Adaptive CMAC-Based Supervisory Control for Uncertain Nonlinear ...

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 2, APRIL 2004

Adaptive CMAC-Based Supervisory Control for Uncertain Nonlinear Systems Chih-Min Lin, Senior Member, IEEE and Ya-Fu Peng

Abstract—An adaptive cerebellar-model-articulation-controller (CMAC)-based supervisory control system is developed for uncertain nonlinear systems. This adaptive CMAC-based supervisory control system consists of an adaptive CMAC and a supervisory controller. In the adaptive CMAC, a CMAC is used to mimic an ideal control law and a compensated controller is designed to recover the residual of the approximation error. The supervisory controller is appended to the adaptive CMAC to force the system states within a predefined constraint set. In this design, if the adaptive CMAC can maintain the system states within the constraint set, the supervisory controller will be idle. Otherwise, the supervisory controller starts working to pull the states back to the constraint set. In addition, the adaptive laws of the control system are derived in the sense of Lyapunov function, so that the stability of the system can be guaranteed. Furthermore, to relax the requirement of approximation error bound, an estimation law is derived to estimate the error bound. Finally, the proposed control system is applied to control a robotic manipulator, a chaotic circuit and a linear piezoelectric ceramic motor (LPCM). Simulation and experimental results demonstrate the effectiveness of the proposed control scheme for uncertain nonlinear systems. Index Terms—Adaptive control, cerebellar model articulation controller (CMAC), chaotic circuit, linear piezoelectric ceramic motor (LPCM), robotic manipulator, supervisory control.

I. INTRODUCTION

T

HERE HAS been considerable attention over the years on researches using the neural-network-based control technique when solving the control problems. Many authors have suggested the neural networks (NNs) as powerful building blocks for a wide class of complex nonlinear system control strategies when there exists no complete model information or, even, a controlled plant is considered as a “black box” [1]–[3]. A comprehensive survey on neural control can be founded in [4]. The most useful property of NNs in control is their ability to uniformly approximate arbitrary input-output linear or nonlinear mappings. Based on this property, the NN-based controllers have been developed to compensate the effects of nonlinearities and system uncertainties in control systems. For feedforward NN, because all the weights are updated during each learning cycle, the learning is essentially global in nature and slow. As well, the ability of function approxManuscript received October 14, 2002; revised February 18, 2003. This work was supported in part by the National Science Council of the Republic of China under Grant NSC 90-2213-E155-016. This paper was recommended by Associate Editor W. Pedrycz. C. -M. Lin is with the Department of Electrical Engineering, Yuan-Ze University, 320 Taiwan R.O.C. (e-mail:[email protected]). Y. -F. Peng is with the Department of Electrical Engineering, Ching-Yun University, 320 Taiwan, R.O.C. Digital Object Identifier 10.1109/TSMCB.2003.822281

imation is sensitive to training data. Thus, the effectiveness of a general multiplayer NN is limited in problems requiring online learning. In the recent developments in [2] and [5]–[7], using adaptive neural-network control, the asymptotic error convergence can be guaranteed. The adaptive neural controllers are based on conventional adaptive control techniques where the NNs are employed to approximate the unknown nonlinear characteristics of the system dynamics. The objective of these controllers is to minimize the trajectory tracking error for the system while the NN parameters are tuned online. The cerebellar model articulation controller (CMAC) is classified as a nonfully connected perceptron-like associative memory network with overlapping receptive-fields [8]. Compared with the multiplayer perceptron with back-propagation algorithm, the CMACs have been adopted widely for the closed-loop control of complex dynamical systems because of its fast learning property, good generalization capability, and simple computation [8]–[12]. It also has similar approximation capability as radial basis function neural network (RBFNN) [2], [13], [14]. The application of CMAC is not only for the control problems but also for model-free function approximation. The CMAC has been already validated that it can approximate a nonlinear function over a domain of interest to any desired accuracy. The contents of these memory locations are referred as weights, and the output of this network is a linear combination of these weights in the memory addressed by the activated inputs [9]. The advantages of using CMAC over NN in many practical applications have been presented in recent literatures [10]–[12], [15], [16]. However, the conventional CMAC uses local constant binary receptive-field basis functions, i.e., its output is constant within each quantized state and the derivative information is not preserved. For acquiring derivative information of input and output variables, Chiang and Lin [17] developed a CMAC with nonconstant differentiable Gaussian receptive-field basis function, and provided the convergence analyzes of this network. This makes CMAC suitable for the nonlinear dynamic system control. The supervisory controllers have been proposed to stabilize the system states around a predefined constraint set [18], [19]. In [18], Mayosky and Cancelo proposed a direct adaptive control strategy for wind energy conversion systems control. It is based on a combination of RBFNN and sliding mode supervisory controller. In [19], a supervisory fuzzy neural network control system is designed to track periodic reference inputs. Since CMAC has simple computation, fast learning ability and good generalization capability compared with the NN’s introduced in [18] and [19], in this paper, an adaptive CMAC-based supervisory control system is proposed for nonlinear systems control.

