Semiglobal stabilization of nonlinear systems using fuzzy control and singular perturbation methods

Fuzzy Sets and Systems 129 (2002) 275–294

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Semiglobal stabilization of nonlinear systems using fuzzy control and singular perturbation methods C.I. Siettos, G.V. Bafas ∗ Department of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Street, Zografou Campus, Athens 157 80, Greece Received 1 September 1999; received in revised form 7 February 2001; accepted 23 May 2001

Abstract Singular perturbation methods based on a Lyapunov approach are implemented for the derivation of su0cient conditions for the semiglobal stabilization with output tracking of nonlinear systems having internal dynamics. The key idea of this approach lies in the decomposition of the full system into a subsystem exhibiting fast dynamics and to a subsystem with slow dynamics, resulting in a substantial simpli4cation of the stability analysis. This two-time scale behavior is achieved by implementing a properly designed fuzzy logic controller, the key element of the proposed model-based composite controller. For the construction of the model-based controller, a new identi4cation technique based on both classical and fuzzy tools is proposed. Results are presented for the semiglobal stabilization of a continuous stirred tank reactor, which c 2002 Elsevier Science B.V. All rights reserved. is a process with high nonlinear behavior.
Keywords: Design; Fuzzy control; Internal dynamics; Nonlinear systems; Semiglobal stabilization; Singular perturbation methods

1. Introduction The main advantage of fuzzy logic techniques over more conventional approaches in solving complex, nonlinear and=or ill-de4ned control problems lies in their inherent capability to incorporate a priori qualitative knowledge and expertise about system behavior and dynamics. This renders fuzzy logic systems almost indispensable for obtaining a more transparent and tactile qualitative insight for systems whose representation with exact mathematical models is a very di0cult or even impossible task. Besides, fuzzy schemes can be used either as self-reliant control solutions or as complementary elements to classical linear or nonlinear model based or black box control structures o;ering therefore a variety of alternatives. Although in the last 20 years, striking results have been obtained by using various fuzzy design methods and despite the fact that in ∗

Corresponding author. Tel.: +30-1-772-3242; fax: +30-1-772-3155. E-mail addresses: [email protected] (C.I. Siettos), [email protected] (G.V. Bafas).

c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 1 ) 0 0 1 3 6 - 1

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many cases the fuzzy control systems outperform other more traditional equivalent approaches, their extensive applicability is limited due to the de4ciency of formal and systematic design techniques which can ful4ll the two essential requirements of a control system: the requirement for robust stability and that of satisfactory performance. Thus, the problem of designing reliable fuzzy control systems in terms of stability and performance has found a remarkable resonance among engineers and scientists. So far, various approaches to this problem have been presented. One of the 4rst contributions to this topic was that of Braae and Rutherford in [2], where they utilized the phase plane method for analyzing the stability of a fuzzy system. But this method is a strictly numerical method and its applicability is limited for systems of order less than two. In [14], Kickert and Mamdani proposed the use of describing functions for the stability analysis of unforced fuzzy control systems. However, this is only an approximate method, while it is restricted to systems with known dynamics. In [18], the authors introduced the notion of the energy of fuzzy relations to investigate the local stability of a free fuzzy dynamic system. However, the need for fuzzy relation matrices and the need for calculation of the characteristic function of energy makes the particular method inadequate for the analysis of real-world plants. As motivated by the work of Tanaka and Sugeno in [35], many schemes have been proposed for analyzing the stability of fuzzy systems [8,17,21,16,37,39]. The main idea behind this approach lies in the decomposition of a global fuzzy model into simpler linear fuzzy models, which locally represent the dynamics of the whole system. But this approach is limited due to the need for systems representation by Takagi–Sugeno fuzzy models, which give only an approximation to the real systems and therefore may lead to wrong stability results. In [15,23] the authors proposed numerical methods for the stability analysis for fuzzy controllers in the sense of Lyapunov’s direct method. These methods give only a quantitative overview of the stability behavior, but no analytical proof. In [9,13] the authors proposed the stability analysis of fuzzy systems using Popov–Lyapunov techniques. However, the Popov approach is restricted to systems whose control processes are known with a degree of uncertainty. In recent years, the problem of designing stable robust and adaptive fuzzy controllers with satisfactory performance based on the sliding mode approach has attracted much attention [3,4,6,25,36,40,42,43] to name but a few. The proposed schemes take advantage of both sliding and fuzzy features. The design of such schemes is based on Lyapunov’s direct method. However, the applicability of the majority of those techniques is limited, since they concern the design problem for a special class of nonlinear systems with no internal dynamics, i.e., nonlinear systems that can be expressed in the form x(n) (t) = f(x; x; ˙ : : : ; x(n−1) ) + g(x; x; ˙ : : : ; x(n−1) )u;

(1) (n−1)

where f(x) and g(x) are nonlinear continuous functions, x = (x; x; ˙ :::;x ) is the state vector of order n, u is the control input. In this paper, we consider the design problem of semiglobal stable fuzzy control with output tracking for a more general class of nonlinear systems having internal dynamics, i.e., systems possessing a relative degree r¡n. In practical control problems, global stabilization is a very heavy, if not an unattainable requirement; what is really needed is to design a convenient control law such that the system is stabilizable in a compact bounded region around the equilibrium. The design of the proposed fuzzy control scheme is based on singular perturbation methods [19,28,30]. We derive su0cient conditions for establishing the semiglobal asymptotic stabilization of the closed loop system by constructing convenient Lyapunov functions. Loosely speaking the principle idea of this approach lies in the decomposition of the full system of order n, into a subsystem of order r exhibiting fast dynamics and to a subsystem of order n − r with slow dynamics. Based on the two-time scale behavior, we 4rst derive su0cient conditions to stabilize each of the subsystems. The semiglobal stabilization of the full system is established by some additional design constraints yielding a composite nonlinear model-based-fuzzy controller. The generation of fast and slow dynamics is realized by implementing a properly designed fuzzy logic controller that is the essential element of the implemented model-based composite controller. The stabilization of the closed

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loop system is accomplished in the framework of a model reference adaptive scheme to guarantee a near zero error tracking performance. Hence, the ultimate objective is to render the desired output a semiglobal attractor. For the needs of construction of the composite control law, a new identi4cation approach based on a combination of fuzzy and well-established classical techniques is proposed. The approach developed in this paper is motivated by the universal approximating property of fuzzy systems. It is assumed that some fuzzy rule based model describing in a qualitative way the system dynamics is either available or can be constructed using the knowledge of experts in the operational domain of interest. Next, the derived fuzzy model is approximated using polynomials and the identi4cation is carried out in real time using recursive least squares (RLS). The paper is organized as follows: in the next section, the su0cient conditions for semiglobal stabilization of nonlinear fuzzy control systems using singular perturbation methods are derived. Also, the structure and the properties of the relevant fuzzy controller are given. Section 3 describes the proposed fuzzy model identi4cation technique. Finally, in Section 4 the fuzzy control scheme is used to control an unstable high-nonlinear chemical process. Since the model identi4cation part comprises a crucial element in the e;ectiveness of the proposed fuzzy control scheme, the approximation e0ciency of the proposed fuzzy based identi4cation method is compared with respect to other more conventional techniques.

