A neural network-based tracking control system

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL. 39, NO. 6, DECEMBER 1992

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A Neural Network-Based Tracking Control System Heng-Ming Tai, Member, IEEE, Junli Wang, a n d Kaveh Ashenayi, Senior Member, IEEE Abstract-This paper presents an application of the backpropagation neural network to the tracking control of industrial drive systems. The merits of the approach lie in the simplicity of the scheme and its practicality for real-time control. Feedback error trajectories, rather than desired and / or actual trajectories, are employed as inputs to the neural network tracking controller. It can follow any arbitrarily prescribed trajectory even when the desired trajectory is changed to that not used in the training. Simulation was performed to demonstrate the feasibility and effectiveness of the proposed scheme.

I. INTRODUCTION

M

AKING the output of a system track a given reference trajectory is a common industrial problem. For example, to move the robot joints [l] or underwater vehicles [2], drive the servomotors [31, assemble and dissemble equipment etc., all require the machine involved to follow a prescribed trajectory. In order to obtain satisfactory tracking performance, the dynamics of controlled systems are usually simple (e.g., linear) and explicitly known so that modern control strategy can be successfully applied. However, when the structure of the plant is unknown or the parameter variation is excessive, the effectiveness of modern control theory diminishes. For instance, when a fixed controller setting is employed in an industrial drive system with widely changing environment, unsatisfactory performance often occurs. Even if it is possible to develop a reasonably accurate model, the resulting control algorithm is so computationally intensive that it becomes infeasible to implement it in a real-time control environment. Erlvisioned on the performance exhibited by an experienced human operator, it is believed that a controller should be designed to have abilities to learn from experience and to use the knowledge gained during the training process. A tracking system is one in which the plant output signal is controlled so that it becomes and remains nearly equal to an externally applied input reference signal. Under this circumstance, the output is said to track or follow the reference input. In motor drive applications, this may require that each motor follow its predetermined position or speed trajectory during starting, speed change, Manuscript received October 2, 1991; revised December 2, 1991. H.-M. Tai and K. Ashenayi are with the Department of Electrical Engineering, University of Tulsa, Tulsa, OK 74104. J. Wang is with the Department of Electrical Engineering, Atlanta, C A 30332. IEEE Log Number 9202512.

and braking but without causing excessive stresses to the entire system hardware and with no excessive inrush current. Several tracking control techniques are evolving such as the sliding mode [41, variable structure [5],and self-tuning and model reference adaptive controls [6], [7]. The sliding mode and variable structure control systems require a valid model and/or dynamics of the plant being controlled. Thus they are not robust in the sense that the controller is, mainly due to the structured uncertainty of controlled plants, sensitive to large parameter variation and noise. Although the adaptive controls are effective in compensating the influence of the structured uncertainty, it is not clear that adaptive means can overcome the unstructured uncertainty. In addition, conventional adaptive control schemes require the information about the plant structure and may not guarantee the stability of the system in the presence of unmodeled dynamics [61, [81. Moreover, most of these adaptive control algorithms are rather complicated and, thus, need excessive computation effort for real-time implementation. In particular, the self-tuning controller did not track the reference signal well when the plant was nonlinear and must be redesigned if the plant is to follow a different reference input [9]. Recently, an emerging technique that mimics the adaptive distributed architecture in the human brain, namely artificial neural networks (A"), provides potential alternatives to tackle the task mentioned above. The applications of artificial neural networks promises a high computation rate provided by the massive parallelism, a great degree of robustness, or fault tolerdnce due to the distributed representation, and the ability of adaptation, learning, and generalization to improve performance. The idea of using neural networks for controlling physical systems has been explored [8]-[141. Such learning control systems typically employ neural networks (mostly with the error back-propagation learning algorithm) to learn the characteristics or inverse dynamics of controlled systems. All of these schemes use the desired iesponse and/or the plant output as inputs to the neural network. Psaltis et al. [lo], for example, proposed a two-stage learning algorithm to train a feedforward neural net controller to act as the inverse of the plant. Network training is performed based on on-line observation of the inputs and outputs of the plant. However, the operation may take much time due to the time-consuming learning of the back-propagation method. In order to circumvent this problem, a model learning scheme using a simple (nominal) dynamical model