1083-4419/04$20.00 © 2004 IEEE

LIN AND PENG: ADAPTIVE CMAC-BASED SUPERVISORY CONTROL FOR UNCERTAIN NONLINEAR SYSTEMS

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This control system consists of an adaptive CMAC and a supervisory controller. The adaptive CMAC includes both a CMAC and a compensated controller and is the main controller used to achieve reference signal tracking. Then, the supervisory controller is used to assist the adaptive CMAC, and will drive the system states to a predefined constraint set based on the uncertainty bounds of the system’s model. If the controlled system tends to instability by using the adaptive CMAC, especially in the transient period, the supervisory controller will be activated to work with the adaptive CMAC to stabilize the closed-loop system. But if the adaptive CMAC works well, the supervisory controller will be idle. It will indicate that the adaptive CMAC equipped with the supervisory controller is globally stable in the Lyapunov sense. The proposed adaptive CMAC-based supervisory control system is then applied to control a robotic manipulator, a Chua’s chaotic circuit and a linear piezoelectric ceramic motor (LPCM). Simulation and experimental results demonstrate the effectiveness of the proposed control scheme for the uncertain nonlinear systems. This paper is organized as follows. Problem formulation is described in Section II. The design procedures of the proposed adaptive CMAC-based supervisory control system are constructed in Section III. Simulation and experimental results are provided to validate the effectiveness of the proposed control system in Section IV. Conclusions are drawn in Section V.

is the tracking error. If the plant dynamics and where , external disturbance are known (i.e., the functions and are known), then the control problem can be solved by the so-called feedback linearization method [20]. In this case, , and are used to construct the ideal the functions control law.

II. PROBLEM FORMULATION

III. ADAPTIVE CMAC-BASED SUPERVISORY CONTROL SYSTEM

Consider the th-order nonlinear dynamic system expressed in the canonical form

.. .

(4) where , in which ( ) are positive constants. Applying control law (4) to system (2) results in the following error dynamics. (5) is chosen such that all roots of the polynomial fall strictly in the open left half of the complex plane. This implies that tracking of the reference for trajectory is asymptotically achieved where any starting initial conditions. However, the functions and are not precisely known in general and the external disturbance is always unknown. Therefore, the ideal control law (4) is unrealizable for practical applications. For the system output to follow a given reference trajectory effectively, an adaptive CMAC-based supervisory control system is proposed in the following sections. Suppose

The configuration of the adaptive CMAC-based supervisory control system, which combines a supervisory controller and an adaptive CMAC, is depicted in Fig. 1. The control law is assumed to take the following form. (6)

(1) or, equivalently the form (2) where and are unknown real continuous functions (in and are the control input general nonlinear), and output, respectively, is the unknown external disturbance, and is a state vector of the system which is assumed to be available. In order for (2) to be controllable, it is required that for all in a certain controllability region . Since is continuous, without loss of generality, it is assumed that for all . First of all, the functions and are assumed to be bounded. The purpose is to design a control system such can track a given reference trajectory that the state . Here, the tracking error vector is defined as (3)

is the supervisory controller; is where and a the adaptive CMAC, which combines a CMAC . The supervisory controller can be compensated controller designed alone to stabilize the states of the controlled system within a predefined constrain set; however, the control performance is omitted. Thus, the adaptive CMAC is introduced to cooperate with the supervisory controller to force the system states within the predefined constraint set and achieve satisfactory tracking performance. A. Supervisory Controller The design of supervisory controller is necessary in case of divergence of states. This controller is used to pull the states back to predefined constraint set. The supervisory controller fires only when the system states leave the predefined constraint set. When the system states stay within the constraint set, only the adaptive CMAC will be utilized to approximate the ideal control law. The design of supervisory controller is presented here and the design of adaptive CMAC will be analyzed in the following subsection. From (2) and (4) and using (6), an error equation is obtained. (7)