2. Design of stable fuzzy control systems Generally speaking, a nonlinear time-continuous system can be represented in state space form as ˙ = fg (x; u; d; t); x(t)

(2a)

y(t) = h(x; t);

(2b)

where the vector x ∈  n denotes the state variables, u ∈  m is the vector of control inputs, d ∈  p is the vector of disturbances, the vector y ∈  q denotes the system outputs and fg :  n →  n , h :  n →  m are smooth vector 4elds. One of the fundamental objectives in the design of control systems is to 4nd feedback control laws ˙ = fg (x; u(x; t); d; t) is stable in a well de4ned u = u(x; d; t), such that the closed loop system given by x(t) compact region around some equilibrium position x∗ , i.e., semiglobal stable. In this paper, we consider the problem of semiglobal stabilization of the class of nonlinear systems having relative degree r in a compact region around the equilibrium point ˙ = fg (x; d) + gg (x; d)u; x(t)

(3a)

y(t) = h(x);

(3b)

where x ∈  n ; (y; u) ∈ ; d ∈  p and f; g :  n →  n , h :  n → ; by de4nition the relative degree r of a system is the number of times that the output has to be di;erentiated to obtain an explicit relation between y and u. Di;erentiating y with respect to time for r times, system (3) takes the form z˙1 = z2 ; z˙2 = z3 ; .. .

z˙r−1 = zr ; z˙r = f(z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d) + g(z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d)u;

(4a)

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z˙r+1 = q1 (z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d); .. .

(4b)

y = h(z1 ; z2 ; : : : ; zr );

(4c)

z˙n = qn−r (z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d);

where [z1 ; z2 ; : : : ; zr−1 ; zr ; zr+1 ; : : : ; zn ] = [y(x); y(x); ˙ : : : ; y(r−1) (x); y(r) (x); 1 (x); : : : ; n−r (x)], y(k) (t) = Lkf h(x); r−1 (r) r k ∀k¡r; y (t) = Lf h(x) + Lg Lf h(x)u and Lf h(x) = Lf (Lfk−1 h(x)) =
(Lfk−1 h(x))f, k = 1; 2; : : : ; r − 1 and Lg Lf h(x) =
(Lf h(x))g. Lg h =
h(x)g(x; d) and Lf h =
h(x)f(x; d) denote the Lie derivatives of h with respect to g and f, respectively. To simplify the notation de4ne z = (z1 ; z2 ; : : : ; zr−1 ; zr ; zr+1 : : : ; zn ),  = (z1 ; z2 ; : : : ; zr−1 ; zr ), W = (zr+1 ; zr+2 ; : : : ; zn ). The vector W is an n − r vector de4ning the internal dynamics (4b) of the system which simply represents the uncontrollable part of the system, while subsystem (4a) describes the controllable part of the system. In fact, many practical control problems 4t to this class of systems. Assuming (a) that the zero dynamics, which are the internal dynamics setting V = 0 in (4b) are globally asymptotically stable, (b) each of the scalar vectors qi in (4b) depends only on z1 and applying the control law given by u=

1 Lg Lfr−1 h(x)

(−Lrf h + );

(5)

it has been proven [12,38], that there exists a smooth static feedback law  = (V; W);

(6)

which globally stabilizes the equilibrium z = 0 of the closed loop system. However, implementing the composite control law (5) requires (a) a perfect model of the system under control and (b) the capability of real-time explicit measurement of the system disturbances. Yet, in real-world problems we can only derive an approximate model of the system (the explicit use of the functions f(x; d) and g(x; d) in the composite control law (5) is unattainable since in practice they are more or less unknown), while in most of the cases the explicit measurement of the disturbances in real time is very di0cult or even impossible. Furthermore, 4nding a stabilizing control law (6) for systems with relative degree greater than one turns to be a rather complex task and the requirement of the qi ’s dependence on only z1 is very restrictive. The aim of this paper is twofold: to 4nd a fuzzy control law that guarantees the semiglobal stability of (4), overcoming the limitation of the qi ’s dependence on only one component of the state vector, (a) in the face of modeling uncertainties and (b) in the presence of structural-unobservable disturbances d, while driving the system output on the desired trajectory. The key idea behind the proposed design methodology is the decomposition of the stabilization problem of the full system (4) into two sub-problems of reduced order; one of order r and one of order n − r. The objective is to formulate simple su0cient conditions for stabilizing the full system based on Lyapunov’s direct method using the tools of singular perturbation techniques. The decomposition of the full system into a fast and a slow subsystem is achieved by implementing a properly designed fuzzy logic controller. 2.1. Semiglobal stabilization of two-dimensional nonlinear systems To begin with, consider the two-dimensional nonlinear system with r = 1: z˙1 = f(z1 ; z2 ) + g(z1 ; z2 )u;

(7a)

z˙2 = q1 (z1 ; z2 );

(7b)

y = z1 = h(z1 ):

(7c)

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The purpose is to 4nd a control law that semiglobally stabilizes the equilibrium of the above system. Assuming uncertainty on the dynamics f and g, the proposed feedback law, which is based on their corresponding approximations, fˆ and gˆ is chosen as u=

1 ˆ 1 ; z2 ) + ] [−f(z g(z ˆ 1 ; z2 )

(8)

with  = K(z1 ):

(9)

The nonlinear function (z1 ) denotes a fuzzy input–output mapping and K is the output scaling factor. Substituting (8), (9) in (7), the system is rewritten as z˙1 =  Of(z1 ; z2 ) + Dg(z1 ; z2 )(z1 );

(10a)

z˙2 = q1 (z1 ; z2 );

(10b)

