Robust Controllability of T–S Fuzzy-Model-Based
Control Systems With Parametric Uncertainties
Shinn-Horng Chen, Wen-Hsien Ho, and Jyh-Horng Chou, Senior Member, IEEE
Abstract—The robust controllability problem for the Takagi– Sugeno (T–S) fuzzy-model-based control systems is studied in this paper. Under the assumption that the nominal T–S fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal T–S fuzzy-model-based control systems has a full row rank for its controllability matrix), a sufficient condition is proposed to preserve the assumed property when the parameter uncertainties are added into the nominal T–S fuzzy-model-based control systems. The proposed sufficient condition can provide the explicit relationship of the bounds on parameter uncertain-ties to preserve the assumed property. Besides, a robustly global controllability condition and the related robustly global stabiliz-ability condition of the uncertain T–S fuzzy-model-based control systems are also presented in this paper. A nonlinear mass–spring– damper mechanical system with parameter uncertainties is given as an example to illustrate the application of the proposed sufficient conditions.
Index Terms—Elemental parameter uncertainties, robust con-trollability, robust stability, Takagi–Sugeno (T–S) fuzzy model.
I. INTRODUCTION
R
ECENTLY, it has been shown that the fuzzy-model-based representation that was proposed by Takagi and Sugeno [1], which is known as the T–S fuzzy model, is a success-ful approach to deal with the nonlinear control systems, and there are many successful applications of the T–S fuzzy-model-based approach to the nonlinear control systems (see [2]–[23] and references therein). All the aforementioned works regarding successful applications of the T–S fuzzy-model-based approach (see [2]–[23] and references therein) are, under the assumption that the nominal T–S fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal T–S fuzzy-model-based control systems has a full row rank for its controllability matrix), to design the fuzzy parallel-distributed-compensation (PDC) controllers.However, in fact, in many cases, it is very difficult, if not impossible, to obtain the accurate values of some system pa-rameters. This is due to the inaccurate measurement, unaccessi-Manuscript received March 24, 2009; accepted July 25, 2009. First published August 21, 2009; current version published December 3, 2009. This work was supported in part by the National Science Council, Taiwan, under Grant E151-031, Grant NSC96-2628-E327-004-MY3, and Grant NSC97-2221-E037-003.
S.-H. Chen is with the Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung City 80778, Taiwan (e-mail: shchen@cc.kuas.edu.tw).
W.-H. Ho is with the Department of Medical Information Management, Kaohsiung Medical University, Kaohsiung City 80708, Taiwan (e-mail: whho@kmu.edu.tw).
J.-H. Chou is with the Institute of System Information and Control, National Kaohsiung First University of Science and Technology, Kaohsiung City 811, Taiwan (e-mail: choujh@ccms.nkfust.edu.tw).
Digital Object Identifier 10.1109/TFUZZ.2009.2030670
bility to the system parameters, or variations in the parameters. These parametric uncertainties may destroy the controllability property of the T–S fuzzy-model-based control systems. Some researchers have studied the controllability problems of vari-ous types of fuzzy systems [24]–[37], but to the authors’ best knowledge, there are no literatures that have studied the issue of robust controllability for the uncertain T–S fuzzy-model-based control systems.
The purpose of this paper is to present an approach to in-vestigate the robust controllability problem of the T–S fuzzy-model-based control systems with parameter uncertainties. Un-der the assumption that the nominal T–S fuzzy-model-based control systems are locally controllable, a sufficient condition is proposed to preserve the assumed property when the param-eter uncertainties are added into the nominal T–S fuzzy-model-based control systems. The proposed sufficient condition can provide the explicit relationship of the bounds on elemental parameter uncertainties to preserve the assumed property. Be-sides, a robustly global controllability condition and the related robustly global stabilizability criterion for the uncertain T–S fuzzy-model-based control systems are also proposed in this pa-per. A nonlinear mass–spring–damper mechanical system with elemental parameter uncertainties is given in this paper to illus-trate the application of the proposed sufficient conditions.
This paper is organized as follows. Section II describes the ro-bust controllability analysis for the uncertain T–S fuzzy-model-based control systems, in which the sufficient criteria for both robustly local controllability and robustly global controllability are presented. Section III derives the sufficient condition of both robustly global controllability and stabilizability for the uncer-tain T–S fuzzy-model-based control systems. In Section IV, an illustrative example is given to demonstrate the applicability of the sufficient criteria proposed in this paper. Finally, Section V offers some conclusions.
