A hybrid adaptive fuzzy control for a class of nonlinear ... - CiteSeerX

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 1, FEBRUARY 2003

A Hybrid Adaptive Fuzzy Control for A Class of Nonlinear MIMO Systems Han-Xiong Li and Shaocheng Tong

Abstract—A hybrid indirect and direct adaptive fuzzy output tracking control schemes are developed for a class of nonlinear multiple-input–multiple-output (MIMO) systems. This hybrid control system consists of observer and other different control components. Using the state observer, it does not require the system states to be available for measurement. Assisted by observer-based state feedback control component, the adaptive fuzzy system plays a dominant role to maintain the closed-loop stability. control and sliding mode Being the auxiliary compensation, control are designed to suppress the influence of external disturbance and remove fuzzy approximation error, respectively. Thus, the system performance can be greatly improved. The simulation results demonstrate that the proposed hybrid fuzzy control system can guarantee the system stability and also maintain a good tracking performance. Index Terms—Adaptive control, fuzzy control, multiple-input–multiple-output (MIMO) nonlinear systems, observer, sliding mode control, stability.

I. INTRODUCTION

F

UZZY control methodologies have emerged in recent years as promising ways to approach nonlinear control problems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge for nonlinear control problems. In very complicated situations, where the plant parameters are subject to perturbations or when the dynamics of the systems are too complex for a mathematical model to describe, adaptive schemes have to be used online to gather data and adjust the control parameters automatically [1]–[6]. However, no stability conditions have been provided so far for these adaptive approaches. Based on the universal approximation theorem [7], stable direct and indirect adaptive fuzzy control schemes were first developed to control unknown nonlinear systems with the closed stability given by Lyapunov function method [8], [9]. Afterwards, several stable adaptive fuzzy control schemes have been introduced for controlling single-input–single-output (SISO) nonlinear systems [10]–[14]. In these adaptive fuzzy control schemes, the controllers are generally composed of two main components. One is fuzzy logic system for the rough tuning. The other is

one kind of robust compensator, such as supervisory control control [10], sliding-mode control [11]–[14], or the [8], combination of the latter two, for the fine-tuning. Recently, several stable adaptive fuzzy control schemes are developed for multiple-input–multiple-output (MIMO) nonlinear systems [15]–[20]. However, these adaptive control techniques are only limited to the MIMO nonlinear systems whose states are assumed to be available for measurement. In many practical situations, state variables are often unavailable in nonlinear systems. Thus, the output feedback or observer-based adaptive fuzzy control is required for such complicated applications. In this paper, indirect and direct adaptive fuzzy controllers are presented for a class of nonlinear MIMO plants with the unavailable state variables based on the previous work [15]–[17]. The adaptive fuzzy logic system is used to approximate the unknown vector functions and then the state observer is constructed, upon which both indirect and direct adaptive fuzzy control system can be developed to control the MIMO system and maintain the control system stability. Being the auxiliary compensation, and the sliding mode control are designed to improve the system performance by suppressing the influence of external disturbance and removing the fuzzy approximation error. Thus, the proposed hybrid adaptive fuzzy control system can guarantee the closed-loop stability, and also attenuate the influence of the matching error and external disturbance to an arbitrarily small level. II. PROBLEM FORMULATION Assume that a nonlinear MIMO system is represented by the following set of differential equations:

Manuscript received February 26, 2002; revised May 15, 2002 and June 27, 2002. The work was supported in part by the RGC of Hong Kong under Grants CityU 1054/00E and CityU 1086/01E, and in part by the National Natural Science Foundation of China. H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong. S. Tong is with the Department of Basic Mathematics, Liaoning Institute of Technology, JinZhou 121001, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2002.806314

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1063-6706/03$17.00 © 2003 IEEE

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(1)

