H Tracking-Based Sliding Mode Control for
Uncertain Nonlinear Systems via an Adaptive
Fuzzy-Neural Approach
Wei-Yen Wang, Member, IEEE, Mei-Lang Chan, Chen-Chien James Hsu, and Tsu-Tian Lee, Fellow, IEEE
Abstract—In this paper, a novel adaptive fuzzy-neural sliding mode controller with tracking performance for uncertain nonlinear systems is proposed to attenuate the effects caused by unmodeled dynamics, disturbances and approximate errors. Because of the advantages of fuzzy-neural systems, which can uniformly approximate nonlinear continuous functions to ar-bitrary accuracy, adaptive fuzzy-neural control theory is then employed to derive the update laws for approximating the uncer-tain nonlinear functions of the dynamical system. Furthermore, the tracking design technique and the sliding mode control method are incorporated into the adaptive fuzzy-neural control scheme so that the derived controller is robust with respect to unmodeled dynamics, disturbances and approximate errors. Compared with conventional methods, the proposed approach not only assures closed-loop stability, but also guarantees an tracking performance for the overall system based on a much relaxed assumption without prior knowledge on the upper bound of the lumped uncertainties. Simulation results have demonstrated that the effect of the lumped uncertainties on tracking error is efficiently attenuated, and chattering of the control input is significantly reduced by using the proposed approach.
Index Terms—Adaptive control, fuzzy-neural approximator, tracking performance, sliding mode control, uncertain nonlinear systems.
I. INTRODUCTION
O
VER the past decade, fuzzy logic has been successfully ap-plied to many control problems [1]–[3]. Parallel to the de-velopment of the fuzzy logic control, neural networks are also applied to several control problems [4]–[7] with satisfactory re-sults. Because both the neural network and fuzzy logic system are universal approximators [8], [9], researches [10]–[12], [29] have been conducted to derive various fuzzy-neural controllers to obtain better control performance. Examples include an adap-tive tracking control method with a radial basis function neuralManuscript received April 20, 2001; revised January 26, 2002. This work was supported by the National Science Council, Taiwan, R.O.C., under Grant NSC 89-2218-E-030-004. This paper was recommended by Associate Editor W. Pedrycz.
W.-Y. Wang is with the Department of Electronic Engineering, Fu-Jen Catholic University, Taipei, Taiwan 24205, R.O.C. (e-mail: wayne@ ee.fju.edu.tw).
M.-L. Chan is with the Department of Electrical Engineering, I-Lan Institute of Technology, I-Lan, Taiwan, R.O.C. (e-mail: mlchan@ilantech.edu.tw).
C.-C. J. Hsu is with the Department of Electronic Engineering, St. John’s and St. Mary’s Institute of Technology, Taipei, Taiwan, R.O.C. (e-mail: jameshsu@mail.sjsmit.edu.tw).
T.-T. Lee is with the Department of Electrical and Control Engi-neering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: ttlee@cn.nctu.edu.tw).
Publisher Item Identifier S 1083-4419(02)03114-X.
network (RBFNN) [13] proposed for nonlinear systems to adap-tively compensate the nonlinearities of the systems, a direct and indirect adaptive control schemes using fuzzy systems and neural networks for nonlinear systems [14] to provide design algorithms for stable controllers, etc. In addition, control systems based on a fuzzy-neural control scheme are augmented with sliding mode control (SMC) [15], [16] to ensure global stability and robustness to disturbances. With the use of the adaptive fuzzy-neural con-trol [10]–[12], [29] and the sliding mode concon-trol [17], [19], [31], two objectives can be achieved. First, modeling impression and bounded disturbance are effectively compensated. Secondly, the stability and robustness of the system can be verified.
