Feedback-Based Iterative Learning Control for MIMO LTI Systems

Transcription

1 International Journal of Control, Feedback-Based Automation, Iterative and Systems, Learning vol. Control 6, no., for. MIMO 69-77, LTI Systems Aril 8 69 Feedback-Based Iterative Learning Control for MIMO LTI Systems Tae-Yong Doh and Jung Rae Ryoo Abstract: This aer rooses a necessary and sufficient condition of convergence in the L - norm sense for a feedback-based iterative learning control (ILC) system including a multi-inut multi-outut (MIMO) linear time-invariant (LTI) lant. It is shown that the convergence conditions for a nominal lant and an uncertain lant are equal to the nominal erformance condition and the robust erformance condition in the feedback control theory, resectively. Moreover, no additional effort is required to design an iterative learning controller because the erformance weighting matrix is used as an iterative learning controller. By roving that the least uer bound of the L -norm of the remaining tracking error is less than that of the initial tracking error, this aer shows that the iterative learning controller combined with the feedback controller is more effective to reduce the tracking error than only the feedback controller. The validity of the roosed method is verified through comuter simulations. Keywords: Convergence, iterative learning control (ILC), L -norm, least uer bound, multiinut multi-outut (MIMO) linear time-invariant (LTI) systems, nominal erformance, robust erformance, structured singular value.. INTRODUCTION Iterative learning control (ILC) has been used in control systems that erform reeated tasks [,]. Just as humans learn skills by trial and error, the ILC system learns the dynamics of the system by reeated trials. Studies on ILC systems, therefore, have been mainly focused on the convergence of roosed learning algorithms. Because ILC is an oen-loo control scheme, it is alied to real systems along with feedback control for enhancing robustness against unreeatable disturbances and for reducing the tracking error in the early stage of learning [3-8]. It is desirable that the least information about the lant is required in design of an iterative learning controller. However, at least a nominal lant model is needed because the convergence of the ILC system is guaranteed based on the mathematical model of the lant. The ILC system is designed to guarantee a convergence condition exressed in terms of the nominal lant model. As a result, the ILC system Manuscrit received August 5, 6; revised July 4, 7; acceted November 6, 7. Recommended by Editor Tae Woong Yoon. Tae-Yong Doh is with the Deartment of Control and Instrumentation Engineering, Hanbat National University, San 6-, Duckmyung-dong, Yuseong-gu, Daejeon 35-79, Korea ( dolerite@hanbat.ac.kr). Jung Rae Ryoo is with the Deartment of Control and Instrumentation Engineering, Seoul National University of Technology, 7 Gongneung-dong, Nowon-gu, Seoul , Korea ( jrryoo@snut.ac.kr). alied to a real lant may not converge because modeling errors are unavoidable to some extent. To solve the roblem, some robust aroaches to feedback-based ILC have been roosed. Moon et al. suggested a robust aroach to ILC design for a unity feedback control system [4]. Doh et al. resented a sufficient condition for robust stability and robust convergence for uncertain linear systems and a method to design a feedback controller and an iterative learning controller simultaneously [5]. Hu et al. derived a necessary and sufficient condition of convergence for linear time-invariant systems with multilicative uncertainties and time delays and showed that the condition is of the same form with the robust erformance condition [9]. Tayebi and Zaremba roosed a result similar to that of Hu et al. []. Most studies on robust ILC with current feedback have focused on single-inut and single-outut (SISO) systems with multilicative uncertainties. In this aer, motivated by the results in [9-], we consider a feedback-based ILC scheme for multi-inut and multioutut (MIMO) lants. We first obtain a necessary and sufficient condition of convergence in the L -norm sense for nominal lants, which is equal to the nominal erformance condition in the feedback control theory. And then a modified convergence condition considering the lant uncertainty is derived, which is described only by the nominal arts and is exactly the same as the robust erformance condition exressed by the structured singular value ( µ ) [- 4]. The iterative inut udating rule is established by