ˆ 1 ; z2 )g(z1 ; z2 )= g(z where Of(z1 ; z2 ) = f(z1 ; z2 ) − f(z ˆ 1 ; z2 ), Dg(z1 ; z2 ) = g(z1 ; z2 )= g(z ˆ 1 ; z2 ) and  = 1=K. For  su0ciently small, (10) can be considered as a singular perturbed system with fast (10a) and slow (10b) dynamics. Setting  = 0 in (10a), we get Dg(z1 ; z2 )(z1 ) = 0 ⇒ (z1 ) = 0 ⇒ z1 = 0; from which the following reduced model is obtained z˙2 = q1 (0; z2 ):

(11)

Eq. (11) presents the zero dynamics of (10). Let us choose a Lyapunov candidate for the full system (10) as (z1 ; z2 ) = aV (z2 ) + (1 − a)W (z1 ):

(12)

Di;erentiating (12) with respect to time we obtain ˙ 1 ; z2 ) = aV˙ (z2 ) + (1 − a)W˙ (z1 ) = a @V z˙2 + (1 − a) @W z˙1 (z @z2 @z1 =a

@V @W q1 (z1 ; z2 ) + (1 − a) [Of(z1 ; z2 ) + KDg(z1 ; z2 )(z1 )] @z2 @z1

=a

@W @V q1 (0; z2 ) + (1 − a) [Of(z1 ; z2 ) + KDg(z1 ; z2 )(z1 )] @z2 @z1

+a

@V [q1 (z1 ; z2 ) − q1 (0; z2 )]: @z2

(13)

The term (@V=@z2 )[q1 (z1 ; z2 ) − q1 (0; z2 )] in (13) can be viewed as a perturbation associated with the zero dynamics of the full system. For the zero dynamics, the following assumption is made. Assumption 1. For a compact bounded set B ⊆  around the equilibrium of the system (10); there exist a Lyapunov function V that satis4es the inequality @V q1 (0; z2 ) 6 −c1 !2 (z2 ); @z2

c1 ¿ 0; ∀z2 ∈ B :

(14)

Turn now to the fast subsystem (10a). Choosing the Lyapunov candidate W (z1 ) = 12 z12 ;

z1 ∈ Bz ; Bz ⊆ 

(15)

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C.I. Siettos, G.V. Bafas / Fuzzy Sets and Systems 129 (2002) 275–294 Table 1 Fuzzy control rule base z1 u

nb pb

nm pm

ns ps

ze ze

ps ns

pm nm

pb nb

Fig. 1. The membership functions for (a) the input and (b) output variables.

and di;erentiating with respect to time, Eq. (15) yields @W W˙ (z1 ) = [Of(z1 ; z2 ) + K Dg(z1 ; z2 )(z1 )] = z1 Of(z1 ; z2 ) + z1 K Dg(z1 ; z2 )(z1 ) @z1 6 |z1 ||Of(z1 ; z2 )| + K Dg(z1 ; z2 )z1 (z1 ) 6 |z1 ||Of(z1 ; z2 )|max + K Dg(z1 ; z2 )z1 (z1 ):

(16)

Assumption 2. In view of (16), there is a fuzzy control law given by (9) that yields |z1 ||Of(z1 ; z2 )|max + K Dg(z1 ; z2 )z1 (z1 ) 6 −c2

2

(z1 );

c2 ¿ 0 and (z1 ; z2 ) ∈ Bz × B :

(17)

For Dg(z1 ; z2 )¿0 and z1 = 0 inequality (17) implies K ¿−

c2

2

(z1 ) + |z1 ||Of(z1 ; z2 )|max : Dg(z1 ; z2 )|min z1 (z1 )

(18)

For Assumption 2 to hold, the fuzzy control action (z1 ) should satisfy the design requirement z1 (z1 ) ¡ 0;

z1 = 0; (0) = 0; ∀z1 ∈ Bz :

(19)

In view of (19) the fuzzy control rules are developed as in Table 1. These rules are expressed in an “If–Then” format. For instance the 1st fuzzy rule may be written as If z1 is nb Then  is pb:

(20)

The associated membership functions for the state variable z1 and the fuzzy control action  are given in Fig. 1a and b, respectively. The membership function with the abbreviation nb is used to de4ne the fuzzy set “negative big”, the term nm is used for “negative medium”, ns for “negative small”, ze for “zero”, ps for “positive small”, pm for “positive medium” and pb for “positive big”. At this point it should be noted that simple analytical functions, that satisfy the design requirement (19), such as (z1 ) = −z1 or (z1 ) = −sgn(z1 ) could be used as a control law. However, (a) using a relay feedback control law such as (z1 ) = −sgn(z1 ) has the drawback of leading the system to the chattering e;ect. In general, this is highly undesirable, since the control action changes rapidly and

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discontinuously near the desired trajectory and therefore may excite high frequency unmodeled dynamics, while (b) using a simple continuous function such as (z1 ) = −z1 has the drawback of resulting in unattainable high control actions, for example, in the face of drastic changes of the desired operating points and/or large-magnitude disturbances. This may lead to the saturation, or even worse to the failure, of the actuators. On the other hand, fuzzy logic o;ers a very facile and transparent way of designing robust controllers utilizing expert knowledge based on heuristics. Fuzzy controllers can be easily so designed as to prevent undesirable high control actions caused by large changes and=or measurement noise, while ensuring that the design requirements are ful4lled in a bounded region around the equilibrium. Furthermore, the proposed fuzzy control action varies continuously with the distance of actual state from the desired trajectory smoothing therefore the control behavior. Assumption 3. For the perturbation term on the right-hand size of (13) ∃c3 ∈ R such that @V [q1 (z1 ; z2 ) − q1 (0; z2 )] 6 c3 (z1 )!(z2 ); @z2

∀(z1 ; z2 ) ∈ Bz × B :

(21)

If Assumptions 1–3 are satis4ed, then (13) becomes ˙ 1 ; z2 ) 6 −ac1 !2 (z2 ) + ac3 (z1 )!(z2 ) − (1 − a)c2 2 (z1 ) (z 

!(z2 ) −0:5ac3 ac1 = −[!(z2 ) (z1 )] : −0:5ac3 (1 − a)c2 (z1 )

(22)

For (22) to be negative de4nite it su0ces 1 ac1 (1 − a)c2 − a2 c32 ¿ 0; 4

or

c2 ¿

a2 c32 : 4ac1 (1 − a)

(23)

Substituting (23) into (18), one 4nally obtains K ¿−

(a2 c32 )=(4ac1 (1 − a)) 2 (z1 ) + |z1 ||Of(z1 ; z2 )|max : Dg(z1 ; z2 )|min z1 (z1 )

(24)

