Robust Control for Nonlinear Motor-mechanism Coupling System Using Wavelet Neural Network - Systems,...

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IEEE TRANSACTIONS ON SYSTEMS MAN AN D CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003 489

Robust Control for Nonlinear Motor-MechanismCoupling System Using Wavelet Neural Network

Rong-Jong Wai , Member, IEEE

Abstract—A robust controlled toggle mechanism, which isdriven by a permanent magnet (PM) synchronous servo motoris studied in this paper. First, based on the principle of com-puted torque control, a position controller is developed for themotor-mechanism coupling system. Moreover, to relax the re-quirement of the lumped uncertainty in the design of a computedtorque controller, a wavelet neural network (WNN) uncertaintyobserver is utilized to adapt the lumped uncertainty online.Furthermore, based on the Lyapunov stability a robust controlsystem, which combines the computed torque controller, the WNNuncertainty observer and a compensated controller is proposedto control the position of the motor-mechanism coupling system.The computed torque controller with WNN uncertainty observeris the main tracking controller, and the compensated controller isdesigned to compensate the minimum approximation error of theuncertainty observer. Finally, simulated and experimental resultsdue to a periodic sinusoidal command show that the dynamicbehaviors of the proposed robust control system are robust withregard to parametric variations and external disturbances.

Index Terms—Computed torque control, permanent magnetsynchronous servo motor, robust control, toggle mechanism,uncertainty observer, wavelet neural network.

I. INTRODUCTION

COMPUTED torqueor inverse dynamicstechnique is a spe-cial application of feedback linearization of nonlinear sys-tems. A number of works related to computed torque control of robotic manipulators have been published [1]. The computed

torque controller is utilized to linearize the nonlinear equation

of robot motion by cancellation of some, or all, nonlinear terms

[1]. However, the objection to the real-time use of such control

scheme is the lack of knowledge of uncertainties, which include

parametric variations and external disturbances of the system.

The adaptive control technique is essential for providing a stable

and robust performance for a wide range of applications (e.g.,

robot control, process control, etc.), and most of the applications

are inherently nonlinear with uncertainties [2]–[4]. Therefore,

several computed torque controllers have tried to circumvent the

problem of uncertainties using adaptive techniques [5]–[7].

Recently, much research has been done on applications of wavelet neural networks, which combine the capability of

artificial neural networks in learning from processes [8]–[11]

and the capability of wavelet decomposition [12]–[15] for iden-

tification and control of dynamic systems [16]–[20]. In Zhang

Manuscript received May 28, 2000. This work was supported by the NationalScience Council of Taiwan, R.O.C., under Grant NSC 90-2213-E-155-014. Thispaper was recommended by Associate Editor P. E. Borne.

The author is with the Department of Electrical Engineering, Yuan Ze Uni-versity, Chung-Li 32026, Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSMCB.2003.811125

and Benveniste [16], the new notation of wavelet network was

proposed as an alternative to feedforward neural networks

for approximating arbitrary nonlinear functions based on the

wavelet transform theory, and a backpropagation algorithm

was adopted for wavelet network training. Zhang et al. [17]

described a wavelet-based neural network for function learning

and estimation, and the structure of this network was similar to

that of the radial basis function network except that the radial

functions were replaced by orthonormal scaling functions.

Zhang [18] presented wavelet network construction algorithms

for the purpose of nonparametric regression estimation. From

the point of view of function representation, the traditionalradial basis function (RBF) networks can represent any function

that is in the space spanned by the family of basis functions.

However, the basis functions in the family are generally

not orthogonal and are redundant. This means that the RBF

network representation for a given function is not unique and is

probably not the most efficient. In this study, the family of basis

functions for the RBF network is replaced by an orthogonal

basis (i.e., the scaling functions in the theory of wavelets) to

form a wavelet neural network [17], [19].

The toggle mechanism has many applications, for example,

clutches, rock crushers, truck tailgates, vacuum circuit breakers,

pneumatic riveters, punching machines, forging machines, and

injection modeling machines, etc. On the other hand, a per-manent magnet (PM) synchronous servomotor has some merit

of compact structure, high air-gap flux density, high power

density, high torque-to-inertia ratio, and high torque capability.

