Stable and convergent iterative feedback tuning of fuzzy controllers for discrete-time SISO systems

Expert Systems with Applications 40 (2013) 188–199

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Stable and convergent iterative feedback tuning of fuzzy controllers for discrete-time SISO systems Radu-Emil Precup a,⇑, Mircea-Bogdan Ra˘dac a, Marius L. Tomescu b, Emil M. Petriu c, Stefan Preitl a a

Department of Automation and Applied Informatics, ‘‘Politehnica’’ University of Timisoara, Bd. V. Parvan 2, RO-300223 Timisoara, Romania Faculty of Computer Science, ‘‘Aurel Vlaicu’’ University of Arad, Complex Universitar M, Str. Elena Dragoi 2, RO-310330 Arad, Romania c School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward, Ottawa, ON, Canada K1N 6N5 b

a r t i c l e

i n f o

Keywords: Convergence Discrete-time input affine SISO systems Iterative feedback tuning PI-fuzzy controllers Real-time experimental results Stability

a b s t r a c t This paper proposes new stability analysis and convergence results applied to the Iterative Feedback Tuning (IFT) of a class of Takagi–Sugeno–Kang proportional-integral-fuzzy controllers (PI-FCs). The stability analysis is based on a convenient original formulation of Lyapunov’s direct method for discrete-time systems dedicated to discrete-time input affine Single Input-Single Output (SISO) systems. An IFT algorithm which sets the step size to guarantee the convergence is suggested. An inequality-type convergence condition is derived from Popov’s hyperstability theory considering the parameter update law as a nonlinear dynamical feedback system in the parameter space and iteration domain. The IFT-based design of a lowcost PI-FC is applied to a case study which deals with the angular position control of a direct current servo system laboratory equipment viewed as a particular case of input affine SISO system. A comparison of the performance of the IFT-based tuned PI-FC and the performance of the PI-FC tuned by an evolutionarybased optimization algorithm shows the performance improvement and advantages of our IFT approach to fuzzy control. Real-time experimental results are included. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The stability analysis of fuzzy control systems has been investigated extensively in the context of nonlinear autonomous/nonautonomous systems in close connection with their stabilization. The current approaches reported in the literature concerning the stable design of fuzzy control systems with Takagi–Sugeno–Kang fuzzy controllers are based mainly on linear matrix inequalities (LMIs) (Blazˇicˇ et al., 2009; Chiang & Liu, 2012; Feng, 2006; Sala, Guerra, & Babuška, 2005; Tanaka, Yoshida, Ohtake, & Wang, 2009; Tsai, 2011; Zhang, Shi, & Xia, 2010) making use of quadratic, piecewise quadratic, non-quadratic, parameter-dependent or polynomial Lyapunov functions (Boulkroune & M’Saad, 2011; Kruszewski, Wang, & Guerra, 2009; Lee, Park, & Joo, 2011; Li & Ge, 2011; Precup & Hellendoorn, 2011). The LMIs are computationally solvable even in relaxed versions, and they require numerical algorithms embedded in well acknowledged software tools or implementations in other programming languages. The design of optimal control systems is of permanent interest because it ensures very good control system performance indices by the minimization of objective functions expressed as integral quadratic performance indices (Bayam, Liebowitz, & Agresti, ⇑ Corresponding author. Tel.: +40 2564032 29/30/40 (lab), 26 (office); fax: +40 256403214. E-mail address: [email protected] (R.-E. Precup). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.07.023

2005; Cazarez-Castro, Aguilar, & Castillo, 2010; Linda & Manic, 2011; Ruano, Fleming, Teixeira, Rodríguez-Vázquez, & Fonseca, 2003; Tikk, Johanyák, Kovács, & Wong, 2011; Vašcˇák & Palˇa, 2012). The Iterative Feedback Tuning (IFT) carries out the gradient-based minimization of the objective functions making use of the input-output data from the closed-loop system in several experiments conducted per iteration (Hjalmarsson, Gevers, Gunnarsson, & Lequin, 1998; Hjalmarsson, Gunnarsson, & Gevers, 1994). A good overview on IFT is given in Hjalmarsson (2002) while ensuring the unbiased estimates of the objective function with respect to controller parameters. Various extensions of IFT to Multi Input-Multi Output (MIMO) systems are investigated in Huusom, Poulsen, and Jørgensen (2009a). The extension of IFT according to Huusom, Poulsen, and Jørgensen (2009a, 2009b) provides additional ways to disturbance rejection, and it improves the convergence. Recent IFT applications to industrial control problems in servo systems and drives are discussed in Kissling, Blanc, Myszkorowski, and Vaclavik (2009), McDaid, Aw, Xie, and Haemmerle (2010), and Ra˘dac, Precup, Petriu, and Preitl (2011). The combination of IFT and fuzzy control aims the fuzzy control system performance enhancement. A combination of IFT and fuzzy control is analyzed in Nafaa, Hadjadj-Aoul, and Mehaoua (2005), and the fuzzy control system enables the run-time adaptation based on IFT and knowledge acquired from past experience. A fuzzy-based supervisor that modifies the parameters of an iteratively