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Fig. 1.


Block diagram of adaptive CMAC-based supervisory control system.

where

.. .

.. .

.. .

.. .

..

.

.. .

such that , the bounds of the funcIn order to design and should be known. Therefore, the following tions assumption is made. , Assumption 1: The bound functions and are known such that and for , where , and . Moreover, the external disturbance is bounded by . Based on Assumption 1 and observing (10), the supervisory is designed as follows. controller

.. .

.. .

(11)

The Lyapunov function is defined as follows. (8) where is a symmetric positive definite matrix which satisfies the following Lyapunov equation. (9)

where

is a sign function, and the operator index if if

in which is a preset positive constant. Substituting (4) and , it is obtained that (11) into (10) and considering the case

is a positive definite matrix. Take the derivative and of the Lyapunov function and use (7) and (9), then

(12)

(10)

Using the supervisory controller shown in (11), the incan be obtained for nonzero value of the equality . As the results from (12), tracking error vector when


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Fig. 2. Architecture of a CMAC.

the supervisory controller is capable to drive the tracking error , to zero. However, owing to the selection of the bounds , and sign function, the supervisory control law will result in excessive and chattering control effort. Furthermore, the transient tracking performance may be unsatisfied. Therefore, the adaptive CMAC will be designed to overcome these drawbacks. B. Implementation of CMAC Fig. 2 shows the architecture of CMAC including input space, association memory space, receptive-field space, weight memory space and output space [8], [21]. The signal propagation and the basic function in each space of CMAC are introduced as follows. 1) Input Space : The is a continuous -dimensional , each input input space. For a given state variable must be quantized into discrete regions (called an element) according to given control space. The number of , is termed as a resolution. In this study, the input elements, state variables are represented as (13) 2) Association Memory Space : Several elements can be accumulated as a block. The number of blocks, , in the denotes an assoCMAC is usually greater than two. The components. ciation memory space with Fig. 3 depicts the schematic diagram of two-dimensional and ( is the number CMAC operations with is divided into of elements in a complete block), where is divided into blocks a, b, and c. blocks A, B, and C, and By shifting each variable an element, different blocks will be obtained. For instance, blocks D, E, and F for , and blocks d, e, and f for are possible shifted elements. In this space, each block performs a receptive-field basis function, which can be formulated as rectangular [8] or triangular or any continuously bounded function (e.g., Gaussian [17], [21] or B-spline [9], [22]). Here, Gaussian function is adopted as the receptive-field basis function, represented as (14) where mean

presents the th block of the th input and variance .

with the

Fig. 3.

CMAC in two-dimension with = 4 and n

= 9.

3) Receptive-Field Space : Areas formed by blocks, , and are called receptive-fields. The named as , is equal to in this study. number of receptive-field, Each location of corresponds to a receptive-field. The multidimensional receptive-field function is defined as

(15) th receptive-field, and . Note that if is a constant for the inside area covered by the th receptive-field and zero for the outside, the scheme discussed above becomes the conventional CMAC. The multidimensional receptive-field function can be expressed in vector notation as where

is associated with the

(16) and . Thus, , and are new receptive-fields for different blocks. In the CMAC network schemes, no receptive-field is formed by the combination of different layers such as “A, B, C” and “d, e, f.” With this kind of quantization and receptive-field composition, each state is covered by different (less than or equal to ) receptive-fields. Suppose the input falls within the th receptive-field, this field becomes active. Then, nearby inputs can activate one or more of the same weights, which can produce similar outputs. This correlation provides a very useful property of the CMAC, namely, local generalization. The output generalization capability of the CMAC is controlled mainly by the width of the blocks. If two inputs are far apart in the input space, there will be no overlap in their elements sets in , i.e., no generalization. where

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4) Weight Memory Space : Each location of to a particular adjustable value in the weight memory space with components can be expressed as .. . .. .