Inequality (24) provides a su0cient design constrain for the semiglobal stabilization of the closed loop system (10). 2.2. Semiglobal stabilization of multi-dimensional nonlinear systems Referring to the n-dimensional system (4), we de4ne a dummy output as &(z1 ; z2 ; : : : ; zr ) = Lfr−1 h(z1 ; z2 ; : : : ; zr ) + kr−2 Lfr−2 h(z1 ; z2 ; : : : ; zr ) + · · · + k0 h(z1 ; z2 ; : : : ; zr ) = zr + kr−2 zr−1 + · · · + k1 z2 + k0 z1 :

(25)

Substituting the composite control law u=

1 ˆ 1 ; z2 ; : : : ; zn ; d) + K[&(z1 ; z2 ; : : : ; zr )]} {−f(z g(z ˆ 1 ; z2 ; : : : ; zn ; d)

(26)

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in (4), one obtains the singular perturbed system z˙1 = z2 ; z˙2 = z3 ; .. . z˙r−1 = zr ;

(27a)

z˙r =  Of(z1 ; z2 ; : : : ; zn ; d) + Dg(z1 ; z2 ; : : : ; zn ; d)K[&(z1 ; z2 ; : : : ; zr )]; z˙r+1 = q1 (z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d); .. . z˙n = qn−r (z1 ; z2 ; : : : ; zr ; zr+1 ; : : : ; zn ; d); &(z1 ; z2 ; : : : ; zr ) = zr + kr−2 zr−1 + · · · + k1 z2 + k0 z1 :

(27b)

Setting  = 0 in (27a), we get Dg(z1 ; z2 ; : : : ; zn ; d)K[&(z1 ; z2 ; : : : ; zr )] = 0 ⇒ [&(z1 ; z2 ; : : : ; zr )] = 0 ⇒ &(z1 ; z2 ; : : : ; zr ) = 0:

(28)

From (25) using (28), one obtains zr |&(V)=0 = −kr−2 zr−1 − · · · − k1 z2 − k0 z1 :

(29)

Let us denote the state vector, which corresponds to the zero dynamics as z|&(V)=0 = (z1 ; z2 ; : : : ; zr−1 ; −kr−2 zr − · · · − k1 z2 − k0 z1 ; zr+1 : : : ; zn ):

(30)

We now choose a Lyapunov function candidate for the full system (27) as (z) = aV (W) + (1 − a)W (V):

(31)

Di;erentiating with respect to time, Eq. (31) yields T ˙ (z) = a
V (W)[q1 (z|&(V)=0 ); : : : ; qn−r (z|&(V)=0 )]T + (1 − a)
W (V)V˙

+a
V (W){[q1 (z); : : : ; qn−r (z)]T − [q1 (z|&(V)=0 ); : : : ; qn−r (z|&(V)=0 )]T }:

(32)

Working with the same procedural reasoning as in the case of two-dimensional systems, suppose that the following hold: Assumption I. For a compact bounded set B ⊆ Rn−r around the equilibrium of system (27), there exist a Lyapunov function V that satis4es the inequality
V (W)[q1 (z|&(V)=0 ); : : : ; qn−r (z|&(V)=0 )]T 6 −c1 !2 (W):

(33)

Choosing the Lyapunov candidate as W (V) = 12 &2 (V)

(34)

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283

for subsystem (27a) and di;erentiating with respect to time, Eq. (34) reads ˙ = &(V)(z˙r + kr−2 z˙r−1 + · · · + k1 z˙2 + k0 z˙1 ) W˙ (V) = &(V)&(V) = &(V)z˙r + &(V)(kr−2 zr + · · · + k1 z3 + k0 z2 ) = &(V)Of(z; d) + &(V)Dg(z; d)K[&(V)] + &(V)(kr−2 zr + · · · + k1 z3 + k0 z2 ) = &(V)[Of(z; d) + (kr−2 zr + · · · + k1 z3 + k0 z2 )] + Dg(z; d)K[&(V)]:

(35)

Referring to the maximum errors of the approximations, the following inequality holds W˙ (V) 6 |&(V)|{|Of(z; d)|max + |(kr−2 zr + · · · + k1 z3 + k0 z2 )|} + Dg(z; d)K&(V)[&(V)]:

(36)

Assumption II. For a compact set Bz ⊆ Rr there exist a fuzzy control law such that the Lyapunov function W (V) satis4es the inequality W˙ (V) 6 −c2

2

(V);

∀V ∈ Bz ; c2 ¿ 0:

(37)

Comparing (37) with (36) one 4nally obtains K ¿−

ac2

2

(V) + |&(V)|{|Of(z; d)| + |(kr−2 zr + · · · + k1 z3 + k0 z2 )|} : Dg(z; d)(&(V)[&(V)]

(38)

For Assumption II to hold, the fuzzy control action should satisfy the following design speci4cation &(V)[&(V)] ¡ 0;

&(V) = 0 and (0) = 0; ∀V ∈ Bz

(39)

while it is assumed that the model approximation is such that Dg(z; d)¿0: The fuzzy controller is designed as in the two-dimensional case, using the rule base given in Table 1, where now the new output &(V) replaces the state variable z1 . The corresponding membership functions for both input and output variable are given in Fig. 1 b, respectively. The next steps in the design of the fuzzy controller refer to the selection of the reasoning and the defuzzi4cation method. In this paper, we take the max-product fuzzy inference method and the center of area defuzzi4cation method [7,44]. Assumption III. For the perturbation term on the right-hand size of (32), ∃c3 ∈ R such that ∀(W; V) ∈ B × Bz ;
V (W){[q1 (z); : : : ; qn−r (z)]T − [q1 (z|&(V)=0 ); : : : ; qn−r (z|&(V)=0 )]T } 6 c3 !(W) (V):

(40)

To this end, the following theorem holds. Theorem. Let Asumptions I–III hold and let the function V () be such that (i) V ()¿0; ∀W ∈ B ; (ii) V (W) → ∞ as W → ∞. Suppose further that the roots si of the polynomial p(s) = sr−1 + kr−2 sr−2 + · · · + k1 s + k0

(41)

satisfy the condition Re[(i ]¡0; i = 1; 2; : : : ; r − 1; i.e.; all the roots of (41) have negative real parts. Then the equilibrium of the closed loop system (27) is globally stable in the region B × Bz for all the values K

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of the ouput-scaling factor of the fuzzy controller satisfying the inequality K ¿−

((a2 c32 )=(4ac1 (1 − a)))

2

(V) + |&(V)|{|Of(z; d)|max + |kr−2 zr + · · · + k1 z3 + k0 z2 )|} ; Dg(z; d)|min &(V)[&(V)]

V = 0: (42)