R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

tuned proportional-integral-derivative (PID) controller is suggested in Sanjuan, Kandel, and Smith (2006). The parameter mapping of IFT-based proportional-integral (PI) controllers onto the parameters of Takagi–Sugeno–Kang proportional-integral-fuzzy controllers (PI-FCs) in terms of the equivalence under certain conditions between fuzzy control systems and linear/linearized control systems is discussed in Precup, Preitl, Rudas, Tomescu, and Tar (2008), Precup et al. (2008,2012). Several structures that combine iterative and soft computing techniques are proposed in Xu and Hou (2009). Improved performance of control systems can be obtained in terms of tuning the fuzzy controllers by evolutionary-based optimization algorithms. The parameter tuning of fuzzy controllers using simulated annealing algorithms is carried out in Haber, Haber-Haber, Jiménez, and Galán (2009) and Precup et al. (2012). A cross-entropy-based approach to tune the parameters of optimal fuzzy controllers is given in Haber, del Toro, and Gajate (2010). Particle Swarm Optimization algorithms are applied in different settings in Bingül and Karahan (2011), Precup, David, Petriu, Preitl, and Ra˘dac (2011) and Valdez, Melin, and Castillo (2011) to improve the parameter tuning performance of fuzzy controllers. An Ant Colony Optimization algorithm characterized by fuzzy pheromone updating and adaptive parameter tuning is used to tune the parameters of the fuzzy sliding-mode controllers for the ball-and-beam system in Chang, Chang, Tao, Lin, and Taur (2012). Attractive applications of Gravitational Search Algorithms are reported in Precup et al. (2011) and Precup, David, Petriu, Preitl, and Ra˘dac (2011). This paper presents two new contributions in addition to the state-of-the-art which includes authors’ recent papers discussed in Nafaa et al. (2005), Sanjuan et al. (2006), Precup, Preitl, Rudas et al. (2008), Precup, Preitl, Tomescu et al. (2008), Xu and Hou (2009) and Precup et al. (2012). First, original fuzzy control system stability results are given employing a simplified formulation of Lyapunov’s direct method for discrete-time systems building upon the formulation given in Precup, Tomescu, Preitl, and Petriu (2009). Our stability analysis results are dedicated to processes modeled by discrete-time input affine SISO systems as a representative class of nonlinear systems. Such processes can be controlled by parameterized controllers tuned by means of IFT. The stability analysis is necessary for IFT-based fuzzy control systems because: – The PI controllers are initially tuned in terms of IFT, and the parameters are next mapped onto the parameters of PI-FCs in terms of the modal equivalence principle accompanied by the nonlinearities specific to fuzzy control systems. – The stability analysis enables the systematic design of the PIFCs to ensure the fuzzy control system performance enhancement. Second, a convergent IFT algorithm is suggested. The convergence is guaranteed by the fulfillment of an inequality-type convergence condition. The convergence condition is derived from Popov’s hyperstability analysis results (Landau, 1979; Popov, 1973) applied here to the parameter update law of the IFT algorithm. With this regard the update law is reformulated as a nonlinear dynamical feedback system considered in the parameter space and iteration domain. This paper includes a thorough discussion of a set of real-time experimental results for a case study that deals with the IFT-based fuzzy control of a servo system. This case study, in which the nonlinearities are not smooth, shows a serious extension of the applications of IFT-based fuzzy control. The nonlinear control system case study included in this paper outlines that our IFT approach to fuzzy control ensures a reduced number of evaluations of the objective function and reduced stochastic characteristics compared to evolutionary-based optimi-

189

zation algorithms. These advantages are proved in the paper by the comparison with a Gravitational Search Algorithm (GSA) algorithm (Rashedi, Nezamabadi-pour, & Saryazdi, 2009; Rashedi, Nezamabadi-pour, & Saryazdi, 2010) as an example of evolutionary-based optimization algorithm. Our new contributions are important and advantageous with respect to the state-of-the-art and to our previous IFT algorithms because: – The stability analysis is applied to fuzzy control systems with Takagi–Sugeno–Kang PI-FCs by the transfer of the dynamics of the PI-FCs to the process dynamics. Therefore, an extended nonlinear process is obtained, and a simple derivation of the stability analysis theorem is offered. There is no need for the separation of the process model to consist of two parts as in the usual stability analysis of nonlinear dynamical systems, i.e., a linear part with dynamics and a static nonlinearity. – The application of Popov’s hyperstability analysis to the convergence of the IFT algorithm does not require knowledge on the global minimum as in the application of Lyapunov’s results. This paper addresses the following topics. The problem formulation is presented in the next section in terms of the main aspects concerning the new IFT algorithm with guaranteed convergence. Section 3 is next focused on original stability analysis results in a general formulation dedicated to discrete-time input affine SISO systems. A design method of Takagi–Sugeno–Kang PI-FCs tuned by IFT and with guaranteed fuzzy control system stability is suggested in Section 4. Section 5 is dedicated to the case study that applies our theoretical approach to the angular position control of a direct current (DC) servo system laboratory equipment. The advantages of the new approach are highlighted by a comparison with a Takagi–Sugeno–Kang PI-FC tuned by GSA. Real-time experimental results are included. The conclusions are pointed out in Section 6.

2. Problem formulation The structure of the linear control system with IFT algorithm is presented in Fig. 1, where r – the reference input, d – the disturbance input (noise), e – the control error, u – the control signal, q – the parameter vector containing the controller parameters, C(q) – the transfer function of the linear (PI) controller to be replaced by the Takagi–Sugeno–Kang PI-FC to ensure the control system performance enhancement, F – the transfer function of the reference model prescribing the desired behavior to be exhibited by the control system, P – the transfer function of the process, y – the controlled output, yd – the desired output (of the reference model), dy = y - yd – the output error, and the vector i contains the performance specifications imposed to the control system, i.e., the desired/imposed values of performance indices including overshoot, settling time, rise time, etc.

Fig. 1. Structure of linear control system with IFT algorithm.

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A simple expression of the objective function J(q) to be minimized by IFT is

JðqÞ ¼ ð0:5=NÞ

N X ½dyðt; qÞ2 ;

ð1Þ

t¼1

where N is the number of samples which set the length of each experiment. A typical objective related to J(q) is to find a parameter vector q⁄ to minimize J(q) and make the error dy tend to zero as t ? 1. That objective is expressed analytically in terms of the optimization problem

q ¼ arg minJðqÞ;

ð2Þ

q2Dq

where several constraints can be imposed. The most important constraint concerns the control system stability, and Dq stands for stability domain in (2). The reference model is usually chosen as a second order transfer function in normalized form where the natural frequency and the damping factor can easily be transferred to performance indices. This model also embodies the behavior of the typical control system structures which act as low-pass filters. It is of course a subject of compromise on how the performances are requested through the reference model because with a certain parameterization of the controller it may be possible that the reference model response is never matched by the control system. The a priori information on the process can be incorporated in the controller design before choosing the reference model. IFT algorithms solve the optimization problem (2) by numerical stochastic approximation algorithms making use of the control signal and of the controlled output measured during the control system operation. The Robbins–Monro stochastic approximation algorithm iteratively approaches a zero of a function affected by stochastic noise without knowledge on its fully expression. IFT algorithms thus hold both an input-output sensitivity functionbased tuning scheme, and a stochastic convergence result which is necessary in an experiment-based environment where the random factors appear every time. IFT is not only dedicated to Model Reference Adaptive Control (MRAC) schemes but also to the more generally formulated Linear-Quadratic Gaussian (LQG) criteria where flexible objective functions can be formulated such that to weight the state variables, the control errors, the control signals and the controlled outputs as well. Therefore, the MRAC is a particular case in this context. Important results that back-up the application of IFT to control of nonlinear processes are presented in Hjalmarsson (1998) and Sjöberg et al. (2003), and applied to processes with smooth nonlinearities. The update law to calculate the next parameter vector qi+1 is

qiþ1 ¼ qi  ci ðRi Þ1 est



 @J i ðq Þ ; Ri > 0; det Ri – 0; @q

where: i, i 2 N – the index of the current iteration, est

ð3Þ h

i

@J ð iÞ @q 0

q

– the

estimate of the gradient vector, ci – the step size, and q – the initial guess of the controller parameters. The usual choice for the sequence {ci}ieN should ensure the convergence of the algorithm in the stochastic sense by reducing the effect of the noise around the local minimum which would otherwise lead to the lack of convergence. A common choice with this regard is (Hjalmarsson, 2002)

