H/spl infin/ fuzzy output feedback control design for nonlinear systems ...

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 3, JUNE 2003

331

Fuzzy Output Feedback Control Design for Nonlinear Systems: An LMI Approach Sing Kiong Nguang, Senior Member, IEEE, and Peng Shi

Abstract—This paper addresses the problem of stabilizing a fuzzy output feedback class of nonlinear systems by using an controller. First, a class of nonlinear systems is approximated by a Takagi–Sugeno (TS) fuzzy model. Then, based on a well-known Lyapunov functional approach, we develop a technique for designing an fuzzy output feedback control law which guarantees the 2 gain from an exogenous input to a regulated output is less or equal to a prescribed value. A design algorithm fuzzy output feedback controller is given. for constructing an In contrast to the existing results, the premise variables of the fuzzy output feedback controller are not necessarily to be the same as the premise variables of the TS fuzzy model of the plant. A numerical simulation example is presented to illustrate the theory development. Index Terms—Fuzzy system models, Lyapunov function, stabilizing controller.

Over the past two decades, there has been rapidly growing interest in approximating a nonlinear system by a Takagi–Sugeno (TS) fuzzy model [1]–[13]. Based on this fuzzy model, a modelbased fuzzy control was developed to stabilize the nonlinear system. This fuzzy modeling approach provides a powerful tool for modeling complex nonlinear systems. Unlike conventional modeling approaches where a single model is used to describe the global behavior of a systems, TS modeling approach is essentially a multimodel approach in which simple submodels (typically linear models) are combined to described the global behavior of the system. Typically, a continuous-time TS fuzzy dynamic model is described by fuzzy IF–THEN rules of the following form: Plant Rule : IF is and and is , THEN

I. INTRODUCTION

I

N RECENT years, the problem of nonlinear control has been extensively studied by a number of researchers; see, for instance, [14]–[18]. This problem can be stated as follows. Given a dynamic system with the exogenous input and measured gain of the mapping output, design a control law such that the from the exogenous input to the regulated output is minimized or no larger than some prescribed level. In general, there are two common approaches for providing solutions to nonlinear control problems. One is based on the dissipativity theory and theory of differential games (see [22] and[14]), and the other is based on the nonlinear version of the classical bounded real lemma as developed by Willems [25] and Hill and Moylan [23]; see, e.g., [24], [18], and [17]. Both of these approaches convert control to the solvability of the the problem of nonlinear so-called Hamilton–Jacobi equation (HJE). A nice feature of results. these results is that they are parallel to the linear Further research along the line of the dissipativity theory and theory of differential games has been attempted (see, e.g., [20], [17], and [18]) where results on disturbance attenuation for nonlinear systems via state feedback and/or output feedback have filtering been provided. In [19], solutions to the nonlinear problem have been obtained. However, until now, it is still very difficult to find a global solution to the HJE. Manuscript received September 25, 2001; revised July 29, 2002 and September 10, 2002. S. K. Nguang is with The Department of Electrical and Electronic Engineering, The University of Auckland, 92019 Auckland, New Zealand (e-mail: [email protected]). P. Shi is with the Land Operations Division Defence Science and Technology Organization, Salisbury SA 5108, Australia (e-mail: peng.shi@dsto. defence.gov.au). Digital Object Identifier 10.1109/TFUZZ.2003.812691

where

are the premise variables, are fuzzy sets that are characterized by membership is the state vector, is the input, functions, and are of appropriate dimensions and is the matrices the number of IF–THEN rules. , by using a singleton fuzzifer, Given a pair product fuzzy inference and weighted average defuzzifier, the final state of the fuzzy system is inferred as follows:

(1.1) where

while is the grade of membership of It is assumed in this paper that

for all . Therefore

1063-6706/03$17.00 © 2003 IEEE

in

.