LI AND TONG: HYBRID ADAPTIVE FUZZY CONTROL

25

Where

as the plant state vector; as the control input; as the output vector; , , ( ) as the smooth . function, is the external disturbance. with respect to time for Differentiating times, respectively, until the inputs appear, one obtains the input–output form of (2) as (2) Define

with

then

Assumption 1: The matrix as previously defined is exists, and bounded for all , nonsingular, i.e., is some compact set. where Assumption 2: The plant is feedback linearizable by staticand state feedback. It has a general vector zero dynamics are exponentially attractive [22]. Control Objectives: Determine a output robust fuzzy conand an adaptive law for adjusting the troller parameter vectors such that the following conditions are met. i) All the signals involved are uniformly bounded and . ii) For a prescribed attenuation level , the following tracking performance is achieved as

Where and

, is an adaptation gain, and

(4) are weighing matrices, .

III. OBSERVER-BASED INDIRECT ADAPTIVE FUZZY CONTROL .. .

.. .

Let MIMO fuzzy logic systems of the form [17]

.. .

and

be

(5) where .. .

Equation (2) can be rewritten as

For the given references rors as

Denote

If the state variables of (1) are available for measurement, the fuzzy control can be chosen as [16], [17] (3)

(6)

, define the tracking er-

and are fuzzy systems to approxwhere imate and , respectively. is a robust control to attenuate the disturbance effect on system outputs; and is the feedback gain vector to make to be Hurwitz. the characteristic polynomial of If the state variables are unavailable for measurement, controller (6) can not be used to control nonlinear system (3). Thus, by replacing the system state and error with their estimates and , controller will be designed as (7)

and

is the same as in (6), is the feedback control for where and need to be designed; and is a sliding-mode control to compensate fuzzy approximation errors. and be of the following form: (8) (9)


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Since

, substituting (7) into (3) yields

Then, the error dynamic (12) of the observation can be formulated as

(10) (16) Thus, the problem has been converted to the classical problem of designing a state observer for estimating the state vector in (10). Design the observer as follows:

According to (12), (13), and (14), (16) can be expressed as

(17) (11) is the observer gain vector to where is make sure that the characteristic polynomial of strict Hurwitz. Define the observation error as and subtracting (11) from (10) results in

, where Assumption 3: i) The matrix , i.e., ii) There exist a constant such that

. is invertible, and ,

,

and

(12) and belong to compact sets It is assumed that , , , and , respectively, which are defined as

,

where , , , and are the designed parameters, and is the number of fuzzy inference rules. , Define the optimal parameter vector

Assumption 4: For the given positive–definite matrices and , there exist positive–definite solutions and for the matrix equations in (18)–(19), as shown at the bottom of the page, where is a prescribed attenuation level, is a matrix, is a positive parameter to be de. signed, and and is assumed to be measurable, take Since the control law as (20) (21) (22)

(13) (14) and fuzzy approximation error

as a sliding gain to be determined. with The parameter adaptive adjusting laws are chosen as (23) (24)

.

(15)

and are two adaptation gains to be dewhere signed. The main property of indirect adaptive fuzzy control scheme is summarized in the following theorem

(18) (19)



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Equation (28) can be formulated in the form of

(29) Design the state observer as

Fig. 1.

(30)

Observer-based indirect adaptive fuzzy control system.

Theorem 1: For the nonlinear system (3), the indirect adaptive fuzzy control law is chosen as (7) with robust and sliding control in (22)–(22) and parameter adaptation in (23) and (24). If Assumptions 3 and 4 are satisfied, then the proposed fuzzy control scheme can guarantee the following properties: , ; 1) , , , tracking per2) For a prescribed attenuation level , the formance (4) is achieved. The proof of Theorem 1 is given in Appendix A. Remark 1: Remark 1 is given in Appendix B. The architecture of the observer-based indirect adaptive fuzzy control system is illustrated in Fig. 1.

is the observer gain vector to where to be guarantee the characteristic polynomial of Hurwitz. , subtracting If the observation error is defined as (30) from (29) results in

