Robustness design of nonlinear dynamic systems via fuzzy linear control [PDF]

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 5, OCTOBER 1999

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Robustness Design of Nonlinear Dynamic Systems via Fuzzy Linear Control Bor-Sen Chen, Senior Member, IEEE, Chung-Shi Tseng, and Huey-Jian Uang

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Abstract—This study introduces a fuzzy linear control design robustness permethod for nonlinear systems with optimal formance. First, the Takagi and Sugeno fuzzy linear model is employed to approximate a nonlinear system. Next, based on the fuzzy linear model, a fuzzy controller is developed to stabilize the nonlinear system, and at the same time the effect of external disturbance on control performance is attenuated to a minimum level. Thus based on the fuzzy linear model, performance design can be achieved in nonlinear control systems. In the proposed fuzzy linear control method, the fuzzy linear model provides rough control to approximate the nonlinear control system, while the scheme provides precise control to achieve the optimal robustness performance. Linear matrix inequality (LMI) techniques are employed to solve this robust fuzzy control problem. In the case that state variables are unavailable, a fuzzy control is also proposed to achieve a robust observer-based optimization design for nonlinear systems. A simulation example is given to illustrate the performance of the proposed design method.

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Index Terms— robust control, linear matrix inequality, nonlinear fuzzy observer, Takagi–Sugeno fuzzy control.

I. INTRODUCTION

T

HE control design of nonlinear systems is a difficult process, and in practical control systems the plants are always nonlinear. Thus many nonlinear control methods have been developed for nonlinear systems to overcome the difficulty in controller design for real systems. However, in these control system designs, the nonlinear systems must have some predictable behaviors. For example, the system must be minimum-phase, it must be sufficiently smooth, and the parameters must be exactly known in order for the feedback linearization method to be applied. Furthermore, these control schemes are so complicated that they are not suitable for practical application. control schemes [19], [20] Recently, the nonlinear have been introduced to deal with the robust performance design problem of nonlinear systems. However, the designer has to solve a Hamilton–Jacobi equation, which is a nonlinear partial differential equation. Only some very special nonlinear systems have a closed-form solution. In general, conventional control schemes are not suitable for practical nonlinear control system design. Manuscript received July 8, 1998; revised July 8, 1999. This work was supported by the National Science Council under Contract NSC 88-2213-E007-069. The authors are with the Department of Electrical Engineering, National Tsing Hua University, 30043, Hsin Chu, Taiwan, R.O.C. Publisher Item Identifier S 1063-6706(99)08729-9.

In the past few years, there has been rapidly growing interest in fuzzy control of nonlinear systems, and there have been many successful applications. Despite the success, it has become evident that many basic issues remain to be addressed. The most important issue for fuzzy control systems is how to get a system design with the guarantee of stability and control performance, and recently there have been significant research efforts on the issue of stability in fuzzy control systems [2], [5], [6], [22]–[24]. In [6], an approach was given for the stability of fuzzy design issues of nonlinear systems. In other studies, a nonlinear plant was approximated by a Takagi–Sugeno fuzzy linear model [1], and then a model-based fuzzy control was developed to stabilize the Takagi–Sugeno fuzzy linear model. They all ignore the approximation error between nonlinear system and fuzzy model. In practice, the effect of approximation error will deteriorate the stability and control performance of the nonlinear systems. Therefore, the stability and control performance of nonlinear control systems still need further study. Recently, based on feedback linearization technique, adaptive neural network or fuzzy control schemes have been introduced to deal with nonlinear systems [7]. In [7], even tracking performance has been guaranteed, though the the complicated parameter update law and control algorithm make this control scheme impractical, especially in the case of considering the projection algorithm for the parameter update law to avoid the singularity of feedback linearization control. For practical control design, a simple fuzzy control design with guaranteed control performance is more appealing for nonlinear systems. In this work, the fuzzy linear model of Takagi and Sugeno is used to approximate a nonlinear system. Then, a hybrid fuzzy controller is introduced to stabilize the nonlinear system, and at the same time eliminate the effects of external disturbance below a prescribed level , so that a control performance can be guaranteed. In this desired approach only a linear fuzzy control design is used, without complicated feedback linearization or parameter update law control as in [7], [13], and [28], although, the same performance is achieved. Finally, a fuzzy control design with robustness optimization is also introduced based on the optimal control scheme by minimizing the attenuation level . This proposed method attempts to combine the fuzzy linear performance to obtain a simple but practical model and algorithm for robust performance control design of nonlinear systems. If the state cannot be directly measured, a fuzzy observer-based control design is developed. This situation is more complex because a fuzzy observer must be designed to performance. achieve the optimal

