model is proposed to represent a nonlinear system. Next, using a linear matrix inequality (LMI) approach, the fuzzy filtering problems are characterized in terms ...
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 11, NOVEMBER 2001
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Fuzzy Estimation for a Class of Nonlinear Discrete-Time Dynamic Systems Chung-Shi Tseng, Member, IEEE, and Bor-Sen Chen, Senior Member, IEEE
Abstract—This paper studies fuzzy filtering design for state estimation of nonlinear discrete-time systems with bounded but unknown disturbances. First, the Takagi and Sugeno fuzzy model is proposed to represent a nonlinear system. Next, using a linear matrix inequality (LMI) approach, the fuzzy filtering problems are characterized in terms of a linear matrix inequality problem (LMIP). The LMIP can be efficiently solved using convex optimization techniques. Simulation examples are given to illustrate the design procedure and the applicability of the proposed method. The results indicate that the proposed method is suitable for practical applications. Index Terms—
fuzzy filtering, LMI.
I. INTRODUCTION
I
N GENERAL, it is difficult to design an efficient filter for state estimation of nonlinear systems. A nonlinear estimation algorithm, known as the extended Kalman filter (EKF), is often used for state estimation. With this algorithm, linearization of the nonlinear systems around the present estimate make it possible to apply the linear Kalman filter theory [1]–[5]. The design of an EKF relies on an exact knowledge of plant dynamics and the second-order statistical properties of external disturbances and measurement noises. In some practical systems, however, the statistical properties of external disturbances filand measurement noises are rarely known. Recently, tering has been used to address the issue of uncertain external disturbances and measurement noises [6], [7]. In [7], model uncertainties are additionally dealt with via a backward Riccati filtering design problem for nonequation. In this paper, the linear systems is studied. filter in comparison with a The advantage of using an Kalman filter is that no statistical assumption on the exogenous filter is designed by minimizing state signals is needed. The estimation error for the worst-case bounded disturbances and filters are more robust than the Kalman noises. As a result, filtering design, the induced filter. Furthermore, with an norm of the operator from the disturbances to the estimation error can be less than a prescribed attenuation level [7], [10]. filter design (or control design) needs to solve Nonlinear a nonlinear Hamilton–Jacobi equation, which cannot be easily solved, except for some special cases [10]–[15] . In [13], an Manuscript received April 21, 2000; revised July 9, 2001. This work was supported by National Science Council under Contract NSC 88-2213-E-007-069. The associate editor coordinating the review of this paper and approving it for publication was Dr. Rick S. Blum. C.-S. Tseng is with the Department of Electrical Engineering, Ming Hsin Institute of Technology, Hsinchu, Taiwan, R.O.C. B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Publisher Item Identifier S 1053-587X(01)09227-3.
estimation algorithm is proposed for nonlinear systems based on a linearized model around the suboptimal state estimate with an adaptive performance bound. The adaptive process increases the computational burden. In [15], necessary and sufficient condisuboptimal tions are given for the solvability of a standard control and estimation problem that involve the solvability of a pair of partial differential equations of the Hamilton–Jacobi filtering problem is studied type. In this paper, the nonlinear using a fuzzy approach. Recently, there have been many applications of fuzzy systems theory in various fields, such as control systems, communication systems, and signal processing. In most of these applications, fuzzy systems are considered to be universal approximators for certain nonlinear systems. The Takagi and Sugeno fuzzy model [16], which has been proved to be a good representation for a certain class of nonlinear dynamic systems, was extensively studied in control systems [16]–[21]. In this study, a Takagi and Sugeno fuzzy model is proposed to interpolate a class of nonlinear systems. Based on the fuzzy model, the fuzzy filtering design problem is characterized by making the prescribed attenuation level as small as possible, subject to some linear matrix inequality (LMI) constraints. This convex feasibility problem is also called the linear matrix inequalities problem (LMIP). LMIP can be efficiently solved by the convex optimization algorithm [22]. discrete-time The paper is organized as follows. An fuzzy filtering design for the state estimation of nonlinear systems is introduced in Section II. In Section III, simulation examples are provided to demonstrate the design procedure. Finally, concluding remarks are made in Section IV. II.