0278-0046/92$03.00

0 1992 IEEE

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TAI er al.: A NEURAL NETWORK-BASED TRACKING CONTROL SYSTEM

for the generalized learning of the neural network was suggested by Ozaki et al. [8]. This procedure is efficient in learning the plant dynamics and is carried out off line. These controllers possess common features which have attracted control engineers; that is, they can respond flexibly to unmodeled dynamics and unexpected situations by using their learning and adaptation capabilities. In contrast, conventional controllers must be programmed in advance so that signals from sensors can be processed properly in response to (usually small) environmental changes. In this paper we propose a tracking control design using neural networks that do not learn the inverse dynamics of the plant as in the aforementioned approaches. The merits of the approach lie in the simplicity of the scheme and its practicality for real-time implementation. The design concept results from the belief that the main objective of the control system design is to determine the controller that can generate the proper control signal to achieve the desired performance in the plant output. Hence it is not necessary to explicitly identify or learn the plant dynamics as long as we can closely and quickly follow any prescribed reference signal. In this design, values of desired signals need not be known explicitly and in advance for the network training in that we employ error signals, the difference between the actual signafs and desired signals, and corresponding command signals, which are required to produce desired control signals, as training data. The neural net is trained using the error back-propagation method [15]. This design allows us to do the training off line and thus to avoid the disadvantage of back-propagation learning, which requires long training time. After the learning, we can implement it fQr real-time operation. In addition, we can change the reference signal level to any value and do not require to repeatedly train the neural network. Therefore, it is well suitable for trajectory tracking control. The objective of this paper is to design a tracking controller using neural networks for the industrial drive system rather than to discuss the theoretical development of neural networks. This tracking controller has the following salient features: 1) the tracking capability is outstanding, 2) it is robust against parameter variation and noise, and 3) it is suitable for high speed real-time implementation. In the following section, we describe the error back-propagation algorithm [15], which is the method used here to adapt the weights in the neural networks. General speaking, this algorithm may be viewed as a procedure for learning decision functions for patterns, based on information gathered from training set examples. Section I11 describes a neural net control scheme to track reference signal. In Section IV, simulation results are given to demonstrate the effectiveness of the proposed control scheme. 11. NETWORKSTRUCTURE AND LEARNING SCHEME

The network used for this study is a three-layer feedforward neural network with the error back-propagation

learning algorithm [15] that consists of input, hidden, and output layers. Each layer contains several processing elements or perceptrons with sigmoidal nonlinearities. Cybenko [16] has shown that this net can be used to approximate arbitrary functions, i.e., it can model any continuous nonlinear transformations. The significant consequence is that a neural net using sigmoidal nonlinearities and multilayer perceptrons with only one hidden layer can form complex disjoint and convex decision regions. In this study, the back-propagation neural network is utilized as pattern classifier instead of acting as the inverse of the controlled plant. The network is feedforward in the sense that each unit receives inputs only from the units in the preceding layer. The idea underlying the design of the feedforward network is that the information going to the input layer units is recorded into an internal representation and the outputs are generated by the internal representation rather than by the inputs. The network then converts input signals according to connection weights. During learning, connection weights are adjusted in a direction to minimize the sum of squared errors between the desired outputs and the network outputs. The errors are then propagated back to assign credits to each connection weight. A back-propagation neural network used herein is shown in Fig. 1. In the following, the subscripts k , j and i refer to any unit in the output, hidden, and input layers, respectively. The total inputs to unit j in the hidden layer or unit k in the output layer is net,

w,,O,

=

r

=

k ,j

s

=j

,i

(1)

S

where wrs is the weight from the sth unit to the rth unit and 0, represents the output of unit s in the hidden and input layers. A sigmoidal nonlinearity is then applied to each unit r to obtain the output as 1

0, =f(net,)