Proof. If Asumptions I–III hold, then using inequalities (33), (37) and (40), Eq. (32) yields ˙ (z) 6 −ac1 !2 (W) − (1 − a)c2 2 (V) + ac3 (V)!(W)
 ac1 −0:5ac3 = −[ !(W) (V) ] [ !(W) −0:5ac3 (1 − a)c2 If c2 ¿

a2 c32 ; 4ac1 (1 − a)

(V) ]T :

˙ then from (43) (z) ¡ 0; z = 0:

(43) (44)

Hence, substituting (44) into (38) it is shown that there exist a critical positive real number K ∗ given by K∗ = −

((a2 c32 )=(4ac1 (1 − a)))

2

(V) + |&(V)|{|Of(z; d)| + |(kr−2 zr + · · · + k1 z3 + k0 z2 )|} Dg(z; d)&(V)[&(V)]

(45)

such that for K¿K ∗ the output of system (27) is guaranteed to be stable in the compact set B × Bz . This implies that limt→∞ [&(V)] = 0. In view of (25), we obtain ˙ + k0 y(t) = 0: y(r−1) (t) + kr−2 y(r−2) (t) + · · · + k1 y(t)

(46)

Taking the Laplace transform of (46) and selecting the coe0cients ki ; i = 0; 1; : : : ; r − 2 such that all the roots of polynomial (41) to possess negative real parts we obtain that the state vector will asymptotically move along (46) to the origin. 3. Nonlinear system identi!cation using truncated Chebyshev series approximation of fuzzy models Model identi4cation comprises a fundamental issue in the design of high performance control systems, since most of them are making an explicit or implicit use of process models. Model identi4cation techniques are well developed for linear systems. However, the increasing demand for more e;ective control structures has rendered the identi4cation of nonlinear models a necessity. But, the performance and the applicability of both classical and fuzzy model-based nonlinear and adaptive control schemes strongly depend on the model adequacy. In the literature, several approaches to the topic of nonlinear model identi4cation have been presented including 4rst-principles modeling, nonlinear time series [11,22], several nonlinear techniques [10,41], neural networks [5,24], fuzzy systems [20,26,32 – 34]. The 4rst-principle approach refers to the construction of a perfect model of the system based entirely on physical insight. However, in the case of real-world systems the derivation of exact mathematical models and=or subsequent analysis are most of the times very di0cult or even impossible tasks. One of relevant advantages of fuzzy systems when compared to other more quantitative in nature nonlinear model identi4cation approaches lies in their inherent capability of incorporating a priori qualitative knowledge and expertise about the system behavior and dynamics in the form of a rule base. However, the lack of mathematical models, which characterize fuzzy systems, limit often their applicability, making various vital tasks such as stability analysis and nonlinear system identi4cation formidable. For that purpose, it is often enviable, in virtue of high speed, reduced computational storage and system analysis using

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285

Table 2 Fuzzy dynamical model u[(k − 1)T ] y[(k − 1)T ]

nb nm ns ze ps pm pb

nb nb nb nb nb nm nm nm

nm nb nb nm nm ns ns ns

ns nm nm nm ns ze ze ze

ze ns ns ns ze ps ps ps

ps ze ze ze ps pm pm pm

pm ps ps ps pm pm pb pb

pb pm pm pm pb pb pb pb

well established methods from classical linear and nonlinear control theory to have analytical expressions for the applied fuzzy systems. Recall that for the construction of the model-based control law (26) an identi4cation technique is needed for the approximation of the unknown functions f and g. At this point a new systematic methodology for nonlinear system identi4cation is introduced. The method facilitates the development of nonlinear analytical models with the aid of fuzzy logic and truncated Chebyshev series [31]. The essence of the proposed method is that it makes possible the construction of analytical models, which contain only few polynomial terms, capable of representing both qualitatively and quantitatively the appropriate process dynamics by incorporating a priori qualitative knowledge given in a fuzzy rule base form. In the rest of this section, it is shown in a concise manner, how the truncated Chebyshev series approximation of fuzzy models can be used for the identi4cation of nonlinear systems. To begin with, suppose, that the dynamical behavior of a process can be qualitatively described by some fuzzy dynamical model which in turn may be viewed in the form of an “If–Then” rule base. Each fuzzy rule can be expressed using the following notation Ri : If u(t − 1) is positive and y(t − 1) is positive Then y(t) is positive big:

(47)

In Table 2, an example of such a rule base using the fuzzy sets negative big (nb), negative medium (nm), negative small (ns), zero (ze), positive small (ps), positive medium (pm) and positive big (pb) for the assignment of each variable is outlined; u(t − 1) is the control action at time t − 1, y(t − 1) is the system output at time t − 1 and y(t) is the system output at the current instant time. It is important to notice that in the general case the fuzzy dynamical model does not approximate in a precise, quantitative way the process dynamics. It only gives, a rough qualitative description of the system behavior within the operating region of interest, based on observation. With the fuzzy dynamical model at hand, the proposed identi4cation algorithm can be stated as follows. Step 1: Set a maximum order m for the Chebyshev polynomials. Use the roots of the mth order Chebyshev polynomial as input values to the fuzzy dynamical system to numerically calculate the input–output mapping in the normalized interval [−1; 1]; using common fuzzi4cation, inference and defuzzi4cation techniques. The value of the maximum order m of the Chebyshev polynomials is chosen so as to yield adequate approximation accuracy in terms of the maximum deviation from the fuzzy model. Step 2: Use least squares method (LS) to approximate the input–output mapping by the Chebyshev polynomials. The LS problem refers to 4nding an estimate of the vector X of the coe0cients of the Chebyshev polynomials that minimize the quantity S(X) =

N  -=1

|yi − CP(xi ; X)|;

(48)

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where yi denotes the output value of the fuzzy model as it is calculated from the ith pair of values of the input variables xi which in the general case is a vector of n past system outputs and m past inputs: T (t − 1) = [−y(t − 1); : : : ; −y(t − n); u(t − 1); : : : ; u(t − m)]:

(49)

Let us denote as X∗ the values of the coe0cients that minimize S(X). Thus, the polynomial CP(xi ; X∗ ) comprises an analytical mathematical approximation to the fuzzy model. Step 3: Perform an analysis of variance to select the polynomial terms that mostly contribute to the variation in the data. Rearrange the polynomial CP(xi ; X∗ ) with the selected terms. Do While (t¿t0 ) { Step 4: Use the reduced polynomial model utilizing well established identi4cation algorithms such as recursive least squares to perform on-line and in real-time estimation of the polynomial coe0cients X. The RLS formulation, which can be regarded as a one step prediction technique, can be written brieSy as [22] X(t) = X(t − 1) + .(t − 1)[y(t) − T (t)X(t − 1)]; .(t − 1) =