II. ROBUSTCONTROLLABILITYANALYSIS
Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of the premise part and the linear uncertain dynamic model of the conse-quent part in the exact T–S fuzzy control model with para-metric uncertainties can be derived from the given nonlin-ear uncertain control model [4]. The parametric uncertainties can be viewed to take different forms, such as structured (el-emental) and unstructured (norm-bounded). Elemental para-metric uncertainties are those for which the elemental infor-mation of the uncertain matrix is utilized and bounds on the individual elements of the uncertain matrix are considered, whereas norm-bounded parametric uncertainties are those for 1063-6706/$26.00 © 2009 IEEE
which only a norm bound on the uncertain matrix is consid-ered [38]–[40]. However, if the elemental information of the parametric uncertain matrices is considered, the results will be less conservative than those results that do not utilize the elemen-tal information of the parametric uncertain matrices [38], [39]. In this paper, we mainly consider the elemental parametric un-certainties. The T–S fuzzy-model-based control system with parametric uncertainties for the nonlinear control system with parametric uncertainties can be obtained as follows:
˜
Ri: IF z1is Mi1 and · · · and zg is Mig
THEN
˙x(t) = (Ai+ ∆Ai(t))x(t) + (Bi+ ∆Bi(t))u(t) (1)
with the initial state vector x(0), where ˜Ri(i = 1, 2, . . . , N )
denotes the ith implication, N is the number of fuzzy rules,
x(t) = [x1(t), x2(t), . . . , xn(t)]T denotes the n-dimensional
state vector, u(t) = [u1(t), u2(t), . . . , up(t)]T denotes the
p-dimensional input vector, zi(i = 1, 2, . . . , g) are the premise
variables, Ai and Bi(i = 1, 2, . . . , N ) are, respectively, the
n× n and n × p consequent constant matrices, ∆Ai(t) and
∆Bi(t)(i = 1, 2, . . . , N ) are, respectively, the time-varying
parametric uncertain matrices that exist in the system matri-ces Ai and the input matrices Bi of the consequent part of
the ith rule due to the inaccurate measurement, unaccessibility to the system parameters, or variation of the parameters, and
Mij(i = 1, 2, . . . , N , and j = 1, 2, . . . , g) are the fuzzy sets.
In many interesting problems, we have only a small number of uncertain parameters, but these uncertain parameters may en-ter into many entries of the system and input matrices [39]–[42]. Therefore, in this paper, we suppose that the time-varying para-metric uncertain matrices ∆Ai(t) and ∆Bi(t) take the forms
∆Ai(t) = m k = 1 εik(t)Aik and ∆Bi(t) = m k = 1 εik(t)Bik (2) where εik(t) are the time-varying elemental parametric
uncer-tainties, and Aik and Bik are, respectively, the given n× n
and n× p constant matrices, which are prescribed a priori to denote the linearly dependent information on the time-varying elemental parametric uncertainties εik(t), i = 1, 2, . . . , N , and
k = 1, 2, . . . , m. For example, consider a two-mass system with
an uncertain stiffness that is described by Sinha [42]. The system matrix A is A = 0 0 1 0 0 0 0 1 −˜k(t) k(t)˜ 0 0 ˜ k(t) −˜k(t) 0 0 = 0 0 1 0 0 0 0 1 −k0 k0 0 0 k0 −k0 0 0 + ε(t) 0 0 0 0 0 0 0 0 −1 1 0 0 1 −1 0 0
where the parametric uncertainty ε(t) enters into four entries of the system matrix. Therefore, the forms in (2) are the structured parametric uncertain forms for the general and practical cases [42].
A. Robustly Local Controllability
In this section, for the uncertain T–S fuzzy-model-based control system in (1), each fuzzy-rule-nominal model ˙x(t) =
Aix(t) + Biu(t), which is denoted by{Ai, Bi}, is assumed to
be controllable (i.e., each fuzzy-rule-nominal model {Ai, Bi}
has a full row rank for its controllability matrix). Due to inevitable uncertainties, each fuzzy-rule-nominal model
{Ai, Bi} is perturbed into the fuzzy-rule-uncertain model
{Ai+ ∆Ai(t), Bi+ ∆Bi(t)}. Our problem in this section is
to determine the condition such that each fuzzy-rule-uncertain model{Ai+ ∆Ai(t), Bi+ ∆Bi(t)} for the T–S
fuzzy-model-based control system (1) is still controllable. Before we investi-gate the robust property of controllability for the uncertain T–S fuzzy-model-based control system (1), the following definition and lemmas need to be introduced.