..

.

.. .

.. . .. .

.. . ..

functions into a partially linear form [23], [24], the expansion of in Taylor series becomes .. .

.. . .

.. .

.. .

.. .

(17) , and where denotes the connecting weight value of the th output associated is initialized from with the th receptive-field. The weight zero and is automatically adjusted during online operation. 5) Output Space : The output of CMAC is the algebraic sum of the activated weights in the weight memory, and is expressed as

.. . .. . (24) ; is the optimal parameter of where is the estimated parameter of ; ; is a vector of higher-order terms;

(18) The outputs of the CMAC can be expressed in a vector notation as

; ; and

are defined as (25)

(19) In a two-dimension case shown in Fig. 3, the output of the , , CMAC is the sum of the value in receptive-fields and , where the input state is (3,3). In this study, the CMAC is utilized to estimate the ideal control law presented in (4), so that a single-output CMAC can be written as follows:

; ;

(26)

Rewriting (24), becomes (27)

(20) Substituting (27) into (23), yields C. Compensated Controller Assume there exists an optimal CMAC such that mate the ideal control law

to approxi-

(21) where is a minimum reconstructional error; are optimal parameters of , , , and , becomes Rewriting

, , and , respectively.

(28) where the uncertain term , and this term is assumed to be bounded with a small positive ). From (23) and (28), the error (7) can constant (i.e., be rewritten as follows.

(22)

(29)

where , , and are the estimates of the optimal parameters. By subtracting (21) from (22), an approximation error is defined as

Theorem1: Consider the nonlinear dynamic system expressed by (2). The adaptive CMAC-based supervisory control law is designed as (6) where the supervisory control law is designed in (11) and the adaptive CMAC is presented in (22). Here, in the adaptive CMAC, the adaptive laws of the CMAC are designed as (30)–(32) and the compensated controller is designed as (33) with the estimation law given in (34), where , , and are strictly positive constants. As a result,

(23) and . Since, the linearization where technique transforms the multidimensional receptive-field basis


the stability of the adaptive CMAC-based supervisory control system can be guaranteed. (30) (31) (32) (33)

Since tion (i.e., implies ,

,

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, , and

is a negative semi-definite func), which are bounded. Let function , and integrate function

with respect to time

(34)

(38) is bounded, and Since is nonincreasing and bounded, the following result can be obtained.

(35)

(39)

Proof: A Lyapunov function is defined as

where the estimation error of the uncertainty bound is defined . Taking the derivative of the Lyapunov function as (35) and using (9) and (29), it is concluded that

is bounded, by Barbalat’s Lemma [25], In addition, since . That is, as . it can be shown that As a result, the stability of the proposed adaptive CMAC-based supervisory control system can be guaranteed. D. Convergence Analyzes of CMAC Although the stability of the adaptive CMAC-based supervisory control system can be guaranteed, the parameters , and in (30)–(32) cannot be guaranteed within a desired bound value. From (20), the output of the CMAC is bounded if the weights, means and variances are bounded. Define the constrain , and for , and , respectively, as sets (40) (41) (42)

(36) From (11) and (30)–(34), (36) can be rewritten as follows.

(37)

if if

if if

where is a two-norm of vector, and , and are positive constants. According to the projection algorithm [20], the adaptive laws (30)–(32) can be modified as follows: a) For , use, shown in (43) at the bottom of the page. b) For , use, shown in (44) at the bottom of the page. c) For , use, shown in (45) at the bottom of the next page. , and defined in Theorem 2: For the constraint sets , (40), (41) and (42), respectively, if the initial values and then the adaptive laws (43)–(45) guarantee that , and for all .

or and

or and

and

and

(43)

(44)

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Proof: Define a Lyapunov function (46)

, Using the property , where , which is according to and ; the supervisory controller shown in (11); and the compensated controller shown in (33), it is obtained that