ci ¼

c0 i

a

; i 2 N; i P 1; 0:5 < a 6 1;

such as the Levenberg–Marquardt algorithm (LMA) as suggested in Huusom, Poulsen, and Jørgensen (2009b) and the Broyden–Fletcher–Goldfarb–Shanno algorithm according to Hamamoto, Fukuda, and Sugie (2003). These algorithms are expected to give very good results when the signal to noise ratio is high. However, since the LMA interpolates between the steepest descent algorithm when far from the minimum and the Gauss-Newton algorithm when close to the minimum, the Gauss-Newton approximation makes use of the first-order derivatives of the cost function which are affected by noise. In the stochastic approximation algorithm (3) this choice of the Hessian approximation can worsen the algorithm if the signal to noise ratio is low. As the noise that enters the closed loop has a lower intensity, it is expected that the conditions approach the deterministic case where the LMA is a better approach. On the other hand, the estimate of the Hessian is also more expensive to compute since it requires extra experiments. A good compromise is the steepest descent with the step scaling sequence chosen as to respect the convergence of the algorithm in the stochastic sense, i.e., the step sequence should tend to zero at infinity but not too fast. In the case study presented in this paper, the noise intensity is low; therefore, the LMA can be employed. Some hyperstability results will be applied as follows to the parameter update law (3) in order to ensure the convergence of our IFT algorithm. That is the reason why the update law (3) is expressed as a dynamical feedback system in the parameter space and iteration domain. In this context consider the feedforward discrete-time linear time-invariant (LTI) block

qiþ1 ¼ Iqi þ Ili ; v i ¼ Iqi ;

ð5Þ

which is completely controllable and completely observable because of the identity matrix I. Let us consider the nonlinear (NL) feedback block

wi ¼ ci ðRi Þ1 est



 @J i ðv Þ : @v

ð6Þ

The blocks LTI and NL are connected according to the block diagram presented in Fig. 2. The feedback structure illustrated in Fig. 2 is used in the hyperstability analysis viz. convergence analysis, and it justifies that (5) and (6) are equivalent to (3). The block NL satisfies the inequality required by Popov’s hyperstability theory (Landau, 1979)

mði0 ; i1 Þ ¼

i¼i1 X

ðwi ÞT v i P e20 ; 8i1 P i0 ; e0 ¼ const;

e0 – 0;

where the superscript T indicates matrix transposition. The necessary and sufficient condition for the nonlinear dynamical feedback system described by (5)–(7) to be hyperstable (Landau, 1979; Popov, 1973) is that the discrete transfer function matrix

HðzÞ ¼ 0 þ ðzI  IÞ1 ¼ diagð1=ðz  1Þ;

1=ðz  1Þ;

...;

1=ðz  1ÞÞ

ð8Þ

must be a positive real discrete transfer function matrix. The particular expression of the matrix H(z) in (8) is positive real according to the definitions given in Hitz and Anderson (1969), so the system

ð4Þ

where the initial step size c0, c0 > 0, is set in order to ensure a compromise to numerical stability and to convergence speed. The matrix Ri can be an estimate of the Hessian, a Gauss–Newton approximation of the Hessian or the identity matrix to simplify the signal processing and reduce the complexity of IFT algorithms. Different other choices for the Hessian approximation are possible

ð7Þ

i¼i0

Fig. 2. Block diagram used in convergence analysis.

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(5)–(7) is hyperstable. Therefore, the convergence of our IFT algorithm with the parameter update law (3) is guaranteed provided that the inequality (7) holds. Our IFT algorithm consists of the following steps:

f; b : Rn  N ! Rn ; fðxðtÞÞ ¼ ½ f1 ðxðtÞÞ f2 ðxðtÞÞ ::: fn ðxðtÞÞ T ; bðxðtÞÞ ¼ ½ b1 ðxðtÞÞ b2 ðxðtÞÞ . . . bn ðxðtÞÞ T ; ð12Þ n

Step 0: Set the initial controller parameters in the parameter vector q0. Step 1: Conduct the two experiments for the considered CS structure and record the input–output data pairs (u1, y1) and (u2, y2). The first experiment is called the normal one, and it corresponds to the usual control system operation. In the second experiment, referred to as the gradient one, the reference input is the control error of the first experiment. Calculate the estimate of the gradient of outh i put error est @dy ðt; qi Þ (Hjalmarsson, 2002; Hjalmarsson, @q Gevers, Gunnarson, & Lequin, 1998; Hjalmarsson, Gunnarsson, & Gevers, 1994)

  @dy 1 @C 1 i  Est ðt; qi Þ ¼ ðq ; q Þ  y2 ðk; qi Þ; @q Cðq1 ; qi Þ @ q

Ri : IF x1 IS X l1 AND x2 IS X l2 AND . . . AND xn IS X ln THEN u ð9Þ

where the subscript 2 highlights the gradient experiment. Step 2: Generate the output of the reference model yd and calculate the output error dy. Step 3: Calculate the estimates of the gradient @@Jq ðqi Þ and eventually of the Hessian Ri(qi) of J making use of est½@dy ðt; qi Þ @q from (9) substituted to

  N  X @J i @dy ðq Þ ¼ ð1=NÞ dyðt; qi Þest ðt; qi Þ ; @q @q t¼1

ð10Þ i

Step 4: Set the step size c to fulfill the sufficient convergence condition (7). If no value can be found for ci to satisfy (7), the classical choice should be made according to (4). The convergence property is still guaranteed if ci is chosen based on (4) without satisfying (7). Step 5: Calculate qi+1 by the update law (3).

The initial step 0 is conducted only once and the other steps are repeated in all iterations till the objective function has decreased sufficiently to meet the performance specifications imposed to the control system. Additional details regarding the IFT algorithms are presented in Hjalmarsson, Gunnarsson, and Gevers (1994), Hjalmarsson, Gevers, Gunnarson, and Lequin (1998), Huusom et al. (2009a), Precup, Preitl, Rudas, Tomescu, and Tar (2008), Precup et al. (2008). 3. Stability analysis approach The process in the fuzzy control system is modeled by the following discrete-time input affine SISO system described by statespace mathematical model (Precup, Tomescu, Preitl, & Petriu, 2009)

yðtÞ ¼ gðxðtÞÞ;

xð0Þ ¼ x0 2 X;

ð13Þ X il ; l

where ¼ 1 . . . n, are the fuzzy sets expressed as linguistic terms afferent to the state variables xl, ui(x) is the control signal produced by rule Ri with the firing strength ai = ai(x), 0 6 ai 6 1, i = 1...nRB, subject to

ai ðxÞ ¼ minðlXl1 ðx1 Þ; lXl2 ðx2 Þ; . . . ; lXln ðxn ÞÞ; 8x 2 X ð14Þ

lXil ; l ¼ 1 . . . n, are the membership functions of the linguistic terms X il ; l ¼ 1 . . . n, and nRB is the number of rules. An active region of rule Ri is defined as the set

i

xðt þ 1Þ ¼ fðxðtÞÞ þ bðxðtÞÞuðtÞ; t 2 N;

