Fuzzy linear pulse-transfer function-based sliding-mode control for

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

187

Fuzzy Linear Pulse-Transfer Function-Based Sliding-Mode Control for Nonlinear Discrete-Time Systems Chih-Lyang Hwang

Abstract—In this paper, a nonlinear discrete-time system in the presence of input disturbance and measurement noise is approxsubsystems described by the linear pulse-transfer imated by functions with the subjection of input disturbance and measurement noise. Although the input disturbance and the measurement noise are unknown, they are modeled as known pulse-transfer functions. The approximation error between the nonlinear discrete-time system and the fuzzy linear pulse-transfer function system is represented by the linear time-invariant dynamic system in every subsystem, whose degree can be larger than that of the corresponding subsystem. Besides the input disturbance and the measurement noise, the uncertainties are caused by the approximation error of fuzzy-model and the interconnected dynamics resulting from the other subsystems. Owing to the presence of input disturbance, measurement noise, or uncertainties, a disadvantageous response occurs. Based on Lyapunov redesign, the switching control in every subsystem is designed to reinforce the system performance. Due to the time-invariant feature for a constant reference input, the operating point can approach the sliding surface in the manner of finite-time steps. The stability of the overall system is verified by Lyapunov stability theory. The simulations are given to confirm the usefulness of the suggested control. Index Terms—Dead-beat control, discrete-time sliding-mode control, fuzzy linear pulse-transfer function, internal model principle, Lyapunov redesign, Lyapunov stability theory.

I. INTRODUCTION

N

ONLINEARITIES of practical control systems always exist. If the phenomenon of nonlinear system is dominant, then it is not suitable to use a linear controller to cope with this kind of control problem. There are many approaches for the control of nonlinear systems, e.g., sliding-mode control [1], [2], neural-network control [3], [4], and fuzzy control [5], [6]. Each method has its own advantages and disadvantages. For example, a sliding-mode control is robust to the uncertainty; however, a chattering control input probably occurs. The approximation of any continuous function using a neural-network seems outstanding and effective. However, a compact set for approximation is not bright to estimate, the number of neuro is not clear to determine, and it is probably time consuming for an online learning. Furthermore, the instability caused by the parameter drift probably happens [6]. The major advantage of fuzzy control is that a mathematical model for the system is not required. Only some input–output data are employed to obtain an effective control or modeling. However, the heuristics-based fuzzy control or modeling is lacking in systematic Manuscript received February 23, 2001; revised July 11, 2001. This work was supported by Basic Research of Tatung University under Grant number B88-1100-05. The author is with the Department of Mechanical Engineering, Tatung University, Taipei, 10451 Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier S 1063-6706(02)02961-2.

design or stability analysis or stability proof. Furthermore, its performance is usually not more excellent than that of the other controllers. Although many adaptive fuzzy controls (e.g., [7], [8]) improve the system performance, it has the similar drawbacks of online neural-network control. Due to the shortcomings of heuristics-based fuzzy control or modeling, a model-based fuzzy control or modeling, e.g., Takagi and Sugeno linear model based control [9]–[15], is a promising method to reinforce the control performance of fuzzy system. linear continuous-time (or discrete-time) stateThey use space or transfer function dynamic subsystems for different fuzzy sets. However, the stability of these fuzzy control systems [9]–[13] is that the solution a common positive definite matrix or the solution of Riccati differential equation is required for every subsystem. Johansen et al. [13] tries to interpret and identify the dynamic Takagi–Sugeno fuzzy model. They define a multiobjective identification problem, namely, the construction of a dynamic model that is a good approximation of both local and global dynamics of the underlying system. However, these objectives are often conflicted. Alternatively, a robust controller is required to obtain a better system performance. Furthermore, the aforementioned approaches [9]–[13] need a state estimator if the state is not accessible. Under the circumstances, the structure of closed-loop system becomes complex. Recently, Xie and Rad [14] used a local transfer function to represent the consequence of the th fuzzy rule. Then a local minimization -optimal) control with online of integral-square-error (or learning of control parameters is designed. However, the paper cannot reject the input disturbance or the measurement noise. For obtaining a zero steady-state error, the reference model must contain the unstable zero [16]. In [14], if the unstable zeros of subsystem are unknown or the error of parameter estimation happens, the steady-state error occurs. Moreover, the aforementioned studies [9]–[14] only use a linear state-feedback control for every subsystem; their robustness is probably poorer than that using a nonlinear control for every subsystem (cf. [15] for the details). Because of the rapid development of personal computer and DSP chip, a discrete-time controller becomes more and more important nowadays [17], [18]. In this paper, the subsystem is expressed as a linear pulse-transfer function with the consideration of the input disturbance and the measurement noise modeled as known pulse-transfer functions. The approximation error between the nonlinear discrete-time system and the fuzzy linear pulse-transfer function is characterized by the linear time-invariant dynamic system in every subsystem, whose degree can be larger than that of the corresponding subsystem. In a similar way of decentralized control, the interconnected dynamics caused by the other subsystems occurs. The uncertainties caused by the approximation error of fuzzy-model and the interconnected