Take the derivative of the Lyapunov function with respect to time (47) Suppose the first line of (43) is true, either or for the conditions and ), thereby ( is guaranteed. On the other hand, when the condiand ) hold, tions ( . can also be assured. Since the initial In this case, value of is bounded (i.e., ), is bounded by for all . Similarly, it can be proved the constraint set is bounded by the constraint set and is that for all . bounded by the constraint set (49)

E. Stability Analyzes Using Projection Algorithm The modified adaptive laws shown in (43)–(45) guarantee that the weights, means and variances are bounded within a preset value. Therefore, the convergence property of the CMAC is also guaranteed, and the asymptotical stability is illustrated by the following theorem. Theorem 3: Consider the nonlinear system expressed by (2). The adaptive CMAC-based supervisory control law is designed as (6) where the supervisory control law is designed in (11) and the adaptive CMAC is presented in (22). Here, in the adaptive CMAC, the adaptive laws of the CMAC are given in (43)–(45) and the compensated controller is designed as (33) with the estimation law given in (34). As a result, the asymptotically stability of the control system is guaranteed. or ( Proof: When the conditions and ), or ( and ) and or ( and ) hold, the stability analysis is the same as Theorem 1. On the other hand, when and ) or ( ( and ) or ( and ) occurs, the derivative of the Lyapunov function shown in (36) can be rewritten as follows.

Using the same discussion shown in Theorem 1, the stability as . property can be also guaranteed since IV. ILLUSTRATIVE EXAMPLES In this section, the proposed adaptive CMAC-based supervisory control system will be applied to control a rigid robotic manipulator, a Chua’s chaotic circuit and a linear piezoelectric ceramic motor. A. Example 1 The dynamic equation of a one-link rigid robotic manipulator is given by [5] (50) where the link is of length and mass , and is the angular and . The above position with initial values dynamical equation can be written as the following. (51) where

, , , , the parameters in (51) are chosen as , and is the external disturbance and is assumed and period . The to be a square wave with amplitude reference model is defined as (52)

(48)

if if

or and

and

(45)


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Fig. 4. Simulation results of supervisory control for the robotic manipulator: (a) Tracking response. (b) Control effort u. (c) Tracking error e.

where and is a periodic rectangular signal. The adaptive CMAC-based supervisory control , and system implemented here needs to have the bounds . In this system, and are chosen. Also

and

by solving (9), it is obtained that

are determined. Then, . The param-

eters are chosen as , , , , , and . Furthermore, the CMAC used in this example can be characterized as , , . The receptive-field basis functions for are chosen as and , 2, 3, 4, 5, 6, 7, 8. Here, the parameters are chosen as and

for all and . The receptive-fields are selected to cover the input along with each input dispace mension, and along with each input dimension, and . Note that, only less than four weights are updated at any given weights are required to deinstant regardless that fine the outputs. Fig. 4 shows the simulation results under which the supervisory controller is implemented alone, and Fig. 4(a) illustrates the tracking response of system state and reference . Also, Fig. 4(b) and (c) depict the associated consignal trol effort and tracking error , respectively. Although favorable tracking responses is obtained, the chattering phenomena of the control efforts due to the switching operation lead to the reduction of tracking accuracy. Fig. 5 presents the simulation results of adaptive CMAC-based supervisory control system, and

Fig. 5. Simulation results of adaptive CMAC-based supervisory control for the robotic manipulator: (a) Tracking response. (b) Control effort u. (c) Supervisory control u . (d) Tracking error e.

Fig. 5(a) illustrates the tracking response of system state and . Also, Fig. 5(b)–(d) shows the associated reference signal control effort , the supervisory control and tracking error , respectively. Note that Fig. 5(c) shows two activation periods: [0, 0.015] and [0.145, 0.175] sec. After 0.175 sec, the supervisory controller is no longer activated. In Fig. 5(d), the tracking error of the proposed control system is much smaller than the control result presented in [5]. B. Example 2 The typical Chua’s chaotic circuit, as shown in Fig. 6, is a simple electronic system that consists of one linear resistor ( ), ), one inductor ( ), and one nonlinear two capacitors ( , resistor ( ) [26], [27]. It has been shown to possess very rich nonlinear dynamics such as chaos and bifurcations. The dynamic equations of Chua’s circuit are written as