¼ ui ðxÞ; i ¼ 1 . . . nRB ;

9i ¼ 1 . . . nRB such that ai – 0;

   T N  X @dy @dy R ðq Þ ¼ ð1=NÞ est ðt; qi Þ est ðt; qi Þ : @q @q t¼1 i

and g:R  N ? R describe the process dynamics, and u is the control signal produced by the fuzzy controller. The Takagi–Sugeno–Kang fuzzy controllers that control the process modeled by (11) and (12) can be considered generally as nonlinear state feedback controllers. They employ the MAX and MIN operators in the inference engine and the weighted sum method for defuzzification. The use of these operators does not restrict the generality of our approach because it does not require the differentiability of fuzzy controller’s input-output map. Other tnorms and s-norms can be used instead of the MAX and MIN operators, respectively. The ith rule in the rule base of the fuzzy controller, referred to as Ri, i = 1. . .nRB, nRB P 2, is expressed as

ð11Þ

where y is the controlled output, x(t) is the state vector, xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ . . . xn ðtÞ T 2 X  Rn , n 2 N; n P 1, X is the universe of discourse, the time variable t (with the initial time moment t0 = 0) will be omitted as follows for simplicity, x0 is the initial state vector, the continuous functions

X Ai ¼ fx 2 Xjai ðxÞ – 0g; i ¼ 1 . . . nRB :

ð15Þ

Since the regions different to (15) do not affect the inference engine, the expression of the control signal produced by the fuzzy controller is

uðxÞ ¼

" nRB X

i

i

a ðxÞu ðxÞ

i¼1

#," nRB X

# i

a ðxÞ :

ð16Þ

i¼1

Let the process be characterized by the state-space model defined in (11) and let the radially unbounded function V:X ? R such that V(x) > 0, 8x 2 X n f0g, V(0) = 0. The first difference of the function V(x(t)) along the trajectory of (11), denoted by DV(x(t)), is supposed to fulfill the condition

DVðxðtÞÞ ¼ Vðxðt þ 1ÞÞ  VðxðtÞÞ < 0:

ð17Þ

Using the notation Vk(x(t)) for the Lyapunov function candidate V(x(t)) which is considered along the trajectory of the system (1) for u(t) = uk(x(t)), the first difference of Vk(x(t)) is DVk(x(t)):

DV k ðxðtÞÞ ¼ V k ðxðt þ 1ÞÞ  V k ðxðtÞÞ; 8x 2 X Ak ; k ¼ 1 . . . nRB :

ð18Þ

Our stability analysis theorem presented as follows is derived on the basis of Lyapunov’s theorem for discrete-time systems starting with the formulation given in (Slotine & Li, 1991). Theorem 1. Let the fuzzy control system be described by the discretetime input affine SISO system modeled in (11), the Takagi–Sugeno– Kang fuzzy controller characterized by (13)–(16), and x = 0 an equilibrium point of (1). Let

V : X ! R; VðxðtÞÞ ¼ xT ðtÞPxðtÞ; where P is an n  n positive definite matrix such that

ð19Þ

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DV k ðxðtÞÞ < 0; 8x 2 X Ak ; k ¼ 1 . . . nRB :

ð20Þ

Then all state vectors x(t) will converge globally asymptotically to the origin x(t) = 0 as t ? 1 . The proof of Theorem 1 is presented in Appendix 1. This theorem offers the sufficient inequality-type conditions (20) for the global asymptotic stability of the equilibrium point at the origin. Theorem 1 proves that if each subsystem is stable in the sense of Lyapunov under a common Lyapunov function, the (overall) closed-loop system is also stable in the sense of Lyapunov. Since no fuzzy model of the plant is involved, the number of subsystems generated is relatively small, and the common Lyapunov function can be found easily. This approach decomposes the stability analysis to the analysis of each rule, so it is not complex. Theorem 1 is applied in this paper in order to set the values of the parameters of the Takagi–Sugeno–Kang PI-FCs in order to offer stable fuzzy control systems. 4. Structure and design of Takagi–Sugeno–Kang PI-fuzzy controllers The Takagi–Sugeno–Kang PI-FC is a discrete-time controller built around the two inputs-single output fuzzy controller (TISO-FC) and the structure presented in Fig. 3(a), where: De(t) = e(t)  e(t  1) – the increment of control error, Du(t) = u(t)  u(t  1) – the increment of control signal, and q1 – the backward shift operator. The Takagi–Sugeno–Kang PI-FCs are characterized by the fuzzification according to Fig. 3(b) (the TISO-FC includes the scaling of inputs and output), the inference engine and defuzzification according to the previous section, and the inference engine is assisted by the following complete rule base (nRB = 9):

R1 : IF eðtÞ IS N AND DeðtÞ IS N THEN DuðtÞ ¼ gK P ½DeðtÞ þ aeðtÞ; 2

R : IF eðtÞ IS N AND DeðtÞ IS ZE THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ; R3 : IF eðtÞ IS N AND DeðtÞ IS P THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ; R4 : IF eðtÞ IS ZE AND DeðtÞ IS NTHEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ;

The consequent parts of rules R2 to R8 are the same and the consequent parts of rules R1 and R9 are the same because the Takagi– Sugeno–Kang PI-FC can be considered as a bumpless interpolator between two separately designed linear PI controllers. Our design method will ensure this behavior of the fuzzy controller. The parameters KP and a are obtained by the continuous-time design of the linear PI controller with the transfer function

CðsÞ ¼ kc ð1 þ T i sÞ=s ¼ kC ½1 þ 1=ðT i sÞ;

where kC, kC = Tikc, is the controller gain and Ti is the integral time constant. Tustin’s method is next applied to obtain the incremental discrete-time linear PI controller in the consequent parts of all rules except R1 and R9, with the parameters

K P ¼ kC ½1  T s =ð2T i Þ; a ¼ 2T s =ð2T i  T s Þ;

xC;1 ðtÞ ¼ uðt  1Þ; xC;2 ðtÞ ¼ eðt  1Þ

xC;1 ðt þ 1Þ ¼ xC;1 ðtÞ þ fTISOFC ðeðtÞ; eðtÞ  xC;2 ðtÞÞ; xC;2 ðt þ 1Þ ¼ eðtÞ; uðtÞ ¼ xC;1 ðtÞ þ fTISOFC ðeðtÞ; eðtÞ  xC;2 ðtÞÞ;

DeðtÞÞ ¼ fTISOFC ðeðtÞ; eðtÞ  xC;2 ðtÞÞ:

R8 : IF eðtÞ IS P AND DeðtÞ IS ZE THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ; R9 : IF eðtÞ IS P AND DeðtÞ ISP THEN DuðtÞ ¼ gK P ½DeðtÞ þ aeðtÞ: ð21Þ The purpose of parameter g in (21) is the reduction of fuzzy control system’s overshoot by offering small absolute values of Du(t) for the same signs of e(t) and De(t) (Precup et al., 2008). The smaller the value of the parameter g is, 0 < g < 1, the smaller the overshoot will be.

