Design of a model identification fuzzy adaptive controller and stability analysis of nonlinear processes

Fuzzy Sets and Systems 121 (2001) 169–179

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Design of a model identi"cation fuzzy adaptive controller and stability analysis of nonlinear processes D.I. Sagias, E.N. Sara"s, C.I. Siettos, G.V. Bafas ∗ Department of Chemical Engineering, National Technical University of Athens, GR-15780, Athens, Greece Received 7 September 1998; received in revised form 7 July 1999; accepted 24 July 1999

Abstract This paper deals with the design of a model identi"cation fuzzy adaptive controller with real-time scaling factors adjustment and the stability analysis of nonlinear distributed parameter systems. The solution branch of such systems frequently contains limit points (or turning points) which represent the boundary between stability and instability of the system. Hence, stability analysis is required for the determination of the stable and unstable operating regions. The performance of the proposed fuzzy self-tuning controller is compared to an equivalent conventional adaptive controller, over a wide range of step disturbances and operating regions. The proposed fuzzy adaptive scheme in comparison with the conventional adaptive scheme exhibits a much robust response, shorter settling times, overshooting less the controlled variable and smaller IAE of the manipulated c 2001 Elsevier Science B.V. All rights reserved. variable for the entire range of step disturbances.
Keywords: Nonlinear system; Limit points; Stability analysis; Model identi"cation fuzzy adaptive controller

1. Introduction What is required for the qualitative and quantitative study of many practical engineering systems is the construction of a set of time-dependent nonlinear partial di=erential equations (PDEs). Generally speaking, the state equations resulting from mass, energy and moment balances are of the form @x = L(x; z; p); @t ∗

(1)

Corresponding author. Tel.: + 30-17723242; fax: + 3017723155. E-mail address: [email protected] (G.V. Bafas)

where L is a partial di=erential operator de"ned over an Euclidean space V ; z is the one, two- or three-dimensional Euclidean space vector, x is the distributed, over z, state vector and p is the vector of structural parameters. A striking characteristic on the solution branch (x(p); p) of such systems is the occurrence of limit points (or turning points) which represent the boundary between stability and instability of the system. It thus follows that the solution branch may contain physically unstable (or unobservable) regions. This paper deals with the stability analysis of a plug Fow tubular reactor, which is a typical time-dependent nonlinear distributed parameter system. Due to the built-in complexity, the analysis of the system requires the implementation of ad hoc computational schemes.

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 1 5 3 - 0

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Here the solution of Eq. (1) all over the operating branch is obtained by applying "nite element basis functions and Galerkin’s method of weighted residuals along with an arc-length continuation method for "nding the limit points. Furthermore, the design procedure of a model identi"cation fuzzy adaptive controller (MIFAC) for real-time scaling factors adjustment is proposed for the control problem. In many cases, adaptation comprises a necessity for the control loop. One reason for this is the requirement of satisfactory performance over a wide range of the operating branch; process time-varying characteristics is another motivation for using adaptive control. Over the last 20 years, a large number of conventional modelling and control methods have been proposed to cope with nonlinear and=or time-varying systems including input-state linearization [10], input–output linearization [9], model predictive schemes [17], various direct and indirect adaptive control schemes [2,3]. However, (a) the poor modelling of system uncertainties, (b) the inherent diLculty of incorporating a priori qualitative information about the system dynamics, limit the eLciency and the applicability of the classical approaches. On the other hand, modelling and control techniques based on fuzzy logic comprise very powerful approaches of handling imprecision and nonlinearity in complex systems, where the derivation of an accurate model is very diLcult or even impossible. One of the relevant advantages of fuzzy approach over more conventional techniques is the eLciency of incorporating a priori qualitative information about the system dynamics; besides fuzzy logic systems have proven to be universal approximators that can be used to approximate to any precision any continuous nonlinear function [23]. To this end, fuzzy logic can be used either as add-on technique to other approaches or as self-reliant methodology providing thereby a plethora of alternative modelling and control structures. The performance of the proposed self-tuning fuzzy controller is compared against an equivalent conventional model identi"cation adaptive controller (MIAC), over a wide range of step disturbances and operating regions. The two schemes are compared with several methods including time integral performance criteria such as Integral of Absolute Error (IAE) [22].

Fig. 1. The case study: a tubular reactor.

