Adaptive Fuzzy Output Tracking Control of MIMO Nonlinear Uncertain ...

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 2, APRIL 2007

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Adaptive Fuzzy Output Tracking Control of MIMO Nonlinear Uncertain Systems Bing Chen, Xiaoping Liu, and Shaocheng Tong

Abstract—In this paper, the adaptive fuzzy tracking control problem is discussed for a class of uncertain multiple-input–multiple-output (MIMO) nonlinear systems with the block-triangular structure. The fuzzy logic systems are used to approximate the unknown nonlinear functions. By using the backstepping technique, the adaptive fuzzy tracking control design scheme is developed, which has minimal learning parameterizations. The adaptive fuzzy tracking controllers guarantee that the outputs of systems converge to a small neighborhood of the reference signals and all the signals in the closed-loop system are semiglobally uniformly ultimately bounded. Two examples are used to show the effectiveness of the approach. Index Terms—Adaptive fuzzy control, backstepping, multiple-input–multiple-output (MIMO) nonlinear systems, output tracking, uncertainty.

I. INTRODUCTION

I

N PRACTICE, most plants are nonlinear and contain uncertainties. During the past years, many people have devoted a lot of effort to both theoretical research and implementation techniques to handle nonlinear control problems. In [13] a genetic-algorithm-based fuzzy modelling approach was proposed to generate Takagi–Sugeno–Kang (TSK) models. A simple but effective fuzzy-rule-based models of complex systems from input–output data was developed in [12]. Fuzzy control methodology has emerged in recent years as a promising way to deal with the control problems of nonlinear systems containing highly uncertain nonlinear functions. It has been shown that fuzzy logic systems can be used to approximate any nonlinear function over a convex compact region [20] and [21]. Based on this observation, many systematic fuzzy controller design methods have been developed to solve output tracking control problems for single-input–single-output (SISO) systems with unknown nonlinearities. Stable direct and indirect adaptive fuzzy control schemes were first developed to control uncertain nonlinear systems by Lyapunov function method [20]. Afterwards, several stable adaptive fuzzy control schemes have been

Manuscript received January 3, 2005; revised October 6, 2005 and January 10, 2006. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, and the Natural Science Foundation of China under Grant 60674055, by the national Key Basic Research and Development Programme of China under Grant 2002CB312200, and by the Taishan Scholar Programs Foundation of Shandong Province. B. Chen is with the Institute of Complexity Science, Qingdao University, Qingdao 266071, P. R. China. X. P. Liu is with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]). S. C. Tong is with the Department of Basic Mathematics, Liaoning Institute of Technology, Jinzhou 121000, P. R. China. Digital Object Identifier 10.1109/TFUZZ.2006.880008

introduced, respectively, for SISO nonlinear systems [2], [3], [15], [16], [19]. In recent years, the corresponding research results have been extended to multiple-input–multiple-output (MIMO) nonlinear systems [18]. The basic idea of these works is to use the fuzzy logic systems to approximate the unknown nonlinear functions in systems and design adaptive fuzzy controllers by using Lyapunov stability theory. All the results mentioned previously are obtained with the restriction that the system is feedback linearizable. This means that the unknown nonlinear functions satisfy the matching conditions. However, in practice, a large class of physical systems may contain unknown nonlinear functions which do not satisfy the matching conditions. In this case, the adaptive fuzzy control approaches mentioned previously fail. Backstepping, which is based on the nonlinear stabilization technique of “adding an integrator” introduced in [17] and [1], and was first used in nonlinear adaptive control in [6], leads to the discovery of a structural strict feedback condition under which the systematic construction of robust control Lyapunov function is always possible. Up to now, backstepping-based adaptive control technique, which is mainly used to deal with the robust control of nonlinear systems with parametric uncertainties, has become one of the most popular design methods for a large class of nonlinear systems [9], [10], [14], [23]. Recently, the adaptive neural control approach based on backstepping design has been developed for nonlinear uncertain systems without the requirement of matching conditions. In [7], [8], and [24], stable neural controller design schemes were proposed for unknown nonlinear SISO systems via backstepping design technique. With the backstepping design technique, neural networks were mostly applied to approximate the unmatched and unknown nonlinearities, and then implement adaptive control using the conventional control technology. In [5], an adaptive neural control approach was proposed for a class of MIMO nonlinear systems with triangular structure in control input. By using the triangular property, an integral-type Lyapunov functions are introduced to construct a Lyapunov-based controller. Particularly, the further results on backstepping-based neural control for more general uncertain MIMO nonlinear systems, in which the unknown system state interconnections appear in every equation of each subsystem, have been proposed in [4]. The advantage of adaptive neural control based on backstepping methodology includes that both the parameters and the nonlinear functions can be unknown and the uncertainties in systems need not satisfy the matching conditions. Similar to neural networks, the fuzzy logic systems can be also used to uniformly approximate unknown nonlinear functions. Compared with neural networks, fuzzy logic systems can achieve the faster