FUZZY FILTERING FOR NONLINEAR DISCRETE-TIME SYSTEMS
Consider a class of nonlinear discrete-time system
(1) denotes where and denote meathe state vector; sured output and a linear combination of the state variables to and are be estimated, respectively; assumed to be bounded external disturbance and measurement , and noise, respectively;
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are constant matrices; and and are vector fields with and . The purpose of this study is to find an estimator: a mapping as well as the observation whose input is the initial estimate and whose output is sequence of signals , , such that the performance estimates
fuzzy system are inferred, respectively, as follows:
(2)
and [7]–[10]. Here, , is an initial weighting matrix that is a is assumed to be symmetric and positive definite, and symmetric semi-positive definite matrix. Note that (2) can be written in the following form: is achieved for all
(5) where
and nally
is the grade of membership of
in
. Fi-
(6) (3) must be less than Remark 1: The meaning of (2) is that , , and from a prescribed level for all possible the energy point of view, i.e., the induced norm of the oper, , and to , must be less ator from than . For stable LTI systems with an infinite-time horizon, norm is identical to the norm in the frethe induced filter design, by protecting against the quency domain [7]. worst case, is more suitable for systems with unknown (or unoptimality certain) driving noise and measurement noise. is achieved with the lowest possible value of in (2). The th rule of the Takagi–Sugeno fuzzy model for the discrete-time nonlinear system in (1) is proposed as the following form [16]–[18]:
Rule If Then
for ; rules;
is
and
In this paper, we assume [16] for and for Therefore, we have for
(7)
and (8) The fuzzy model in (5) can be interpreted as an interpolation of linear systems through the membership function to approximate the nonlinear system in (1). Therefore, the nonlinear system in (1) can be described as
is (4)
, where is the fuzzy set; ; is the number of If–Then are the premise variables; and . The state dynamic and the output of the
(9)
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FUZZY ESTIMATION FOR A CLASS DYNAMIC SYSTEMS
and
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are the premise variables; initial estimate (the exact interval of concern is ); and . depend on Remark 4: In general, the premise variables depend on the the state variables, and the premise variables , , estimated state variables. If , then , , . The overall fuzzy estimator is written as (10)
where
, and (11)
(12) denote the approximation (or interpolation) errors between the nonlinear system in (1) and the fuzzy model in (5). Assumption: There exist and such that (13) and (14) . for all trajectories Remark 2: The assumptions of and are related to the assumptions in (13) and (14). Under these assumptions, the assumptions in (13) and (14) are still valid . In other words, the system under consideration for is not universally applicable and restricted on the system with and . Remark 3: The T–S fuzzy model is a piecewise interpolation of several linear models at different operating points through fuzzy membership functions. It is only an approximation model for a nonlinear system. The approximation error between the fuzzy model and the original nonlinear system does exist. The idea behind the assumptions in (13) and (14) is that if the approximation error can be covered by an upper bound, then we fuzzy filter for the nonlinear system to can design a robust tolerate the approximation error based on the upper bound. The upper bound can be thought of as the worst-case approximation error. By the assumptions in (13) and (14), we obtain an upper bound of the fuzzy approximation error as follows:
(17) Then, from (9) and (17), the augmented system can be written as the following form:
(18)
Let us define
(15) Based on the fuzzy model (4), the following fuzzy estimator is proposed to deal with the state estimation for the nonlinear system in (1) Estimation Rule : is and is If (16) Then where rule
is the fuzzy estimator gain for the th estimation ;
Then, the augmented system in (18) can be rewritten as the following form:
(19)
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Therefore, the modified as follows:
estimation performance in (3) can be
performance in (20) is guaranteed for a prescribed state. Proof: From (20), we obtain
(20) where
is an initial weighting matrix, and
The following well-known lemmas are useful for our design. Lemma 1 (Schur Complements for Nonstrict Inequalities)[22]: The linear matrix inequality (LMI) (21) where on , is equivalent to
,
, and
depends
and (22) denotes the Moore–Penrose inverse of . where Lemma 2 (S-Procedure for Nonstrict Inequalities) : [22]: Consider the following condition on for all
such that (23)
If there exists
such that (24)
then (23) holds. Then, we get the following result. Theorem 1: In the nonlinear system (1), if there exist and a symmetric positive definite matrix such that the matrix inequalities
(25) and
hold for , where
, then the
estimation
By (19) and with
, we get
at steady
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FUZZY ESTIMATION FOR A CLASS DYNAMIC SYSTEMS
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(26)
and
From (26), if
(27)
then, by the property of membership function in (7), (8), we obtain
(28) is an initial weighting matrix. where estimation performance is achieved with a Hence, the prescribed . Under this situation, we will encounter the consatisfying (27) and the constraint straint on the variables in (15). Note that (15) is equivalent to
(29)
where
.