(53)

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Fig. 6. Chua’s chaotic circuit.

where the voltages , and current are state variables; is a constant; and denotes the nonlinear resistor, which is a function of the voltage across two terminals of . The is defined as a cubic function as (54) The state equations in (53) are not in the canonical form in (1). Therefore, a linear transformation is needed to transform them into the form of (1). According to [27], the dynamic equations of transformed Chua’s circuit can be rewritten as

(55) Fig. 7. (a) Phase plant trajectory of the transformed Chua’s circuit with u(t) = 0. (b) State x of the transformed Chua’s circuit. (c) State x of the transformed

where

Chua’s circuit. (d) State x of the transformed Chua’s circuit.

(9), , and is the external disturbance and is assumed to be and period . And, this a square-wave with amplitude system selects

and . To compare with numerical results in [27], the system has the same reference trajectory . Fig. 7 illustrates the simulation results of the transformed Chua’s circuit without the control system. But the following employs the adaptive CMAC-based supervisory control system to control the transformed Chua’s circuit to follow the desired reference trajectories. First, choose and

. Then, by solving

is determined. Next, choose the

, , , , parameters , and . In addition, the , CMAC used in this example can be characterized as , . The receptive-field basis functions are chosen as for , , 2, 3, 4, 5, 6, 7, 8. As well, the parameters are 2, 3 and and chosen as

for all and . The receptive-fields are selected to cover the input space along with each input dimension. Note that, only less than four weights are updated at any given weights are required to define instant regardless of the outputs. Fig. 8 shows the simulation results under the supervisory controller alone, and Fig. 8(a) illustrates the trajectory of


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Fig. 8. Simulation results of supervisory control for the transformed Chua’s circuit: (a) Phase plant trajectory (x ; x ). (b) The state x and its reference trajectory y . (c) The state x and its reference trajectory y _ . (d) The state x and its reference trajectory y . (e) The control effort u. (f) The tracking error e.

the system state in the phase plane. Also, Fig. 8(b)–(d) show the states , and , and their reference trajectories , and , respectively. Fig. 8(e) and (f) depict the associated control effort and tracking error , respectively. The tracking performance can be achieved, yet there exists the undesirable control chattering. Fig. 9 shows the simulation results of the adaptive CMAC-based supervisory control system, and Fig. 9(a) illustrates the trajectory of the system state in the phase plane. Also, Fig. 9(b)–(d) show the states , , and , and their reference trajectories , , and , respectively. Fig. 9(e)–(g) depict and the associated control effort , the supervisory control the tracking error , respectively. Fig. 9(f) shows that the supervisory controller is activated in the initial 0.135 sec. However, after 0.135 sec, the supervisory controller is no longer activated. Finally, in Fig. 9(g), the tracking error under the proposed control scheme is much smaller than the control result proposed in [27].

where indicates the position of the moving table of the LPCM; denotes a nonlinear dynamic function related to the comexpresses the ponents of stress, strain and electric field; is the control input, control gain of LC resonant inverter; represents the pre-load force and friction force. The and above dynamical motion equation can be rewritten as the following state equation (57) where , . Both the nominal case and the parameter variation case are provided in the experiment. The parameter variation case is the addition of one iron disk with 2.7 kg weight to the mass of the moving table. The reference model with rise time 0.42 sec, damping ratio set at 1 is chosen as

C. Example 3

(58)

The experiment for the LPCM position control is implemented with the proposed adaptive CMAC-based supervisory control system and its configuration is shown in Fig. 10. The adopted LPCM is a HR4 motor manufactured by Nanomotion with 15 W 270 Vrms 320 mArms 14 N. A servo control card is installed in the control PC, which includes multi-channels of D/A, A/D, PIO and encoder interface circuits. The position of the moving table is fed back using a linear scale. The dynamic motion equation of LPCM is given by [28] (56)

where and is a periodic step reference trajectory with amplitude 3.5 cm. First, and

are determined. Then, by solving (9), is obtained. The parameters are chosen as

, , , , , and . Also, the CMAC , , used in this example can be characterized as . The receptive-field basis functions are

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Fig. 9. Simulation results of adaptive CMAC-based supervisory control for the transformed Chua’s circuit: (a) Phase plant trajectory (x ; x ). (b) The state x and its reference trajectory y . (c) The state x and its reference trajectory y_ . (d) The state x and its reference trajectory y . (e) The control effort u. (f) The supervisory control u . (g) The tracking error e.

dimension. Therefore, the parameters are chosen as and

Fig. 10.