2. Mathematical modelling and discretization The case under study is shown in Fig. 1 and concerns the control of a jacketed plug Fow tubular reactor where a "rst-order exothermic, irreversible reaction A → B takes place. Assuming constant temperature for the coolant, the state modelling equations are given by the following partial di=erential equations [22]: 
@CA @CA E +v = − k0 exp − CA ; (2a) @t @z RT cp A

@T @T + cp vA @t @z


E ACA (2b) = UAt (TC − T ) + (−DHR )k0 exp − RT

with the boundary conditions: CA (0; t) = C0 ;

T (0; t) = T0 ;

CA (z; 0) = CPro"le ; 0¡z¡l;

T (z; 0) = TPro"le ; (2c)

where CA (z; t); T (z; t) are the concentration and temperature pro"le, respectively, inside the reactor,  is the mass density, v represents the Fuid Fow velocity, cp for speci"c heat capacity, − DHR denotes the heat of reaction, E is the activation energy, k0 is the Arrhenius factor, R is the thermodynamic constant, l is the length of reactor, U is the overall heat transfer coeLcient, At is the heat transfer unit area, A is the cross section, TC is the coolant temperature. Clearly, the set of equations (2a) and (2b) consist a time dependent nonlinear distributed parameter system with time delay; the governing equations are discretized in space and in time by a combination

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of Galerkin’s method of weighted residuals [5], "nite element basis functions and the di=erential algebraic system solver DASSL [18]. The domain is subdivided into a "nite number of node elements. The solution CA (z; t) and T (z; t) is being approximated numerically by a set of quadratic polynomials (basis functions): CA (z; t) =

N 

CAi (t)’i (z);

(3a)

i=1

T (z; t) =

N 

Ti (t)’i (z);

(3b)

i=1

where N is the number of nodes, ’i (z) are the basis functions which are taken to be unity at the ith node and zero at all other nodes, CAi and Ti are the values of the solution at the ith node. Galerkin’s method seeks for the solution of Eq. (1), setting the weighted residuals equal to zero:   
 @x − L(x; z; t) ’i dz = 0; Ri = @t V i = l; : : : ; N: (4) The weighting factors are taken to be the basis functions, which are used for the construction of the approximate solution (Eq. (3)). Substituting Eq. (3a) and (3b) in Eq. (4), the set of nonlinear partial di=erential equations reduces to a large system of coupled "rst-order ordinary di=erential equations of the form Dx˙ = +(x; p);

(5a)

where x = [x1 ; x2 ; : : : ; xN ] is the set of unknowns at the nodes, p is the set of structural parameters and xi = [CAi ; Ti ]; D is a 2N ×2N matrix with the following structure:   B 0 D= : (5b) 0 B B is a N ×N matrix with elements given by bij = V ’i ’j dz; + is a N × 1 time-dependent vector which corresponds to the set of Galerkin’s weighted residuals calculated at steady state, i.e. setting @x=@t = 0 in Eq. (1). To this end, the discretized problem given by Eq. (5) is solved in time using the DASSL.

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3. Stability analysis Depending on the values of the set of structural parameters p; the solution path of the tubular reactor contains limit points. An equilibrium point (x; p) is said to be a limit point (or regular turning point), when it satis"es the conditions [13]: (a) the Jacobian of L(x; p) with respect to x i.e. Lx (x; p) is singular, (b) the system Lx (x; p) = ’Lp (x; p) has no solution; Lp (x; p) is the Jacobian of L(x; p) with respect to p; (c) the Jacobian Lx (x; p) at (x; p) has only one zero eigenvalue. Newton’s iteration method for solving the linearized steady-state equation set, Rx x = − R

(6)

fails to converge as the solution approaches limit points, since the Jacobian Rx comes very close to singularity. Instead of using Newton’s method explicitly, the solution branch through the limit points is traced by solving the linearized equation set (6) by an arc-length continuation method [1]. Stability of solution changes across a limit point and is determined by monitoring its response to small disturbances. If the disturbance grows with time then the solution is unstable; otherwise the solution is stable. Let x be a disturbance with in"nitesimal amplitude, then linearizing L(x; p) around x0 one obtains Lx (x0 ; 0; p)x + Lx˙(x0 ; 0; p)x˙ = 0:

(7)

The stability of the above linearized system is governed by the generalized eigenvalue problem Lx (x0 ; 0; p)Ci = i [− Lx˙(x0 ; 0; p)]Ci = 0;

(8)

where i are the eigenvalues, Ci represent the modes of the solution derived by Eq. (8), Lx (x0 ; 0; p) is the Jacobian matrix as it is calculated by the weighted residuals at a given value p; using parameter continuation, Lx˙(x˙0 ; 0; p) ≡ B is the overlap matrix of the "nite element basis functions. The Jacobian which has the following form in block notation:   @LCA @LCA  @CA @T   (9) Lx (x0 ; 0; p) =   @LT @LT  @CA