ð25Þ

where the nonlinear input-output map of the TISO-FC is

fTISOFC : R2 ! R; DuðtÞ ¼ fTISOFC ðeðtÞ;

R : IF eðtÞ IS ZE AND DeðtÞ IS P THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ;

ð24Þ

as illustrated in Fig. 4. Eq. (24) and Fig. 4 result in the discrete-time state-space model of Takagi–Sugeno–Kang PI-FC

6

R7 : IF eðtÞ IS P AND DeðtÞ IS N THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ;

ð23Þ

where Ts is the sampling period. Two important aspects are highlighted in relation with the rule base (21). First, the number of rules in this complete rule base can be reduced further to support the low-cost implementation, and convenient rule reduction and interpolation techniques can be applied with this regard (Baranyi, Korondi, Hashimoto, & Wada, 1997; Baranyi et al., 2002; Johanyák, 2010). Second, in order to apply Theorem 1, the dynamics of Takagi–Sugeno–Kang PI-FC is moved to process’s dynamics as follows. The state variables xC,1 and xC,2 are defined for the Takagi–Sugeno–Kang PI-FC

5

R : IF eðtÞ IS ZE AND DeðtÞ IS ZE THEN DuðtÞ ¼ K P ½DeðtÞ þ aeðtÞ;

ð22Þ

ð26Þ

Eqs. (11), (25), and (26) lead to the expression of the state-space model of Takagi–Sugeno–Kang PI-FC

xC;1 ðt þ 1Þ ¼ xC;1 ðtÞ þ DuðtÞ; xC;2 ðt þ 1Þ ¼ rðtÞ  gðxðtÞÞ;

ð27Þ

uðtÞ ¼ xC;1 ðtÞ þ DuðtÞ: The models (11) and (25) are next merged in the following discretetime state-space model of extended process, i.e., the process extended with the dynamics of Takagi–Sugeno–Kang PI-FC:

xðt þ 1Þ ¼ fðxðtÞÞ þ bðxðtÞÞ½xC;1 ðtÞ þ DuðtÞ; xC;1 ðt þ 1Þ ¼ xC;1 ðtÞ þ DuðtÞ; xC;2 ðt þ 1Þ ¼ rðtÞ  gðxðtÞÞ;

ð28Þ

yðtÞ ¼ gðxðtÞÞ: Using the model (29), the Takagi–Sugeno–Kang PI-FC is replaced by the TISO-FC with the two input variables

eðtÞ ¼ rðtÞ  gðxðtÞÞ; DeðtÞ ¼ rðtÞ  gðxðtÞÞ  xC;2 ðtÞ;

Fig. 3. Structure (a) and input membership functions (b) of PI-FC.

ð29Þ

and the output variable Du(t). Therefore, this transformation of the models leads to the expression of the rule base (21) as a particular case of (3). The design method dedicated to the accepted class of Takagi– Sugeno–Kang PI-FCs consists of the steps I to III to obtain the parameters of the PI-FCs Ts, KP and a (specific to the linear design), and Be, BDe and g (specific to the fuzzy design):

R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

193

Fig. 4. Structure of PI-FC that includes the definitions of the state variables.

Step I. Apply a design method to tune the continuous-time linear PI controller, set Ts, apply (23) to calculate the initial parameter vector q0 ¼ ½ K 0P a0 T , set the reference model structure and its parameters according to the performance specifications imposed to the control system. Step II. Conduct the steps 0 to 5 of our IFT algorithm presented in Section 3 to obtain the optimal parameter vector q ¼ ½ K P a  T . Step III. Express the discrete-time state-space model of the extended process, set the values of the parameters Be and g according to the performance specifications and to the stability analysis approach such that to fulfill the stability conditions (20) in Theorem 1, and apply the modal equivalence principle to map the linear controller parameters onto the Takagi–Sugeno–Kang PI-FC ones:

BDe ¼ aBe :

ð30Þ

The steps I and II correspond to the linear design, and the step III corresponds to the fuzzy design. The value of the parameter Be depends on the reference input such that to ensure the firing of all rules. 5. Case study, results and discussion The theoretical approach is applied to nonlinear servo system control. The experimental setup is built around a DC servo system with backlash laboratory equipment (Fig. 5). It is characterized by rated amplitude equal to 24 V, rated current equal to 3.1 A, rated torque equal to 15 N cm, rated speed equal to 3000 rpm, and inertial load mass equal to 2.030 kg. The position controllers are implemented digitally on a PC making use of an FPGA-based A/D-D/A interface connected by USB to the PC. The nonlinear process used in the angular position control is characterized by the nonlinear continuous-time state-space model

8 0; if juðtÞj 6 ua ; > < mðtÞ ¼ ku;m ½uðtÞ  ua sgnðuðtÞÞ; if ua < juðtÞj < ub ; > : if juðtÞj P ub ; ku;m ðub  ua ÞsgnðuðtÞÞ; " # " # 0 0 1 _ xðtÞ þ mðtÞ; xðtÞ ¼ kP1 =T R 0 1=T R

ð31Þ

yðtÞ ¼ ½ 1 0 xðtÞ; where t is the independent continuous time argument, t 2 R, t P 0, the control signal u is the duty cycle of a pulse width modulated (PWM) signal, m is the output of the saturation and dead zone static nonlinearity represented by the first equation in (31), the state vector x(t) is

xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ T ¼ ½ aðtÞ xðtÞ T ;

ð32Þ

x1(t) = a(t) is the first state variable that represents the angular position, and x2(t) = x(t) is the second state variable that represents the angular speed. The disturbance inputs and the initial conditions were omitted in (31) for the sake of simplicity. The parameters of the linear dynamics represented by the second and third equation in (31) are the gain kP1 = 121.6956 and the small time constant TR = 0.9198s. The parameters of the saturation and dead zone static nonlinearity in (31) are identified by nonlinear least squares as ku,m = 1.149, ua = 0.13 and ub = 1 . The process has an input nonlinearity related to the actuator, i.e., a saturation and dead zone static nonlinearity. However, this is not included in the following simplified model of the process expressed as the transfer function P(s)

PðsÞ ¼ kP =½sð1 þ T R sÞ;