(50)

1 P(t − 1)T (t); T (t)P(t − 1)(t)

P(t) = [I − .(t − 1)T (t)]P(t − 1);

(51)

where y(t) is the system output at the time t, T (t) is the nonlinear regression vector whose elements are the terms of the polynomial CP(xi ; X), P(t0 ) = (T (t0 )(t0 )) is a positive de4nite matrix; the real time identi4cation algorithm is used for t¿t0 . The time instant t0 is chosen such that the matrix P(t0 ) becomes nonsingular for the 4rst time. Step 5: Estimate the nonlinear function fg as follows: Using a single step integration method with time step T , the solution of Eq. (2) is obtained by solving  y[(k + 1)T ] = h[x(kT ) +

(k+1)T (kT )

fg (x; u; d; /) d/]:

(52)

It is assumed that the function fg remains unchanged in the interval [kT; (k + 1)T ]. Implementing a forward rectangular technique for the approximation of the integral, Eq. (52) reads y[(k + 1)T ) = h[x(kT ) + fg (x; u; d)T ]:

(53)

By direct comparison of (53) and the derived prediction model, y(t + 1) = y[(k + 1)T ] = CP(T (t); X) and considering a linear invertible output function h, one obtains,   T ˆf g (x; u; d) = h−1 CP( (t); X) − y(kT ) ; T where fˆg (x; u; d) is the approximation of fg (x; u; d). End While }

(54)

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287

Fig. 2. The continuous stirred tank reactor.

4. A simulation example: semiglobal stabilization with ouput tracking of a nonlinear process 4.1. Process model and bifurcation analysis The proposed methodology presented in Section 2 will be used for the control of a continuous stirred tank reactor (CSTR) (Fig. 2), where an exothermic irreversible reaction with 4rst order kinetics takes place. The dynamic behavior of the process of interest is represented by the following dimensionless set of nonlinear ordinary di;erential equations resulting from mass and energy balances into the reactor: x(t) ˙ = (1 − x − xs ) − De−.=(y+ys ) (x + xs );

(55a)

y(t) ˙ = (1 − y − ys ) + DBe−.=(y+ys ) (x + xs ) − 0(y + ys ) + 0us + 0u;

(55b)

where x, y are, respectively, the dimensionless concentration and temperature inside the reactor, xs ; ys are the dimensionless nominal values of concentration and temperature, D is a Damkohler number, B is the dimensionless heat of reaction, . is the dimensionless activation energy, 0 is the dimensionless heat transfer coe0cient, u s is the dimensionless nominal coolant temperature, u is the dimensionless coolant temperature (manipulated variable). Simulation results were obtained for the following values of the variables: xs = 0:8255, ys = 0:9415; 0 = 2:3244, B = 209:205, D = 7:21010 , . = 25, u s = 0:8714. Fig. 3 illustrates graphically the steady state dependence of y versus 0. As it is shown the steady state behavior of the system exhibits some very interesting nonlinear phenomena. We see a branching diagram containing both stationary stable, thus physically observable (branches a–b, d–e) and unstable, thus physically unobservable (branches b–c, c–d) solutions. At 0 = 2:0559 there is a Hopf bifurcation revealing an exchange of stability from stable motionless equilibria to stable equilibria with periodic motion; for parameter values in the interval 2:0599¡0¡2:3604 we have a soft generation of limit cycles, that is the amplitude of oscillation grows continuously for increasing values of 0. Fig. 4 shows clearly on the (y; x) phase plane a limit cycle for 0 = 2:18, a value in the middle of the periodic branch. Upon increasing 0 beyond c there is a discontinuous jump from high temperatures and large values of amplitude to zero amplitude and low temperatures. That is, the system jumps to the lower branch d–e, where an exchange of stability from limit cycles to stationary equilibria occurs. For every value of 0 in

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Fig. 3. Steady state equilibrium solutions (y; 0).

Fig. 4. Limit cycle: (y; x) phase plane projection for 0 = 2:18.

the interval between the two turning points c, d a stable, thus observable equilibrium and an unstable, thus unobservable equilibrium coexist. Stability results are obtained by monitoring the eigenvalues of the Jacobian of the system [29], while the branch tracing is achieved by performing a parameter continuation method [1]. 4.2. Controller design The problem that arises is the control of the reactor temperature at 0 = 2:3244. As it was described above, at this value of 0 there are two steady states: the open loop stable state l where ys = 0:9415, xs = 0:8255 and open loop unstable state m where ym = 1, xm = 0:5. The desired operating point is the open loop unstable equilibrium since the conversion at l is considered very low. Thus the control objective is the semiglobal stabilization of the reactor temperature at m, following a desired trajectory from l to m state; it is assumed that the reactor is subjected to unknown disturbances.

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289

The desired performance of the closed loop system is speci4ed by the following reference model: y˙ d (t) = −yd (t) + (ym − ys ):

(56)

Subtracting (56) from (55b), in relevance to (7) one 4nally obtains the following system of equations: z˙1 = f(z1 ; z2 ) + yd − (ym − ys ) + g(z1 ; z2 )u

(57a)

z˙2 = q1 (z1 ; z2 );

(57b)

y = z1 = h(z1 );

(57c)

where z2 = x, z1 = y − yd , q1 (y; x) = (1 − x − xs ) − Be−.=(y+ys ) x − Be−.=(y+ys ) xs , f(y; x) = (1 − y − ys ) + DBe−.=(y+ys ) x − 0y − 0ys + 0us

and

g(z1 ; z2 ) = 0:

Implementing the design procedure of Section 2, we 4rst choose as a Lyapunov candidate for the full system the function (y; x) = 12 x2 + 12 z12 :

(58)

Then, it can be easily shown that Assumption I holds for c1 = (1 + Be−.=ys ) and !2 (x) = x2 . The perturbation term on the right-hand size of (13) reads @V [q1 (z1 ; x) − q1 (0; x)] = xB(xs + x)(e−.=ys − e−.=(y+ys ) ): @z2

(59)

But, xB(xs + x)(e−.=ys − e−.=(y+ys ) )6c3 |x| |z1 |, ∀x; z1 ∈ [−0:4; 0:4] × [−0:1; 0:1], c3 = 20. Designing the fuzzy logic controller as described in Section 2 with the requirement that its output satis4es the inequality |K(z1 )| ¿