1063-6706/02$17.00 © 2002 IEEE

188

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

dynamics often result in a poor performance or even instability. Due to the time-invariant feature for a constant reference input and the existence of finite-order system, the operating point reaches to the sliding surface in finite-time steps. Due to the presence of input disturbance or measurement noise or uncertainty, a huge transient response or a poor steady-state response occurs. In this situation, the fuzzy switching control based on Lyapunov redesign is designed to improve the system performance. It is not necessary to assume that the matching condition of uncertainties must be satisfied. The upper bound of the unmodeled dynamics for the design of fuzzy switching control has a limit [19]. For a time-varying trajectory tracking, the overall system is not linear time-invariant. Its operating point is only in the vicinity of sliding surface; however, the performance of the present paper is good enough as compared with other controllers. In summary, the proposed fuzzy linear pulse-transfer function based sliding-mode control (FLPTFBSMC) includes a fuzzy equivalent control and a fuzzy switching control. The purpose of this paper is to combine the advantages of the model-based fuzzy control (or modeling) and the robust control concept (e.g., sliding-mode control, internal model principle, and Lyapunov redesign) to develop an easy and effective fuzzy robust controller for a class of nonlinear discrete-time systems. Finally, the stability of the nonlinear discrete-time system controlled by the proposed control is verified by Lyapunov stability theory. The simulations for constant and sinusoidal reference inputs are arranged to verify the effectiveness of the proposed control.

where and denote the system output, the conrepresents troller output and the system input, respectively, an unknown input disturbance but modeled as pulse-transfer represents a nominal nonfunction, linear continuous function operator achieved from the approximation of nonlinear continuous system (e.g., bilinear transformation) or system identification using input–output data (e.g., denotes the unleast-squares parameter estimation), and certainty operator. A fuzzy dynamic model to represent local linear input–output relations of nonlinear discrete-time systems is described by the fuzzy IF–THEN rules. The th rule of this fuzzy dynamic model for the nominal nonlinear discrete-time system is expressed as the following form: is

System Rule

and

is

THEN (2) is the known delay where ) for , where is the number of (i.e., denotes the output from the th IF–THEN IF–THEN rules, are premise variables that are funcrules, and tions of . Assume that and for are known and coprime. The output of the overall fuzzy system is inferred as follows:

II. MATHEMATICAL PRELIMINARIES Throughout this paper, a discrete-time signal at sampling (i.e., ) of continuous signal is represented interval . The notation denotes a fuzzy set by In addition, denotes a fuzzy term of premise variable denotes a scalar of selected for rule . multiplication. A polynomial representation is defined as fol, where for lows: denote bounded real coefficients, stands for ), and the system degree (i.e., if denotes the backward-time shift operator (i.e., ). The notations of denote the stable , respectively. Without loss of genand unstable part of is assumed to be monic (i.e., erality, the polynomial ) to obtain a unique factorization. The upper sub, represents the polynomial script of a polynomial, e.g., is of the th subsystem. The symbol ess. is used. The notation is the Kronecker delta: , defined [20]. The symbol , otherwise. if III. PROBLEM FORMULATION Consider the following nonlinear discrete-time systems:

(1)

(3) , and . In addition, . Assume that the fuzzy controller shares the same fuzzy sets with the fuzzy system (2)

where

Control Rule

IF is THEN

and

is

(4) stands for the real system output for feedback where denotes an unknown measurement noise but control, can be modeled as a pulse-transfer function, the polynoand are found to obtain the mials equivalent control of the th subsystem (2) such that the th subsystem (2) is stable and the specific trajectory tracking is represents the switching control of accomplished, and the th subsystem (2) to reinforce the system performance of the th subsystem (2). Then, the overall fuzzy control is inferred as follows:

(5)

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

189

Fig. 1. Control block diagram.