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Applying the S-procedure in Lemma 2, (27) and (29) are such that equivalent to the existence of
(30)
For the convenience of design, only the steady-state case (i.e., ) is considered in this study. Then, let for (and thus, ). In this situation, (30) becomes
(32)
For the convenience of design, let (33) where With
. , (32) is equivalent to
(31)
This completes the proof. From the analysis above, the most important work associfiltering problem consists of solving ated with the fuzzy from (25). In general, it is not easy to analytifor (25). cally determine the common solution Fortunately, (25) can be converted into linear matrix inequality problem (LMIP) [22]. The LMIP can be solved in a computationally efficient manner using a convex optimization technique such as the interior point method. First, the matrix inequalities in (25) are converted into LMIs using Schur complements as in Lemma 1 by the following procedures. From (25), we obtain
(34)
where
and
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By the Schur complements in Lemma 1, (34) is equivalent to the following LMIs:
(39) . By the choice of and in (38) and (39), a for all worst-case design philosophy is adopted, and the assumptions in (13) and (14) can be guaranteed. Remark 7: The problem of choosing and in the fuzzy plant rules can be considered to be that of constructing a Takagi–Sugeno fuzzy model for nonlinear systems and can be described by the following two steps. Step 1) Specify the number of fuzzy sets and membership functions for the premise variables in advance. Step 2) The weighted recursive least-squares algorithm is applied to determine the parameters of the fuzzy model [27]–[30]. and to construct the With the above steps, one can obtain fuzzy model. III. SIMULATION EXAMPLES Example 1: Consider the following discrete-time nonlinear chaotic system [31]:
(35)
(40) To obtain a better robust filtering performance (against disturbances), the attenuation level can be reduced to the minimum possible value such that (3) is satisfied. Therefore, the design procedure is summarized as follows. Design Procedure: Step 1) Construct the fuzzy model in (4) Step 2) Evaluate and , given an initial attenuation level . (thus, Step 3) Solve the LMIP in (35) to obtain and ). obtain Step 4) Decrease the attenuation level , and repeat Step 3)–Step 4) until a positive definite is not obtained. Step 5) Obtain the fuzzy estimator from (17). Remark 5: The LMIP can be solved very efficiently by a convex optimization technique such as the interior point algorithm [22]–[26]. Remark 6: and can be evaluated by off-line simulations. Note that if and are chosen as diag
and
diag
(36)
then
(37) Therefore,
and
can be chosen as follows: (38)
. In this example, for the conwhere , venience of simulation, it is assumed that , and is normal distribution noise with zero mean and variance 0.01, respectively. By simulations, we and . The design purpose is to esobserve that timate the state variables of the chaotic system. Following the design procedure in the above section, the fuzzy filtering design for the discrete-time system in (40) is given by the following steps. Step 1) Rule 1: IF is about 1.0 and is about 1.0. THEN , . Rule 2: is about 1.0 and is about 0. THEN IF , . Rule 3: is about 1.0 and is about 1.0. THEN IF , . Rule 4: is about 0 and is about 1.0. THEN IF , . Rule 5: is about 0 and is about 0. THEN IF , . Rule 6: is about 0 and is about 1.0. THEN IF , .
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Rule 7: is about 1.0 and IF , Rule 8: is about 1.0 and IF , Rule 9: is about 1.0 and IF , Here
is about
1.0. THEN .
is about 0. THEN . is about 1.0. THEN .
0
Fig. 1. Estimation error (squared) of (a) (x x^ ) (b) (x proposed method (solid line) and EKF (dashdot line).
0 x^ )
[the
TABLE I RATIOS OF ERROR TO STATE
and Triangular membership functions are used for Rules 1 to 9 in this example. and are chosen acStep 2) cording to Remark 6. Steps 3 and 4) Solve the LMIP. In this case, we obtain
The initial condition in the simulation is assumed to be . Fig. 1 and shows the estimation errors (squared) for using the proposed fuzzy filter and the EKF. In this example, the statistical conditions and are used for the design of the EKF. The ratios of error to state of the proposed fuzzy filter and the fuzzy filter is EKF are listed in the Table I. The proposed obtained without any information about external disturbances and measurement noise, as long as they are bounded. In fact, and change with time the statistical properties of and are rarely known beforehand. Obviously, the proposed fuzzy filter is more robust. Example 2: A trajectory estimation of re-entry vehicles (RV) by radar is shown in Fig. 2. The RV flies over several hundred kilometers along a ballistic trajectory above the Earth. The RV centered at the model in radar coordinates radar site can be expressed as [32], [33]
and the fuzzy estimation gains are found to be
where ,
velocity components along , , and , respectively; , , and disturbances; and air density and ballistic coefficient, respectively; [ 9.8 (m/sec )] gravity force. and are defined as follows:
Step 5) Construct the
, and
fuzzy estimator as follows: In this example, ( 2440 K /m ) is assumed to be a constant. The air density is a function of altitude and described as .