2 7 Tae-Yong Doh and Jung Rae Ryoo using the weighting erformance matrix as an iterative learning controller. Owing to these results, there is no need to design an iterative learning controller if a feedback controller is designed to satisfy the robust erformance condition. Moreover, using the similar aroach in [], we demonstrate that the least uer bound of the L -norm of the remaining tracking error is less than that of the initial tracking error in case of MIMO lants. Finally, a simulation study on a two-mass/sring/damer system is erformed to verify the effectiveness of the roosed method. Throughout this aer, signals in the time domain are denoted by lower-case letters and their caitals denote their own Lalace transforms, for examle, L { f ( t)} = F( s). If there is no other definition, caitals such as Gs ( ) or G stand for transfer matrices. The subsace of real rational matrices in H (resectively, H ) is denoted by RH (resectively, RH ). σ() and σ() signify the singular value and the largest singular value, resectively. The Lalace variable s and the angular frequency ω will be omitted when these do not lead to any confusion. A lower linear fractional M M transformation (LFT) on with M = M M I M is ( ) F l ( M, ) := M + M M [4]. In a similar manner, an uer LFT on with M is ( ) ( ) u M M M I M F, := + M.. MAIN RESULTS Consider the feedback control system in Fig.. In this figure, yd ( t) is the desired trajectory, q yt () is the lant outut, and ut ( ) is the feedback control inut. C(s) is the feedback controller that stabilizes the feedback control system. G(s) is the lant with q inuts and oututs. To aly ILC technique to the feedback system, we consider an ILC system shown in Fig. with the additional inut vk () t of which iterative udating rule is given by Vk+ () s = W()( s Vk() s + Ek()) s () with V () s = where the tracking error Ek ( s ) is given by E () s = Y () s Y () s () k d k and W ( s ) is a stable erformance weighting matrix with the following tye: Fig.. Feedback control system. Fig.. Feedback-based ILC system. w () s W () s =. w ( s) (3) Here, the subscrit k means the number of iterations. () is similar to the laws roosed by Kavli [5] and Roh et al. [6] and generates a modified desired trajectory by learning. In the formulation of the ILC roblem, the following assumtions are made: Assumtion : The desired trajectory yd () t is bounded within the tracking interval, i.e., Yd ( s) RH. Assumtion : Without loss of generality, the zeroinut resonse Y () s of the lant Gs ( ) is invariant with resect to iteration and equal to zero, i.e., Y () s = Yk () s = for all k where Yk () s is the zero-inut resonse of the lant at the k th iteration. The convergence condition in the L -norm sense is summarized as the following theorem. Theorem : For the ILC system shown in Fig., the roosed learning control algorithm () converges uniformly to ( ) k d k v ( t) = lim v ( t) = L I W S W SY (4) as k, in the L -norm sense if and only if W () s S() s <, (5) where ( ) Ss () = I + G() s C() s is the sensitivity matrix of the feedback control system revealed in Fig.. As the iteration tends to infinity, the remaining error is given by