Based on the approximation of the fuzzy linear model (e.g., [9]–[15]), (1) is approximated by the overall fuzzy model with approximation error; i.e., there exist the following uncertain and , where and polynomials , such that

where (9) (10) denotes the nominal closed-loop characteristic The represents the polynomial of the th subsystem, the uncertainties of the th subsystem caused by the approximation ) and the interconnected error (i.e., ]. The dynamics resulting from the other subsystems [i.e., in (8) must be equal to zero, i.e., term in braces

(6) (11)

where , and Substituting (5) into (6) yields

is coprime.

Equation (11) denotes the output of the closed loop of the th subsystem. It is assumed that the reference input, the input disturbance, and the measurement noise are modeled as follows:

(12) (7)

dewhere notes an interconnected dynamics of the th subsystem caused by the other subsystems. After some mathematical manipulations

(8)

and are coprime pairs. Define the following sliding surface:

where

(13) and is a stable monic polywhere nomial. The objectives of this paper are described as follows (see Fig. 1): i) a fuzzy control based on a fuzzy linear pulse-transfer function (2) is constructed to stabilize the nonlinear system (6) subject to input disturbance, measurement noise, and uncertainties; ii) the response for constant reference input approaches the sliding surface in a manner of finite-time steps; iii) based on the Lyapunov redesign, the switching control of the th subis designed to improve the system perforsystem, i.e., mance. Comparison between the proposed control and the proposed control without the fuzzy switching control is also given.

190

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

IV. FUZZY EQUIVALENT CONTROL The fuzzy subsystem (2) controlled by the proposed control is considered in this section. (4) with A. Dead-Beat Sliding Surface of the th Nominal Subsystem First, the input disturbance and the measurement noise are . For the dead-beat assumed to be zero, i.e., must have the to the sliding surface, the sliding surface following form: (14) is a polynomial of the th subsystem with the where degree which is the same as the number of dead-beat steps. and the polynomial Remark 1: Because in (12) is the same for every subsystem, the following exists. fact that Substituting (11) and (12) into (13) gives

Because the control system (1) and the proposed control (5) are autonomous and finite-order, for a constant reference input the steady-state responses of system output and control input are constants. Suppose there exists a finite such that as constant for . Then, the overall linear time-invariant system (22) is dead-beat to the sliding surface

(22) where . Hence, the trajectory reaches the sliding surface in the same way as finite-time steps. B. Subjection of Input Disturbance and Measurement Noise Consider Section 4-A with the input disturbance and the mea. From (11)–(13) and surement noise, i.e., (14), (23) is achieved, as shown at the bottom of the page. Let (24) (25)

(15)

Then

Comparing (14) and (15) yields (26) (16) where is stable, the denominator of the right-hand Because side of (16) must be stable. Therefore, the following equations are achieved: (17)

(27) Then, the characteristic polynomial of the nominal closed-loop system becomes

(18) where monic polynomial and the polynomial Assume that

and are coprime, the has the degree contains the minimum degree.

(28) or

(19) From (9) and (16)–(19), the following equations are accomplished:

(20) (21) where

is a stable polynomial with the degree .

(29) where

and . In addition,

is the same as (21). Theorem 1: Consider the system (6) and the controller (5) . The polynomials with and are achieved from (24), (25), (29) and (21). Then, the response for a constant reference input arrives to the sliding surface at most

(23)

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

191

steps (i.e., as ), denotes a time step that input and output of the th where is subsystem approach steady-state values. In addition, as . bounded and is stable, is constant Proof: Because and the closed-loop system is time invariant with finite order, the input and output of the th subsystem approach steady-state values. From the result of Section 4-A, and as a time-invariant overall system (22) is achieved. Based on the definition of Kronecker delta operator [17], [18], ; i.e., the trajectory of th subsystem approaches the sliding time steps. Then the total time surface at most for the trajectory to reach the sliding surface is at most steps (i.e., , as ). Because is stable, as , and is bounded. Remark 2: The polynomials and denote and , respectively. For example, the modes of is a periodic signal with period where represents the . In a similar way, control cycle time. Then (Hz) the fola sinusoidal input disturbance with frequency lowing polynomial is set. Furthermore, the measurement noise (i.e., ) is typically of high frequency [17]. One way to deal with it is to make .