332


for all . For the convenience of notations, let and ; then the final state of the fuzzy system can be represented as

Assumption 2.1: (2.2) Assumption 2.2:

(1.2) (2.3) For the fuzzy controller design, it is supposed that the fuzzy system is locally controllable. First, the local state feedback con: trollers are designed as follows, based on the pairs Controller Rule : IF is and and is , THEN

The resulting fuzzy system model is inferred as the weighted average of the local models of the form

(2.4)

then, the final fuzzy controller is

Definition 2.1: Suppose is a given positive real number. A gain less than system of the form (2.4) is said to have or equal to if In practice, it is not always the case that all the state are measureable, hence, it is necesary to design an observer to estimate the state. In [8] and [9], by restricting the premise variables to be measurable, a fuzzy observer has been developed. This restriction enables the authors to select the premise variables of the fuzzy observer be the same as the premise variables of the TS model of the plant. However, in most cases the premise variables of a general TS model are not measureable. Especially, the case when a TS model is derived from nonlinear plant equations (offline modeling). If this is the case, then the premise variables of the fuzzy observer can not be selected to be the same as the premise variables of the TS model of the plant. In general, it is extremely difficult to derive an accurate fuzzy systems model by imposing that all the premise variables are measurable. Hence, the results given in [8] and [9] are very restrictive. What we intend in this paper is to design a fuzzy output feedback controller by allowing its premise variables to different from be the same as the premise variables of the TS model of the plant. II. SYSTEM DESCRIPTION AND DEFINITION

(2.5) In this paper, we consider the following back: Controller Rule : IF is and and is

fuzzy output feed-

, THEN

for (2.6) are the premise variables of the controller, where is the estimated state vector, is the estimated is the estimated controlled output, measurement, are the observer gains, are the controller gains, and is the number of IF–THEN rules. fuzzy output feedback is inferred as the The final weighted average of the local models of the following form:

The class of nonlinear time systems under consideration is described by the following fuzzy system model: Plant Rule : IF is and and is , THEN

for (2.1) are fuzzy sets, is the where is the input, is the disturstate vector, is the measurement, is the conbance, , , , , and trolled output, the matrices , , are of appropriate dimensions, and is the number of IF–THEN rules. Throughout this paper, we adopt the following standard assumptions.

(2.7) Remark 2.1: In [8] and [9], the premise variables of the fuzzy output feedback controller are assumed to be the same as the premise variables of the fuzzy systems model. This actually means that the premise variables of the fuzzy system model are assumed to be measurable. However, in general, it is extremely

NGUANG AND SHI:

FUZZY OUTPUT FEEDBACK CONTROL DESIGN

333

difficult to derive an accurate fuzzy systems model by imposing that all the premise variables are measurable. In this paper, we do not impose that condition, we choose the premise variables of the controller to be different from the premise variables of the fuzzy system model of the plant. Here, the premise variables of the controller are selected to be the estimated premise variables of the plant. , design an Problem Formulation: Given a scalar fuzzy output feedback controller of the form (2.7) such that (2.5) holds. . In the sequel, without loss of generality, we assume Let us denote the estimation error as (2.8) By differentiating (2.8), we get

(2.11) where (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18)

(2.9) The estimation dynamics of the be represented as follows:

fuzzy output feedback can

III. FUZZY OUTPUT FEEDBACK CONTROL DESIGN In this section, we convert the problem of fuzzy output feedback control to the solvability of the linear matrix inequalities (LMIs). Theorem 3.1: Given the closed system (2.11) satisfying Assumptions 2.1 and 2.2, suppose there exist a positive–definite symmetric matrix and a positive constant such that for , the following matrix inequalities”

(3.1) hold with (2.10) Using (2.9) and (2.10), we get the closed-loop system of the following form:

and

are as in (2.12) and (2.15), and

334


Then, the control performance of (2.5) is guaranteed and the closed-loop system (2.11) is quadratically stable. Proof: Let us choose a Lyapunov function for the closed system (2.11) as

Differentiating (3.2), we obtain (3.3), as shown at the bottom of the page. Let us examine the last two terms of (3.3)

(3.2)

(3.3)

NGUANG AND SHI:


335

(3.4)

where

and (3.6) Using (3.1), we get

(3.7) Integrating (3.7) from

to

yields

(3.8) , Knowing that , (3.8) reduces to

and

(3.9) control performance (2.5) is acheived. Therefore, the Next, we need to show that the closed-loop system (2.11) is is zero. quadratically stable. Assume that the disturbance From (3.7), we have (3.5)

where

,

,

. Employing (3.3)–(3.5) becomes

and (3.10) Therefore, the closed system (2.11) is quadratically stable. Remark 3.1: In general, (3.1) are nonconvex, hence it is not easy to analytically determine common solution for (3.1). Fortunately, (3.1) can be transformed into a minimization problem subject to some LMIs called eigenvalue problem (EVP) [26]. The EVP can be solved in a computationally effi-

336


cient manner using a convex optimization technique such as the interior point method. output feedback control, In the spirit of the conventional the following technique is employed to decouple the fuzzy observer design from the controller design. We assume