3 Feedback-Based Iterative Learning Control for MIMO LTI Systems 7 e ( t) = lim e ( t) k k {( I T( I WS) ) Yd} { I W S I WS Yd} = L = L ( ) ( ), where T s ( ) (6) () = I + G() s C() s G() s C() s is the comlementary sensitivity matrix of the feedback control system shown in Fig.. Proof: ( ) In Fig., the tracking outut at the k th iteration is ( I GC) ( ) ( ) Y = + GC V + Y = T V + Y. (7) k k d k d Using (), (), (7), and the relationshi between S and T, i.e., S + T = I, the udated learning inut at the ( k + ) th iteration is given by Vk+ = WSVk + WSYd. (8) Similarly, V = W SV + W SY. (9) k k d From (8) and (9), we get ( ) V V = W S V V. () k+ k k k By Parseval s theorem, () becomes v ( t) v ( t) = V V k+ k k+ k W S V V. () k k Consequently, it is clear that if (5) is satisfied, the udated learning inut v () t converges to v () t k = L { V ( s)} in the L -norm sense as k aroaches infinity. The fixed value V ( s) in (4) is obtained from (8) by substituting Vk ( s ) and Vk + () s with V (). s With the hel of () and (4), the remaining error E ( s) is also obtained as (6). ( ) The necessity can be roved using contradiction. Let ω be such that { } WS = σ W ( jω ) S( jω ). () Denote the singular value decomosition of W ( j ω ) S ( j ω ) as W ( j ω ) S ( j ω ) { } r + { } = σ W ( jω ) S( jω ) g ( jω ) h ( jω ) i= i W j S j gi j hi j σ ( ω ) ( ω ) ( ω ) ( ω ), W j S j and g, where r is the rank of ( ω ) ( ω ) have unit length. Suose the g( j ω ) is written as jθ jθ jθ T g ( jω ) = α e α e α e, (3) where α i and θ i, ( π ]. Let β i be such that βi θi = βi jω + jω with β i = if θi = and let V () s be given by β α β β s + s s α = β + s Vˆ V () s () s β s α β + s s with relacing β i βi + s V ˆ () s is chosen so that i h i (4) (5) if θi =. A scalar function c if ω ω < εor ω+ ω < ε Vˆ ( jω) = (6) otherwise, where < ε and c= c π/ ε. Then V ( s) Trace[ V ( jω) V ( jω)] dω (7) = π = c. In case of V3() s V(), s 3 V () s V () s = W () s S()( s V () s V () s c = [σ{ W ( jω ) S( jω ) } π π + { W ( jω ) S( jω ) } { W j S j } σ π] W s S s c c = σ ( ω ) ( ω ) = () () c = V () s V (), s (8) because W S is larger than and (). V s = Similarly, it can be shown that

4 7 Tae-Yong Doh and Jung Rae Ryoo V () s V () s V () s V () s (9) k+ k k k for all k. Hence, the roof is comleted. Remark : The convergence condition (5) is well known as the nominal erformance condition in the feedback control theory [3]. In other words, the weighting matrix W ( s ) to secify the erformance of the feedback system lays a role of iterative learning controller and also determines the convergence of the ILC system. Because the convergence condition shown in Theorem includes the lant uncertainty, it is not roer to aly this condition to real lants. To examine the convergence of the ILC system with lant uncertainty, therefore, a modified convergence condition is required, which is exressed with only the known arts of the system. Now, let the lant Gs ( ) be an element of the set of uncertain lants { G s s s } G = F ( ( ), ( )) ( : ) M ( ), () u o where Go () s is a lant free from the uncertainty, an ( n+ ) ( n+ q) transfer matrix, and artitioned as Go () s G () o s Go () s =, Go () s G () o s ( s ) is a model uncertainty, and are defined as with () M ( ) and M( ) := { () RH : ( s ) () for all s + }, = {diag ( δir,, δ SIr,,, ) δ S F : i, (3) mj mj } j S F r i i + m j j = n = [3], resectively. For = n n M, the structured singular value µ ( M ) is defined as µ ( M ) = min σ( ) det( ) { :, I M = } unless no makes I M singular, in which case µ ( M ):= [,3]. The following lemma, which is widely known as the robust erformance test in the feedback control theory, will be used to derive the modified convergence condition. Lemma [3]: Let β >. Suose that P( s ) is a stable, real-rational, roer transfer matrix with n+ q inuts and n+ oututs and is artitioned as P() s P() s Ps () =. P() s P () s (4) For all ( s) M ( ) with < / β, F l ( P, ) is well-osed, internally stable, and F l ( P, ) β if and only if su µ ( P ( j ω)) β ω P q where P := {, f : f } diag( ). (5) The uncertainty-free convergence condition is summarized as Theorem. Theorem : Let < ρ <. For the ILC system shown in Fig., the roosed learning control algorithm () converges uniformly to (6) as k, in the L -norm sense for all ( s) M ( ) with < / ρ if and only if ω where { } su µ F ( ( ω) ( ω)) ρ P l M j, C j < (6) o G o o o o o G M = W G W W G. G I G (7) As the iteration tends to infinity, the remaining error converges to (6). Proof: ( ) Using the similar aroach in the roof of Theorem, the relationshi between consecutive udated learning inuts is given by k+ k u o k k V V = W ( I + F ( G, ) C) ( V V ), (8) which leads to V V k+ k W I Fu Go C Vk Vk ( + (, ) ). Under Assumtions and, if ( + (, ) ) (9) W I F G C < (3) u o is satisfied, the udated learning inut converges to v () t given in (4) in the L -norm sense as k. W ( ( ) ) I + F u Go, C is equal to the transfer matrix from w to z in the block diagram shown in Fig. 3 and can be rearranged as an LFT with z = Fu( F l( MC, ), ) w. Hence, the condition (3) can be reresented as Fu( F l( MC, ), ) <. Finally, according to Lemma, (3) is ensured if (6)

5 Feedback-Based Iterative Learning Control for MIMO LTI Systems 73 Fig. 3. Block Diagram of ( ). is met for all ( s) M ( ) with < / ρ. ( ) Let the condition (6) not be satisfied, that is, let ω be such that { } µ F ( ( ω ) ( ω )) P l M j, C j < (3) for all ( s) M ( ) with < / ρ. Then, by Lemma, F ( F ( MC, ), ) = W( I+ F ( G, ) C u l u o ). From this result, as shown in the roof of Theorem, it is easily obtained that Vk + ( s) Vk( s) Vk( s) Vk( s) for all k. Therefore, (6) is the necessary condition for the convergence in the L -norm sense. Remark : The convergence condition obtained in Theorem is equal to that of robust erformance by structured singular value. Therefore, if the feedback control system satisfies the robust erformance condition, the convergence of the feedback-based ILC system is automatically ensured, and vice versa. Accordingly, the ILC design roblem is equivalent to finding a feedback controller Cs ( ) satisfying (6). Remark 3: Using the so-called D-K iteration, we can obtain a controller Cs ( ) which not only stabilizes the overall feedback system but also satisfies the condition (6) [3]. The uer bound on the model uncertainty can be easily found using several methods in the books related with robust control [3,7]. Remark 4: Let the lant Gs ( ) be a SISO system with the multilicative uncertainty, i.e., Gs () = (+ ( sw ) ()) s G(), s (3) W I + F ( G, ) C where Gn ( s ) is the nominal lant, W () s is a known stable weighting function, and ( s ) is an unknown stable transfer function satisfying < / ρ. Then, as in [9] and [], (6) boils down to the robust erformance condition n u o WSn + WTn ρ, (33) where Sn = /( + CGn) and Tn = CGn/ ( + CGn). As Theorem, the tracking error converges uniformly to the remaining error given in (6) in the L -norm sense as the iteration tends to infinity. However, no one can guarantee that the remaining error is less than the initial tracking error. Therefore, a condition is needed that the tracking error converges to a less value than the initial one, which is given in the following theorem. Theorem 3: For the ILC system shown in Fig., if there exists a W (s) such that I W < and if W S <, (34) where W = W/ ( I W ), then i) the tracking error is bounded for all k and converges uniformly to the remaining error (6) as k in the L -norm sense; ii) the least uer bound of the L -norm of the remaining error is less than the least uer bound of the L -norm of the initial tracking error, i.e., e α, e α, with α < α ; iii) the infinity norm of the remaining error is less than that of the initial tracking error, i.e., E < E. Proof: i) If W = I, (34) is equal to condition (5) and the roof of convergence of the tracking error is the same as that of Theorem. Consider W I. Since I W <, (34) becomes W S < I W, (35) which imlies that the convergence condition (5) is satisfied and then the tracking error is bounded for all iterations. By Theorem, the remaining error can be also given by (6). ii) The initial tracking error is obtained by setting V () s =. Let α and α be the least uer bounds of the remaining error and the initial tracking error, resectively: e = E (36) ( I W) S( I W) yd = α, e = E S y d = α. (37) From the condition (35), we get the following inequality: I W < W S,. ω (38) We have