ð33Þ

with the process gain kP = ku,mkP1 = 139.88, is used in the IFT algorithm. However, the case study is treated from a linear perspective in the initial step when the PI controller is tuned and the control system performance indices are obtained with this simple controller. The model given in (33) can be viewed as a simplified model of servo systems which belong to control systems in various applications (Dankovic´, Nikolic´, Milojkovic´, & Jovanovic´, 2009; Fazel Zarandi, Türksßen, & Torabi Kasbi, 2007; Horváth & Rudas, 2004; Iglesias, Angelov, Ledezma, & Sanchis, 2010; Jiao & Yan, 2011; Škrjanc, Blazˇicˇ, & Agamennoni, 2005). The design method presented in Section 4 is applied as follows. The continuous-time linear PI controller has been obtained in the step I by the frequency domain design imposing the phase margin of 60o resulting in the controller tuning parameters kC = 0.01036 and Ti = 3.1043s. Setting Ts = 0.01s the initial discrete-time linear PI controller parameters calculated in terms of (23) are

q0 ¼ ½ q01 q02 T ¼ ½ K 0P a0 T ¼ ½ 0:01034 0:0032 T :

ð34Þ

The continuous-time transfer function of the reference model with oscillatory dynamics (to reduce the number of iterations because of the process nonlinearities) is

FðsÞ ¼ 1=ð1 þ 0:6s þ s2 Þ: Fig. 5. Experimental setup.

ð35Þ

A filter was introduced on the reference input in order to alleviate the overshoot that is motivated by the presence of the integrator

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components in both the process (since it is a servo system for position control) and the controller. The filter’s continuous-time transfer function is Fr(s) = 1/(1 + 1.5s). The discrete-time forms of reference model and of reference input filter were used in the simulation and in the real-time experiments as well. All the above settings were applied in both the simulations and the experiments. The simulation results are first presented. The IFT algorithm was applied in step II of the design method. The parameters obtained after 35 iterations for Ri = I2 and ci = 5  109, i = 0 . . . 35 (that satisfy (7) for e0 = 1 at all iterations), are T 35 q35 ¼ ½ q35 q35 a35 T ¼ ½ 0:02255 0:0004 T : 1 2  ¼ ½ KP

ð36Þ

The step III starts with the derivation of the discrete-time statespace model of the extended process. Accepting that the control signal u and the reference input r are changing at the discrete sampling intervals the discrete-time state-space model of the extended process becomes then (28), where T

xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ xC;1 ðtÞ xC;2 ðtÞ  ;   x1 ðtÞ þ T R ½1  expðT s =T R Þx2 ðtÞ fðx; tÞ ¼ ; ½expðT s =T R Þx2 ðtÞ   kP ½T s þ T R expðT s =T R Þ  T R  mðtÞ; bðx; tÞ ¼ kP ½1  expðT s =T R Þ gðxðtÞÞ ¼ x1 ðtÞ; 8 if juðtÞj 6 ua ; > < 0; mðtÞ ¼

> :

ð37Þ Fig. 6. Simulation results: controlled output (angular position) and control signal versus time for the linear control system with the PI controller before IFT (CS – control system).

ku;m ½uðtÞ  ua sgnðuðtÞÞ; if ua < juðtÞj < ub ;

ku;m ðub  ua ÞsgnðuðtÞÞ;

if juðtÞj P ub :

For comparison reasons, a fuzzy control system is designed from the initial PI controller before tuning, and another one is designed from the resulting PI controller obtained by IFT-based tuning. A good choice of Be (to ensure the firing of all three input linguistic terms) for a constant reference input of r = 40rad is Be = 20 for both Takagi–Sugeno–Kang PI-FCs and the other parameter of the Takagi–Sugeno–Kang PI-FC, namely BDe, results from (30). The values of the parameter g and of the parameter BDe were g = 0.65 and BDe = 0.064 for the initial fuzzy controller, and g = 0.99 and BDe = 0.008 (showing practically crisp membership functions for the linguistic terms N and P and singleton for the linguistic term ZE) for the final one. The Lyapunov function candidate that fulfils the stability conditions (20) for this fuzzy control system is defined in (19), where

P ¼ diagð1; 1; 1; 1Þ; x ¼ ½ x1

x2

x3

x4  T ¼ ½ x1

x2

xC;1

xC;2 T : ð38Þ

The stability check on the basis of the stability condition (20) assisted by (19) and (38) is not difficult as it is carried out by the digital simulation of fuzzy control system behavior in important operating regimes. A band-limited white noise of variance 0.01 has been fed to the disturbance input d in the real-time experiments. All controllers were also tested on the simplified linear process model (33) in order to outline the differences between the behavior of the control systems with the simplified model and the behavior of the control systems with the nonlinear process model. The digital simulation results are presented in Figs. 6–9. The fuzzy controller developed from the initial PI controller deals better with the dead zone that causes the large overshoot in Fig. 6. After IFT-based tuning, the linear PI controller offers an aperiodic CS response that is closer to the reference model response in the sense given by the objective function. Both control systems with the IFT-based tuned PI controller and with the subsequent Takagi–Sugeno–Kang PI-FC derived from it offer a faster response than the initial fuzzy control ssytems, and these two controllers also

Fig. 7. Simulation results: controlled output (angular position) and control signal versus time for the fuzzy control system with the Takagi–Sugeno–Kang PI-FC before IFT (FCS – fuzzy control system).

deal with the process nonlinearity. The values of the objective functions in the four cases that correspond to Figs. 6–9 were evaluated to 16.1623, 11.0007, 2.8807 and 2.6803, respectively. The intermediate step with IFT tuning proves to be useful since the fuzzy control system with the final IFT-based tuned fuzzy controller is better than the fuzzy control system with the initial one due to the smaller objective function.

R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

195

fuzzy controllers were tuned starting with the initial PI controller and with the final PI controller after IFT. The experiments were conducted with the same reference model and reference input filter. The results are presented in Figs. 10–13. For this experimental scenario the IFT-based tuning started with the same initial parameters given in (34). The parameters obtained after nine iterations for Ri = I2 and c0 = 5  109, ci = c0/i, i = 1 . . . 9 (that also satisfy (7) for e0 = 1 at all iterations), are

q9 ¼ ½ q01 q02 T ¼ ½ K 0P a0 T ¼ ½ 0:01199 0:0028 T :

Fig. 8. Simulation results: controlled output (angular position) and control signal versus time for the linear control system with the PI controller after IFT (CS – control system).