@T

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Fig. 2. Steady-state paths (CA2 (con2), con2) for v∗ ¡v = 1:8; v = v∗ = 1:7335; v∗ ¿v = 1:7.

is a large, sparse and nonsymmetric matrix of size 2N ×2N . Hence, stability analysis in the vicinity of an equilibrium solution is reduced to the solution of the generalized eigenvalue problem given by Eq. (8). What is really required, is "nding only the lower modes of the generalized eigenvalue problem, i.e., eigenvectors corresponding to eigenvalues with algebraically largest real parts. If all the eigenvalues of eigenproblem (8) lie in the left-half complex plane, the system is stable; otherwise, if there is at least one eigenvalue with positive real part, the system is unstable. The information gathered by "nding eigenvalues with algebraically smallest real parts are in practice of minor signi"cance. In this paper, the determination of the most dangerous modes of Eq. (8) is attained using standard Arnoldi algorithm [21]. 3.1. Steady state solution branches The steady-state solution branches (x(con2); con2); for "xed values of Fuid velocity are shown in Fig. 2; they were obtained by taking the following values of the parameters: k0 = 1:34 × 108 =min; E=R = 8700 K; CA1 = 1:6 kmol=m3 ; T1 = 423 K; T2 = 440 K;

A = 0:002 m2 ; At = 0:01 m; DH R = −44 000 kcal= kmol; TC = 293 K; U = 25 Kcal m2 =min=grad; cp = 25 kcal=kmol K;  = 47 kmol=m3 ; l = 5 m; con2 = UAt =(cp A). As it is shown in Fig. 2, the equilibrium solution of system (2a) and (2b) exhibits some very interesting nonlinearities. Particularly, there is a value of the Fuid velocity, say v∗ for which the steady-state solution of L(x(con2); con2) = 0 has two for v¿v∗ ; one for v = v∗ or no limit points for v¡v∗ . The computed value of v∗ was v∗ = 1:7335 m=s. Across limit points an odd number of eigenvalues of the Jacobian changed sign indicating that they crossed the imaginary axis in the complex plane; this implies that the stability of solution changes. Assume that the Fuid velocity is taken to be v¿v∗ ; hence the solution branch contains two limit points. When con2¿con2u or con2¡con2l the system is stable. Between con2l¡con2¡con2u the solution branch exhibits a hysteresis behaviour: as con2 decreases following the solution branch towards con2l from the equilibrium state D to the equilibrium state C, the system remains stable until con2 equals con2l. At con2 = con2l a limit point occurs; imposing a small decrement in con2, the system loses stability and the system jumps discontinuously to the state A,

D.I. Sagias et.al., / Fuzzy Sets and Systems 121 (2001) 169–179

where it re-gains stability. Increasing con2 towards con2u, the solution recovers its observability if the branch is followed from state A to state B. Further increment of con2 beyond limit point B results in a discontinuous jump of the system to state D. The hysteresis behaviour renders the solution branch from B to C physically unstable, thus unobservable and uncontrollable. The system becomes observable all over the solution branch by changing the Fuid velocity to a value v¡v∗ . The stability of the linearized system is monitored by computing the eigenvalues with algebraically largest real parts. Far enough from the hysteresis region, Arnoldi’s eigensolver "nds that the algebraically largest real part corresponds to eigenvalue max = 0 indicating that the linearized system is marginally stable. Moving towards limit points, the real parts of the eigenvalues of smallest magnitude come closer to zero. Along the branch BC Arnoldi’s eigensolver "nds some eigenvalues with positive real parts. For example, at the parameter value con2 = 0.2067 the eigenvalues with algebraically largest real part are the complex conjugate 1; 2 ≈ 1:7 ± 14:9i. The above implies that the solution branch BC is unstable. 4. Model identication fuzzy adaptive control design A major problem encountered in nonlinear or=and time-dependent systems is the degradation of the closed-loop performance as the system shifts away from the initial operational settings. This drawback imposes the need of using adaptive controllers, i.e., controllers which adjust their parameters optimally, according to some objective criteria. So far, many schemes have been proposed for fuzzy adaptive control, including self-organizing control [19,20], membership functions adjustment [4,24] and scaling factors adjustment [14,8,16,12,7]. In [14], Maeda and Murakami and Daugherity et al. in [8], proposed adjustment mechanisms for the tuning of scaling factors by evaluating the control result based on system performance indices such as overshoot, rising time, amplitude, settling time. In [16], Palm addressed the method of adjusting optimal scaling factors by measuring on-line the linear dependence between