(c32 =4c1 )|z1 | + |Of(z1 ; z2 )|max + |yd − (ym − ys )| ; Dg(z1 ; z2 )|min

(60)

i.e., Assumption 2 holds for 2 (z1 ) = z12 , then according to Theorem 1, the closed loop system (59), is globally stable in the region of interest (x; z1 ) ∈ [−0:4; 0:5] × [−0:1; 0:1]. 4.3. Simulation results for the identi;cation problem The estimates of the unknown functions f, g are obtained by using the methodology of Section 3. It is assumed that the fuzzy dynamical model that describes the system behavior is the same model given in Section 3. The approximation is carried out using Chebyshev polynomials. As motivated from the requirement to obtain the lowest order of the Chebyshev polynomials that give adequate approximation accuracy, we worked out with the following truncated Chebyshev polynomial y(kT ) = c0 + c1 y[(k − 1)T ] + c2 [4y[(k − 1)T ]3 − 4y[(k − 1)T ]] + b1 u[(k − 1)T ]:

(61)

For the implementation of the control law (8) it is desirable to have some approximations of the functions f and g. Therefore, in the 4rst stage the identi4cation of the process was carried out o;-line about the stable ˆ and Fig. 6b shows the function g along state l. Fig. 6a shows the function f along with the estimated f,

290

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Fig. 5. System identi4cation using a square wave input signal.

Fig. 6. Identi4cation of the f(a) and g(b) unknown functions using the fuzzy model approximation.

with the estimated gˆ after 20 000 time steps with a sampling time of T = 0:001 min. The input to the process was the square wave shown in Fig. 5. The selection of the amplitude and the square wave frequency were selected such that the system dynamics are adequately excited, while avoiding undesirable phenomena such as high amplitude oscillations that can take the system signi4cantly away from its operating region and=or

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Table 3 Implemented NARMAX models add[k(T − 1)] M1 M2 M3 M4 M5

0 y[(k y[(k y[(k y[(k

− 2)T ] − 1)T ]y[(k − 2)T ] − 1)T ]2 − 2)T ]2

Table 4 Input sequences for identi4cation

Switching frequency

Quatratic wave

21

22

23

1:4 min

p = 0:01

p = 0:1

p=1

cause instability. As it is shown the estimations are almost identical to the original functions resulting in |Of(z1 ; z2 )|max = 0:2

and

Dg(z1 ; z2 ) =

g(z1 ; z2 ) = 0:95: g(z ˆ 1 ; z2 )

For comparison purposes, the e0ciency of the derived model was compared to other 4ve nonlinear classical models, imposing four di;erent classes of input sequences. The classical models follow the general NARMAX form y(kT ) = c0 + c1 y[(k − 1)T ] + c2 u[(k − 1)T ] + c3 add[(k − 1)T ];

(62)

where the 4ve di;erent modes of the additive term add(kT − 1) are given in Table 3, while Table 4 gives the di;erent input sequences. The input sequences are de4ned as u(t) = u(t − 1) + ‘p (t);  0 with probability p; ‘p (t) = rand(t) with probability (1 − p);

(63) (64)

where rand(t) is a random variable uniform in the interval [−5; 5]. For demonstration purposes, Table 5 compresses the results of the comparative analysis with respect to the maximum error of approximation |Of(z1 ; z2 )|max . As it is shown, the proposed methodology outperforms the other nonlinear models for low switching frequencies. For the switching probability p = 0:1 all types seem to be equivalent, while for high switching frequencies the classical models seem to give better approximations. This may be explained as follows: for low switching frequencies the process has more time to respond to input changes rendering the nonlinear dynamic behavior more clear, while for high switching frequencies the process cannot follow the input changes producing therefore low amplitude outputs; in that case the system behavior can be better approximated by linear models. 4.4. Simulation results for the control problem After an o;-line identi4cation is carried out, both control and identi4cation are implemented. Fig. 7a shows the system response without disturbances and the responses when the reactor is subjected to disturbances in the structural parameter 0 ranging from −10% to −30%. Fig. 7b gives the corresponding manipulated responses.

292

C.I. Siettos, G.V. Bafas / Fuzzy Sets and Systems 129 (2002) 275–294 Table 5 Results of the comparative analysis of the proposed and NARMAX models |Of(z1 ; z2 )|max Model Proposed M1 M2 M3 M4 M5

Input signal Quatratic wave

21

22

23

0.20 3.51 19.70 0.60 0.60 0.61

11.70 81.40 45.0 19.00 18.70 18.71

15.70 102.80 47.80 15.00 15.50 13.30

1.06 0.003 0.008 0.02 0.008 0.02

Fig. 7. System responses: (a) Controlled variable responses for load changes in Â, (b) manipulated variable responses for load changes in Â: (− −) without load changes, (− −) −10%, (− −) −20%, (− −) −30%. (c) Controlled variable responses for load changes in input temperature, (d) manipulated variable responses for load changes in input temperature: (− −) −15%, (− −) −7%, (− −) +7%, (− −) +15%.

Fig. 7c gives the system responses for step disturbances of the inlet temperature ranging from −15% to 15%, while the corresponding responses of the manipulated variable are given in Fig. 7d. As it is shown the output tracking is almost perfect leading to a stable closed loop system. The resulting control scheme has rendered the state m physically stable and thus observable.