V. FUZZY SWITCHING CONTROL Like convectional SMC, the fuzzy switching control is employed to cope with the unmodeled dynamics such that the system performance is improved. The proposed fuzzy switching control is designed as follows:

(34) is the same as (28), is a causal stable where will be discussed later. rational weighting function and Substituting (11), (24), (26), (28), (29) and (34) into (13) gives

(35) . The first term where in of (35) is obtained from the last section. The signal , it must be decomposed (35) includes the effect of , into two parts for the stability analysis. One includes and the other does not contain it, i.e., from (10) and (34) (36a) where

C. The Stability of the Closed Loop of the th Subsystem First, the sensitivity functions of the nominal and real closed-loop systems of the th subsystem are given as follows:

(36b)

(30)

(36c)

and denote the nominal and real loop where pulse-transfer functions of the th subsystem (31)

(36d) Define the difference of

as follows:

(32) (37) The following theorem about the robustness of the closed-loop of the th subsystem is introduced. Theorem 2 [17], [20]: Consider the closed-loop , where is stable. The system and (or number of zero of and ) in is same. If the following condition: , (or ) is stable. is satisfied, then Under the conditions in Theorem 2, the following characteristic polynomial of the real closed-loop system of the th subsystem is stable: (33)

Then, from (34), (35), and (37)

(38) where (39) (40)

192

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

According to (40), (36), and (13), the upper bound of estimated as follows:

is (41)

First, the case:

, is considered. Suppose , where . Then, the following equation is achieved by using (37), (42), (47), (48), and Remark 1

where , satisfying the following inequality: on

(42)

. Then the control in where (43), as shown at the bottom of the page, is employed to deal , where with the

(44) The switching gain of (43) satisfies the following inequality: (45) where (46)

(52)

(47)

(48) Theorem 3: Consider the system (6) and the controller in (34) and in (43). The conditions of (5) with Theorem 2, (41) and (42) are satisfied. Then are bounded and the following tracking performance (49) is achieved.

where for Because

. If , then

(or

(53) and for the inequalities , are obtained. Substituting (41) into (53) gives

(54)

(49)

From (53) and (54), the results (43) and (44) are accomplished. In summary, the switching gain chosen from (45) makes . Then, are bounded and the performance (49) is accomplished. , can be obtained. Similarly, the case

Proof: Define the following Lyapunov function:

as

, ).

(50)

VI. SIMULATIONS AND DISCUSSIONS Consider a nonlinear discrete-time system described by the is and following nine system rules: System Rule : IF is , THEN

The change rate of (50) is given as follows:

(51)

(55)

if otherwise

(43)

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

193

where (56)

and one extra order with large coefficients. The premise variable and represent the physical position signal and velocity signal, respectively. The fuzzy sets of and are defined as the modified Gaussian membership functions shown in (58) and (59) at the bottom of the page, denotes and . where The corresponding premise membership functions are depicted in Figs. 2(g) and (h). . A constant reference input is assigned as follows: and . The equivaLet lent control design contains the modes of the reference input, the input disturbance and the measurement noise: and . In addition, the sliding surface with the coefficients: and in Diophantine equation are selected. uses the following The fuzzy switching control control parameters: , (where ) and . The corresponding responses are shown in Fig. 2. The solid and dotted lines in Figs. 2(a), (b), and (c) denote the responses of the proposed , i.e., without control and the proposed control with the fuzzy switching control, respectively. From Fig. 2(a) and (b), one realizes that the tracking performance are excellent and the control input are smooth enough. The trajectory reaches the sliding surface in 22 or 25 time steps (cf. Fig. 2(c)). Indeed, the fuzzy switching control improves the system performance. Moreover, the overall parameters of the fuzzy system (22) (i.e.,