173

each input and output signal of the fuzzy controller. According to the above method the scaling factors are expressed in terms of input–output cross-correlation functions. However, these methods cannot be used to tune the scaling factors in real-time since the adjustment is made at the end of the control interval. Jung et al. in [12], proposed a real-time tuning of the scaling factors, based on a variable reference tuning index and an instantaneous system fuzzy performance according to system response characteristics. In [7], the authors presented an algorithm for the adjustment of the scaling factors using tuning rules, which are based on heuristics. In this paper, a model identi"cation fuzzy adaptive controller (MIFAC) for scaling factors adjustment in real time is proposed. The developed adaptive algorithm combines features of both classical and fuzzy control techniques. The structure of the proposed fuzzy adaptive control mechanism is shown in Fig. 3. It consists of a fuzzy logic-based controller (FLC), a system identi"cation unit, the controller synthesis unit and the process desired closed-loop performance. Recursive least squares (RLS) are used for parameter estimation in the identi"cation unit, while the determination of the scaling factors of the fuzzy controller is obtained in the controller synthesis unit by pole placement [11]. The key idea is to obtain a desired closed-loop performance by appropriate placement of the scaling factors of the fuzzy controller, based on a model that approximates the process behaviour. The fuzzy logic controller is a two input–single output controller with the following variables: e(t) = r(t) − y(t); ce(t) = e(t) − e(t−1); cu(t) = u(t) − u(t − 1); where r(t) is the set point at time t; y(t) is the process output at time t; e(t); ce(t) are the error and the change of error at time t; cu(t) is the change of controller output at time t. The structure of the rule base is constructed heuristically and provides negative feedback control. The control rules are built using a typical heuristic approach based on the qualitative desired behaviour of the closed-loop system in a step response [12,6,25]. These control rules are of the form: IF e is P (Positive) and ce is N (Negative) then cu is N (Negative);

(10a)

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D.I. Sagias et.al., / Fuzzy Sets and Systems 121 (2001) 169–179

Fig. 3. Basic structure of the Fuzzy Adaptive Control Mechanism. Table 1 Fuzzy logic rule base

posed as a linear discrete equation y(t) + a1 y(t − 1) + a2 y(t − 2) + · · · + an y(t − n)

ce

e

NB NM NS ZE PS PM PB

NB ZE NS NS NM NM NB NB

NM PS ZE NS NS NM NM NB

NS PS PS ZE NS NS NM NM

ZE PM PS PS ZE NS NS NM

PS PM PM PS PS ZE NS NS

PM PB PM PM PS PS ZE NS

PB PB PB PM PM PS PS ZE

IF e is N (Negative) and ce is P (Positive) then cu is P (Positive):

(10b)

The above control rules can be represented in the general form of IF e is Ai and ce is Bj then cu is Ck :

(10c)

Table 1 compresses the design of the control rules for the fuzzy sets [NB: negative big, NM: negative medium, NS: negative small, ZE: zero, PS: positive small, PM: positive medium, PB: positive big] of the fuzzy variables [e; ce, cu]. The input variables are laid out along the axes, and each matrix element represents the output variable. System identi"cation is carried out monitoring input–output data. In a general manner, the model of the system at the neighborhood of equilibrium can be

= b0 u(t) + · · · + bm u(t − m) + c0 d(t) + · · · + cl d(t − 1);

(11)

where y(t) is the system output at time t; u(t) is the control action at time t; d(t) is the disturbance imposed in the system at time t; with m6n; l6n. The parameters of the model are estimated in real time using recursive least squares. Assume that the error and change of error at time t are given by e (t); ce (t). The derivation of the controller output is obtained utilizing max-dot fuzzy inference method and centroid defuzzi"cation method [25]:  i; j 'ij uij cu =  ; (12) i; j 'ij where 'ij = Ai (e(t))Bj (ce(t)):

(13)

In a general form the incremental control action cu can be represented as a nonlinear function of the input variables e(t), ce(t): cu = f(e ; ce ; t) = f(GE e; GCE ce; t):

(14)

For small perturbations e; ce around equilibrium, the above equation is approximated by the linearized equation     @f @f cu = e + ce: (15) @e ce @ce e