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To deal with this fuzzy observer-based control design, it needs to simultaneously solve multiple quadratic matrix inequalities. These quadratic matrix inequalities can be transformed to equivalent linear matrix inequalities (LMI’s), and these LMI’s are combined into a standard LMI problem (LMIP). This standard LMIP can be solved to complete fuzzy control design. Finally, the optimal the optimal design of fuzzy control system is formulated as a socalled eigenvalue problem (EVP) to minimize the maximum eigenvalue of a matrix that depends on a variable, subject to the LMI constraints. The main contribution of this paper is twofold: i) robust design of nonlinear systems is dealt with via a fuzzy observerbased linear controller and ii) both the stability of the fuzzy attenuaobserver-based control system and the optimal tion of the effect of the external disturbance on the control performance are guaranteed. A simulation example is provided to illustrate the design procedure and performance of the proposed methods. In the proposed hybrid fuzzy control method, the fuzzy linear method provides rough tuning to approximate the nonlinear control scheme provides precise tuning system, while the optimal to achieve an optimal robustness performance. The simulation results show that the optimal robustness performance can be achieved by the proposed method. The paper is organized as follows: the problem description is presented in Section II. Robust stabilization and optimal performance design via fuzzy linear control are described in Section III. In Section IV, a fuzzy observer-based control with robustness optimization is introduced. In Section V, a simulation example is provided to demonstrate control design and to confirm the effectiveness of fuzzy the robust performance. Finally, concluding remarks are made in Section VI. Note that this paper is a modified version of the conference paper in [26].

A fuzzy dynamic model has been proposed by Takagi and Sugeno [1] to represent local linear input/output relations of nonlinear systems. This fuzzy linear model is described by fuzzy If–Then rules and will be employed here to deal with the control design problem of the nonlinear system (1). The th rule of this fuzzy model for the nonlinear system (1) is of the following form [1], [2], [6]: Plant Rule : is If Then

and

and

is (2)

where is the fuzzy set, , is the number of If–Then rules, and , are the premise variables. The overall fuzzy system is inferred as follows [1], [6], [16]:

for

,

(3) where

and is the grade of membership of In this paper, we assume

for

in

.

and

II. PROBLEM DESCRIPTION Consider the following nonlinear system: (1)

for all . Therefore, we get [2], [6]

where

(4) for

and

denotes the state vector, (5) denotes the control input,

denotes the unknown disturbances with a known upper bound , and and depend on . Definition 1 [17]: The solutions of a dynamic system are said to be uniformly ultimately bounded (UUB) if there exist there is positive constants and , and for every , such that a positive constant

Therefore, from (1) we get

(6)

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Suppose that there exist bounding matrices such that

where

and

(12) and

denotes the approximation error between the nonlinear system (1) and the fuzzy model (3). Suppose the following fuzzy controller is employed to deal with the above control system design: Control Rule : is If Then for given by

(13) and the bounding matrices for all trajectory can be described by

and

(14) and

and

is (7)

and where Remark 1: 1) If we assume

, for

[15].

and

. Hence, the overall fuzzy controller is

then the plant rule can be represented as (8)

is defined in (4) and (5) and are the control where ). parameters (for Substituting (8) into (6) yields the closed-loop nonlinear control system as follows:

Plant Rule : If Then

is

and

and

is (15)

2) Obviously, according to assumption above

and

are the worst case representations for the approximation and error if there exist such that (12) and (13) hold for some and (for ). According to assumption above, we get (9) where (10) and

