , b

jf = [b

jf i ]1in, and H

ji is a fuzzy set describing the

neighbourhood ofxi around the j th functioning point.

It is easy to see that Eq.(17)can be considered as a fuzzy rule of a TakagiSugeno fuzzy

system. By using the algebraic product as an inference engine and the centre average for

defuzzification, the following system is obtained:

x(n) =

rfj =1

wjf n

i=1ajf ixi + bjf n u

rf

j =1

wjf

(18)

where wjf =

ni=1Hji

(xi ), rf is the number of functioning points and Hji(xi ) is the degree

of membership ofxi in the fuzzy set Hji .

If

ff0(x)=

rf

j =1 w

j

f[

ni=1 a

j

f ixi ]

rfj =1

wjf


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Fig. 2. The variation of in the state space.

The fuzzy implication uses the product operation rule. The connective AND is implemented

by means of the minimum operation, whereas fuzzy rules are combined by algebraic addition.

Defuzzification is performed using the centroid method, which generates the gravity centre of the

membership function of the output set. Since the membership functions that define the linguistic

terms of the output variable are singletons, the output of the fuzzy system is given by

=

mi=1

in

j =1

ji

mi=1

nj =1

ji

whereji is the degree of membership ofH

ji , andm is the number of fuzzy rules used.

The objective of this fuzzy supervisor is to determine the weighting factor, , which gives the

participation rate of each control signal. Indeed, when the absolute values of the tracking errore

and its time derivatives e,e, . . . , en1 are small (near to zero), the plant is governed by the H

controller given by(8),( = 1). Conversely, if the error and its derivatives are large, the plantis governed by the SMC given by(6)and =0. The variation of for the case wheren =2 is

depicted inFig. 2.

The control action,u , is determined by

u =(1 )uSMC+ u. (19)

Remark. In the case of a large rule base, some techniques can be employed to significantly

reduce the number of rules activated at each sampled time by using the system position in the state

space [2,10,14]. Indeed, Boukezzoula et al. [2] have demonstrated that using a strict triangularpartitioning allows guaranteeing that, at each sampling time, each input variable is described with

two linguistic terms at the most. Thus, the output generated by the fuzzy system withn inputs is

then reduced to that produced by the subsystem composed of the 2n fired rules.


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3.1. Stability analysis

The theorem of Wong et al. [20] will be used to prove the global stability of the system

governed by the control law(19).Using SMC and Hcontrol, this theorem can be rewritten as

follows:

Theorem 1. Consider a combined fuzzy logic control system as described in this work. If

(1) there exist a positive definite, continuously differentiable, and radially unbounded scalar

function V ,

(2) every fuzzy subsystem gives a negative definite V in the active region of the corresponding

fuzzy rule,

(3) the weighted sum defuzzification method is used, such that for any output u we have

min(uSMC, u) u max(uSMC, u),

(4) then the resulting control u, given by (19),guarantees the global stability of the closed loopsystem.

Satisfying the two first conditions guarantees the existence of a Lyapunov function in the

active region which is a sufficient condition for ensuring the asymptotic stability of the system

during the transition from the sliding mode control to the H one.

So, let us consider the following Lyapunov function:

V =1

2eT Pe

where P is a solution of the Riccati equation.Section2.3has shown that the synthesized H control ensures the decrease of the Lyapunov

functionV.

From Eq.(7)in the sliding mode case, we have

1

2

n1i=1

2i [e(i1)]2 +

1

2[en1]2

1

2sTs.

Since P is a positive definite matrix, we have

V = 12

eT Pe 12

maxeTe

wheremax is the maximal eigenvalue ofP .

Thus to satisfy the second condition of the theorem, it is enough to choosei such that

max min[(2i)1in1, 1]. (20)

This condition guarantees that in the neighbourhood of the steady state (Hcontrol), the value

of the Lyapunov function(7)is greater than that of(11)which corresponds to the control acting

in the transient state.

To guarantee min(uSMC, uinf) u max(uSMC, uinf)(third condition), the balancing term takes its values in the interval [0 1].