Variable structure control with a sliding mode [36]–[39] has attracted great interest because of the essential property of the nonlinear feedback control, which has a discontinuity on one or more manifolds in the state space. It is particularly suited to the deterministic control of uncertain and nonlinear systems [36], [38]. Although sliding mode control has long being known for its capabilities in achieving robust control, however, it also suffers from large control chattering that may excite the unmodeled high frequency response of the systems due to the discontinuous switching and imperfect implementations. In general, there is a trade-off between chattering and robustness. Various controllers incorporating the sliding mode control and fuzzy control have been proposed [18]–[26] to reduce the chattering in sliding mode control. Most of the proposed methods, however, require that nonlinear functions of the dynamical system are known, which is impractical in real applications. Furthermore, sliding mode control rejects uncertainties and disturbances provided matching conditions are satisfied. The key assumption is that the matching uncertainties or disturbances are bounded, and bounds on norm of the uncertainties are available for design. However, due to the complexity of the structure of uncertainties, uncertainty bounds may not be easily obtained. Based on estimated upper bounds of the matching uncertainties, sliding mode controllers [33], [34] are proposed to guarantee asymptotic stability. However, in practical applications, the exact upper bounds of the uncer-tainties cannot be obtained in general. Though being solved to some extent through the abovementioned approaches, design problems for the uncertain nonlinear dynamical system are not well addressed.
To relax the assumption, we adopt the tracking design technique [29], [32] in this paper because the lumped uncer-tainty is bounded rather than explicitly known. The fuzzy-neural appromximator is first used to approximate the unknown non-linear functions of the dynamical systems through tuning by 1083-4419/02$17.00 © 2002 IEEE
the derived update laws. Subsequently, the tracking de-sign technique and the sliding mode control method are incor-porated into the adaptive fuzzy-neural control scheme to derive the control law. As a result, the overall system by using the tracking-based adaptive fuzzy-neural sliding mode controller is robust with respect to unmodeled dynamics, disturbances, and approximate errors. Compared with conventional fuzzy sliding mode control approaches which generally require prior knowl-edge on the upper bound of the uncertainties, the proposed ap-proach not only assures closed-loop stability, but also guaran-tees a desired tracking performance for the overall system based on a much relaxed assumption. Moreover, control chat-tering inherent in conventional sliding mode control is signifi-cantly reduced by using the proposed approach.
This paper is organized as follows. Section II gives a brief description of the sliding mode control method and fuzzy-neural approximator, which form the basis to derive the tracking-based adaptive fuzzy-neural sliding mode controller with tracking performance in Section III. Examples are illustrated in Section IV. Conclusions are drawn in Section V.
II. PRELIMINARIES
Consider the th-order nonlinear dynamical system of the form
(1)
where is
a vector of states which are assumed to be measurable, and are the control input and system output, respectively,
is the bounded external disturbance, i.e., and are smooth uncertain nonlinear functions, is assumed strictly positive, i.e., . It is assumed that there exists a solution for (1) and the order of the nonlinear system (1) is known.
A. Sliding Mode Control
Sliding mode control generally assumes that is measurable and that are given. Define a switching surface as
(2) where are chosen such that is a Hurwitz poly-nomial. Equation (2) implies
(3)
If , the dynamics-reduced th-order
system of (1) becomes
(4) where
..
. ... ... . .. ...
Following similar derivations in [35], we can obtain a control law for (1) by using the sliding mode control method shown in Lemma 1.
Fig. 1. Configuration of a fuzzy-neural approximator.
Lemma 1: Consider the nonlinear system (1) with given
non-linear functions and . Suppose that control input is chosen as
(5)
and that satisfies the Lyapunov
matrix equation
(6) where is the sliding surface defined in (2), are elements
of , and is given. Then
and as .
Proof: Given in Appendix A.
In practical applications, however, and are gener-ally uncertain rather than given. The controller of (5) derived in Lemma 1 is not always obtainable. Therefore, a new con-troller needs to be designed taking account the unknown non-linear functions, which will be adequately approximated by a fuzzy-neural approximator.
B. Fuzzy-Neural Approximator
As shown in Fig. 1, the fuzzy-neural network [11], [12] con-sisting of fuzzy IF-THEN rules and a fuzzy inference engine is used as a function approximator. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an
input linguistic vector to an output
linguistic variable . The th fuzzy IF-THEN rule is written as
if is and and is then is
where and are fuzzy sets with membership functions and , respectively. By using product inference, center-average, and singleton fuzzifier, output from the fuzzy-neural approximator can be expressed as
(7)
where is the membership function of the fuzzy vari-able is the number of the total IF-THEN rules, and is
the point at which . is an
ad-justable parameter vector, and is a fuzzy
basis vector, where is defined as
(8)
To approximate the uncertain nonlinear functions and in (1), adaptive update laws to adjust the parameter vector in (7) of the fuzzy-neural approximator need to be developed. Let and be the estimation functions for the uncertain nonlinear functions and , respectively. By using the fuzzy-neural approximator in (7), the estimation functions and can be obtained from the outputs of the fuzzy-neural approximator, which are defined as follows:
(9) and
(10) where and are adjustable parameter vectors. The fuzzy-neural approximator is valid under the following assumptions.