=

1 + exp { -(net, - 0,)/6,}

(4

where 6, serves as a threshold of unit r and 6, determines the slope of the activation function f. In this study, we assume that 6, = 1. Hence, each layer communicates with all successive layers, there is no feedback within the network either between layers of individual units nor can units communicate with other units in the same layer. In the learning process, the network is presented with a pair of patterns, an input pattern, and a corresponding desired output pattern. Learning comprises changing the weights and thresholds so as to minimize the mean squared error between the actual outputs and the desired output patterns in a gradient descent manner. The activity of each unit is propagated forward through each layer of the network by using (1) and (2). The resulting output pattern is then compared with the desired output pattern, and an

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methodology is different than many other neural net control scheme in that the neural network is used as a classifier rather than an inverse of the plant. Control knowledge and information are in essence embedded into the internal representation of the neural net. The control signal can be generated virtually from a table look-up procedure rather than multiplication operations as in the Input Layer Hidden Layer Output Layer or self-tuning and model reference approaches. It has the h 6 merit of not necessarily to retrain the neural net whenFig. 1. Neural net classifier ever the desired trajectory is changed to that not used in the training. The proposed neural network tracking control system is error 8, for each output unit is calculated as shown in Fig. 2. Training involves using the error signals, e between the plant output signals and the desired signals as inputs to the neural network. The neural network tracking controller (Fig. 3.) contains four units: preprocessing, neural network classifier, look-up table, and servo where tk is the desired output and 0, is the actual drive unit. The preprocessing part will scale the error output. The error at the output is then back-propagated signal into the range of [ - 1 , +1] and partition it into recursively to each lower layer as follows: several groups. Each group clusters those error signals for which an appropriate control action would correct. These (4) errors serve as inputs to the neural net classifier. A neural net classifier is a feedforward three-layer back-propagation net as described in Section 11, which consists of the In order for the network to learn, the value of each weight input layer with several units, one hidden layer with and threshold has to be incrementally adjusted in propor- various number of units, and six neurons in the output tional to the contribution of each unit to the total error. layer. The number of units in the input layer are deterThe change in each weight and threshold is calculated as mined by the particular application and the number of trajectories to be followed. After the learning process, it performs the function of classification and/or mapping. Awr,(f + 1 ) = $rO, + a A w r , ( f ) r = k ,j ; s = j , i Outputs of the neural net classifier will be rounded off ( 5 ) into "1" or "0," in which "1" indicates the abnormal case. The look-up table will determine whether to increase (or to reduce) the control signal based on the output decision where 7 controls the rate of learning and f denotes the produced from the neural net classifier. In particular, it number of times for which a set of input patterns have relates the possible error ranges and their corresponding been presented to the network. The parameter (Y de- control actions. This signal is then fed through the servo termines the effect of previous weight changes on the drive unit to generate the proper control signal V, to current direction of movement in weight space. The con- drive the plant to correct the deviation. The servo drive nection weights and thresholds of the network were ini- unit contains the D/A converter, amplifier, trigger, and tialized to small random values uniformly distributed be- SCR. The error signals was scaled down to values within tween -0.5 and 0.5. This was done to prevent the hidden [ - 1, + 11 according to the specified operation range. The units from acquiring identical weights during learning. corresponding error regions for classification purpose is One characteristic of error back-propagation nets is depicted in Fig. 4. In particular, if the scaled error e is long learning time. Learning times are typically longer between -0.05 and +0.05, we categorize that the plant when complex decision regions are required and when works normally; hence no control action is needed. If the networks have more hidden layers. Nevertheless, they scaled error is -0.3, then the neural net will send a exhibit extremely low memory and computation require- message to point out that the plant is off track. In such a ments during classification. Hence, it is very suitable for circumstance, the tracking controller will make some efreal-time processing and control. fort to direct the plant back to the right track. Table IV gives information about decision rules, error 111. NEURALNETWORK MODELSFOR TRACKING ranges, and corresponding cpntrol actions employed in CONTROL Simulation 2 in Section IV. In the control actions column, The basic concept behind the neural network approach A denotes the scaling factor and the amount of is to generate an approximation to the classification re- increase/decrease of the proportional terms is expressed gions from input-output measurements (based on the by three coefficients k , , k , , and k,. The values of the empirical data) and to use them as feedforward rules to scaling factor and parameters k , , k,, and k , are detercalculate the appropriate control signal. This control mined by the particular characteristics of the controlled Lon~l\~5