(57) The nominal subsystem is assumed to be the second-order system with the constant coefficients but different delays (cf. , where . The (57)); those are nominal systems include different delays, nonminimum phase and unstable features. In addition, the real nonlinear system is different from the nominal system by 20% parameter variations

indeed converge to constant values (cf. Fig. 2(d)). The poles and zeros of the overall fuzzy system (22) are depicted in Fig. 2(e) and (f); their steady-state values are and , respectively. The system is stable and nonminimum phase ). (i.e., Similarly, the responses by using the weighting function are shown in Fig. 3. For further demonstration of the effectiveness of the fuzzy switching control, the different initial conditions of the output is substituted for the original zero initial conditions. The comparative output responses for different initial conditions are

otherwise otherwise

(58)

otherwise otherwise

(59)

194

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

0

0

Fig. 2. The responses for r(k ) = 1 with the subjection of v (k ) = 1:5; w(k ) = ( 1) 1:5. (a) y (k ) for the proposed control (—) and the control with (k ) = 0( ). (b) u(k ) for the proposed control (—) and the proposed control with u (k ) = 0( ). (c)  (k ) for the proposed control (—) and the proposed control with u (k ) = 0( ). (d) a (k )(–); a (k )(. . .); a (k )(–); b (k )( ); b (k )( ); b (k )(+); (b )(k )(- -). (e) Overall poles. (f) Overall zeros. (g) Fuzzy membership function for premise variable z (k ). (h) Fuzzy membership function for premise variable z (k ).

u

111

111

3

1

111

1

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

Fig. 3.

:

(1 15

The output responses of Fig. 2 case for  (q ).

0 0:46q

:

= (1 31

0 0:32 )=

(a)

195

ferent initial conditions such that the system performance is improved. Similarly, a sinusoidal reference input is assigned as follows: , where Hz and sec. Let and . The fuzzy equivalent control design includes the modes of the reference input, the input disturbance and the measurement noise: and . The other control parameters are the same as the aforementioned case is replaced of constant reference input except that . A larger switching gain is selected to deal with by the larger uncertainties caused by the tracking of a sinusoidal trajectory [cf. Figs. 2(e), (f) and 5(e), (f)]. The responses for the are shown in Fig. 5. Fig. 5(a) proposed control with points out that the tracking performance is good enough except the transient response. Because the premise variables defined and those in (58) and (59) are functions of are not constant values in the steady state, an overall linear time-invariant dynamic system (22) can’t be obtained. Hence, the sliding surface can’t approach to zero in finite time [cf. Fig. 5(c)]. Steady-state responses of the overall system parameters in Fig. 5(d) are periodic. Therefore, the overall poles and zeros are time-variant [cf. Fig. 5(e) and (f)]. Comparisons between Fig. 2(e), (f) and Fig. 5(e), (f) give that the dynamics of time-varying reference input is more complex than that of constant reference input. Furthermore, the replacement of by makes the control result using only the fuzzy equivalent control unstable. Under the circumstances, the output responses for the proposed control with and are shown in Fig. 6. The transient response is much improved (cf. Fig. 5(a) and Fig. 6). The output response using the proposed control with for the Fig. 5 case except is shown in Fig. 7. In short, the fuzzy switching control indeed reinforces the system performance that can’t be obtained from the traditional fuzzy linear feedback control [15]. If the frequency of the sinusoidal reference input is small enough, the operating point almost reaches the sliding surface in the sense of finite-time steps [e.g. Fig. 5(c)]. Due to space limits, it is omitted. VII. CONCLUSION

(b) Fig. 4. The output responses of Fig. 2 case for different initial conditions by using the proposed control (—) and the proposed control with u (k ) = 0( ). (a) y (0) = 1. (b) y (0) = 1.