Assumption 1 [27]: Let belongs to a compact set , and is a designed parameter. It is known that optimal parameter vectors and lie in some convex regions
(11) and
(12) where the radii and are constants
(13) and
(14)
Assumption 2 [28]: The parameter vector is chosen such that is bounded away from zero.
Therefore, the fuzzy-neural approximator in the form of (7) can be used as a linearly parameterized approximator to approx-imate the uncertain nonlinear functions and to arbi-trary accuracy [9] as Lemma 2.
Lemma 2 [9]: For any given real continuous function on a compact set and arbitrary , there exists a fuzzy-neural approximator in the form of (9) such that
III. TRACKING-BASEDADAPTIVEFUZZY-NEURAL SLIDINGMODECONTROLLER
As mentioned earlier, the controller of (5) can not be obtained by Lemma 1 if and are uncertain nonlinear func-tions. To solve this problem, the fuzzy-neural approximator is used to approximate the uncertain nonlinear functions by using
the update laws derived to tune the adjustable parameter vector . In what follows, the tracking design technique and the sliding mode control method are incorporated into the adaptive fuzzy-neural control scheme so as to attenuate the adverse ef-fects caused by the unmodeled dynamics, disturbance, and ap-proximate errors.
To approximate the uncertain nonlinear functions and , (1) becomes
(15)
where is the lumped
uncertainty. It is assumed that there exist optimal parameter es-timates defined as (13) and (14), such that the approxi-mation error is minimal. To facilitate the design process of the controller, the lumped uncertainty is generally assumed to have an upper bound.
Assumption 3: There exists a positive constant , such that .
Based on Assumption 3, a controller which assures asymp-totic stability for the uncertain nonlinear system can be obtained from Lemma 3 below.
Lemma 3: Consider the nonlinear system (1) with uncertain
nonlinear functions and , which is approximated as (15). Suppose Assumptions 1–3 are satisfied and control input is chosen as
(16) where and are the estimate of and , respectively, and are elements of in (6), and the update laws are chosen as
(17)
where is the adaptation gain matrix, ,
and is the sliding surface defined in (2). Then and
as .
Proof: Given in Appendix B.
As shown in Lemma 3, needs to be determined in advance to construct the control input . In practical appli-cations, however, the exact upper bound cannot be obtained in general. Given that the upper bound can be chosen so as to attenuate the uncertainties, large control chattering neverthe-less occurs. Case 1 of the illustrative examples in this paper will show this effect for different selected. To relax the imprac-tical constraint, a new control law is designed by using the tracking design technique based on a much relaxed assumption below.
Assumption 4 [29], [32]: The lumped uncertainty is
as-sumed such that .
To this end, we can proceed to introduce the main theorem to derive a control law, which guarantees an tracking per-formance for the overall system without prior knowledge on the upper bound of the lumped uncertainties of the uncertain non-linear system.
Theorem 1: Consider the nonlinear system (1) with uncertain
nonlinear functions and , which is approximated as (15). Suppose Assumptions 1, 2, and 4 are satisfied and control input is chosen as
(18) and the update laws as (17), where is the design constant serving as an attenuation level, is the sliding surface defined in (2), and are elements of in (6). Then the tracking performance [29], [32] for the overall system satisfies the fol-lowing relationship:
(19)
where , and .
Proof: Given in Appendix C.
As shown in Theorem 1, the design constant serving as an attenuation level is specified by designers during the design process. The constraint to specify an upper bound of the un-known lumped uncertainties required in Lemma 3 is therefore removed. Furthermore, chattering effect of the control input is substantially reduced by using this approach, as will be demon-strated in Case 2 of the illustrative examples in this paper. The desired effect comes at no surprise because the term
accounting for the control chattering in the control law of (16) is replaced by a much smoother term in the derived con-trol law of (18).
Remark 1: If a set of initial conditions
and can be obtained, and
, then control performance of the overall system satisfies (20)
where . That
is, an arbitrary attenuation level can be obtained, if is ade-quately chosen.
Design Algorithm:
Step 1) Select control parameters such that matrix is a Hurwitz matrix. Determine and
.