Definition [5]: The T–S fuzzy-model-based control system in (1) is locally controllable if each fuzzy rule model {Ai+ ∆Ai(t), Bi+ ∆Bi(t)}(i = 1, 2, . . . , N) is
controllable.
Lemma 1 [43]: The system model ˙x(t) = Ax(t) + Bu(t) is
controllable if and only if the n2× n(n + p − 1) matrix, i.e.,
Q = In 0 • • • 0 0 • • • 0 B −A In • • • 0 0 • • • B 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • In 0 • • • 0 0 0 0 • • • −A B • • • 0 0 (3) has rank n2, where A∈ Rn×n, B∈ Rn×p, and I
n denotes the
n× n identity matrix.
Lemma 2 [44]: The matrix measures of the matrices ¯W and
¯
V , namely, µ( ¯W ) and µ( ¯V ), respectively, is well defined for
any norm and have the following properties: 1) µ(±I) = ±1, for the identity matrix I;
2) − ¯W ≤ −µ(− ¯W )≤ Re(λ( ¯W ))≤ µ( ¯W )≤ ¯W, for
any norm• and any matrix ¯W ∈ Cn×n;
3) µ( ¯W + ¯V )≤ µ( ¯W ) + µ( ¯V ), for any two matrices
¯
W , ¯V ∈ Cn×n;
4) µ(γ ¯W ) = γµ( ¯W ), for any matrix ¯W ∈ Cn×n and any
nonnegative real number γ;
where λ( ¯W ) denotes any eigenvalue of ¯W , and Re(λ( ¯W ))
denotes the real part ofλ( ¯W ).
Lemma 3: For any γ < 0 and any matrix W¯ ∈ Cn×n, µ(γ ¯W ) =−γµ(− ¯W ).
Proof: This lemma can be immediately obtained from
prop-erty 4) in Lemma 2. Q.E.D.
Lemma 4: Let ¯N ∈ Cn×n. If µ(− ¯N ) < 1, then det(I +
¯
N )= 0.
Proof: From property 2) in Lemma 2, and since µ(− ¯N ) < 1,
we can get that Re(λ( ¯N ))≥ −µ(− ¯N ) >−1. This implies that λ( ¯N )= −1. Therefore, we have the stated result. Q.E.D. From Lemma 1, it is known that for the uncertain T–S fuzzy-model-based control system (1), each fuzzy-rule-uncertain model {Ai+ ∆Ai(t), Bi+ ∆Bi(t)} in (1) is controllable if
and only if the n2× n(n + p − 1) matrix ˜ Qi= Qi+ m k = 1 εik(t)Eik (4)
has full row rank n2, where
Qi= In 0 • • • 0 0 • • • 0 Bi −Ai In • • • 0 0 • • • Bi 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • In 0 • • • 0 0 0 0 • • • −Ai Bi • • • 0 0 (5) and Eik= 0 0 • • • 0 0 • • • 0 Bik −Aik 0 • • • 0 0 • • • Bik 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • 0 0 • • • 0 0 0 0 • • • −Aik Bik • • • 0 0 . (6) Let the singular-value decomposition of Qibe
Qi= Ui[ Si 0n2×n(p−1)] ViH (7)
where Ui∈ Rn
2×n2
and Vi∈ Rn (n + p−1)×n(n+p−1)are the
uni-tary matrices, VH
i denotes the complex-conjugate transpose
of matrix Vi, Si= diag[σi1, . . . , σin2], and σi1 ≥ σi2 ≥ · · · ≥
σin2 > 0 are the singular values of Qi.
Remark 1: From the works of Tsakalis and Ioannou [45]
and Rugh [46], we know that, for a linear time-varying system ˙x(t) = A(t)x(t) + B(t)u(t), as well as for t belonging to a time interval [t0, tf], if A(t) is (n− 2) times differentiable, B(t) is
(n− 1) times differentiable, and the matrix in (3) has rank n2
for t∈ [t0, tf], then the linear time-varying system is
control-lable. Therefore, in this paper, the rank condition on controlla-bility of the matrix in (4) is suitable only for the following cases: 1) The uncertain matrices ∆Ai(t) and ∆Bi(t) are, respectively,
assumed to be (n− 2) times and (n − 1) times differentiable, and 2) ∆Ai(t) = ∆Ai and ∆Bi(t) = ∆Bi are time-invariant
uncertain matrices, where i = 1, 2, . . . , N.
In the following, we present a sufficient criterion to ensure that the uncertain T–S fuzzy-model-based control system in (1) is robustly locally controllable.