(11)

(16)

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and

on , and is a positive-definite weighting matrix. The on physical meaning of (19) is that the effect of must be attenuated below a desired level from the viewpoint is, i.e., the gain from of energy, no matter what to must be equal to or less than a prescribed value . In general, is chosen as a positive small value less than for attenuation of . The inequality in (19) can be seen as bounded-disturbance and bounded-state but with a prescribed gain . If the initial condition is also considered, the inequality (19) can be modified as

(20)

(17) i.e., the approximation error in the closed-loop nonlinear system is bounded by the specified structured bounding matrices and . Stability is the most important issue in the control systems. Obviously, it is appealing for control engineers to specify in the fuzzy controller (8) such that control parameters the stability for the closed-loop nonlinear system (9) can be guaranteed is unknown but bounded. In this study, we assume that will deteriorate the control perforHowever, the effect of mance of fuzzy control system. Therefore, how to eliminate to guarantee the control performance is an the effect of control is important issue in the control systems. Since the most important control design to efficiently eliminate the on the control system, it will be employed to effect of deal with the robust performance control in (9). Let us consider control performance [7], [13]: the following

(18)

or (19) denotes the terminal time of the control, is a where prescribed value which denotes the worst case effect of

where is some symmetric positive-definite weighting matrix. From the analysis above, the design purpose of the proposed fuzzy control system is to specify a linear fuzzy control (8) such that both the stability of fuzzy linear control system control performance in (20) with a prescribed and the attenuation level are guaranteed. in The robustness optimization is to achieve a minimum . For (20) to obtain maximum elimination of the effect of nonlinear system (1), this design problem is how to specify subject to a stabilizable fuzzy control in (8) to minimize the constraint (20). III.

CONTROL DESIGN VIA FUZZY LINEAR CONTROL

From the description in the above section, the design purpose of this study is how to specify a fuzzy linear control law in (8) for the nonlinear system in (9) with the guaranteed control performance in (20). Let us choose a Lyapunov function for the system (9) as (21) where the weighting matrix The time derivative of

. is (22)

By substituting (9) into (22), we get

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From (24), we get

(26) From (25) and (26), we get

(27) (23) Then, we get the following result. Theorem 1: If the fuzzy controller (8) is employed in the nonlinear system (1) and there exists a positive-definite matrix such that the following matrix inequalities:

in (4) and (5), the above From the properties of inequality can imply the following inequality:

(24) then the closed-loop are satisfied for control performance nonlinear system (9) is UUB and the of (20) is guaranteed for a prescribed . Proof: From (23), we get

(28) Since

, we get

(29) (the minimal eigenvalue of ). where , . According to Whenever a standard Lyapunov extension [21], [25], this demonstrates that the trajectories of the closed-loop system (9) are UUB. to yields Integrating (28) from

(30) From (21), we get

(31) control performance is achieved This is (20) and the with a prescribed . Corollary 1: In the case of , if the fuzzy controller (8) is employed in the closed-loop nonlinear system (9) and such that there exists a positive-definite matrix the matrix inequalities in (24) are satisfied, then the closedloop system (9) is quadratically stable. , from (28) we get Proof: In the case of (25)

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Therefore, the closed-loop system (9) is quadratically stable. This completes the proof. In general, it is not easy to analytically determine a common for (24). Furthermore, the solution solution may not be unique. Fortunately, (24) can be reformulated into the linear matrix inequality problem (LMIP) [15]. The LMIP can be solved in a computationally efficient manner using a technique such as the interior point method. First, the matrix inequalities in (24) are transformed to the equivalent LMI’s by the following procedures. and , By introducing new variables (24) is equivalent to the following matrix inequalities:

The state dynamic of fuzzy system is the same as (3) and the output of the fuzzy system is inferred as follows: (37) Therefore, the output of the nonlinear system in (35) can be rearranged as the following equivalent system:

(38) (32) By the Schur complements [15], (32) is equivalent to the following LMI’s:

where (39)