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convergence because the fuzzy models perform a fuzzy blending of local models, and fuzzy logic systems are capable of accommodating both numerical data and expert knowledge as they can be represented by linguistic IF–THEN rules. However, compared with neural control, there are only a few results available in the literature on adaptive fuzzy control via backstepping design approach. More recently, the pioneering works on adaptive fuzzy control for unknown nonlinear systems via backstepping method was proposed in [22], where the T–S type fuzzy logic systems are used to approximate the unknown nonlinear functions, and the backstepping-based adaptive fuzzy tracking control methodology has been developed for strict-feedback SISO nonlinear systems. An adaptive fuzzy tracking controller has been constructed by using backstepping design technique. The advantage of the scheme proposed in [22] is that it does not require matching conditions for the unknown nonlinear functions, and has less adaptive parameters. So far, there is no result on adaptive fuzzy output tracking control for MIMO nonlinear systems with unmatched nonlinear functions. In this paper, we consider adaptive fuzzy control of a class of uncertain MIMO nonlinear systems with block-triangular forms. MIMO nonlinear systems with block-triangular forms was first proposed in [4]. The purpose of this paper is to develop an adaptive fuzzy control design method for output tracking control problems of MIMO nonlinear systems. By using the backstepping approach, a new systematic procedure is developed for the synthesis of the stable adaptive fuzzy output tracking controllers for MIMO nonlinear systems with block-triangular structure. The fuzzy logic systems are used to approximate the unknown nonlinear functions. The adaptive fuzzy controllers are constructed by using backstepping design technique. The proposed design scheme achieves semiglobal uniform ultimate boundedness of all the signals in the closed-loop systems. The tracking error is proven to converge to a small neighborhood of the origin. The adaptive laws in the paper are designed based on norms of unknown parameters in fuzzy logic approximators, so the number of the adaptive laws is reduced to the number of unknown nonlinear functions. As a result, adaptive controllers require less computation time.

, then the corresponding variable Remark 1: If does not exist and does not appear in system (1). If , stands for the maximum number of state variables of the th subsystem which are embedded in th subsystem. Remark 2: Note that the state variables of the th subsystem appear in the th equation of the th subsystem in the form with , which implies that , so will not appear in the first equations the state variable of the th subsystem. Therefore, system (1) is called to be of the block-triangular form. MIMO nonlinear systems with the block-triangular form has been first proposed in [4]. It is clear that system (1) is equivalent to the system in [4]. See [4] for more details. Just by using the properties of block-triangular form, the output tracking control problem can be solved for uncertain MIMO nonlinear systems via backstepping approach. The control objective is to design adaptive fuzzy controllers for system (1) such that: i) all the signals in the closed-loop system remain uniformly ultimately bounded, and ii) the output follows a given reference signal . From a mathematical point of view, fuzzy logic systems can be used as practical function approximators. Thus, the following fuzzy logic system is used to approximate a continuous function defined on some compact set. : IF is and, , and is , THEN is into fuzzy logic system

where , is the total number of fuzzy rules. and

and and are fuzzy sets. is the membership of

. Let and The fuzzy logic system above can be expressed as follows.

, .

II. PRELIMINARIES AND PROBLEM FORMULATION Consider a class of uncertain MIMO nonlinear systems described by the following differential equations:

(2) If all memberships are chosen as Gussian functions, then we have the following lemma [20]. is a continuous function defined on a Lemma 1: Let , there exists compact set . Then for any given constant a fuzzy logic system (2) such that

(1) , , are the state variables of the th subwhere and are the control input and output of system, the th subsystem, respectively. and are unknown nonlinear functions, , and are positive integers. For simplicity, throughout this paper, the following notations are used: , and with being the order of the th subsystem.

