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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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494 IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS—PART B: CY BERNETI CS, VOL. 33, NO. 3, JUNE 2003
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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WAI: ROBUST CONTROL FOR NONLINEAR MOTOR-MECHANISM COUPLING SYSTEM 495
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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496 IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS—PART B: CY BERNETI CS, VOL. 33, NO. 3, JUNE 2003
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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WAI: ROBUST CONTROL FOR NONLINEAR MOTOR-MECHANISM COUPLING SYSTEM 497
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.