(33) . for , If the LMI’s in (33) have a positive-definite solution for , then the closed-loop system is stable and the control performance in (20) is guaranteed for a prescribed . optimization design for fuzzy control Therefore, the system of (1) is formulated as the following constrained optimization problem:

denotes the approximation error between the output of nonlinear uncertain system (35) and the fuzzy output (37). Suppose the following fuzzy linear observer is proposed to deal with the state estimation of nonlinear system (1): Observer Rule : is If

and

and

is

Then

(40)

and are the observer where ). parameters (for The overall fuzzy observer can be defined as follows:

minimize subject to

and (33)

(34)

This problem is also called eigenvalue problem (EVP) and can be solved very efficiently by convex optimization algorithm. More details will be discussed in the next section. IV. FUZZY OBSERVER-BASED

(41)

CONTROL

In the previous sections, we assumed that all the state variables are available. In practice, this assumption often does not hold. In this situation, we need to estimate state vector from output for state feedback control. Suppose the nonlinear system to be controlled is of the following form: (35) denotes the output of the system. where The th rule of the fuzzy model for the nonlinear system (35) is of the following form: Plant Rule : is If Then where

Control Rule : is If Then

and

and

is (42)

Hence, the overall fuzzy controller is given by (43)

and

and

is (36)

for

Remark 2: In this situation, the estimated state variables . may be specified as the premise variables, i.e., Suppose the following fuzzy controller is employed to deal with the above control system design:

.

are the control parameters (for where Let us denote the estimation errors as

). (44)

CHEN et al.: ROBUSTNESS DESIGN OF NONLINEAR DYNAMIC SYSTEMS VIA FUZZY LINEAR CONTROL

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Let us denote

By differentiating (44), we get

(49) Therefore, the augmented system defined in (48) can be expressed as the following form:

(45) is defined as (39), where is modified as

(50)

is the same as (10), and

Assume that a bounding matrix such that

where

exists where (51)

(46) for all trajectories. Then, the augmented system is equivalent to the following form:

where

and

(47)

(52) After manipulation, (47) can be expressed as the following form: where

(48)

for

and

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for , then the closed-loop system (50) is control performance of (54) is guaranteed UUB and the for a prescribed . Proof: From (56), we get

(53) where Therefore, the follows:

for . control performance can be modified as

(54) denotes the terminal time of the control, is a where prescribed value which denotes the worst case effect of on , and and are some positive-definite weighting matrix. for the augmented system in (50) Define the Lyapunov as (55) The time derivative of

(58)

is From (57), we get

(59) From (58) and (59), we get

(60) in (4) and (5), the above From the properties of inequality can imply the following inequality:

(56) Then we get the following result. Theorem 2: Suppose the fuzzy control law (43) is employed is the common in the nonlinear system (50), and solution of the following matrix inequalities:

(61) , we get

Since

(62) (57) where

.

CHEN et al.: ROBUSTNESS DESIGN OF NONLINEAR DYNAMIC SYSTEMS VIA FUZZY LINEAR CONTROL

Whenever , . By the argument as in Theorem 1, this demonstrates that the trajectories of the closed-loop system (50) are UUB. to yields Integrating (61) from

where

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and multiplying it into (66), we get

(63) From (55), we get

(64) control performance is achieved This is (54) and the with a prescribed . Corollary 2: In the case of , suppose the fuzzy control law (43) is employed and there exists a common such that the matrix inequalities in (57) solution are satisfied then the closed-loop system (50) is quadratically stable. Proof: The proof is trivial. From the above analysis, the most important task to deal with the fuzzy observer-based state feedback problem is to solve a common solution from the matrix inequalities (57). For the convenience of design, and are chosen in the following form:

(68) Therefore, (68) is equivalent to

(65) , . By substituting (65) into (57), we get

where

,

, and

(69) , and , the matrix inequalities With (69) can be rearranged as the following form:

(70) where

(66) By introducing a new matrix (67) The analysis above shows that when dealing with the fuzzy

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observer-based fuzzy control system, the most important task and is to solve common solutions from the matrix inequalities (70). Since the variables and are cross-coupled, there are no effective algorithms for solving this matrix inequalities as yet. However, we easily check that the matrix inequalities (70) imply , i.e.,

(71) . for , By the Schur complements, (71) can be transformed into the following linear matrix inequalities (LMI’s) for a prescribed :

Fig. 1. Fuzzy sets of x1 .