Assumption 1: There exist constants and for

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and

such that

CHEN et al.: ADAPTIVE FUZZY OUTPUT TRACKING CONTROL OF MIMO NONLINEAR UNCERTAIN SYSTEMS

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It is obvious that Assumption 1 requires the unknown funcare not zero. Without loss of generality, it is assumed tions . that Lemma 2: Let be a real-valued continuous function and satisfy with and being two constants. Define function as follows: (3)

where and are real-value functions with . has the following properties. Then the integral function 1)

A simple calculation gives (4), as shown at the bottom of the , the following page. By using the transformation result is obtained:

2) (5) Similarly

(6) Therefore, Conclusion 2 follows immediately from substituting (4)–(6) into (3). Proof: Conclusion 1 is obtained immediately from the in. To get the equalities gives second conclusion, differentiating

III. ADAPTIVE FUZZY CONTROL DESIGN In this section, the backstepping design technique is used to design tracking controllers for all the subsystems of (1). Note that all the subsystems in (1) are interconnected, the stability analysis of whole closed-loop MIMO system becomes difficult. But with the block-triangular property, it is feasible to design

(4)


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a full state feedback controller and prove the stability of the closed-loop system. For the th subsystem of (1), the backstepping design procedure contains steps. At the Step , fuzzy logic systems will be employed to approximate the unknown nonlinear function, and then an intermediate feedback control will be developed, which guarantees the stability of the th subsystem in the th subsystem with respect to a Lyapunov . The tracking controller will be designed at the function step. In the following, we will give the procedure of backstepping design for the th subsystem. 1) Step 1: Define the tracking error variable . Then from the first differential equation in the th subsystem, the following differential equation can be obtained:

where

According to Lemma 1, for a given , there exists a fuzzy logic system , which can be employed to , such that approximate (10) where

(7) are the unknown functions of the variables , and , . , the corresponding state variable does If not appear in (7). Consider a Lyapunov function candidate as where

and

and is the approximation error. It is evident that (10) can be rewritten as (11)

(8)

with

. Substituting (11) into (9) produces

where (12) . By using the well-known inequality with a constant , the following inequality can be obtained: Now, let

specified later, and time derivative of

is an unknown parameter, which will be is its estimation. Then, by Lemma 2 the is given by

(13) where and are positive constants. Note that , substituting (13) into (12) leads to

(9)


(14)


Now, choose the intermediate virtual control input

as

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Then, take a Lyapunov function candidate as (18)

(15) where is the design constant. Then, by using (15), (14) can be expressed as

where and

.

According to Lemma 2 the derivative of is given by (19) and (20), as shown at the bottom of the page. Define (16) 2) Step 2: Define

, then (17)

where

and

are the functions of the variables , and .

(19)

(20)


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By Lemma 1, for constant , a fuzzy logic system can be used to approximate the unknown funcsuch that tion

and suppose that the fuzzy logic system to approximate the unknown function

is used

where

Similar to (11) in step 1),

can be rewritten as (21)

such that

. Substituting (21) into (20) yields

with

where (22) Define obtained:

. Then the following inequality can be

with

, and

is

a given positive constant. By choosing

(23) Furthermore, it follows from substituting (16) and (23) into (22) that

(26) the Lyapunov function

(24) satisfies the following inequality:

(25) Now, choose

(27) where 4) Step : Define gives ating

Then, (25) becomes

with

. . Then, differenti-

(28) . Consider

with the following Lyapunov function candidate:

where the equality 3) Step k:

is used. In general, let

(29) where and


.


By Lemma 2, the derivative of

293

is given by (30)

Fig. 1. Structure of the u .

Substituting (34) into (33) gives

(31) where

(35) By choosing

Since system

is unknown, by Lemma 1, for , a fuzzy logic is used to approximate such that

(36) and using

, (35) becomes

which can be expressed as follows: (32)

(37)

where . Then, by taking (27) into account with , and substituting (32) into (31), the following inequality can be obtained:

The design procedure of the controller can be visualized from the block diagram shown in Fig. 1. are introduced to Remark 3: Note that the parameters of the fuzzy estimate the norms of the unknown parameters , which reduces the number logic approximators of adaptive laws to the number of unknown nonlinear functions in the original system. As a result, the computation burden is reduced considerably. IV. ANALYSIS OF STABILITY

(33) Define that

The backstepping-based fuzzy adaptive control design scheme has been developed in above section. In this section, we will propose the following adaptive laws

. Then, a simple calculation shows (38)

(34)

and analyze the stability of the closed-loop system. The main result is summarized in the following theorem. Theorem 1: Consider the system (1), together with Assump; , the packtion 1. Suppose that for can be approximated by the fuzzy aged unknown functions