With and and by the Schur complements, the matrix inequality (3.18) can be rearraged as the following form:

(3.22)

(3.11) This choice is suitable for designing the fuzzy observer and the controller separately. This may lead to a more conservative result. Subsituting (3.11) into (3.1), we get

where

(3.23)

(3.12)

(3.24) where (3.25) (3.26) (3.13) (3.14) (3.15) and

(3.27) and need to be deThere are four parameters termined from (3.22) which is still nonconvex LMIs. Note that (3.22) implies that

(3.28)

(3.16) By introducing a new matrix (3.17) and multiplying both sides of (3.12) by (3.17), we obtain

(3.18) where

By the Schur complement, it can be transformed to the convex LMI form, as shown in (3.29) at the bottom of the page, for . Now (3.29) are convex LMIs, hence, the and (thus, observer parameter ) parameters can be solved by a convex optimization technique such as the interior point method. Substituting and into (3.22), (3.22) and (thus, the control becomes convex LMIs. Similarly, ) can be easily solved. parameter IV. SIMULATION EXAMPLE The following model is used in this simulation:

(3.19) (3.20) (4.1) and (3.21)

, and . A fuzzy system model where for the previous system is given as follows.

(3.29)

NGUANG AND SHI:


337

Fig. 1. Ratio of the estimator error energy to the disturbance energy.

Rule 1: IF

is

Rule 1: IF

, THEN

is

, THEN

(4.2) Rule 2: IF

is

, THEN

and

(4.3) where membership functions

, and are , respectively, and

(4.4)

, the , and Rule 2: IF

is

, THEN

(4.5) where

Remark 4.1: Note that the premise variable of the aforemenwhich is unmeasurable. Hence, tioned fuzzy model is as its premise we can not selected a fuzzy controller with varaible. Therefore, the results given in [8] and [9] cannot be employed. , an fuzzy output Using Theorem (3.1) and with feedback was found to be the following.

Remark 4.2: The premise variable of the above fuzzy controller is selected to be the estimated of

338


Fig. 2.

Histories of the state variables x(t) and the estimated state variables x ^ with w (t) = 0 and w (t) = 0.

Fig. 3.

Disturbance input, w (t).

. Simulation results for the ratio obtained by using the controller for the system (4.1) is depicted in Fig. 1. Fig. 2 shows and the estimated state the histories of the state variables, and . Both of these state variables, with variables converge to zero. This reveals that the closed-loop system is a stable system. Figs. 3 and 4, respectively, show the and which were used input disturbance signals during the simulation. After 300 s, the ratio of the energy of to the energy of tends to a , constant value which is about 0.07. So which is less than the prescribed value 1.

V. CONCLUSION This paper has examined the problem of stablizing a class of fuzzy output feedback confuzzy system models using an troller. A nonlinear sysyem is first approximated by a TS fuzzy model. Then based on a well-known Lyapunov functional apfuzzy output feedback proach, a technique for designing an control law which guarantees the -gain from an exogenous input to a regulated output is less or equal to a prescribed value has been developed. In contrast to the results given [8], [9], the fuzzy output feedback controller premise variables of the are allowed to be different from the premise variables of the TS fuzzy model of the plant.

NGUANG AND SHI:

Fig. 4.


339

Disturbance input, w (t).

REFERENCES [1] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135–156, 1992. [2] K. Tanaka, “Stability and stabilizability of fuzzy neural linear control systems,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 438–447, Nov. 1995. [3] K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability control theory, and linear matrix inequality,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 1–13, Feb. 1996. [4] 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. [5] S. G. Cao, N. W. Ree, and G. Feng, “Quadratic stability analysis and design of continuous-time fuzzy control systems,” Int. J. Syst. Sci., vol. 27, pp. 193–203, 1996. [6] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst. Man. Cybern., vol. SMC-15, pp. 116–132, Feb. 1985. [7] C. L. Chen, P. C. Chen, and C. K. Chen, “Analysis and design of fuzzy control system,” Fuzzy Sets Syst., vol. 57, pp. 125–140, 1993. [8] X. J. Ma, Z. Q. Sun, and Y. Y. He, “Analysis and design of fuzzy controller and fuzzy observer,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 41–51, Feb. 1998. [9] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed = fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 249–265, Apr. 2000. [10] S. K. Nguang and P. Shi, “Stabilization of a class of nonlinear time-delay systems using fuzzy models,” in Proc. Conf. Decision Control, Sydney, Australia, 2000, pp. 4415–4419. [11] M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 133–142, Feb. 1999. [12] S. H. Zak, “Stabilizing fuzzy system models using linear controllers,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 236–240, Apr. 1999. [13] L. X. Wang, A Course in Fuzzy Systems and Control. Upper Saddle River, NJ: Prentice-Hall, 1997. [14] J. A. Ball and J. W. Helton, “ control for nonlinear plants: Connection with differential games,” Proc. 28th IEEE Conf. Decision Control, pp. 956–962, 1989. [15] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory. New York: Academic, 1982. [16] A. J. van der Schaft, “ -gain analysis of nonlinear systems and nonlinear state feedback H control,” IEEE Trans. Automat. Contr., vol. 37, pp. 770–784, June 1992. [17] A. Isidori and A. Astolfi, “Disturbance attenuation and —Control via measurement feedback in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1283–1293, Sept. 1992.