Step 2) Choose an appropriate to solve the Lyapunov ma-trix equation (6).
Step 3) Construct membership functions of the fuzzy sets to approximate the uncertain nonlinear functions
and .
Step 4) Choose an appropriate adaptation gain matrix to establish the Lyapunov function.
Step 5) Obtain the update laws from (17), and control laws from (16) or (18), respectively, depending on dif-ferent assumptions on the lumped uncertainties.
Remark 2: This paper investigates mainly on SISO systems.
However, it can be easily extended to MIMO systems via an input-output linearizaton technique [36]. A brief description on the derivations toward a similar design approach for MIMO sys-tems is given in Appendix D.
IV. ILLUSTRATIVEEXAMPLES
Example 1: Consider the following nonlinear dynamical
system [35]:
(21) where denotes the uncertain nonlinear function, and is the disturbance. Let the sliding surface be
defined as , and . We obtain
by solving the Lyapunov matrix equation. A set of membership functions are constructed for
as
for
Let I, and initial states . Control
laws will be derived by using Lemma 3 (Case 1) and Theorem 1 (Case 2), respectively, depending on different assumptions on the lumped uncertainties.
Case 1) Assume that the upper bound of the lumped un-certainty is known, i.e., , and
is chosen as 0.3 and 0.5, respectively. According to Lemma 3, the control input can be obtained as ,
with update law .
Case 2) Assume that the upper bound of the lumped un-certainty is unknown. The design constant , which serves as an attenuation level, is chosen as 0.1 and 0.2, respectively. According to Theorem 1, the con-trol input can be obtained as
, with update law .
Figs. 2 and 3 show the time responses of the states and control input with of Case 1 in Example 1 by using Lemma 3, assuming that the upper bound of the lumped uncer-tainties is available for design. As clearly demonstrated in Fig. 2, the time responses of the states are oscillatory due to the distur-bance . This comes at no surprise because the improper selection of cannot effectively suppress the distur-bance. If is selected, the time responses of the states are satisfactory as shown in Fig. 4. Although the impact of the disturbance is alleviated as shown in Fig. 4 with the selection of a better , the problem of control chattering, however, becomes much serious as clearly demonstrated in Fig. 5, in com-parison to that of Fig. 3. In general, the control law obtained by
Fig. 2. State responses in Case 1 withk = 0:3 in Example 1.
Fig. 3. Time response of the control input in Case 1 withk = 0:3 in Example 1.
Fig. 4. State responses in Case 1 withk = 0:5 in Example 1.
Lemma 3 to compensate the lumped uncertainties results in se-rious control chattering. In general, there is a trade-off between chattering and robustness.
On the other hand, Figs. 6 and 7 show the time responses of the states and control input with of Case 2 in
Fig. 5. Time response of the control input in Case 1 withk = 0:5 in Example 1.
Fig. 6. State responses in Case 2 (the proposed method) with = 0:2 in Example 1.
Fig. 7. Time response of the control input in Case 2 (the proposed method) with = 0:2 in Example 1.
Example 1 by using the proposed method (Theorem 1), under Assumption 4 without prior knowledge on the upper bound of the lumped uncertainties. As demonstrated in Fig. 7, the control chattering is significantly reduced while maintaining satisfac-tory state responses, compared with those shown in Figs. 3 and
Fig. 8. State responses in Case 2 (the proposed method) with = 0:1 in Example 1.
Fig. 9. Time response of the control input in Case 2 (the proposed method) with = 0:1 in Example 1.
5 with various by Lemma 3. If the design parameter , which serves as an attenuation level, is further reduced to , better state responses can be obtained as shown in Fig. 8. Note that the problem of control chattering does not occur as demon-strated in Fig. 9.
Example 2: Consider the uncertain nonlinear system having
the same model as (21), where the uncertain nonlinear
func-tion and external disturbance are and
, respectively. If the same design parameters ( or ) as Case 1 in Example 1 are taken for computer simulation by Lemma 3 for this example. We find that the distur-bance cannot be effectively suppressed because the upper bound of the lumped uncertainties has changed due to the change of the uncertain nonlinear function. Therefore, the previously selected parameter can not be applied to different systems in general. It is therefore a typical trial-and-error process to determine a suit-able , which satisfies Assumption 3 that the upper bound of the lumped uncertainties is available for design as required by Lemma 3. Figs. 10 and 11 show the time responses of the states and control input with of Case 1 in Example 2 by using Lemma 3 via a trial-and-error process. As demonstrated
Fig. 10. State responses in Case 1 withk = 1:5 in Example 2.