(31) is

Similarly as in the previous section, fuzzy system expressed as

(32) IV. OBSERVER-BASED DIRECT ADAPTIVE FUZZY CONTROL In this section, it is assumed that is a known constant matrix. If the state variable is available for measurement, the direct fuzzy controller is chosen as

where and are assumed to belong to compact sets and , respectively

,

(25) where fuzzy logic system the following control law:

aims to directly approximate

Define the optimal parameter vector (33)

(26) and the fuzzy approximation error is a robust control to attenuate effects of both fuzzy and approximation errors and the external disturbance. If the state variable is not available for measurement, the direct fuzzy controller (25) has to be modified as

, (34)

The dynamic equation of the observation error (31) can be written as

(27) is a robust control, is a feedback for the estiwhere is a sliding control to cope with the fuzzy mated errors, and approximation errors. Substituting (26) and (27) into (3) results in

(35) According to (31), (34) can be expressed as

(36) (28)

where

is an error parameter.


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Assumption 5: There exist a constant such that , , i.e., . and Assumption 6: For the given positive–definite matrices and , there exist positive–definite matrixes and for the (37) and (38), respectively; see (37)–(38), as shown at the bottom of the page. The direct adaptive control scheme is designed as (39) (40)

Fig. 2.

Observer-based direct adaptive fuzzy control system.

(41) (42)

Rewrite (44) in the following form:

(43) where is a prescribed attenuation level, is a matrix, is a positive parameter to be designed, and . Theorem 2: For the nonlinear MIMO system (3), the fuzzy controller law is chosen as (39)–(42) with parameter adjusting law (43). If Assumptions 5 and 6 are satisfied, the direct adaptive fuzzy control scheme proposed can guarantee the following properties: , ; 1) , , , tracking performance (4) is achieved for a pre2) the scribed attenuation level . The proof of Theorem 2 is given in Appendix C. Remark 2: Remark 2 is similar to Remark 1 that is given in Appendix B. The architecture of the observer-based direct adaptive fuzzy control system is illustrated in Fig. 2.

(45)

where relative degree

V. SIMULATION A. Observer-Based Indirect Adaptive Fuzzy Control Approach Consider the nonlinear differential equation given by

The feedback and observer gain matrices are chosen as

(44)

(37) (38)



29

(a)

(b)

(c) Fig. 3. Trajectories of state variables and their estimates for indirect adaptive fuzzy control (a) State x and its estimation x ^ . (b) State x and its estimation x^ . (c) State x and its estimation x ^ .

Membership functions are chosen as the following:

Defining seven fuzzy rules to be of the following form: If

is

and

is

and


then

is

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Fig. 4.


Control inputs

u

and u .

B. Observer-Based Direct Adaptive Fuzzy Approach

Denote

Consider the following nonlinear systems:

In the same way as [17], onstructing fuzzy systems and and by using observer (17), we obtain the estiand , mated , thus, we have fuzzy systems respectively. , , , , . Adaptation Let , and . The adjusting factors are chosen as initial conditions are chosen as

(46) The fuzzy basis functions and fuzzy logic system are determined similar to those in Section V-A. The tracking reference signals are chosen as the same as those in Section V-A. The initial conditions are chosen as

and the tracking reference signals as Select

Given the prescribed attenuation level , taking . According to Appendix B, , are chosen to make ( ) be SPR. , two positive–definite For the given matrices are solved from (18) and (19)

The simulation results are shown in Figs. 3 and 4. A good tracking performance can be achieved as shown in Fig. 3 where a perfect estimate of states is obtained.

and For the given prescribed attenuation level and definite , positive–definite matrices matrixes are solved from (37) and (38)

The simulation results are illustrated in Figs. 5 and 6. A good tracking performance can be achieved as shown in Fig. 5 where a perfect estimate of states is obtained.



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(a)

(b)

(c) Fig. 5. Trajectories of state variables and their estimates for direct adaptive fuzzy control. (a) State x and its estimation x ^ . (b) State x and its estimation x^ (c) State x and its estimation x ^ .