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logic systems in the sense that the approximating error are bounded. Then the fuzzy adaptive controllers in (36), the interin (26) and the adaptive laws in (38) mediate virtual control for can make all signals in the closed-loop system remain , the controller bounded. Furthermore, given any scalar . parameters can be tuned such that Proof: For stability analysis, consider the Lyapunov candidate function

(39) From (37) and (38), the time derivative of (39) can be expressed as

(40) Note that

which implies that for

Furthermore, all

and

belong to the compact set . This proves that and

are bounded. Note that , , and are design parameters, and , and are constants. Thus for any , the incan be obtained by apequality propriately choosing these design parameters. In addition, according to Assumption 2 and Lemma 2, for each

Furthermore, ,

which

implies that

Substituting this inequality into (40) produces

(41) It is clear that (41) can be rewritten as

For the proof of the boundedness for the original state vari, see [4]. ables Remark 4: The previous analysis shows that tracking errors , , , and . Because , and depends on are unknown, an explicit estimation of the tracking errors is im, and , possible. However, it is clear that reducing , will lead to smaller tracking errors. meanwhile increasing V. SIMULATIONS

(42) ,

Since Assumption 2 implies that it follows from Lemma 2 that

In this section, the adaptive fuzzy approach is applied to the following two examples to verify its effectiveness. 1) Example 1: Consider a two continuous stirred tank reactor process, which is described by the following differential equations [11]:

(43) Consequently, it follows from (42) and (43) that (45) where (44) Define

and .

Then,

(44) becomes



295

0

Fig. 2. Fuzzy rule base on [ 10; 10].

with , , and the values of the process parameters are provided as follows:

The reference signals are assumed to be

The control objective is to design adaptive fuzzy controllers for , under the such that the outputs follows condition that in the system (45) the parameters and the func( ; ) are completely unknown. In the tions simulation, eleven fuzzy sets are defined over interval for all , , , , , , and by choosing the parti, , , , the fuzzy tioning points as membership functions are given as follows:

The fuzzy rule base on the interval Fig. 2. Authorized licensed use limited to: Jan Vascak. Downloaded on February 24, 2009 at 04:01 from IEEE Xplore. Restrictions apply.

is shown in

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Let

Then Fig. 3. Output y (“ ”) follows the reference y

(“ ”).

Furthermore, the fuzzy controllers and adaptive laws are constructed as follows:

Fig. 4. Control input u .

The design parameters are chosen as

with

The simulation was carried out with the disturbance and the initial conditions

The simulation results are shown in Figs. 3–10. From the observation of the simulation results, it is clear that even though the exact information on the nonlinear functions in the system is not available, the adaptive fuzzy controllers guarantee the good tracking performance.



Fig. 5. Output y (“ ”) follows the reference y

Fig. 7. Output y (“ ”) follows the reference y

(“ ”).

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(“ ”).

Fig. 8. Control input u . Fig. 6. Control input u .

and . To this end, the fuzzy membership functions are chosen as the same as in Example 1. Define

2) Example 2 [4]: In the first example, for and , that is, all are constant. Now, let us consider the following MIMO nonlinear system:

(46) and are unknown nonlinear functions with where and . The control objective is to design fuzzy controllers such that the outputs of (46) follow the given signals Authorized licensed use limited to: Jan Vascak. Downloaded on February 24, 2009 at 04:01 from IEEE Xplore. Restrictions apply.

298


Fig. 9. State variables x

(“ ”), x

Fig. 10. Adaptive variables 10^ (“ ”).

(“ ”) and x

(“ ”), ^

Fig. 11. Output y

(“ ”).

(“ ”), 15^

(“:”), and ^

(“ ”) follows the reference y


with

Then

Furthermore, the adaptive fuzzy controllers are constructed as follows: and the adaptive laws are given by


(“ ”).


Fig. 13. Output y

(“ ”) follows the reference y

(“ ”).

Fig. 15. State variables x

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(“ ”) and x

(“ ”).


The reference signals are generated by the following system:

The design parameters are chosen to be

As [4], the nonlinear functions in the system (46) are chosen as follows:

Fig. 16. Adaptive variables 10^ (“ ”).