M IIIY~UII\

COnslP~

nimoidd neurons

C0"SISLS 01

sigmoidalncurnm

507

TA1 et ai.: A NEURAL NETWORK-BASED TRACKING CONTROL SYSTEM

Neural Network Tracking Controller

- 4

I

Fig. 2. Block diagram of the neural network-based tracking control system.

e Re+processing -F

Shaft power Terminal voltage Armature current Speed Armature resistance No-load speed

= 17 kw =220 V =88.9 A =3000 r/min =0.316 R = 1600 r/min

Its dynamics can be approximately represented by a transfer function 23 w(s) =

S 2 + 33s

+ 185

Fig. 3. Neural network tracking controller.

A. Simulation 1

plant and the tracking performance criterion. Parameters k , , k , , and k , play an important role in the whole tracking process because they determine the range of variation of each control effort. In general, in order to obtain good tracking performance, the values k , , k , , and k , should be in ascending proportion and with 2 k , and 3 k , slightly greater than k , and k , , respectively. This control scheme is very similar to the behavior of an experienced operator. Once the operator sees irregular motion, he or she would immediately decide, based on experience, what the next move is and how much effort should be made to correct the abnormal condition. It should be noticed that the proposed approach is conceptually analogous to the fuzzy logic control [17] in the sense that certain expert control rules are provided for making proper control decision. The differences lie in that no membership function and fuzzy logic operation are given. The neural network classifier itself contains the knowledge.

Three sensor signals (Speed, current, and tension) were utilized. There are six neurons in the input layer of the neural net classifier, two units each are designated to speed, current, and tension trajectories, respectively. One unit denotes the negative error signal and another unit denotes the positive error signal. Training sets consisting of 7, 19, and 31 patterns were employed, in which error signals were picked randomly from a simple three error groups, depicted in Fig. 5. Networks with 10, 15, 20, and 30 hidden units were trained and tested using identical training and testing sets and their performance was compared. 200 test patterns were generated randomly. After the training phase, these test patterns were used to test the generalization capability of the network. Table I lists the seven training patterns taught to neural net classifier. This is a very simple and fuzzy training set. Pattern 1 shows that if all the error signals are in normal conditions, then no irregular situation will be pointed out. This means that the motor is operated in the satisfactory condition. Pattern 3 indicates that the speed is too high. I v . SIMULATION AND RESULTS Table I1 displays the simulation results on the number of To investigate the performance of the control algorithm iteration cycles versus the number of units in the hidden described in Section I11 for the tracking control, computer layer with 19 training patterns. This result indicates the simulations were conducted for a dc motor drive system. ability of the neural net to discriminate the training In the first simulation, we test the generalization and patterns. This tells us about how quick the neural net collective processing capabilities of the tracking neural classifier will learn to classify all the training patterns. We network tracking strategy. Simulation 2 demonstrates the may say that the number of units in the hidden layer effectiveness of the proposed tracking control system. represents the learning or intelligence capacity of a neural Since values of the error signals to the input layer of the net. We see that the 20 units hidden layer will give the neural net classifier are continuous over [ - 1, + 11, infinite fastest learning. The 30 units hidden layer is slow in that number of inputs can be expected. We like to see whether classification is complicated and possible interference and a few finite number of training data will produce satisfac- oscillation may occur during the training phase. The simutory tracking results. Several parameters employed in the lation results on the test of the generalization and collecsimulation are listed as follows. The learning tolerance E tive processing capabilities of the network are shown in used to terminate the training process was set to be 0.01. Table 111. It can be seen that the more learning patterns The other parameters of the learning process were learn- used for training, the better the classification. The success ing rate 7 = 0.65, momentum a = 0.6, and 0, = 1. Initial rate is high if we consider individual trajectory only. When weights and thresholds were randomly selected in the all three sensor signals were taken into consideration at interval [ - 0.5,0.5]. The simulation program was written the same time, the highest success rate can only reach in FORTRAN 77, and run on the SUN workstation. The 74%, which was not good. Based on the experience gained test system consists of a shunt connected dc motor with from the simulation, one possible way to improve the

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1

-0 74 -0 65

-043 -0 35

- 0 . 1 3 - 0 05

0.05 0 13

0

0 35 0 43

0 65 0 . 7 4

1

Fig. 4. Normalized error ranges for classification.