Fig. 11. Time response of the control input in Case 1 withk = 1:5 in Example 2.
in Fig. 11, the problem of control chattering becomes serious in order to obtain the acceptable state responses shown in Fig. 10. On the contrary, the proposed method (Theorem 1) uses the design parameter as an attenuation level under Assumption 4 without prior knowledge on the upper bound of the lumped uncertainties. Figs. 12 and 13 show the time responses of the states and control input with of Case 2 in Example 2 by using Theorem 1. As demonstrated in Fig. 13, the control chattering is significantly reduced, compared with that shown in Fig. 11, in which is used by Lemma 3.
In summary, a design parameter serving as an attenuation level, rather than an estimated upper bound, for the lumped uncertainties can be specified by the designer, so that a de-sired system performance can be obtained via the proposed tracking-based adaptive fuzzy-neural sliding mode controller.
V. CONCLUSION
In this paper, an adaptive fuzzy-neural control scheme incorporating both the tracking design technique and the sliding mode control method for uncertain nonlinear systems has been developed, in which a fuzzy-neural model is used to approximate the uncertain nonlinear functions of the dynamical
Fig. 12. State responses in Case 2 (the proposed method) with = 0:1 in Example 2.
Fig. 13. Time response of the control input in Case 2 (the proposed method) with = 0:1 in Example 2.
system. To facilitate the design process, an design algorithm, which can be computerized to derive the tracking-based adaptive fuzzy-neural sliding mode controller for the uncertain nonlinear system, is also presented. As shown in this paper, the proposed tracking-based adaptive fuzzy-neural sliding mode controller not only attenuates the lumped uncertain-ties caused by the unmodeled dynamics, disturbances, and approximate errors associated with the uncertain nonlinear system, but also significantly reduces the control chattering inherent in conventional sliding mode control. Furthermore, the constraint demanding prior knowledge on upper bounds of the lumped uncertainties is removed through the design algorithm of the proposed approach. As demonstrated in the illustrated examples, the tracking-based adaptive fuzzy-neural sliding mode controller proposed in this paper can achieve a better control performance over the conventional methods.
APPENDIX A
Proof of Lemma 1: Consider the Lyapunov function
(22)
The time derivative of (22) is
(23)
Apply (5) to (23) and let . We have the
following relationship:
(24)
We conclude and as . This
completes the proof.
APPENDIX B
Proof of Lemma 3: Consider the Lyapunov function
(25)
where , and . The time derivative of
(25) is
(26)
Apply (16) and (17) to (26) and let . We
have the following relationship:
(27) By using Barbalat’s lemma in [30] and Theorem 2 in [11], (27)
implies and as . This completes
the proof.
APPENDIX C
Proof of Theorem 1: Consider the Lyapunov function
(28) The time derivative of (28) is
Substituting (17) and (18) into (29), we have
(30) By Assumption 4, we integrate (30) from to , and obtain
(31)
Substituting (28) into (31), we have the tracking perfor-mance, satisfying
(32) This completes the proof.
APPENDIX D
A. Derivations of a Similar Design Approach for MIMO Systems
According to [36], the input-output linearization of MIMO systems can be obtained by differentiating the outputs of MIMO systems, until at least one of the inputs appears. Consider the MIMO nonlinear dynamical system
(33) where is a vector of states, represents a vector of the external bounded disturbances,
and are the control inputs and
system outputs, respectively, and ,
and are unknown and smooth vector functions.
Input-output linearization of MIMO systems is obtained by differentiating the outputs , until at least one of the inputs appears. We have
(34) where is the smallest integer such that at least one of the inputs appears in , and the operator denotes the Lie derivatives with respective to . We define
. Then the input-output form of (33) can be described as
.. .
(35)
where we get the equation shown at the top of the next page. Specifically, (35) can be also rewritten as
.. . .. . .. . .. . .. . (36) Equation (36) is basically a set of SISO nonlinear dynamical systems with different order similar to (1) in the paper. Design methodology developed in the paper for SISO systems can then be applied to the MIMO system.
In order to derive a control law for the MIMO system by using the input-output linearizaton technique, the following assump-tions are required.