(“ ”), 5^

(“ ”), 5^

(“:”), and ^

The simulation was performed under the initial conditions , , , , , , , and . The simulation results are shown in Figs. 11–16. By comparing with the simulation results in [4], it is seen that the adaptive fuzzy controllers in this paper achieve the better tracking performance than adaptive neural network controllers, but the control effort is larger. In addition, the number of the adaptive laws in [4] is equal to the dimension of the unknown vector in the nonlinear function approximator multiplied by the number of the unknown nonlinear functions, whereas the number of the adaptive laws in this paper is equal to the number of the unknown nonlinear functions. As a result, the adaptive controllers in this paper require less computation


300


time than [4]. However, it should be pointed out that the design idea in this paper can be also used to reduce the computation time for neural networks. VI. CONCLUSION In this paper, the output tracking control problem has been considered for a class of uncertain nonlinear MIMO systems in block-triangular form. The unknown functions in systems are not linearly parameterized and have no priori knowledge of the bounded functions. Fuzzy logic systems are used to approximate these unknown nonlinear functions. By means of backstepping design technique, the adaptive fuzzy tracking control scheme has been developed for nonlinear MIMO systems. The proposed controllers guarantee that the outputs of the closed-loop system follow the reference signals, and achieve uniform ultimate boundedness of all the signals in the closed-loop system. The main feature of the control scheme proposed in this paper is that the number of the adaptive parameters is independent of the number of the state variables and the selection of fuzzy rules, which results in less computation burden. ACKNOWLEDGMENT

[14] B. Schwartz, A. Isidori, and T. J. Tarn, “Global normal form for MIMO nonlinear systems with applications to stabilization and disturbance attenuation,” Math. Control, Signals, Syst., vol. 12, pp. 121–142, 1999. [15] J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, no. 4, pp. 339–359, Aug. 1996. [16] C. Y. Sue and Y. Stepanenko, “Adaptive control of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, no. 1, pp. 285–295, Feb. 1994. [17] J. Tisias, “Sufficient Lyapunov like conditions for stabilization,” Math. Control Signals Syst., vol. 2, pp. 343–357, 1989. [18] S. C. Tong and H. H. Li, “Fuzzy adaptive sliding model control for MIMO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 3, pp. 354–360, Jun. 2003. [19] S. C. Tong, T. Wang, and J. T. Tang, “Fuzzy output tracking control of nonlinear systems,” Fuzzy Sets Syst., vol. 111, pp. 169–182, 2000. [20] L. X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least squares learning,” IEEE Trans. Neural Netw., vol. 3, no. 5, pp. 807–814, Sep. 1992. [21] L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability. Englewood Cliffs, NJ: Prentice-Hall, 1994. [22] Y. S. Yang, G. Feng, and J. S. Ren, “A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems,” IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 34, no. 3, pp. 406–420, May 2004. [23] B. Yao and M. Tomizuka, “Adaptive robust control of MIMO nonlinear systems in semi-strict feedback forms,” Automatica, vol. 37, no. 9, pp. 1305–1321, 2001. [24] Y. Zhang, P. Y. Peng, and Z. P. Jiang, “Stable neural controller design for unknown nonlinear systems using backstepping,” IEEE Trans. Neural Netw., vol. 11, no. 5, pp. 1347–1359, Sep. 2000.

The authors would like to thank the reviewers for a number of constructive comments that have improved the presentation of this paper.

Bing Chen received the B.S. degree in mathematics from Liaoning University, P. R. China, the M.S. degree in mathematics from Harbin Institute of Technology, P. R. China, and the Ph.D. degree in electrical engineering from Northeastern University, P. R. China, in 1982, 1991, and 1998, respectively. Currently, he is a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, P. R. China. His research interest includes nonlinear control systems, robust control, and fuzzy control theory.

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H

Xiaoping Liu received the B.Sc., M.S., and Ph.D. degree in electrical engineering from Northeastern University, P. R. China, in 1984, 1987, and 1989, respectively. He spent more than ten years with the School of Information Science and Engineering, Northeastern University, P. R. China. In 2001, he joined the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada. His research interests are nonlinear control systems, singular systems, and robust control. Dr. Liu is a member of the Professional Engineers of Ontario.

Shaocheng Tong received the B.S. degree from the department of mathematics, Jinzhou Normal College, P.R. China, the M.S. degree in fuzzy mathematics from Dalian Marine University, P.R. China, and the Ph.D. degree in fuzzy control from Northeastern University, P.R. China, in 1982, 1988, and 1997, respectively. He is currently a Professor in the Department of Mathematics and Physics, Liaoning Institute of Technology, P. R. China. His current research interests include fuzzy control, nonlinear adaptive control, and intelligent control.