TABLE I TRAINING SETPATTERNS EMPLOYED IN SIMULATION 1 Input Current

Speed

Tension

Speed

Pattern Number

-

+

-

+

-

+

H

L

1 2 3 4 5 6 7

0 0 -1 0 0 0 0

0 +1 0 0 0 0 0

0 0 0 0 -1 0 0

0 0 0 +1 0 0 0

0 0 0 0 0 0 -1

0 0 0 0 0 +1 0

0 0 1 0 0 0 0

0 1 0 0 0 0 0

Low

Normal

High

+-----I-~ -1

-0.2 0 0.2

1

Fig. 5 . Normalized error regions for classificationused in Simulation 1.

TABLE I1 NUMBER OF ITERATION CYCLES NEEDED FOR THE NEURAL NETWORK TRACKING CLASSIFIER T o CONVERGE WHENTHE LEARNING TOLERANCE Is 0.01; IN THISSIMULATION, 19 TRAINING PATTERNS WEREUSED Number of Units In The Hidden Layer

Number of Iteration Cycles

10 15 20 30

20,806 19,055 12,536 12,685

performance of proposed scheme is to separate the neural net classifier into two cascade nets. One is trained with a few learning patterns; this will provide the function similar to the fuzzy control technique. The second network is trained with much more learning patterns than the preceding one such that accurate classification can be achieved. The other possibility is to partition the error into more groups like the one in Simulation 2. B. Simulation 2 Only the speed tracking (see Fig. 6) was considered in this study. The neural net classifier is shown in fig. 7. The neural network based tracking controller was trained off line to learn various error groups and the associated control signals as described in Table IV. It is evident that no a priori knowledge about the dc motor (plant) and its dynamics is required. Hence the performance will not be affected and degraded whether the plant is linear or nonlinear, although this test system is linear. In this simulation, we assume that errors frequently occur in the neighbors of points-0.87, -0.54, -0.24,

output Current H L 0 0 0 0 1 0 0

0 0 0 1 0 0 0

Tension H L 0 0 0 0 0 0 1

0 0 0 0 0 1 0

TABLE I11 SUCCESS RATEOF THENEURAL NET-TRACKING CLASSIFIER WITH 200 RANDOMLY GENERATED TEST PATERNS Number of Learning Patterns

Sensor Signals

7

19

31

Speed Current Tension Total

0.855 0.790 0.895 0.600

0.920 0.840 0.870 0.665

0.985 0.875 0.845 0.740

0,0.24,0.54, and 0.87. Thus we partitioned the error range into seven groups as shown in Fig. 4. Each error range index has a corresponding desired output pattern and control action. Within each error group, four points were randomly selected as the training patterns. In case the error signal falls in the error group 5, the neural network model will classify it to pattern (000 100) and the plant controller will look up in the table to generate a control signal, c , + k , A , where A = 19.2. This control signal actually increases the supplied armature voltage to speed up the motor to follow the reference track. The values of proportionality parameters k , , k , , and k , were chosen as 0.7, 1.0, and 1.8, respectively. In real-world application, process disturbance and component variation often occur so that control effort developed under the nominal (perfect) condition may be degraded. To investigate the robustness and disturbance suppression capability of the tracking scheme, two types of disturbance are taken into consideration during the simulation process. One is the random disturbance, another is the system disturbance. The former is usually due to the environmental and white noises, whereas the latter mainly results from the component and structure variation. In the first case, we add the disturbance of amount -0.2 at the moment of 10, 30, and 50 s, respectively. On the other case, we consider the situation that the error signal level was down 0.2 during the run time. The results are plotted in Fig. 8. The solid line is for the reference track and the line with symbol "*" represents the tracking

TAI ef al.: A NEURAL NETWORK-BASED TRACKING CONTROL SYSTEM

509

Control Voltage Tracking Controller

Fig. 6. The tracking control system used in Simulation 2.