Moreover, compared with an induction servomotor, a PM syn-

chronous servomotor has such advantages as higher efficiency,

due to the absence of rotor losses and lower no-load current

below the rated speed, and its decoupling control performance

is much less sensitive to the parameter variations of the motor.

Thus, the PM synchronous servomotor plays a vitally important

role in motion-control applications in the low-to-medium power

range. In the past year, a fuzzy neural network (FNN) control

and a variable structure model-following control (VSMFC) on

the toggle mechanism actuated by a PM synchronous servo-motor have been presented in [21], [22]. In Lin et al. [21], an

FNN controller with adaptive learning rates was implemented

to control a motor-toggle servomechanism. However, this

control strategy lacks a stability analysis, and many rules and

a pre-training process are needed for the FNN to achieve the

best control performance. In Wai et al. [22], a VSMFC system,

which was designed based on the principles of the adaptive

model-following control and the variable structure control,

was developed to control the position of a slider of the toggle

mechanism servo system. Although the stability property of

1083-4419/03$17.00 © 2003 IEEE

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490 IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS—PART B: CY BERNETI CS, VOL. 33, NO. 3, JUNE 2003

Fig. 1. Toggle mechanism driven by PM synchronous servo motor.

the VSMFC system can be guaranteed, some prior knowledge

about the controlled plant must be known, e.g., the bound of thevariation of system parameters and external load disturbance.

Therefore, the motivation of this study is to design a control

scheme to cope with the disadvantages mentioned above. Not

only is the stability property guaranteed, but also, no prior

knowledge of the controlled plant is required.

A toggle mechanism actuated by a PM synchronous servo

motor drive using a robust control system is described in this

study. First, based on the principles of computed torque control,

a position controller is developed. However, the general problem

in the design of a computed torque controller for the mechanical

systems is that the exact lumped uncertainty of the mechanical

system are difficult to be obtained in advance for practical ap-

plications. Therefore, an WNN uncertainty observer is utilizedto adapt the lumped uncertainty on line. Furthermore, based on

the Lyapunov stability a robust control system, which combines

the computed torque controller with the WNN uncertainty ob-

server and a compensated controller, is proposed to control the

position of a slider of the motor-toggle servomechanism. The

computed torque controller with the WNN uncertainty observer

acts as the main tracking controller, and the compensated con-

troller is designed to compensate the minimum approximation

error of the uncertainty observer. Finally, simulated and exper-

imental results due to a periodic sinusoidal command are pro-

vided to validate the effectiveness of the proposed robust control

scheme.

II. MOTOR-MECHANISM COUPLING SYSTEM

A toggle mechanism driven by a PM synchronous motor is

depicted in Fig. 1 [21], [22], in which the most important pa-

rameters that affect the control performance of the toggle mech-

anism are the external force and the parameter variation of

the mass of slider , known as . Moreover, , , and

are the mass of links 2, 3 and 5, respectively; , , and

are the length of link 2, respectively; and are the length

of links 3 and 5, respectively; , , and are the angle of

links 2, 3 and 5, respectively; and are the forces acting

on sliders and , respectively; is the height between the

two horizontal guides where sliders and move along; is

an offset between links 2 and 3. Hamilton’s principle and theLagrange multiplier have been used in [21] and [22] to derive

the differential-algebraic equation for the toggle mechanism.

The implicit method must be employed to solve the differen-

tial-algebraic equation of mechanical motion. The result is a set

of differential equations with only one independent generalized

coordinate.

The motor-mechanism coupling system shown in Fig. 1 can

be represented by the following equation:

(1)

where is a variable vector, denotes a

nonlinear dynamic function, is a control gain,

represents disturbance and friction function, and isthe con-

trol effort. Moreover, , and include

the uncertainties introduced by system parameters and external

disturbance. In addition, the control gain is a constant

negative-sign function and invertible [21]. Now, assume that the

parameters of the system are well known and rewrite (1) as

(2)

where and are the nominal values of

and , and is the nominal value of

, in which the external disturbance . If the

uncertainties occur, i.e., the parameters of the system are

deviated from the nominal value or an external disturbance isadded into the system, the dynamic equation of the coupling

system can be modified as

(3)

where , and denote uncertainties, and is

called the lumped uncertainty and is defined as

(4)

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WAI: ROBUST CONTROL FOR NONLINEAR MOTOR-MECHANISM COUPLING SYSTEM 491

Fig. 2. Block diagram of motor-toggle servomechanism using robust control system.