PC-based LPCM experimental control system.

chosen as for , 2 and , 2, 3, 4, 5, 6, 7, 8. The receptive-fields are selected to cover along with each input the input space

for all and . The experimental results for a periodic step command with two test cases are shown in Fig. 11. The tracking responses are shown in Fig. 11(a) and (d); the associated control efforts are shown in Fig. 11(b) and (e); and the tracking errors are shown in Fig. 11(c) and (f) for nominal case and parameter variation case, respectively. Fig. 11(g) shows that the supervisory controller is only activated at the third sec for the parameter variation case. The experimental results indicate that the high-accuracy motion tracking responses can be obtained for both nominal and parameter variation cases. V. CONCLUSIONS An adaptive cerebellar-model-articulation-controller (CMAC)-based supervisory control system, including an adap-


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Fig. 11. Experimental results of adaptive CMAC-based supervisory control for LPCM. (a) Tracking response for nominal case. (b) Control effort u for nominal case. (c) Tracking error e for nominal case. (d) Tracking response for parameter variation case. (e) Control effort u for parameter variation case. (f) Tracking error e for parameter variation case. (g) Supervisory control u for parameter variation case.

tive CMAC and a supervisory controller, has been proposed for uncertain nonlinear systems. In the adaptive CMAC, the CMAC is used to mimic an ideal control law and a compensated controller is designed to recover the residual part of approximation error. The supervisory controller is designed based on the uncertainty bound of the controlled system to force the system states staying within a predefined constraint set. The adaptive CMAC is used to achieve desired control performance. However, if the adaptive CMAC cannot maintain the system states within the constraint set, the supervisory controller will work to pull the states back to the constraint set. Otherwise, the supervisory controller will be idle. For the system stability to be guaranteed, all the adaptive laws are derived in terms of Lyapunov function. Furthermore, by applying the projection algorithm, the parameter convergences are also guaranteed.

From the simulation and experimental results, the proposed control system not only attenuates the tracking error caused by the modeling uncertainties and external disturbances, but also eliminates the chattering phenomena in the control efforts.

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Chih-Min Lin (SM’99) received the B. S. and M. S. degrees in control engineering, and the Ph.D. degree in electronics engineering from National Chiao Tung University, Taiwan, R.O.C., in 1981, 1983, and 1986, respectively. He was with the Chung Shan Institute of Science and Technology as a Deputy Director of system engineering of missile systems from 1986 to 1992. He also served concurrently as an Associate Professor at Chiao Tung University and Chung Yuan University, Taiwan. He joined the faculty of the Department of Electrical Engineering, Yuan-Ze University, Taiwan, in 1993, and is currently a Professor and the Chairman of the Department of Electrical Engineering. He has also served as the Committee Member of Chinese Automatic Control Society and the Deputy Chairman of IEEE Control Systems Society, Taipei Section. He was the honor research fellow in the University of Auckland, New Zealand from 1997–1998. His research interests include fuzzy neural network, cerebellar model articulation control, guidance and flight control and systems engineering.

Ya-Fu Peng was born in Taoyuan, Taiwan, R.O.C., in 1969. He received the B.S. degree from Yuan-Ze University, Chung-Li, Taiwan, R.O.C., the M.S. degree from National Taiwan Ocean University, Keelung, Taiwan, R.O.C., and the Ph.D. degree from Yuan-Ze University, Chung-Li, Taiwan, R.O.C., in 1994, 1996, and 2003, respectively, all in electrical engineering. Since 1996, he has been with the Department of Electrical Engineering, Ching-Yun University, Chung-Li, Taiwan, R.O.C., where he is currently an Associate Professor. His research interests include motor control, adaptive control, intelligent control, and flight control systems.