Step 7) Check the assumptions of

(72) and and (thus ) from (72) Note that solving is a standard linear matrix inequality problem (LMIP). By , , and into solving the LMIP in (72) and substituting (70), (70) become standard linear matrix inequalities (LMI’s). and (thus Then, solve the LMIP in (70) to obtain ). If the LMI’s in (70) and (72) have a positive-definite and , respectively, then the closed-loop solution for control performance in (54) system (50) is stable and the is achieved for a prescribed . optimization design for fuzzy observerTherefore, the based control system of (50) is formulated as the following constrained optimization problem: minimize subject to (70) and (72)

If they are not satisfied, adjust (expand) the bounds , , and and then for all elements in repeat Steps 3–6. Step 8) Construct the fuzzy observer in (41). Step 9) Obtain fuzzy control rule in (43). Remark 3: , , , 1) The procedures for determining , , are described by the following simple example. and Assuming that the possible bounds for all elements in , , and are

(73)

until This problem can be solved by decreasing and cannot be found in (70) and (72). optimization design procedures of the fuzzy conThe trol systems are summarized as follows. Design Procedures: Step 1) Select fuzzy plant rules and membership function for nonlinear system (1). , and bounding Step 2) Select weighting matrix and and matrices . and solve the LMIP Step 3) Select the attenuation level and (thus in (72) to obtain can also be obtained). , , and into (70) and then solve Step 4) Substitute and (thus the LMIP in (70) to obtain can also be obtained). and repeat Steps 3–5 until and Step 6) Decrease cannot be found.

and

where

, for some

,

, and

, and and

, , , . One possible description for the bounding matrices , , and is

where for , ,

, .

, and

CHEN et al.: ROBUSTNESS DESIGN OF NONLINEAR DYNAMIC SYSTEMS VIA FUZZY LINEAR CONTROL

Fig. 2.

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The trajectories of stataes x1 and x2 (including estimated states x ^1 and x^2 ). (x1 : solid line; x2 : dashed line; x^1 : dotted line; x^2 : dash–dot line.)

Note that , , and can be chosen by other forms , , and . Then as long as follow the design procedures and check (12), (13), and (46) in the simulation. If they are not satisfied, adjust , , and (expand) the bounds for all elements in , and repeat the design procedures until (12), (13), and (46) hold. 2) In general, it is not easy to solve the EVP or LMIP analytically. Fortunately, the EVP or LMIP can be solved very efficiently using convex optimization techniques such as interior point algorithm [15]. Software packages such as LMI optimization toolbox in Matlab are developed for this purpose [27]. V. SIMULATION EXAMPLE To illustrate the fuzzy linear control approach, a balancing problem of an inverted pendulum on a cart is considered. For this example, the state equations of the inverted pendulum is given by

gravity constant, is the mass (kg) of the pendulum, is is the friction factor (N/rad/s) the mass (kg) of the cart, of the pendulum, is the length (m) from the center of mass of the pendulum to the shaft axis, is the moment of inertia (kg m ) of the pendulum, is external disturbance, and is the force (N) applied the cart. The pendulum parameters (kg), (kg), (m), are chosen as (kg m ), and (N/rad/s). Now, following the design procedure in the preceding section, the robust performance design is given by the following steps. Steps 1 and 2: To use the fuzzy linear control approach, we must have a fuzzy model which represents the dynamics of the nonlinear plant. Therefore, we first represent the system (74) by a Takagi–Sugeno fuzzy model. To minimize the design effort and complexity, we try to use as few rules as possible. Hence, we approximate the system by the following four-rule fuzzy model: Rule 1: IF

is about

THEN Rule 2: IF

is about

THEN Rule 3: IF (74) where vertical,

denotes the angle (rad) of the pendulum from the is the angular velocity (rad/s), 9.8 m/s is the

is about

THEN Rule 4: IF THEN

is about

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Fig. 3. The control input.