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Fig. 2. Tactical ballistic missile geometry.
Let the state vector be , where , , and denote the RV position , , and , respectively. The RV is measured by along [sampling time a precision radar with a sampling rate of 4 (s)]. The nonlinear discrete-time state equation can be written as
where
and for stand for the process noises [32]. For conventional air defense missile systems, the ground radar is the major instrument for detecting the RV. The measurement equation is
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where , and denotes the measurement noise and jamming. The purpose of this problem is to produce an estimate from radar output . of the trajectory Observe that the state variables corresponding to the nonlinear terms are , , , and . It is assumed that , , and , and . Therefore, the fuzzy rules are given as follows: Rule 1: IF is about 0 and is about 800 and is about 800 and is about 2000. THEN , . Rule 2: IF is about 0 and is about 800 and is about 800 and is about 0. THEN , . Rule 3: IF is about 0 and is about 800 and is about , 0 and is about 2000. THEN . Rule 4: IF is about 0 and is about 800 and is about is about 0. THEN , 0 and . Rule 5: IF is about 0 and is about 0 and is about 800 is about 2000. THEN , and . is about 0 and is about 0 and is about Rule 6: IF 800 and is about 0. THEN , . is about 0 and is about 0 and is about 0 Rule 7: IF is about 2000. THEN , and . is about 0 and is about 0 and is about Rule 8: IF is about . THEN , 0 and . is about 30 000 and is about 800 and Rule 9: IF is about 800 and is about 2000. THEN , . is about 30 000 and is about 800 and Rule 10: IF is about 800 and is about 0. THEN , . Rule 11: IF is about 30 000 and is about 800 and is is about 2000. THEN about 0 and , . Rule 12: IF is about 30 000 and is about 800 and is , about 0 and is about 0. THEN . is about 30 000 and is about 0 and Rule 13: IF is about 800 and is about 2000. THEN , . is about 30 000 and is about 0 and is Rule 14: IF is about 0. THEN about 800 and , . is about 30 000 and is about 0 and is Rule 15: IF is about 2000. THEN about 0 and , . is about 30 000 and is about 0 and is Rule 16: IF , about 0 and is about 0. THEN . and (for ) are shown in ApWhere , , pendix A, , and
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. For the convenience of simulations, the disturbances , the process noises , and meaare assumed as follows [32]: surement noises for for for where represents the normal distribution with mean and variance . Triangular membership functions are also used for Rules 1 and are chosen according to 16. to Remark 6. Solving the corresponding LMIP, we obtain
Estimation error (squared) of (a) (x 0 x ^ ) , (b) (x 0 x ^ ) , (c) 0 x^ ) , (d) (x 0 x^ ) , (e) (x 0 x^ ) , and (f) (x 0 x^ ) [the proposed
Fig. 3. (x
method (solid line) and EKF (dashdot line).
and the fuzzy estimation gains (for ) are shown in Appendix B. Therefore, we can construct the fuzzy estimator as follows:
In this example, the proposed fuzzy filter is employed to deal with the nonlinear estimation problem in radar detecting systems. The estimation errors (squared) of the proposed fuzzy filtering and extended Kalman filtering are shown in Fig. 3. In this example, the statistical conditions , , and are used for the design of the EKF. The ratios of error to state of the proposed fuzzy filter and the EKF are listed in Table II. From the results of this simulation, it also shows that the performance of the proposed method is also more robust than that of EKF. IV. CONCLUSIONS In this paper, based on a Takagi and Sugeno fuzzy model, fuzzy filtering problems for nonlinear discrete-time systems with uncertain noises are studied. fuzzy filtering An LMI-based design procedure for the problems of the nonlinear discrete-time systems is developed. The proposed design procedure is very simple. Simulation examples are given to illustrate the design procedure, and the results are satisfactory. Therefore, the proposed method is very suitable for practical filtering applications.