TABLE IV LOOK-UPTABLEFOR ERRORRANGES AND CORRESPONDING OUPUTPATTERNS AND CONTROL ACTIONS Error Index

Error Ranges

1

2

1 2 3 4 5 6 7

[ - 1.OO, - 0.741 [ - 0.65, - 0.431 [ - 0.35, - 0.131 [ - 0.05, 0.051 [ + 0.13, + 0.351 [ + 0.43, + 0.651 [ + 0.74, + 1.001

1

0 0 0 0 0 0

0 1 0 0 0 0 0

+

Output Patterns 3 4 0 0 1 0 0 0 0

0 0 0 0 1 0 0

5

6

0 0 0 0 0 1 0

0 0 0 0 0 0 1

Control Actions V, - k 3 A V, - k2A V, - k l A

v, + +

V, k l A V, k 2 A V, + k 3 A

60

-" .

-1

3 v) P

Fig. 7. Neural net classifier used in Simulation 2.

performance with no disturbance. The dash line is for the tracking performance with the random disturbance and the dash-dotted line with the system disturbance. It can be seen that this neural net tracking system follows the trapezoidal curve well. With random disturbance added, the trace was temporarily out of the desired reference trajectory but moved back quickly. With the system disturbance added, the actual trajectory was down a bit from the desired trajectory. Next, we show that how this tracking control scheme performs for the set point tracking (Fig. 9). In case k , , k , and k , are chosen as 0.5, l,.O, and 1.5, respectively, a fast and almost no-overshoot response was obtained. However, if k , , k , , and k , are selected as 0.4, 1.1, and 1.8, respectively, tracking response is oscillated over a bounded range and never converges to the set point. Therefore it is important to pick the right values for these proportional parameters to achieve good results. V. CONCLUDING REMARKS A neural network control scheme has been proposed for achieving the practical and real-time tracking of any arbitrarily prescribed trajectory with high degree of accuracy. The proposed controller classifies the feedback error signal and generates the appropriate control action to

i

0

10

20

30

40

50

60

70

Time (seconds)

Fig. 8. DC motor speed tracking on the stages of starting, running, and braking. reference trajectory; -*-*- trajectory without disturbance; _ _ _ _ _ _ trajectory with random disturbance; trajectory with system disturbance. ~

rectify the plant output, instead of acting as the inverse of the controlled plant. The generation of the tracking control is in essence a table look-up procedure that is well suitable for real-time application. Moreover, it performs well in the presence of uncertainties in plant parameters and control process. Simulation results indicate that this neural net tracking controller can learn and synthesize error knowledge effectively, thereby providing a framework for implementing tracking scheme in a variety of industrial drive applications. It can be seen that if the neural net classifier unit is replaced by fuzzy control rules and take change-in-error signals into consideration, we have a fuzzy control system [17] provided that fuzzy modeling of the plant is also given. The analogy between these two schemes is worth

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L. G. Kraft and D. P. Campagna, “A comparison between CMAC neural network control and two traditional adaptive control systems,” IEEE Control Systems Mag., vol. 10, no. 3, pp. 36-43, 1990. D. Psaltis, A. Sideris, and A. Yamamura, “A multilayered neural network controller,” IEEE Control Systems Mag., vol. 8, no. 2, pp. 17-21, April, 1988. T. Yamada and T. Yabuta, “An extension of neural network direct controller,” in h o c . Int. Workshop Intelligent Robots Syst., Tsuchiura, Japan, 1990, pp. 619-626. M. Kawato, K. Kurukawa, and R. Suzuki, “A hierarchical neuralnetwork model for control and learning of voluntary movement,” Biol. Cybem., vol. 57, pp. 169-185, 1987. A. Barto, R. Sutton, and C. Anderson, “Neuronlike adaptive elements that can solve difficult, learning control problems,” IEEE Trans. Syst. Man, Cybem., vol. SMC-13, pp. 834-846, 1983. A. Guez and J. Selinsky, “A trainable neuromorphic controller,” J . Robotic Syst., vol. 5, pp. 363-388, 1988. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation in parallel distributed processing,” Explorations in the Microstmctures of Cognition, Vol. 1: Foundtions. Cambridge, M A The MIT Press, 1986, pp. 318-362. C. Cybenko, “Approximations by superpositions of a sigmoidal function,” Math. Contr., Signal, Syst., vol. 2, pp. 303-314, 1989. M. Sugeno, “An introductory survey of fuzzy control,” Information Sci., vol. 36, pp. 59-83, 1985.