III. ROBUST CONTROL SYSTEM

The proposed robust control system is depicted in Fig. 2,

where , , , and are the command slider position,

slider position, command angle of link 2, and angle of link 2,

respectively. Since is the desired control objective and is

the state of the motor-mechanism coupling system, the andshould be transformed to and using the one-to-one

relationship [21], [22] as follows:

(5)

A linear scale detects the position signal of the slider . Now,

the robust control law is defined as follows:

(6)

where is a computed torque controller and is a com-pensated controller. The computed torque controller with the

WNN uncertainty observer is the main tracking controller, and

the compensated controller is designed to compensate the min-

imum approximation error of the WNN uncertainty observer.

A. Computed Torque Controller

The control problem is to find a control law such that the state

can track the desired trajectories in the presence of the

uncertainties. Let the tracking error vector be

(7)

If the lumped uncertainty of the controlled system is well

known, the computed torque control law can be defined as

follows:

(8)

where the control gain , and and are

positive constants. Substituting (8) into (3) and using (7), it canbe obtained that

(9)

If the control gain is selected properly such that all roots of

the polynomial (9) are in the open left-half complex plane, it im-

plies , that is, the state can track the desired

trajectories asymptotically. Since the lumped uncertainty

is unknown in practical applications, a WNN uncer-

tainty observer is proposed to adapt the value of the lumped

uncertainty . The purpose of the uncertainty observer is to

determine a value of for the computed torque controller to

compel the state to follow the desired trajectories underthe occurrence of uncertainties. From (8), the computed torque

controller is now defined as

(10)

B. WNN Uncertainty Observer

A four-layer WNN [17], [19], which is comprised of an input

(the layer), a mother wavelet (the layer), a wavelet (the

layer), and an output layer (the layer), is adopted to imple-

ment the proposed WNN uncertainty observer. The inputs of the

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Fig. 3. Simulated responses of robust control system. (a)–(c) Case 1. (d)–(f) Beginning at Case 1 and changing to Case 2 at 5 s. (g)–(i) Beginning at Case 1 andchanging to Case 3 at 5 s.

Using the property, which is according to ,

and the compensated controller shown in (22), then(32)

By use of Barbalat’s lemma [2], [3], as shown in Section III-C,as . Thus, the stability property also can

be guaranteed. The effectiveness of the proposed robust con-trol system for the control of motor-toggle servomechanism will

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Fig. 3. (Continued). Simulated responses of robust control system. (a)–(c) Case 1, (d)–(f) Beginning at Case 1 and changing to Case 2 at 5 s; (g)–(i) Beginningat Case 1 and changing to Case 3 at 5 s.

be demonstrated by the following numerical and experimentalresults.

IV. NUMERICAL AND EXPERIMENTAL RESULTS

By use of Runge–Kutta fourth-order numerical integrationmethod, (1) is solved for the motor-mechanism couplingsystem. For numerical simulations, the parameters of the togglemechanism are designed as follows:

kg kg kg

kg kg

m m m

m m

m m m

(33)

in which is the lead of the screw, is the mass of slider ,and and are the friction coefficient and gravity acceleration,respectively. In addition, the gains of the robust control systemare given as

(34)

For simplicity, the matrix in this study is selected as anidentity matrix. All the gains in the robust control system arechosen to achieve the best transient control performance inboth simulation and experimentation considering the limitationof the control effort and the requirement of stability. Moreover,the initialization of the network parameters described in [19]is adopted to initialize the parameters of the wavelets in thisstudy. The WNN has two, 14, seven, and one neurons at theinput, mother wavelet, wavelet, and output layer, respectively.Three simulation cases due to a periodic sinusoidal commandare addressed as follows:

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Fig. 4. Experimental results of robust control system: (a)–(c) Nominal case; (d)–(f) Parameter variation case.