D.I. Sagias et.al., / Fuzzy Sets and Systems 121 (2001) 169–179

Substituting Eq. (12) in Eq. (14) one obtains "nally the simpli"ed discretized equation [15] cu(t) = GE e(t) + GCE ce(t)

(16a)

which gives the control output at time t; GE, GCE are the scaling factors for the error and change of error, respectively. For the backward m time instances Eq. (16a) yields cu(t − 1) = GE1 e(t − 1) + GCE1 ce(t − 1);

(16b)

cu(t − 2) = GE2 e(t − 2) + GCE2 ce(t − 2);

(16c)

175

1. Acquisition of system output at time t. For t = n; n + 1; n + 2; : : : 2. Perform system identi"cation by a discrete model of the form y(t) + a1 y(t − 1) + a2 y(t − 2) + · · · + an y(t − n) = b0 u(t) + · · · + bm u(t − m): Introducing the z −1 backward shift operator, the above equation reads A(z −1 )y(t) = B(z −1 )cu(t):

···

Summing Eqs. (16b)–(16d) and banding a scaling factor GCUi for each control output cu(t − i), the following di=erence equation is obtained:

The estimation of the vector of parameters T = [a1 ; : : : ; an ; b0 ; : : : ; bm ] at time t is carried out using RLS for tracking parameter changes [22]. The RLS formulation can be written as

GCU1 cu(t − 1) + GCU2 cu(t − 2)

ˆ − 1) + *(t − 1)[y(t) − (t) = (t

cu(t − m) = GEm e(t − m) + GCEm ce(t − m): (16d)

*(t − 1)

= (GE1 + GCE1 )e(t − 1)

= P(t) (t)

+ (GE2 + GCE2 − GCE1 )e(t 2)

=

+ · · · + (GEm + GCEm − GCEm−1 )e(t − m) − GCEm e(t − m − 1):

(17)

Using the z-transform, Eq. (17) can be represented in the Z-domain as

T (t)P(t

1 P(t − 1) (t); − 1) (t) + 1

P(t) = [I − *(t − 1) T

T

(t)]P(t − 1);

(t − 1) = [−y(t − 1); : : : ; −y(t − n); u(t − 1); : : : ; u(t − m)] :

(18a) ˆT

where D = GCU1 + GCU2 z −1 + · · · + GCUm z −(m−1) ; (18b) F = (GE1 + GCE1 ) + (GE2 + GCE2 − GCE1 )z −1 + · · · + (GEm + GCEm − GCEm−1 )z −(m−1) + GCEm z −m :

ˆ (t)(t)]

where

+ · · · + GCUm cu(t − m)

D cu(t − 1) = F e(t − 1);

T

(18c)

To summarize, the self-tuning of the fuzzy controller is as follows:

The vector  (t−1) = [aˆ1 ; : : : ; aˆn ; bˆ0 ; : : : ; bˆm ] denotes the estimates of the parameters obtained at time t − 1. 3. Determine the scaling factors solving the pole assignment identity R = DA + z −1 BF;

(19)

in terms of D and F; R is the desired closed-loop characteristic equation represented in the Z-domain as R = 1 + r1 z −1 + r2 z −2 + · · · + rp z −p ; D = GCU1 + GCU2 z −1 + · · · + GCUm z −(m−1) ;

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D.I. Sagias et.al., / Fuzzy Sets and Systems 121 (2001) 169–179

F = (GE1 +GCE1 )+(GE2 +GCE2 −GCE1 )z −1 + · · · + (GEm +GCEm − GCEm−1 )z

−(m−1)

+ GCEm z −m ; Eq. (19), has unique solution if n = m, and p6 n + m. 4. Update scaling factors GEi ; GCEi ; GCUi ; i = 1; 2; : : : ; m. 5. Implement control using the sum of the control actions as they derived from the fuzzy controller at time instances t − 1; t − 2; : : : ; t − m, i.e.: cu (t) =

m 

GCUi cu(t − i)

(20)

i=1

End For 5. The control problem: MIFAC implementation. Simulation results and discussion The control objective here is to maintain the control variable, which is the composition of the reacting mixture at the output of the reactor, within the desired operational settings and particularly to keep the A reactant concentration CA2 at the outlet, below its nominal steady state value, eliminating mostly concentration disturbances at the inlet. The manipulated variable is taken to be the coolant temperature TC . The performance of the proposed fuzzy adaptive controller is compared with a conventional model identi"cation adaptive controller. Seven fuzzy sets were used for the input–output variables, with "xed symmetrical triangular-shaped membership functions de"ned over the normalized universe of discourse [−1; 1] (Fig. 4). The control rules are given in Table 1.