where

as follows:

Step 7: The assumptions of

and the bounding matrices are chosen as and

, , , (for ). Membership functions for Rules 1–4 are shown in Fig. 1. Select are satisfied (refer to Figs. 4 and 5). Step 8: Then, we construct the observer as Steps 3–6: The optimal is found after several iterations using the LMI optimization toolbox in Matlab. In this case, we obtain the common solution for (70) and (72)

CHEN et al.: ROBUSTNESS DESIGN OF NONLINEAR DYNAMIC SYSTEMS VIA FUZZY LINEAR CONTROL

Fig. 4. The plots of

Fig. 5. The plots of (solid line).

f (x(t)) 0

4 i=1

4 i=1

hi (x1 (t))Ai x(t)

hi (x1 (t))

4 j =1

(dashed line) and

hj (x1 (t))(g(x(t)) 0 Bi )Kj x^(t)

4 i=1

hi (x1 (t))1Ai x(t)

(dashed line) and

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(solid line)

4 i=1

hi (x1 (t))

4 j =1

hj (x1 (t))1Bi Kj x^(t)

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Step 9: Therefore, we obtain the control law

Figs. 2–5 present the simulation results. Initial condi, , , tion is assumed to be , and the external disturbance is assumed in the to be periodically square wave with amplitude and simulations. Fig. 2 shows the trajectories of the states (including estimated states and ). The control input is presented in Fig. 3. The simulation results show that the fuzzy observer-based controller can balance the inverted pendulum performance can with large external disturbance and the be achieved. Due to persistent periodical external disturbance, and still have small vibration around state variables zero at steady state. VI. CONCLUSION In this paper a fuzzy linear control technique and an attenuation technique have provided a rough tuning and a precise tuning, respectively, and are combined to achieve robust performance for nonlinear dynamic systems. If the state variables are unavailable, a fuzzy observer-based control performance. scheme has been also proposed to achieve Furthermore, the stability of the fuzzy control systems is also guaranteed in this work. Actually, the proposed fuzzy linear control can be applied to any robust control design of nonlinear systems. With the aid of linear fuzzy approximation algorithm and control design can be exLMI technique, the robust tended from linear systems toward nonlinear systems. A robust optimization technique is also developed. By emattenuation technique, the performance of ploying the linear fuzzy control design for nonlinear systems can be significantly improved. Furthermore, the robust fuzzy control scheme is also developed to eliminate as much as possible the effect of the external disturbance. Therefore, the proposed design algorithm is appropriate for practical control design of mechanical systems with bounded external disturbances. The proposed design method is simple and the number of membership functions for the proposed control law can be extremely small. However, because of the use of fuzzy apcontrol scheme, the results proximation technique and are less conservative than the other robust control methods. Based on the interior point optimization technique, a design procedure is proposed for the fuzzy observer-based control to achieve the robust optimization design of the nonlinear systems. Simulation results indicate that the desired robust performance for nonlinear systems can be achieved via the proposed method. REFERENCES [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, pp. 116–132, Jan./Feb. 1985.