ð39Þ

The values of the parameters of Takagi–Sugeno–Kang PI-FCs were set to Be = 20 for both fuzzy controllers, and the parameter BDe of Takagi–Sugeno–Kang PI-FCs was obtained using (30). The values of the parameter g and of the parameter BDe were g = 0.9 and BDe = 0.064 for the initial fuzzy controller, and g = 0.8 and BDe = 0.057 for the final one. The Lyapunov function candidate that fulfils the stability conditions (20) for this fuzzy control system is defined by means of (19) and (38). The fuzzy control system with the fuzzy controller that exhibits the experimental results presented in Fig. 11 can cope with the process nonlinearity better than the other control systems, but it still has nonzero steady-state control error. Both control systems with the IFT-based tuned PI controller and with the resulting Takagi–Sugeno–Kang PI-FC offer a slightly faster response, and the latter also ensures the zero steady-state control error. The values of the objective function in the four cases that correspond to Figs. 10–13 were evaluated to 6.6451, 5.1081, 3.2231 and 2.4960, respectively. Therefore, it is shown that the fuzzy control system with the final IFT-based tuned fuzzy controller is better than the fuzzy control system with the initial fuzzy controller because the objective function is reduced. The evolutions of the objective function throughout the iterations of the simulation case study and of the experimental case study are illustrated in Figs. 14 and 15, respectively. The nonlinearity of the process is reflected in the objective function’s decrease in

Fig. 9. Simulation results: controlled output (angular position) and control signal versus time for the fuzzy control system with the Takagi–Sugeno–Kang PI-FC after IFT (FCS – fuzzy control system).

The same scenario was applied in the real-time experiments with the servo system laboratory equipment. The same design method was used starting with the same initial PI controller that was designed using the simplified linear process model (33). The

Fig. 10. Experimental results: controlled output (angular position) and control signal versus time for the linear control system with the PI controller before IFT (CS – control system).

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R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

Fig. 11. Experimental results: controlled output (angular position) and control signal versus time for the fuzzy control system with the Takagi–Sugeno–Kang PI-FC before IFT (FCS – fuzzy control system).

Fig. 13. Experimental results: controlled output (angular position) and control signal versus time for the fuzzy control system with the Takagi–Sugeno–Kang PI-FC after IFT (FCS – fuzzy control system).

Fig. 14. Objective function versus iteration index in the simulations.

Fig. 12. Experimental results: controlled output (angular position) and control signal versus time for the linear control system with the PI controller after IFT (CS – control system). Fig. 15. Objective function versus iteration index in the experiments.

the simulation case study where it is apparent that the objective function stops decreasing after 10 iterations, but further decreases after 20 iterations. That is also the first reason why in the case of real-time experiments completely different results were obtained after IFT than in case of simulation experiments (different values of the parameters and different values of the numbers of iterations), some increases of the objective functions during the itera-

tions in the real-time experiments can be observed. The second reason for different results in the case of real-time experiments compared to the simulation experiments is the stochastic framework specific to real-world processes and generally accounted for in IFT. The injected noise does not influence too much in this case the evaluation of the objective function in the experimental study, the

R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

random factors that affect the evaluation of the objective functions are visible, they are of different nature than the injected noise, namely due to the asymmetric friction in the motor axis. However, the objective function decreases after several iterations. Since the evaluation of the objective function on the real process has a strong variance, and that depends on the applications involved (Ferreira & Ruano, 2009; Hladek, Vašcˇák, & Sincˇák, 2009; Kasabov & Hamed, 2011; Thomsen, Hoffmann, & Fuchs, 2011), the more cautious steepest descent is considered in the IFT algorithm, with small steps in order to ensure the convergence in the framework of the step 4 in our IFT algorithm. As mentioned in Section 1, for additional comparison reasons, the Takagi–Sugeno–Kang PI-FC is tuned by the GSA proposed in Precup et al. (2011). The GSA is considered for comparison to show the performance of our IFT algorithm. The GSA offers an evolutionary-based solution to the optimization problem (2) where the vector variable q is the parameter vector defined as

q ¼ ½ q1 q2 q3 T ¼ ½ b Be g T :

ð40Þ

The simulation of fuzzy control system behavior accepting the same simulation and experimental scenario leads to the following domain Dq, defined in Precup et al. (2011):

Dq ¼ fbj3 6 b 6 17g  fBe j20 6 Be 6 40g  fgj0:55 6 g 6 1g: ð41Þ The simulation of fuzzy control system behavior in several operating regimes shows that Dq guarantees the fulfillment of the stability condition (20). The parameters of GSA were set to N = 20 for the number of agents, to e = 104 for the small constant in the denominator of the weighted sum of all forces, to g(k0) = 100 for the initial value of gravitational constant accepting a linear depreciation law of the gravitational constant, and to maximum 100 iterations. These parameter settings ensure a tradeoff to convergence speed and search accuracy. The parameter values obtained after an average value of 8608.6 evaluations of the objective function until finding its minimum value for the best five runs of the algorithm (to reduce the effects of the stochastic characteristics) are Be = 39.9756, g = 1, BDe = 0.1334 and b = 3.2614. The objective function was evaluated to 4.3607. The comparison between our IFT-based approach to fuzzy controller tuning and the GSA as a representative evolutionary-based algorithm shows that: – The IFT-based approach ensures, as mentioned in Section 1, a strong reduction of the number of evaluations of the objective function. – The IFT-based approach also ensures the reduction of the stochastic characteristics of GSA. Several runs of the GSA are needed to get the solution compared to the IFT where there is no need to repeat the experiments. – The calculation of the estimated gradient of the objective function using experiments conducted on the real-world control systems compensates for model uncertainties and nonlinearities which cannot be embedded in the simulations specific to evolutionary-based optimization algorithms. 6. Conclusions The paper has suggested a three-step stable design method for fuzzy control systems with Takagi–Sugeno–Kang PI-FCs. The method is based on the combination of IFT and fuzzy control, and it aims discrete-time input affine SISO processes. Starting with a poor process model and using a linear controller, the control system performance can be improved in two steps. The first step concerns the

197

IFT, and the second step is related to the use of fuzzy control. The tuning of fuzzy controllers is done in offline manner. The control laws and the stability analysis have been formulated in the discrete-time domain which is essential where the sampling rates are quite low. This discrete-time formulation is required by IFT although convenient continuous-time stability analysis approaches for such systems are given in the literature. The stability analysis results presented here can be extended without difficulties to the design of Mamdani fuzzy controllers with singleton consequents. The application of the stability analysis is relatively simple for practitioners because it makes use of quadratic terms (19) in the definition of the Lyapunov function candidate. The case study treated in this paper shows very good results in the control of a nonlinear process using the application of IFT to a simplified linear model of the process. The minimum of the objective function cannot be guaranteed, but our experiment- and databased tuning proves the improvement of the control system performance indices including the objective function values. The control system performance can be improved further in terms of the fuzzy logic-based compensation of the nonlinearity of the process, but this would lead to discontinuous input-output maps of the fuzzy controllers that do not allow the systematic tuning by means of stability and convergence analysis. The application of the hyperstability theory produces an elegant way to solve the problem of convergence analysis which is not a simple task. Therefore a relatively simple, general and easily algorithmic convergent IFT algorithm is proposed in this paper. However the two indices like speed of convergence and magnitude of oscillations in the dynamics of the controller parameters are not analyzed resulting in the first limitation of the paper. Measures to assess and / or impose analytically those two indices are necessary. The second limitation of the paper is that it produces sufficient inequality-type stability and convergence conditions. Although they are more transparent than the LMIs, a natural objective is to make them stronger. However the finding of the Lyapunov function depends on the process model. Future research will be focused on the extension of the theoretical approaches to nonlinear chaotic systems and to MIMO systems. The reduction of the conservativeness of the stability analysis will be tacked in terms of offering systematic methods to find the matrix P. The inclusion of gradient estimates in evolutionary-based optimization algorithms will be targeted in order to reduce the number of objective function evaluations. Acknowledgements This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-2011-3-0109, and by a grant from the NSERC of Canada. Appendix 1. Proof of Theorem 1 This appendix presents the proof of Theorem 1 dedicated to the global asymptotic stability of the equilibrium point at the origin of the fuzzy control systems. This theorem is supported by the following well acknowledged result based on Lyapunov’s direct method for discrete-time systems (Slotine & Li, 1991): let the process be characterized by the state-space model (11). If there exists a continuous radially unbounded Lyapunov function candidate V : X ? R such that V(x) > 0, 8x 2 X n f0g, V(0) = 0, and