Fig. 4. The membership functions for the input and output variables.

For the performance evaluation of the two adaptive controllers, step-change disturbances in input concentration, ranging from −5% up to −15% are imposed into the process. Simulation results for v = v∗ , are presented for two di=erent operating points: either for the upper point u, or for the middle point m where con2 = con2∗ (Fig. 2). It is assumed that the dynamics of the process are poorly known; therefore the control loop (Fig. 3) involves real-time process identi"cation for both fuzzy and conventional controllers. The best performance of the process identi"cation is obtained taken the values of the degrees of the process polynomials A; B as n = m = 3 and the degrees of the controller polynomials D; F as m = 3. Hence, the process is modelled by the di=erence equation (1 + a1 z −1 + a2 z −2 + a3 z −3 )y(t) = (b1 z −1 + b2 z −2 + b3 z −3 )u(t);

(21)

where the structure of the conventional controller is taken as (1 + f1 z −1 + f2 z −2 + f3 z −3 )y(t) = −(g0 + g1 z −1 + g2 z −2 + g3 z −3 )u(t):

(22)

The parameters of the model are obtained using the recursive least-squares estimation, while the determination of the controller coeLcients is achieved using the pole assignment identity given by Eq. (19). The desired closed-loop pole set in Eq. (19) is speci"ed by the polynomial R = 1 + r1 z −1 + r2 z −2 + r3 z −3 + r4 z −4 + r5 z −5 + r6 z −6 + r7 z −7 ; where r1 = − 4:192; r2 = 8:111; r3 = − 9:734; r4 = 8:028; r5 = − 4:602; r6 = 1:686; r7 = − 0:295. The solution of the identi"cation problem for the upper point u, gives: a1 = − 2:338; a2 = 1:927; a3 = −0:580; b1 = 7:9498×10−5 ; b2 = 3:560×10−4 ; b3 = 6:561×10−5 ; f1 = −1:066; f2 =0:752; f3 = − 0:508; g0 = −14:75; g1 = 35:110; g2 = −27:435; g3 = 9:979 while for the middle point m, gives: a1 = − 2:327, a2 = 1:938; a3 = − 0:600; b1 = 7:787 × 10−5 ; b2 = 2:61 × 10−4 ; b3 = 8:756 × 10−5 ; f1 = − 0:991; f2 = 0:788; f3 = − 0:510; g0 = 24:894; g1 = − 60:998; g2 = 49:770; g3 = − 13:508.

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177

Fig. 5. System response on the upper operating point u: — MIFAC, - - - - MIAC. (a) −5% step disturbance in input concentration, (b) −10% step disturbance in input concentration, (c) −15% step disturbance in input concentration and (d) performance comparison via the IAE criterion.

The gains of the fuzzy controller are derived using the proposed algorithm of Section 4. For the upper point u, the algorithm gives: GCU1 =− 4:255; GCU2 = 4:455; GCU3 = − 2:285; GE1 =0:013; GE2 =0:035; GE3 = − 0:315; GCE1 =0:579; GCE2 =1:578; GCE2 =− 0:335; while for the middle point m, the algorithm gives: GCU1 = − 4:220; GCU2 =4:450; GCU3 = − 2:280; GE1 = 0:013; GE2 =0:036; GE3 =− 0:319; GCE1 = 0:586; GCE2 =1:597; GCE3 =− 0:339. Fig. 5 depicts the responses of the controlled variable using the fuzzy or the conventional adaptive scheme on the upper operating point and Fig. 6 depicts the responses of the controlled variable on the

middle operating point. As is shown, the performance for all the range of step disturbances of the proposed fuzzy adaptive controller is better than that of the equivalent conventional scheme, for both the upper and middle operating points. The proposed controller exhibits a much robust response, with up to 90% faster settling time (Fig. 6c), overshooting less up to 8% of the controlled variable (Fig. 6b). In addition to these simulation results, Figs. 5d and 6d depict in summary the Integral of Absolute Error (IAE) dynamic performance criterion for the manipulated variable on the upper and middle points, respectively. Based on Figs. 5d and 6d, the proposed

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D.I. Sagias et.al., / Fuzzy Sets and Systems 121 (2001) 169–179

Fig. 6. System response on the middle operating point m: — MIFAC, - - - - MIAC. (a) −5% step disturbance in input concentration, (b) −10% step disturbance in input concentration, (c) −15% step disturbance in input concentration and (d) performance comparison via the IAE criterion.

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