[2] G. C. Hwang and S. C. Lin, “A stability approach to fuzzy control design for nonlinear systems,” Fuzzy Sets Syst., vol. 48, pp. 279–287, 1992. [3] J. J. Buckley, “Theory of fuzzy controller: An introduction,” Fuzzy Sets Syst., vol. 51, pp. 249–258, 1992. [4] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller—Part I and Part II,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–435, Mar./Apr. 1990. [5] H. Ying, W. Silver, and J. J. Buckley, “Fuzzy control theory: A nonlinear case,” Automatica, vol. 26, pp. 513–520, 1990. [6] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996. tracking design of linear [7] B. S. Chen, C. H. Lee, and Y. C. Chang, “ systems: Adaptive fuzzy approach,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 32–43, Nov. 1996. [8] B. R. Barmish, “Necessary and sufficient conditions for quadratic stability of an linear systems,” J. Optim. Theory Appl., vol. 46, no. 4, pp. 399–408, 1985. [9] P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust stabilization of linear systems: Quadratic stability and control theory,” IEEE Trans. Automat. Contr., vol. 35, no. 3, pp. 356–361, Mar. 1990. [10] B. A. Francis, A Course in Control Theory, Lecture Notes in Control Information Science. Berlin, Germany: Springer-Verlag, 1987, vol. 8. [11] A. Stoorvogel, The Control Problem: A State Space Approach. Englewood Cliffs, NJ: Prentice-Hall, 1992. [12] J. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State space solution to standard control problems,” IEEE Trans. Automat. Contr., vol. 34, no. 8, pp. 831–847, Aug. 1989. [13] B. S. Chen, T. S. Lee, and J. H. Feng, “A nonlinear control design in robotic systems under parameter perturbation and external disturbance,” Int. J. Contr., vol. 59, pp. 439–461, 1994. [14] B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1990. [15] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [16] 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, pp. 1–10, 1997. [17] H. K. Khalil, Nonlinear Systems. London, U.K.: Macmillan, 1992. [18] Y. M. Cho and R. Rajamani, “A systematic approach to adaptive observer synthesis for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 42, pp. 534–537, Apr. 1997. [19] A. Isidori and A. Asolfi, “Disturbance attenuation and control via measurement feedback in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1283–1293, Sept. 1992. [20] A. Isidori, “ control via measurement feedback for affine nonlinear systems,” Int. J. Robust Nonlinear Contr., 1994. [21] J. P. LaSalle, “Some extensions of Lyapunov’s second method,” IRE Trans. Circuit Theory, pp. 520–527, Dec. 1960. [22] 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, Aug. 1996. [23] X. Ma, Z. Sun, and Y. He, “Analysis and design of fuzzy controller and fuzzy observer,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 41–51, Feb. 1998. [24] C. M. Cheng and N. W. Rees, “Stability analysis of fuzzy multivariable systems: Vector Lyapunov function approach,” Proc. Inst. Elect. Eng.—Contr. Theory Appl., vol. 144, no. 5, Sept. 1997. [25] K. Narendra and A. M. Annaswamy, “A new adaptation law for robust adaptation without persistent excitation,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 134–145, Feb. 1987. [26] C. S. Tseng, B. S. Chen, and H. J. Uang, “A robustness design of uncertain nonlinear dynamic systems via fuzzy linear control,” in IEEE World Congress Computat. Intell., Anchorage, AK, May 1998, pp. 428–433. [27] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: MathWorks, 1995. [28] B. S. Chen, H. J. Uang, and C. S. Tseng, “Robust tracking enhancement of robot systems including motor dynamics: A fuzzy-based dynamic game approach,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 538–552, Nov. 1998.

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CHEN et al.: ROBUSTNESS DESIGN OF NONLINEAR DYNAMIC SYSTEMS VIA FUZZY LINEAR CONTROL

Bor-Sen Chen (M’82–SM’89) received the B.S. degree from Tatung Institute of Technology in 1970, the M.S. degree from National Central University, Taiwan, R.O.C., in 1973, and the Ph.D degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology from 1973 to 1987. He is now a Professor at National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C. His current research interests include control and signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council and the Chair of the Outstanding Scholarship Foundation.

Chung-Shi Tseng received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1984, and the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, in 1987. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, National Tsing Hua University, Hsin Chu, Taiwan, R.O.C. His current research interests include robotics, robust control, and fuzzy systems.

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Huey-Jian Uang received the B.S. degree from the Department of Electrical Engineering, Feng-Chia University, Taichung, Taiwan, R.O.C., and the M.S. degree from National Tsing-Hua University, HsinChu, Taiwan, R.O.C., in electrical engineering. He is currently working toward the Ph.D. degree in electrical engineering, National Tsing Hua University. His current research interests include robotics, robust control, and fuzzy systems.