Consequently, the three conditions of the above theorem are satisfied, and both the global

stability of the system and the error convergence towards zero are guaranteed.


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Fig. 3. The inverted pendulum system.

Remark. We can note that the proof of stability in our case is similar to that used for switching

system theory [5]. Indeed the energy of the system corresponding to the Hcontroller is less

than that for SMC, guaranteeing the stability of the closed loop system during the transition from

SMC to H.

In the event of large external disturbance, leading the system back to a transient mode, theproposed controller adjusts the weighting factor such that the system remains stable in this new

configuration until returning to the steady state, which implies a new variation of the control

signal. This will be illustrated in the simulation section.

3.2. Design procedure

In order to minimize the on-line computing time of the proposed method and to simplify

its real time implementation, the design procedure implies an off-line processing step, and an

on-line step during control execution. In the off-line step, the gains ki are defined in order to

satisfy the Hurwitz criterion. Then by fixing the sliding term, and using the upper bound of theuncertainties (f(x)) and (g(x)) and the maximal value of the admissible control, the positive

constant D is determined. For ease of computation of the Hcontrol and in order to satisfy the

stability condition(20), we advise simplifying the Riccati equation by choosingr = 2, which

leads to a simple Lyapunov function. Then, the i gains are fixed and the matrix Q chosen to

satisfy condition(20).After computing the solution P , the desired attenuation level is imposed.

The supervisor design is essentially based on the available information of the process under

study. Indeed, when a sufficient amount of information is available, it becomes possible to reduce

the number of inputs and the fuzzy rules.

In order to construct the fuzzy supervisor, we define firstly the fuzzy sets for each input andoutput (the error and its derivatives); then the rule base is elaborated.

For the on-line step, the error vector is computed and then injected in the supervisor to

determine the value of to deduce the signal(6) and(8) and, consequently, to apply the global

control signal.

4. Simulation

A simulation of the inverted pendulum depicted in Fig. 3 will be used to illustrate the proposed

approach.

Let x1 = and x2 = . The dynamic equation of the inverted pendulum is given by

Wang [18]:

x1 = x2


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Fig. 4. The structure of the proposed fuzzy supervisor.

x2 =gsinx1

ml x22cos(x1) sin(x1)

mc +m +

cos(x1)mc +m

u

l( 43 mcos2(x1)

mc+m )

y = x1

where g is the acceleration due to gravity, m c is the mass of the cart, m is the mass of the pole,

l is the half-length of the pole, the force u represents the control signal, and d is the external

disturbance. Let us choosem c =1 kg, m = 0.1 kg andl =0.5 m in the following simulations.

The reference signal is assumed here to be yr(t) = (/30) sin(t), and the system is subject to

external disturbances:d(t)= 0.1. sin(t). For the SMC, letk1

=25, and D =7.To compute the Hcontrol law, we fix T = [2 1]T, and Q =diag(1, 1). Furthermore, for

ease of computation, the equality r = 22 is chosen. To obtain a good attenuation level, is

considered as =0.75.

The fuzzy supervisor is constructed by using three fuzzy sets zero, medium, and large for

the tracking error and its time derivative. The corresponding membership functions are triangular,

as shown in Fig. 4. For the output, five singletons are selected; very large (VL), large (L),

medium (M), small (S), and zero (Z), corresponding to 1, 0.75, 0.5, 0.25, and 0, respectively.

The fuzzy rule base is depicted in Fig. 4. Rules are defined by a table; for example, a rule in the

table can be stated as follows: If the absolute value of the error is medium AND the absolute

value of the error derivative is large, THEN is zero. Note that we started with five fuzzy setsfor each input which leads to 25 fuzzy rules. By using the fact that the rule base obtained is in

symmetric form in the state space, we can reduce it to nine rules by using the absolute value of

the inputs. Hence the computing time is considerably reduced.Simulation results related to the application of the three control laws are given inFigs. 59.