Vðxðt þ 1ÞÞ < VðxðtÞÞ;

ð42Þ

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R.-E. Precup et al. / Expert Systems with Applications 40 (2013) 188–199

then the equilibrium point at the origin xðtÞ ¼ 0 ¼ ½ 0 0 . . . 0T 2 X  Rn of the system (11) will be globally asymptotically stable. The hypothesis (20) of Theorem 1 leads to

DV k ðxðtÞÞ ¼ V k ðxðt þ 1ÞÞ  V k ðxðtÞÞ < 0; 8x 2

X Ak ;

k ¼ 1 . . . nRB :

T

DVðxðtÞÞ ¼ f ðxðtÞÞPfððtÞÞ  xT ðtÞPxðtÞ T

þ b ðxðtÞÞPbðxðtÞÞu2 ðtÞ h i T T þ f ðxðtÞÞPbðxðtÞ þ b ðxðtÞÞPfðxðtÞÞ uðtÞ: The following inequality is next obtained from (50) and (51):

ð43Þ The term x(t + 1) is next substituted from (11) into (43)

DV k ðxðtÞÞ ¼ V k ðfðxðtÞÞ þ bðxðtÞÞuk ðtÞÞ  V k ðxðtÞÞ

T

DVðxðtÞÞ < f ðxðtÞÞPfðxðtÞÞ  xT ðtÞPxðtÞ ( ),( ) nRB nRB X X T 2 þ b ðxðtÞÞPbðxðtÞÞ ½ak ðxðtÞÞuk ðtÞ ak ðxðtÞÞ

¼ ½fðxðtÞÞ þ bðxðtÞÞuk ðtÞT P½fðxðtÞÞ þ bðxðtÞÞuk ðtÞ

h

k¼1

i þ f ðxðtÞÞPbðxðtÞ þ b ðxðtÞÞPfðxðtÞÞ uðtÞ:

T

 xT ðtÞPxðtÞ ¼ f ðxðtÞÞPfðxðtÞÞ  xT ðtÞPxðtÞ T

T

ð52Þ

T

T

þ b ðxðtÞÞPfðxðtÞÞuk ðtÞ < 0:

ð44Þ

The multiplication of (44) by ak(x(t)), and the calculation of the sum result in nRB X T T ½f ðxðtÞÞPfðxðtÞÞ  xT ðtÞPxðtÞ ak ðxðtÞÞ þ b ðxðtÞÞPbðxðtÞÞ k¼1 nRB X T T  ½ak ðxðtÞÞu2k ðtÞ þ ½f ðxðtÞÞPbðxðtÞÞ þ b ðxðtÞÞPfðxðtÞÞ k¼1 nRB X

½ak ðxðtÞÞuk ðtÞ < 0:

k¼1

T

PnRB

k¼1

ak ðxðtÞÞ > 0 and the sums are

T

f ðxðtÞÞPfðxðtÞÞ  xT ðtÞPxðtÞ þ b ðxðtÞÞPbðxðtÞÞ ( ),( ) nRB nRB X X  ½ak ðxðtÞÞu2k ðtÞ ak ðxðtÞÞ þ ½f T ðxðtÞÞPbðxðtÞÞ k¼1

k¼1

( ),( ) nRB nRB X X þ b ðxðtÞÞPfðxðtÞÞ ½ak ðxðtÞÞuk ðtÞ ak ðxðtÞÞ < 0: T

k¼1

k¼1

ð46Þ The inequality (46) is next transformed accounting for (16): T

T

f ðxðtÞÞPfðxðtÞÞ  xT ðtÞPxðtÞ þ b ðxðtÞÞPbðxðtÞÞ ( ),( ) nRB nRB X X ½ak ðxðtÞÞu2k ðtÞ ak ðxðtÞÞ þ ½f T ðxðtÞÞPbðxðtÞÞ  k¼1

k¼1

T

þ b ðxðtÞÞPfðxðtÞÞuðtÞ < 0:

ð47Þ

Cauchy–Buniakovski–Schwarz’s inequality results in

( )( ) nRB hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 nRB hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 X X ak ðxðtÞÞ ak ðxðtÞÞuk ðtÞ k¼1

k¼1

( ) nRB hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 X P ak ðxðtÞÞ ak ðxðtÞÞuk ðtÞ ;

ð48Þ

k¼1

which is equivalent to nRB X ½ak ðxðtÞÞu2k ðtÞ P k¼1

( nRB X

)2 ,( nRB X

½ak ðxðtÞÞuk ðtÞ

k¼1

The division of (49) by

)

ak ðxðtÞÞ :

ð49Þ

k¼1

PnRB

k¼1

ak ðxðtÞÞ > 0 using (16) leads to

( ),( ) nRB nRB X X 2 ½ak ðxðtÞÞuk ðtÞ ak ðxðtÞÞ k¼1

k¼1

(( ),( ))2 nRB nRB X X ½ak ðxðtÞÞuk ðtÞ ak ðxðtÞÞ ¼ u2 ðtÞ: P k¼1

k¼1

But, the expression of DV(x(t)) results from (11) and (19):

Eqs. (47) and (52) result finally in

DVðxðtÞÞ < 0:

ð53Þ

Therefore, the equilibrium point at the origin x = 0 will be globally asymptotically stable. The proof is now complete. Concluding, Theorem 1 offers sufficient stability conditions concerning the class of fuzzy control systems defined in Section 3. References

ð45Þ

Eq. (45) is next divided by manipulated as follows:

k¼1

T

þ b ðxðtÞÞPbðxðtÞÞu2k ðtÞ þ ½f ðxðtÞÞPbðxðtÞ



ð51Þ

ð50Þ

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