TABLE II RATIOS OF ERROR TO STATE
APPENDIX A
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(for
). APPENDIX B
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REFERENCES [1] A. P. Sage and J. L. Melsa, Estimation Theory with Application to Communication and Control. New York: McGraw-Hill, 1971. [2] M. S. Grewal and A. P. Andrews, Kalman Filtering. Englewood Cliffs, NJ: Prentice Hall, 1993. [3] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [4] A. Gelb, Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974. [5] N. U. Ahmed and S. M. Radaideh, “Modified extended Kalman filtering,” IEEE Trans. Automat. Contr., vol. 39, pp. 1322–1326, June 1994. [6] K. Nishiyama, “Robust estimation of a single complex sinusoid in white filtering approach,” IEEE Trans. Signal Processing, vol. 47, noisepp. 2853–2856, Oct. 1999. [7] R. S. Mangoubi, Robust Estimation and Failure Detection: A Concise Treatment. New York: Springer, 1998. [8] R. M. Agustin, R. S. Mangoubi, R. M. Hain, and N. J. Adams, “Robust failure detection for reentry vehicle attitude control systems,” J. Guid., Contr. Dynam., vol. 22, no. 6, pp. 839–845, Nov.–Dec. 1999. nonlinear filtering of discrete-time pro[9] U. Shaked and N. Berman, “ cesses,” IEEE Trans. Signal Processing, vol. 43, pp. 2205–2209, Sept. 1995. [10] L. Xie, C. E. De Souza, and Y. Wang, “Robust filtering for a class of approach,” Int. J. discrete-time uncertain nonlinear systems: An Robust Nonlinear Contr., vol. 6, pp. 297–312, 1996. [11] N. Berman and U. Shaked, “ nonlinear filtering,” Int. J. Robust Nonlinear Contr., vol. 6, pp. 281–296, 1996. [12] M. R. James and I. R. Petersen, “Nonlinear state estimation for uncertain systems with an integral constraint,” IEEE Trans. Signal Processing, vol. 46, pp. 2926–2937, Nov. 1998. estimation for nonlinear systems,” IEEE Signal Pro[13] J. B. Burl, “ cessing Lett., vol. 5, pp. 199–202, Aug. 1998. [14] J. A. Ball, P. Kachroo, and A. J. Krener, “ tracking control for a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 44, pp. 1202–1206, June 1999. [15] A. J. Krener, “Nonlinear worst case ( ) control,” in Proc. IEEE 33rd Conf. Decision Contr., Lake Buena Vista, FL, Dec. 1994, pp. 3450–3451. [16] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, 1985. [17] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250–265, Apr. 1998. [18] B. S. Chen, C. S. Tseng, and H. J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 571–585, Oct. 1999. [19] , “Mixed =H fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 249–265, June 2000. [20] , “Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 381–392, June 2001. [21] J. Buckley, “Sugeno type controllers are universal controller,” Fuzzy Sets Syst., vol. 53, pp. 199–303, 1993. [22] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [23] R. Nikoukhah, F. Delebecque, and L. El Ghaoui, LMITOOL: A Package for LMI Optimization in Scilab. Rocquencourt, France: INRIA , 1995. [24] L. El Ghaoui, R. Nikoukhah, and F. Delebecque, “LMITOOL: A front-end for LMI optimization in Matlab,”, 1995. [25] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev., vol. 38, pp. 49–95, 1996. [26] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The MathWorks, 1995. [27] M. Sugeno and K. Tanaka, “Successive identification of fuzzy model,” Fuzzy Sets Syst., vol. 28, pp. 15–33, 1988. [28] E. Kim, M. Park, S. Ji, and M. Park, “A new approach to fuzzy modeling,” IEEE Trans. Fuzzy Syst., vol. 5, no. 3, pp. 328–337, Aug. 1997. [29] C. C. Wong and C. C. Chen, “A clustering-based method for fuzzy modeling,” IEICE Trans. Inform. Syst., vol. E82-D, no. 6, pp. 1058–1065, June 1999. [30] M. Sugeno and T. Yasukawa, “A fuzzy-logic-based approach to qualitative modeling,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 7–31, Feb. 1993.
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[31] S. H. Strogatz, Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley, 1994. [32] S. C. Lee and C. Y. Liu, “Trajectory estimation of reentry vehicles by use of on-line input estimation,” J. Guid., Contr., Dynam., vol. 22, no. 6, Nov.–Dec. 1999. [33] P. Zarchan, Tactical and Strategic Missile Guidance. Washington, DC: AIAA, 1994.
Chung-Shi Tseng (M’01) received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, and the Ph.D. degree in electrical engineering, from National Tsing-Hua University, Hsinchu, Taiwan. He is now an Associate Professor at Ming Hsin Institute of Technology and Commerce, Hsinchu. His research interests are in signal processing and control.
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Bor-Sen Chen (M’82–SM’89) received the B.S. degree from Tatung Institute of Technology, Taipei, Taiwan, R.O.C., in 1970, the M.S. degree from National Central University, Taipei, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology from 1973 to 1987. He is now a Professor at National Tsing Hua University, Hsinchu, Taiwan. His current research interests include control and signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council and the Chair of the Outstanding Scholarship Foundation.