VI. CONCLUSION Both indirect and direct adaptive fuzzy controls in hybrid structure are proposed for a class of nonlinear MIMO processes.

Using the state observer, it does not require the system states to be available for measurement. Aided by observer-based state feedback control, the adaptive fuzzy system behaves a primary control system to maintain a stable control of the


32


Fig. 6.

Control inputs

u

and u .

MIMO process. Being the auxiliary compensation, the control and the sliding mode control components are designed to improve the system performance without disturbing the control will suppress the influence original stability. The from external disturbance and the sliding control will remove the fuzzy approximation error. The proposed hybrid adaptive fuzzy system not only guarantees the closed-loop stability, but tracking performance. also achieves the desired APPENDIX A (A4)

A. Proof of Theorem1 Since

Consider the Lyapunov function candidate (A1) The time derivative of

is

(A2) (A5)

Substituting (11) and (19) into (A2) yields and

(A6) Substituting (A5) and (A6) into (A4) results in (A3) Via parameter adaptive law (23) and (24), (A3) can be formulated as

(A7)



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Choosing and using Lyapunov (18) and Riccati-like (19), (A7) becomes

with (B4) and

(A8) Denoting equality (A8) becomes

and

is chosen to be a proper stable transfer function mato be a proper SPR transfer trix and to make function matrix. Then the state-space realization of (B4) can be written as

, then the in(A9)

, it is included that , , , , [8]. Thus, there are . Since and . to Integrating inequality (A9) from

Since

and and , there is

(B5) where

yields (A10) After this transformation, ( tems. Let cati-like (19) to be

, inequality (A10) implies

Since

(A11) tracking performance (4) is achieved for a prescribed The . attenuation level

) is SPR systo change Ric-

(B6) with

APPENDIX B is a Hurwitz matrix with a propRemark 1: Since , there exists a positive–definite matrix for erly chosen (18) in Assumption 4. Actually, Theorem 1 depends on whether for Riccati-like (19). there exists a positive–definite matrix • If a positive–definite exists for (19), we have . The states estimate and are available to make the proposed adaptive fuzzy control scheme realizable. does not exist for (19), then (17) • If a positive–definite ( , , ) can be converted to the strictly positive-real (SPR) system by using the same way as in [21]. The details are as follows. First, the output error dynamics of (17) can be formulated as (B1)

Since ( matrix

) is SPR, for the given positive , a positive–definite solution exists for (B6). APPENDIX C

A. Proof of Theorem 2 Consider Lyapunov function as (C1) The derivative of

is (C2)

Substituting (30) and (35) into (C2) yields

where (B2) is a known stable transfer funcThe transfer function tion matrix. In order to use the SPR-Lyapunov design approach, (B1) can be written as (B3)


(C3)

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Substituting sliding control (42) and parameter adaptive law (43) into (C3) results in