H

[18] A. Isidori, “Feedback control of nonlinear systems,” in Proc. 1st European Control Conf., 1991, pp. 1001–1012. filtering,” Automatica, [19] S. K. Nguang and M. Fu, “Robust nonlinear vol. 32, pp. 1195–1199, 1996. [20] J. A. Ball, J. W. Helton, and M. L. Walker, “H control for nonlinear systems with output feedback,” IEEE Trans. Automat. Contr., vol. 38, pp. 546–559, Apr. 1993. [21] S. Suzuki, A. Isidori, and T. J. Tarn, “ control of nonlinear systems with sampled measurements,” J. Math. Syst., Estimat., Control, vol. 5, pp. 1–12, 1995. [22] T. Basar, “Optimum performance levels for minimax filters, predictors and smoothers,” Syst. Control Lett., vol. 16, pp. 309–317, 1991. [23] D. J. Hill and P. J. Moylan, “Dissipative dynamical systems: Basic input–output and state properties,” J. Franklin Inst., vol. 309, pp. 327–357, 1980. [24] A. J. van der Schaft, “A state-space approach to nonlinear H control,” Syst. Control Lett., vol. 16, pp. 1–8, 1991. [25] J. C. Willems, “Dissipative dynamical systems part I: General theory,” Arch. Rational Mech. Anal., vol. 45, pp. 321–351, 1972. [26] S. Boyd, L. El Ghaoui, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994.

H

H

H H

H

L

H

Sing Kiong Nguang (M’97–SM’00) graduated (with first class honors) from the Department of Electrical and Computer Engineering, the University of Newcastle, Newcastle, Australia, in 1992, and received the Ph.D. degree from the same university in 1995. He is currently holding a Senior Lectureship position in the Department of Electrical and Electronic Engineering, the University of Auckland, Auckland, New Zealand, where he is the Cluster Coordinator of Systems Modeling and Control Research Cluster at the School of Engineering. He has been working closely with industry and has published over 50 journal papers and over 30 conference papers/presentations on nonlinear control design, nonlinear H-infinity control systems, nonlinear time-delay systems, nonlinear sampled-data systems, biomedical systems modeling, fuzzy modeling and control, biological systems modeling and control, and food and bioproduct processing. Dr. Nguang has served as Associate Editor of the IEEE Conference Editorial Board, and was listed in the 16th Edition of Marquis Who’s Who in the World.

340

Peng Shi received the B.S. degree in mathematics from Harbin Institute of Technology, China, the M.E. degree in modern control theory and applications from Harbin University of Engineering and Heilongjiang Institute of Applied Mathematics, China, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1982, 1985, and 1994, respectively. He also received the Ph.D. degree in mathematics from the University of South Australia (UniSA), Australia, in 1998. He was with the Institute of Applied Mathematics at Heilongjiang University, China, from 1985 to 1989. He held a Postdoctoral Research Associate Position at the Centre for Industrial and Applicable Mathematics at UniSA from 1995 to 1997. He was a Lecturer of Mathematics in UniSA from 1997 to 1999, and then joined the Defence Science and Technology Organization, Australia, in 1999 as a Research Scientist. His research interests include robust control and filtering of sampled-data systems, hybrid systems, Markovian jump systems, fuzzy systems, time-delay systems, singularly perturbed systems, and mathematical modeling and optimization techniques and their applications to defence industry. He has authored or coauthored over 80 journal publications. Dr. Shi is currently serving as Associate editor for the IEEE Control System Society Conference Editorial Board, and is a Member of SIAM.