6 74 Tae-Yong Doh and Jung Rae Ryoo = I + W S W S W S + I W S, ω (39) and therefore W S I W S,. ω (4) Finally, from (38) and (4), the following inequality I W < I W S, ω (4) is obtained, which imlies that ( I W ) S( I W S) < S. (4) Therefore, α < α is established. iii) (4) also yields that d d ( I W ) S( I W S) Y < SY, (43) which means that E < E. Remark 5: Theorem 3 shows the boundedness of the tracking error. Under some conditions, the iterative learning controller diminishes the tracking error below, which only the feedback controller generates. In other words, by adding an iterative learning controller with a simle structure, we can reduce the tracking error more effectively than when only the feedback controller is used. If the lant with multilicative uncertainty, i.e., G = ( + W ) Gn is considered, the conditions for the boundedness of the tracking error are modified to be more or less conservative as shown in Theorem of []. 3. SIMULATION RESULTS Consider a two-mass/sring/damer system as given in Fig. 4 [3]. The dynamical system can be described by the following differential equations: x () t = Ax() t + Bu(), t yt () = Cxt () with xt = [ x t x t x t x t ] () () () () () T, 3 4 T ut () = u() t u () t, yt () = y() t y() t, k k b b A =, m m m m k k+ k b b+ b m m m m T Fig. 4. A two-mass/sring/damer system. B =, C = m. m Suose that the transfer matrix from ( u, u ) to ( x, x) is the nominal lant Gn ( s ) and suose that arameters k =, k = 4, b =., b =., m =, and m = with aroriate units. The lant G(s) is described in the following multilicative uncertain form: Gs () = G()( s I+ ( sw ) ()), s (44) n where W ( s ) is given as u s + 5 s + 5 Wu () s =. s + 5 s + 5 u (45) A method to find Go () s from Gs ( ) is introduced in [5]. To imrove the tracking erformance of y() t = x() t and y () t = x () t in a frequency range ω, the erformance weighting matrix W ( s ) is given as s/ + W () s =. s/ + (46) Using the µ -Analysis and Synthesis Toolbox of C() s C() s Matlab [8], the controller Cs () = C() s C() s satisfying the robust erformance condition that can be obtained where

7 Feedback-Based Iterative Learning Control for MIMO LTI Systems 75 C C C C ( s+ 5)( s+ 5. 5)( s +. 6s+. 3)( s s+. 54) () s = ( s+ 3. 5e4)( s )( s+ )( s s+. 37)( s s+ 8. 7) ( s+ 5)( s )( s+. 8)( s. 8)( s +. 37s ) () s = ( s+ 3. 5e4)( s )( s+ )( s s+. 37)( s s+ 8. 7) ( s+ 5)( s )( s+. 3)( s. 5)( s +. s ) () s = ( s+ 3. 5e4)( s )( s+ )( s s+. 37)( s s+ 8. 7) 36. ( s+ 5)( s )( s +. 4s+. 56)( s s ) () s =. ( s+ 3. 5e4)( s )( s+ )( s s+. 37)( s s+ 8. 7) The largest structured singular value with this controller is ω { F } su µ ( ( ω) ( ω)) 93 P l M j, C j =. as shown in Fig. 5. In simulations, let the desired T trajectory yd () t = yˆd() t yˆd() t, t 5 be the following signal shown in Fig. 6: t /, t < 5 5 yˆ d( t) = ( t ) / 5 +, 5 t <, t 5. (47) We erformed a simulation using the obtained controller and the roosed iterative udating rule. The learning control inut was initialized as v () t =, thus the tracking error at the first iteration was only the result of the feedback control. Fig. 7 indicates the desired trajectory, tracking oututs, and tracking errors e() t = yˆ d() t y(), t e () t = yˆ d() t y (), t at k =, k =, k =, and k =. Fig. 8 resents the root mean square (rms) values of the tracking errors versus the number of iterations, where the erformance evolution by the added iterative learning controller is clearly seen. More quantitatively, the rms values of e () t and e () t decrease steadily to aroach around.37 and.3, 5. % and 6.5 % of the initial rms values,.7 and.7, Fig. 5. µ { ( ( ω), ( ω)) } F P l M j C j versus frequency. Fig. 6. Desired trajectory. Fig. 7. Desired trajectory (dotted line), tracking oututs, and tracking errors at k = (solid line), k = (dashed line), k = (dashed and dotted line), and k = (bold line).