These figures show that SMC and the combined controller provide a fast dynamic response com-

pared to H, and that Hand the combined controller provide a smooth variation of the con-

trol signal without chattering. Hence, the proposed control set-up eliminates the disadvantages

of both Hand SMC, and benefits from their advantages in terms of tracking performance and

the robustness to external perturbations, which is ensured by H control in the steady state.

Thus, we obtain an intermediate dynamics whose advantage is to have a compromise betweenthe settling time and the actuator solicitations. Figs. 57, representing respectively the angular

position, the tracking error, and the phase plane of the inverted pendulum for each control law,

show that the proposed controller ensures a good convergence towards the desired trajectory.


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Fig. 5. The angular position of the inverted pendulum obtained using the three control laws.

Fig. 6. The tracking error obtained using the three control laws.

Figs. 8 and 9 represent the applied efforts (control). The zoom depicted in Fig. 9, which

highlights the short time interval representing the transient state of the system, shows that the

combination of SMC and Hcontrol entails a lower solicitation of the actuators.

To further illustrate the robustness of the closed loop system and the satisfaction of the stability

theorem, we apply a sinusoidal reference signal with time-varying frequency and amplitude. The

chosen desired trajectory forces the system, at each signal variation, to switch from the steady

state to a new transient mode. This hybrid configuration leads the fuzzy supervisor to manage

the combination of the two controllers when the next mode is entered. As given inFig. 10, thesystem output exhibits good tracking performance despite consecutive variations of the reference

signal.Fig. 11presents the corresponding energyV1obtained using sliding mode control and V2obtained using H, and the evolution of the factor which manages the combination such that


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Fig. 7. The phase plane of the inverted pendulum obtained using the three control laws.

Fig. 8. The applied efforts obtained using the proposed method.

the applied control signal forces the system to remain stable and to attain the desired trajectory.

The variations and the abrupt changes of the frequency and the amplitude of the reference signal,

depicted inFig. 11, correspond to the system evolution from the transition to the steady state

mode. In this case the fuzzy supervisor favours H to reach the steady state with a fast dynamic.

In this way the term approaches its upper bound and the energy involved with the sliding mode

controller exceeds that for the Hone. In the approaching phase near the steady state, the

term tends toward zero to favour the Hcontroller. Furthermore,Fig. 11shows that the energyinduced by the sliding mode controller is always exceeding the one induced by H. Hence the

conditions ofTheorem 1are satisfied and the system global stability is guaranteed despite the

configuration changing.


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Fig. 9. The applied efforts obtained using the three control laws (zoom).

Fig. 10. Output tracking.

5. Conclusion

In this work, we have developed a hybrid robust controller for a class of nonlinear and

disturbed systems. The main idea is the use of a fuzzy supervisor to manage efficiently the

action of two controllers based on SMC and H, such that the system remains stable and

robust despite the plant switching from one mode to a new one. Furthermore, this structure

allows us to take advantage of both controllers and to efficiently eliminate their drawbacks.

Simulation results showed the efficiency and the design simplicity of the proposed approach.

Indeed, the SMC provides good performances in the transient state (a fast dynamic response,enlarged stability limits of the system), while the Hcontrol acts mainly in the steady state to

reduce chattering and the effect of the external disturbances. Moreover, for a sinusoidal reference

signal with time-varying frequency and amplitude, the tracking performances are not degraded


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Fig. 11. Evolution of the system energy and the coefficient .

and the system stability is guaranteed. This work can be generalized to multiple controllers, more

than two, managed by the same fuzzy supervisor. Indeed, the structure of the fuzzy supervisor

allows partitioning the state into different substates. An adequate controller can be defined for

each substate to ensure the desired performances. The rule base of the fuzzy supervisor will

be reconstructed so that the premise part defines the subspace and the conclusion part the

corresponding control law. Thus the applied control signal will be a weighted sum of all the

controllers used. Our future work concerns the extension of the proposed approach to uncertain

and multi-input multi-output systems and the use of state observers to overcome the availability

of the state variables for measurement.

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Synthesis of Switching Controllers a Fuzzy Supervisor

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