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Based on the above results, it is clear to see that the implemented control scheme is very e;ective and results in a stable and almost perfect tracking of the reference model. 5. Conclusions A new method for the semiglobal stabilization of nonlinear systems in the presence of internal dynamics was presented. The method is based on the decomposition of the full system into two separate subsystems exhibiting di;erent time-scale dynamic behavior, resulting in a substantial simpli4cation of the overall stability analysis. This is achieved by the implementation of a composite model-based control scheme, whose key element is a fuzzy controller. Singular perturbation methods have been used to derive su0cient stability conditions for the design of the fuzzy controller. For the implementation of the proposed control scheme, a new fuzzy technique for real-time identi4cation that puts classical and fuzzy tools into a common framework was presented. The main advantage of the proposed fuzzy approach over other classical nonlinear identi4cation techniques is the inherent capability of incorporating a priori qualitative knowledge and expertise about the system behavior and dynamics. The proposed methodology is used for the semiglobal stabilization with temperature tracking of a continuous stirred tank reactor. The case under study is a high nonlinear process, exhibiting limit cycles and open-loop physically unstable operating regions. The obtained results demonstrate the e;ectiveness of the method. References [1] P.J. Abbot, An e0cient algorithm for the determination of certain bifurcation points, J. Comput. Appl. Math. 4 (1978) 19–26. [2] M. Braae, D.A. Rutherford, Selection of parameters for a fuzzy logic controller, Fuzzy Sets and Systems 2 (1979) 185–199. [3] C.L. Chen, M.H. Chang, Optimal design of fuzzy sliding-mode control: a comparative study, Fuzzy Sets and Systems 93 (1998) 37–48. [4] C.S. Chen, W.L. Chen, Analysis and design of a stable fuzzy control system, Fuzzy Sets and Systems 96 (1998) 21–35. [5] S. Chen, S. Billings, Neural networks for nonlinear dynamic system modelling and identi4cation, Internat. J. Control 56 (1992) 319–346. [6] X. Chen, T. Fukuda, Robust adaptive quasi-sliding mode controller for discrete-time systems, Systems Control Lett. 35 (1998) 165–173. [7] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [8] G. Feng, S.G. Cao, N.W. Rees, C.K. Chak, Design of fuzzy control systems with guaranteed stability, Fuzzy Sets and Systems 85 (1997) 1–10. [9] C.C. Fuh, P.C. Tung, Robust stability analysis of fuzzy control systems, Fuzzy Sets and Systems 88 (1997) 289–298. [10] M. Henon, On the numerical computation of PoincarXe maps, Physica D 5 (1982) 412–414. [11] E. Hernadez, Y. Arkun, Control of nonlinear systems using polynomial ARMA models, A.I.Ch.E. J. 39 (1993) 446–460. [12] A. Isidori, Nonlinear Control Systems, 3rd Edition, Springer, Berlin, 1995. [13] A. Kandel, Y. Luo, Y.Q. Zhang, Stability analysis of fuzzy control systems, Fuzzy Sets and Systems 105 (1999) 33–48. [14] W.M. Kickert, E.H. Mamdani, Analysis of a fuzzy logic controller, Fuzzy Sets and Systems 1 (1) (1978) 29–44. [15] H. Kiendl, J.J. Ruger, Stability analysis of fuzzy control systems using facet functions, Fuzzy Sets and Systems 70 (1995) 275–285. [16] W.C. Kim, S.C. Ahn, W.H. Kwon, Stability analysis and stabilization of fuzzy state space models, Fuzzy Sets and Systems 71 (1995) 131–142. [17] K. Kiriakidis, A. Grivas, A. Tzes, Quadratic stability analysis of the Takagi–Sugeno fuzzy model, Fuzzy Sets and Systems 98 (1998) 1–14. [18] J.B. Kiszka, M.M. Gupta, P.N. Nikiforuk, Energetistic stability of fuzzy dynamic systems, IEEE Trans. Systems Man Cybernet. 15 (6) (1985) 783–791. [19] P.V. Kokotovic, R.E. O’Malley, P. Sannuti, Singular perturbation ad order reduction in control theory-an overview, Automatica 12 (1976) 123–132. [20] E.G. Laukoven, K.M. Pasino, Training fuzzy systems to perform estimation and identi4cation, Eng. Appl. Artif. Intell. 8 (5) (1995) 499–514. [21] F.H.F. Leung, H.K. Lam, P.K.S. Tam, Design of fuzzy controllers for uncertain nonlinear systems using stability and robustness analyses, Systems Control Lett. 35 (1998) 237–243.

294

C.I. Siettos, G.V. Bafas / Fuzzy Sets and Systems 129 (2002) 275–294

[22] L. Ljung, System Identi4cation: Theory for the User, Prentice-Hall, Englewood Cli;s, NJ, 1987. [23] K. Michels, Numerical stability analysis for a fuzzy or neural network controller, Fuzzy Sets and Systems 89 (1997) 335–350. [24] K.S. Narendra, K. Parthasarathy, Identi4cation and control of dynamical systems using neural networks, IEEE Trans. Neural Networks 1 (1) (1990) 4–27. [25] R. Palm, Sliding mode fuzzy control, IEEE Internat. Conf. on Fuzzy Systems, San Diego, 1992, pp. 519 –526. [26] M. Park, S. Ji, E. Kim, M. Park, A new approach to the identi4cation of a fuzzy model, Fuzzy Sets and Systems 104 (1999) 169–181. [27] W. Rudin, Principles of Mathematical Analysis, 2nd Edition, McGraw-Hill, New York, 1964. [28] A. Saberi, H. Khalil, Quadratic-type Lyapunov function for singularly perturbed systems, IEEE Trans. Automat. Control 29 (1984) 542–550. [29] D.I. Sagias, E.N. Sara4s, C.I. Siettos, G.V. Bafas, Design of a model identi4cation fuzzy adaptive controller and stability analysis of nonlinear processes, Fuzzy Sets and Systems 121 (2001) 169–179. [30] V.R. Saksena, J. O’Reilly, P.V. Kokotovic, Singular perturbations in control theory—a survey, Automatica 20 (1984) 173–193. [31] C.I. Siettos, G.V. Bafas, A.G. Boudouvis, Truncated Chebyshev series approximation of fuzzy systems for control and nonlinear system identi4cation, Fuzzy Sets and Systems 126 (2002) 89–104. [32] M. Sugeno, G.T. Kang, Structure identi4cation of fuzzy model, Fuzzy Sets and Systems 28 (1988) 15–23. [33] M. Sugeno, T. Yasukawa, A fuzzy logic based approach to qualitative modelling, IEEE Trans. Fuzzy Systems 1 (1) (1993) 7–31. [34] T. Takagi, M. Sugeno, Fuzzy identi4cation of systems and its application to modelling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116–132. [35] K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (1992) 135–156. [36] Y. Tang, N. Zhang, Y. Li, Stable fuzzy adaptive control for a class of nonlinear systems, Fuzzy Sets and Systems 104 (1999) 279–288. [37] M.A. Thathachar, P. Viswanath, On the stability of fuzzy systems, IEEE Trans. Fuzzy Systems 5 (1) (1997) 145–151. [38] J. Tsinias, Su0cient Lyapunov-like conditions for stabilization, Math. Control Signals Systems 2 (1989) 343–357. [39] H.O. Wang, K. Tanaka, M.F. Gri0n, An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE Trans. Fuzzy Systems 4 (1) (1996) 14–23. [40] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cli;s, NJ, 1994. [41] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285–317. [42] S.Y. Yi, M.J. Chung, Systematic design and stability analysis of a fuzzy logic controller, Fuzzy Sets and Systems 72 (1995) 271–298. [43] X. Yu, Z. Man, B. Wu, Design of fuzzy sliding-mode control systems, Fuzzy Sets and Systems 95 (1998) 295–306. [44] H.J. Zimmermann, Fuzzy Set Theory and its Applications, 3rd Edition, Kluwer, Dordrecht, 1996.