tracking design of uncer[10] B. S. Chen, C. H. Lee, and Y. C. Chang, “ tain nonlinear SISO systems: Adaptive fuzzy approach,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 32–43, Feb. 1996. [11] J. T. Spooner and K. M. Passino, “Stable adaptive control of a class of nonlinear systems and neural network,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 339–359, Aug. 1996. [12] C. Y. Sue and Y. Stepanenko, “Adaptive control of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 285–294, Apr. 1994. [13] S. C. Tong and T. Y. Chai, “Fuzzy adaptive control for a class of nonlinear systems,” Fuzzy Sets Syst., vol. 101, pp. 31–39, 1999. [14] T. Y. Chai and S. C. Tong, “Fuzzy adaptive control for a class of nonlinear systems,” Fuzzy Sets Syst., vol. 103, pp. 379–389, 1999. control for [15] Q. G. Li and S. C. Tong, “Adaptive neural network MIMO nonlinear system,” in Proc. Int. Symp. Intelligent Control, Istanbul, Turkey, 1997, pp. 1711–1715. [16] S. C. Tong, J. J. T. Tang, and T. Wang, “Fuzzy adaptive control of multivariable nonlinear systems,” Fuzzy Sets Syst., vol. 111, pp. 153–167, 2000. [17] Y. C. Chang, “Robust tracking control of nonlinear MIMO systems via fuzzy approaches,” Automatica, vol. 36, pp. 1535–1545, 2000. [18] R. Ordonez and K. M. Passino, “Stable multi-output adaptive fuzzy/neural control,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 345–353, June 1999. [19] H. G. Zhang and Z. B. Bie, “Adaptive fuzzy control of MIMO nonlinear systems,” Fuzzy Sets Syst., vol. 115, pp. 191–204, 2000. tracking control [20] Y. C. Chang and B. S. Chen, “A nonlinear adaptive design in robotic systems via neural networks,” IEEE Trans. Control Syst. Technol., vol. 5, pp. 13–29, Jan. 1997. [21] Y. G. Leu and W. Y. Wang, “Observer-based adaptive fuzzy -neural control for unknown nonlinear dynamical systems,” IEEE Trans. Syst., Man, Cybern., vol. . 29, pp. 583–591, Oct. 1999. [22] C.-C. Liu and F.-C. Chen, “Adaptive control of nonlinear continuous-time systems using neural networks-general relative degree and MIMO cases,” Int. J. Control, vol. 58, no. 2, pp. 317–335, 1993.

H

H

(C4) From (42), (37), (38), and Assumption 5, inequality (C5) becomes

(C5) Denoting

, and

, (C5) becomes (C6)

Similar to the proof of Theorem1, we can complete the proof of Theorem 2. REFERENCES [1] T. Procyk and E. Mamdani, “A linguistic self-organizing process controller,” Automatica, vol. 15, no. 1, pp. 15–30, 1979. [2] C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller- Part1; Part 2,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–435, Mar. 1990. [3] D. Driankov, H. Hellendoorn, and M. M. Reinfrank, An Introduction to Fuzzy Control. Berlin, Germany: Springer-Verlag, 1993. [4] J. R. Layne and K. M. Passino, “Fuzzy model reference learning control for cargo ship steering,” IEEE Control Syst. Mag., vol. 3, pp. 23–34, 1993. [5] W. A. Kwong and K. M. Passino, “Dynamically focused fuzzy learning control,” IEEE Trans. Syst., Man, Cybern., vol. 26, pp. 28–44, Jan. 1996. [6] A. E. Gegov and P. M. Frank, “Hierarchical fuzzy control of multivariable systems,” Fuzzy Sets Syst., vol. 72, pp. 299–310, 1995. [7] L. X. Wang and J. M. Mendel, “Fuzzy basis function, universal approximation, and orthogonal least square learning,” IEEE Trans. Neural Networks, vol. 3, pp. 807–814, Oct. 1992. [8] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 146–155, Feb. 1993. [9] , Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Upper Saddle River, NJ: Prentice-Hall, 1994.

H

Han-Xiong Li received the B.E. degree from the National University of Defence Technology, China, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1982, 1991, and 1997, respectively. He has been with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, since 1997. His academic experience includes a Research Engineer position at the University of Auckland from 1993 to 1996 and a Research Associate position at the Delft University of Technology in 1992. His industrial experience includes a Senior Process Engineer position for die-bonding processes at ASM, Hong Kong, in 1997, a Project Manager position in the Feasibility Study Department, CITIC, Beijing, China, from 1987 to 1988, and a Software Engineer position at Peking Analysis Center, Beijing, China, from 1982 to 1987. His research interests include intelligent control and systems, integrated design and control for industrial process, and E-control and maintenance, with emphasis on electronic packaging. Dr. Li currently serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: PART B.

Shaocheng Tong received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, China, and the Ph.D degree in fuzzy control from Northeastern University, China, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning Institute of Technology, Jinzhou, China. His research interest includes fuzzy control theory, nonlinear adaptive control, and intelligent control.