Case 1) nominal case ( kg and );Case 2) parameter variation case (7.4 kg weight is added to

the mass of slider and );Case 3) external disturbance case ( kg and

).The control objective is to control the slider to move 0.02 mperiodically. In addition, the initial position of is 0.1216 m,and thecontrolledstroke ofthe slider , is equal to 0.02m.Substituting the slider position into (5), the angle of link 2can be obtained.

In the simulation, the robust control system shown in Fig. 2 is

considered under different simulated conditions. The simulated

results at Case 1, beginning at Case 1 and changing to Case 2

at 5 s, and beginning at Case 1 and changing to Case 3 at 5 s,

are all depicted in Fig. 3. The responses of the position of slider

are depicted in Fig. 3(a), (d), and (g); the associated control

effort are depicted in Fig. 3(b), (e), and (h); the theoretic and

estimated lumped uncertainty are depicted in Fig. 3(c), (f), and

(i). From the simulated results, the proposed control scheme is

robust under the occurrence of parameter variation and external

disturbance. In addition, some experimental results are provided

here to further demonstrate the effectiveness of the robust con-trol system. The control algorithms are implemented using a

Pentium computer with a 2-ms sampling interval. Two test con-

ditions are provided, which are the nominal case and the param-

eter variation case. In the experimentation, two iron disks with

7.4 kg weightto the massof the slider are added tothe param-

eter variation case. The responses of the robust control system

at two test conditions due to a periodic sinusoidal command are

depicted in Fig. 4, in which theresponses of thepositionof slider

are depicted in Fig. 4(a) and (d); the associated control effort

are depicted in Fig. 4(b) and (e); the estimated lumped uncer-

tainty are depicted in Fig. 4(c) and (f). From the experimental

results, the tracking errors converge quickly, and the robust con-


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trol characteristics of the robust control system under the occur-

rence of parameter variation can be clearly observed.

V. CONCLUSIONS

This study successfully demonstrates the application of a ro-

bust control system to control the position of a slider of the

motor-toggle servomechanism. First, the mathematical model

of the motor-toggle servomechanism was introduced. Then, the

theoretical bases of the proposed computed torque controller,

WNN uncertainty observer, and compensated controller were

described in detail. Moreover, simulation and experimentation

were carried out using a periodic sinusoidal reference trajectory

to test the effectiveness of the proposed robust control system.

The major contributions of this study are i) the successful de-

velopment of a robust control methodology, in which an WNN

is utilized to adapt the lumped uncertainty on line and a com-

pensated controller is designed to compensate the minimum ap-

proximation error of the WNN uncertainty observer, and ii) the

successful application of the robust control system to control the

slider position of the motor-mechanism coupling system consid-ering the existence of uncertainties.

ACKNOWLEDGMENT

The author would like to express his gratitude to the referees

and Associate Editor for their kind comments and suggestions.

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Rong-Jong Wai (M’00) was born in Tainan, Taiwan,

R.O.C., in 1974. He received the B.S. degree inelectrical engineering and the Ph.D. degree in elec-tronic engineering from the Chung Yuan ChristianUniversity, Chung-Li, Taiwan, in 1996 and 1999,respectively.

Since 1999, he has been with the Departmentof Electrical Engineering, Yuan Ze University,Chung-Li, where he is currently an AssociateProfessor. He is the chapter author of Intelligent

Adaptive Control: Industrial Applications in the

Applied Computational Intelligence Set (Boca Raton, FL: CRC, 1998) and theco-author of Drive and Intelligent Control of Ultrasonic Motor (Chung-Li,Taiwan: Tsang-Hai, 1999) and Electric Control (Chung-Li, Taiwan: Tsang-Hai,2002). He has authored numerous published journal papers in the area of intelligent control applications. His research interests include motor servodrives, mechatronics, nonlinear control, and intelligent control systems,including neural networks and fuzzy logic.

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