111

0

then shown in Fig. 4. It reveals that the objective of the fuzzy switching control is to cope with the uncertainties including dif-

Based on internal model principal, a fuzzy equivalent control is used to deal with the input disturbance and the measurement noise. The approximation error of fuzzy-model is represented by the linear time-invariant dynamic in every subsystem. Then the operating point of the finite-order system for a constant reference input can reach the sliding surface in a manner of finite-time steps. The uncertainties caused by the approximation error of fuzzy-model and the interconnected dynamics resulting from the other subsystems are not necessarily small. Owing to the subjection of input disturbance or measurement noise or uncertainties, a fuzzy switching control based on Lyapunov redesign is employed to reinforce the system performance. Although the tracking of time-varying reference input cannot have

196

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

(a)

(b)

(c)

(d)

(e)

(f)

0

Fig. 5. (a)–(e) The responses for r (k ) = sin(0:02k); v (k ) = 0:8; w(k ) = ( 1) 0:5 by using the proposed control with u (k ) = 0.

an overall linear time-invariant system, the trajectory is in the neighborhood of the sliding surface; the performance is still acceptable. In addition, the proposed control does not require a state estimator or the solution of a common positive–definite

matrix for every subsystem. The delay of subsystem is not necessarily the same. The proposed control scheme amalgamates the advantages of the model-based fuzzy control (or modeling) and the robust control (e.g., sliding-mode control, internal

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002

197

model principle, Lyapunov redesign) to accomplish an effective controller for a class of nonlinear discrete-time systems. The further work is to use the proposed control for the trajectory tracking of the piezoelectric actuator systems. The extension to multivariable systems with application to robot is also under investigation. REFERENCES

(a)

(b) Fig. 6. The output responses using the proposed control with different weighting functions. (a)  (q ) = 1. (b)  (q ) = (1:31 0:32q )= (1:15 046q ).

0

0

Fig. 7. The output response using the proposed control with  (q ) = : 0:32q )=(1:15 0:46q ) for the Fig. 5 case except v (k ) = 0:85.

(1 31

0

0

[1] K. D. Young, V. I. Utkin, and Ü. Özgüner, “A control Engineer’s guide to sliding mode control,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 328–342, May 1999. [2] K. Furuta and Y. Pan, “Sliding-mode control with sliding sector,” Automatica, vol. 36, no. 3, pp. 211–228, 2000. [3] A. U. Levin and K. S. Narendra, “Control of nonlinear dynamical systems using neural networks: Controllability and stabilization,” IEEE Trans. Neural Networks, vol. 4, pp. 192–206, Apr. 1993. [4] S. Jagannathan and F. L. Lewis, “Multilayer discrete-time neural-net controller with guaranteed performance,” IEEE Trans. Neural Networks, vol. 7, pp. 107–130, Feb. 1996. [5] Y. H. Joo, L. S. Shieh, and G. Chen, “Hybrid state-space fuzzy modelbased controller with dual-rate sampling for digital control of chaotic systems,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 394–408, Aug. 1999. [6] C. L. Hwang and C. Y. Kuo, “A stable adaptive fuzzy sliding-mode control for nonlinear systems with application to four-bar-linkage systems,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 238–252, Apr. 2001. [7] J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 339–359, June 1996. [8] J. Campos and F. L. Lewis, “Deadzone compensation in discrete time using adaptive fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 697–707, Dec. 1999. [9] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issue,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996. [10] B. S. Chen, C. S. Tsen, and H. J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 571–585, Oct. 1999. [11] S. J. Wu and C. T. Lin, “Optimal fuzzy controller design: Local concept approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 171–185, Apr. 2000. [12] E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 523–534, Oct. 2000. [13] T. A. Johansen, R. Shorten, and R. Murray-Smith, “On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 297–313, June 2000. [14] W. F. Xie and A. B. Rad, “Fuzzy adaptive internal model control,” IEEE Trans. Ind. Electron., vol. 47, pp. 193–202, Feb. 2000. [15] G. Feng, S. G. Cao, N. W. Rees, and C. K. Chak, “Design of fuzzy control systems with guaranteed stability,” Fuzzy Sets Syst., vol. 85, no. 1, pp. 1–10, 1997. [16] C. L. Hwang and B. S. Chen, “Adaptive control of optimal model matching in H -norm space,” in Proc. IEE D, Contr. Theory, Applicat., vol. 135, Apr. 1988, pp. 295–301. [17] K. J. Aström and B. Wittenmark, Computer-Controlled Systems—Theory and Design, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1997. [18] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [19] I. Haskara and V. Utkin, “Variable structure control for uncertain sampled data systems,” in Proc. 36th IEEE Conf. Decision Control, San Diego, CA, 1997, pp. 3226–3231. [20] M. Vidyasagar, Optimal Control System Synthesis. Upper Saddle River, NJ: Prentice-Hall, 1986.