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k2 = 1 0

k 3 - 1 5

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. Simulated

- Reference Track

1,

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Time (sec)

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Stmulaled Reference Track

Heng-Ming Tai (M87) was born in Taiwan, Re-

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Time (sec)

Fig. 9. Set point tracking with various proportional parameter values.

further investigating to fully exploit their advantages. Performance comparisons between them are under way and the result will be presented elsewhere. Future research on this subject focuses on finding the criterion or a learning scheme to select the right values for parameters ki’s so that perfect tracking can be guaranteed. ACKNOWLEDGMENT The authors are grateful to the reviewers for their valuable comments.

REFERENCES 111 T. C. Hsia, T. Lasky, and Z. Guo, “Robust independent joint [21 [31

[51 161 171 181

controller design for industrial robot manipulators,” IEEE Trans. Znd. Electron., vol. 38, pp. 21-25, 1991. D. Yoerger and J. Slotine, “Robust tracking control of underwater vehicles,” IEEE J . Oceanic Eng., vol. 10, pp. 462-470, 1985. M. A. El-Sharkawi and M. Akherraz, “Tracking control technique for induction motors,” IEEE Trans. Energy Conversion, vol. 4, pp. 81-87, 1989. H. Hashimoto, K. Maruyama, and F. Harashima, “A microprocessor-based robot manipulator control with sliding mode,” IEEE Trans. Ind. Electron. vol. IE-34, pp. 11-18, 1987. M. El-Sharkawi and C. Huang, “ Variable structure tracking of dc motor for high performance applications,” IEEE Trans. Energy Conversion, vol. 4, pp. 643-650, 1989. C. J. Harris and S. A. Billings, Self-Tuning and Adaptwe Control: Theory and Applications. London: Peter Peregrinus Ltd., 1985. H. Naitoh and S. Tadakuma, “Microprocessor based adjustable speed dc Motor drives using model reference adaptive control,” IEEE Trans. Industry Applications, vol. IA-23, pp. 313-318, 1987. T. Ozaki et al., “Trajectory control of robotic manipulators using neural networks,” ZEEE Trans. Ind. Electron., vol. 38, pp. 195-202, 1991.

public of China, in 1957. He received the B.S. degree from National Tsing-Hua University, Hsinchu, Taiwan, and the M.S. and Ph.D. degrees in 1987 from Texas Tech University, Lubbock, TX, all in electrical engineering. He is presently an Assistant Professor in the Department of Electrical Engineering at the University of Tulsa, Tulsa, OK. His research interests are in adaptive and learning control, fuzzy logic, and neural networks and their applications. Dr. Tai is a member of Eta Kappa Nu. He is listed in Who’s Who in Science and Engineering in the South and Southwest.

Junli Wang received the B.S. degree in 1983 and the M.S. degree in 1986 from the Department of Precision Instrument Engineering at Tianjin University, People’s Republic of China, and the M.S.E.E. degree in 1991 in electrical engineering from the University of Tulsa. From 1986 to 1988, he was a lecturer in the Department of Precision Instrument Engineering at Tianjin University. He is currently a doctoral student in the Department of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA. His research interests are in the areas of industrial automation and control.

Kaveh Ashenayi (SM’86) received the B.S. and Ph.D. degrees in electrical engineering from Oklahoma State University, Stillwater, OK, in 1981 and 1986 respectively. He is currently an Associate Professor of Electrical Engineering at The University of Tulsa since September 1986. His research interests are in the areas of power systems, electrical cable design and testing, artificial neural networks, modeling and design of conventional and renewable energy systems, and automation and robotics. He is a member of the International Neural Network Society (INNS), the International Solar Energy Society (ISES), and Robotics International (RI). He has served as the chairman of the Tulsa chapter of Robotics International. He is currently the national chairman of the Robotics and Automation division of INNS.

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