8 76 Tae-Yong Doh and Jung Rae Ryoo Fig. 8. Root mean square values of tracking errors versus the number of iterations. resectively, which verifies the benefit of iterative learning control. Remark 6: By iteratively erforming D-K iteration or selecting a new erformance weighting matrix W (), s we can obtain a lower ρ in the condition (6) and a controller with higher gain. Then, it is ossible to imrove the initial tracking erformance when only the feedback controller is alied. 4. CONCLUDING REMARKS In this aer, the convergence for a MIMO feedback-based ILC was considered. It was shown that convergence in the L -norm sense is closely related with the nominal erformance condition and the robust erformance condition in the feedback control theory. Therefore, the design roblem of the iterative learning controller for an uncertain MIMO system is equivalent to designing a feedback controller C(s) or selecting a erformance weighting function W (s) to satisfy the robust erformance condition. Moreover, it was shown that the least uer bound of L -norm of the remaining tracking error is less than that of the initial error. From the result, it was verified that the iterative learning controller is useful in reducing the tracking error. Simulation on a -mass/sring/damer system was erformed and its results were resented to validate the effectiveness of the roosed method. REFERENCES [] S. Arimoto, S. Kawamura, S, and F. Miyazaki, Betterment oeration of robots by learning, J. Robotic Systems, vol., no.,. 3-4, 984. [] K. L. Moore, Iterative learning control - an exository overview, Alied and Comutational Controls, Signal Processing and Circuits, vol., no., , 998. [3] T. Y. Kuc, J. S. Lee, and K. Nam, An iterative learning control theory for a class of nonlinear dynamic systems, Automatica, vol. 8, no. 6,. 5-, 99. [4] J.-H. Moon, T.-Y. Doh, and M. J. Chung, A robust aroach to iterative learning control design for uncertain systems, Automatica, vol. 34, no. 8,. -4, 998. [5] T.-Y. Doh, J.-H. Moon, K. B. Jin, and M. J. Chung, Robust ILC with current feedback for uncertain linear systems, Int. J. Systems Science, vol. 3, no., , 999. [6] D. de Roover, O. H. Bosgra, and M. Steinbuch, Internal-model-based design of reetitive and iterative learning controllers for linear multivariable systems, Int. J. Control, vol. 73, no., ,. [7] M. Norrlöf and S. Gunnarsson, Disturbance asects of iterative learning control, Engineering Alication of Artificial Intelligence, vol. 4, ,. [8] P. B. Goldsmith, On the equivalence of causal LTI iterative learning control and feedback control, Automatica, vol. 38, ,. [9] Q. Hu, J.-X. Xu, and T. H. Lee, Iterative learning control design for Smith redictor, Systems and Control Letters, vol. 44,. -,. [] A. Tayebi and M. B. Zaremba, Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust erformance condition, IEEE Trans. on Automatic Control, vol. 48, no.,. -6, 3. [] T.-Y. Doh, Comments on Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust erformance condition, IEEE Trans. on Automatic Control, vol. 49, no. 4, , 4. [] J. C. Doyle, Analysis of feedback systems with structured uncertainties, IEE Proceedings, Part D, vol. 33, , 98. [3] K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice-Hall, Inc., 998. [4] J. C. Doyle, A. Packard, and K. Zhou, Review of LFTs, LMIs and µ, Proc. of the 3th IEEE Conf. Decision and Control,. 7-3, 99. [5] T. Kavli, Frequency domain synthesis of trajectory learning controller for robot maniulator, J. Robotic Systems, vol. 9, no. 5, , 99. [6] C. L. Roh, M. N. Lee, and M. J. Chung, ILC for nonminimum hase system, Int. J. Systems Science, vol. 7, no. 4, , 996.

9 Feedback-Based Iterative Learning Control for MIMO LTI Systems 77 [7] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, Maxwell Mcmillan, 99. [8] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, µ -Analysis and Synthesis Toolbox for Use with Matlab, Mathworks, 998. Tae-Yong Doh received the B.S. degree in Electronics Engineering from Kyungook National University in 99, and the M.S. and Ph.D. degrees in Electrical Engineering from the Korea Advanced Institute of Science and Technology in 994 and 999, resectively. In, he joined the faculty of the Deartment of Control and Instrumentation Engineering, Hanbat National University, Korea, where he is currently an Associate Professor. From 997 to, he worked for Samsung Electronics Co., Ltd as a Senior Research Engineer. His research interests include iterative learning control, reetitive control, robust control, embedded control systems, and intelligent robotics. Jung Rae Ryoo received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from the Korea Advanced Institute of Science and Technology in 996, 998, and 4, resectively. In 5, he joined the faculty of the Deartment of Control and Instrumentation Engineering, Seoul National University of Technology, Korea, where he is an Assistant Professor. From 4 to 5, he worked for Samsung Electronics Co., Ltd as a Senior Engineer. His research interests include robust motion control, otical disk drive servo control, and DSP-based digital control systems.