Assumption (5): can be realizable for
, where represents the relative degree
of , and is finite and known. Moreover, .
Assumption (6): is bounded away from singularity over a compact set . Moreover, is bounded.
Based on the above-mentioned assumptions and the design procedures proposed in the paper, similar results extended for the MIMO systems can be easily obtained.
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Wei-Yen Wang (M’00) was born in Taichung,
Taiwan, R.O.C., in 1962. He received the M.S. and Ph.D. degrees in electrical engineering from Na-tional Taiwan University of Science and Technology, Taipei, in 1990 and 1994, respectively.
Since 1990, he has served concurrently as a Patent Screening Member of the National Intellectual Prop-erty Office, Ministry of Economic Affairs, Taiwan. In 1994, he was appointed Associate Professor in the Department of Electronic Engineering, St. John’s and St. Mary’s Institute of Technology, Taipei. From 1998 to 2000, he was with the Department of Business Mathematics, Soochow University, Taiwan. Currently, he is with the Department of Electronics Engineering, Fu-Jen Catholic University, Taipei. His current research interests and publications are in the areas of fuzzy logic control, robust adaptive control, neural networks, computer-aided design, and digital control.
Mei-Lang Chan received the B.E. degree from the
Department of Industrial Education, National Taiwan Normal University, Taipei, Taiwan, R.O.C., in 1979, and the M.S. and Ph.D. degrees from the Department of Electrical Engineering, National Taiwan Univer-sity of Science and Technology, Taipei, in 1992 and 2000, respectively.
He is now an Associate Professor at National I-Lan Institute of Technology, I-Lan, Taiwan. His current research interests include sliding mode control, adap-tive control, and fuzzy systems.
Chen-Chien James Hsu was born in Hsinchu,
Taiwan, R.O.C. He received the B.S. degree in electronic engineering from the National Taiwan University of Science and Technology, Taipei, in 1987, the M.S. degree in control engineering from National Chiao-Tung University, Hsinchu, in 1989, and the Ph.D. degree from the School of Microelectronic Engineering, Griffith University, Brisbane, Australia, in 1997.
Before commencing his Ph.D. study, he was a Sys-tems Engineer with IBM Corporation for three years, responsible for the information planning and application development for infor-mation systems. He is currently an Assistant Professor of electronic engineering at St. John’s and St. Mary’s Institute of Technology, Taipei. His research inter-ests include digital control systems, neural-fuzzy control systems, genetic algo-rithms, and expert systems.
Tsu-Tian Lee (M’87–SM’89–F’97) was born in
Taipei, Taiwan, R.O.C., in 1949. He received the B.S. degree in control engineering from the National Chiao-Tung University (NCTU), Hsinchu, Taiwan, in 1970, and the M.S. and Ph.D. degrees in electrical engineering from the University of Oklahoma, Norman, in 1972 and 1975, respectively.
In 1975, he was appointed Associate Professor and in 1978, Professor and Chairman of the Department of Control Engineering, NCTU. In 1981, he became Professor and Director of the Institute of Control En-gineering, NCTU. In 1986, he was a Visiting Professor and in 1987, a Full Pro-fessor of Electrical Engineering at University of Kentucky, Lexington. In 1990, he was a Professor and Chairman of the Department of Electrical Engineering, National Taiwan University of Science and Technology (NTUST). In 1998, he became the Professor and Dean of the Office of Research and Development, NTUST. Since 2000, he has been with the Department of Electrical and Control Engineering, NCTU, where he is now a Chair Professor. He has published more than 170 refereed journal and conference papers in the areas of automatic con-trol, robotics, fuzzy systems, and neural networks. His professional activities include serving on the Advisory Board of Division of Engineering and Applied Science, National Science Council, serving as the Program Director, Automatic Control Research Program, National Science Council, and serving as an Ad-visor of Ministry of Education, Taiwan, and numerous consulting positions. His current research involves motion planning, fuzzy and neural control, optimal control theory and application, and walking machines.
Dr. Lee received the Distinguished Research Award from National Science Council, R.O.C., in 1991–1992, 1993–1994, 1995–1996, and 1997–1998, re-spectively, and the Academic Achievement Award in Engineering and Applied Science from the Ministry of Education, Republic of China, in 1998. He is a Fellow of the IEE and the New York Academy of Sciences.