Theorem 1: Suppose that each fuzzy-rule-nominal model {Ai, Bi} is controllable. The uncertain T–S fuzzy-model-based
control system in (1) is robustly locally controllable if the fol-lowing conditions hold simultaneously:
m k = 1 εik(t)φik < 1 (8) where i = 1, 2, . . . , N , and φik= µ(−S−1 i UiHEikVi[In2, 0n2×n(p−1)]T), for εik(t)≥0 −µ(S−1 i UiHEikVi[In2, 0n2×n(p−1)]T), for εik(t) < 0.
The matrices Eik, Si, Ui, and Vi(i = 1, 2, . . . , N ) are,
respec-tively, defined in (6) and (7), and In2 denotes the n2× n2
iden-tity matrix.
Proof: Since each fuzzy-rule-nominal model {Ai, Bi}
(i = 1, 2, . . . , N ) is controllable, from Lemma 1, we have that the matrix Qiin (5) has full row rank (i.e., rank(Qi) = n2). We
know that
rank(Qi) = rank(S−1i UiHQiVi). (9)
Thus, instead of rank( ˜Qi), we can discuss the rank of
[ In2 0n2×n(p−1)] +
m
k = 1
εik(t)Rik (10)
where Rik = Si−1UiHEikVi, for i = 1, 2, . . . , N , and k =
1, 2, . . . , m. Since a matrix has at least rank n2 if it has at
least one nonsingular n2× n2submatrix, a sufficient condition
for the matrix in (10) to have rank n2is the nonsingularity of
Gi= In2+ m k = 1 εik(t) ¯Rik (11) where ¯Rik = Si−1UiHEikVi[In2, 0n2×n(p−1)]T, for i = 1, 2, . . . , N.
Using the properties in Lemmas 2 and 3, as well as from (8), we have µ − m k = 1 εik(t) ¯Rik = µ − m k = 1 εik(t)(Si−1UiHEikVi[In2, 0n2×n(p−1)]T) ≤ m k = 1 µ(−εik(t)(Si−1UiHEikVi[In2, 0n2×n(p−1)]T)) = m k = 1 εik(t)φik < 1. (12)
Thus, from Lemma 4, we have det(Gi) = det In2+ m k = 1 εik(t) ¯Rik = 0. (13) Hence, the matrix Gi in (11) is nonsingular, i.e., the matrix
˜
Qi in (4) has full row rank n2. Therefore, from the results
mentioned earlier and from Lemma 1, the local controllability of the uncertain T–S fuzzy-model-based control system in (1) is ensured. Thus, the proof is completed. Q.E.D.
Remark 2: The proposed conditions in (8) can give the explicit
relationship of the bounds on εik(t) to preserve local
control-lability. In addition, the bounds that are obtained by using the proposed sufficient criterion on εik(t) are not necessarily
sym-metric with respect to the origin of the parameter space regarding
εik(t), where i = 1, 2, . . . , N , and k = 1, 2, . . . , m.
Remark 3: This paper studies the problem of robust
con-trollability analysis. If the proposed conditions in (8) are sat-isfied simultaneously, each rule of the uncertain T–S fuzzy-model-based control system in (1) is guaranteed to be robustly
locally controllable. This implies that in the fuzzy PDC con-troller design, if the proposed conditions in (8) are satisfied simultaneously, the PDC controller of each control rule can control every state variable in the corresponding rule of the un-certain T–S fuzzy-model-based control system in (1). However, here, it should be noticed that although the PDC controller of each control rule can control every state variable in the corre-sponding rule under the presented conditions in (8) being held, the PDC controller gains should be determined using global design criteria that are needed to guarantee the global stability and control performance [4], where many useful global design criteria have been proposed by some researchers in the litera-tures (see, for example, [2]–[9], [12], [13], [17], and [23] and references therein).
However, the following uncertainty forms are also considered in this paper for the time-varying parametric uncertain matrices ∆Ai(t) and ∆Bi(t) [4]:
∆Ai(t) = MA∆(t)NAi and ∆Bi(t) = MBi∆(t)NBi
(14) for i = 1, 2, . . . , N, where MAi, MBi, NAi, and NBiare known
constant real matrices with appropriate dimensions, and ∆(t) is an unknown matrix function that has
∆(t)∈ Ω := {∆(t)|∆(t) ≤ 1
the elements of ∆(t) are Lebesgue measurable}. Following the same proof procedures that are given in Theorem 1, we can get the following corollary to ensure that the uncertain T–S fuzzy-model-based control system in (1), with the uncertainty forms given by (14), is robustly locally controllable.
Corollary 1: Suppose that each fuzzy-rule-nominal model {Ai, Bi} is controllable. The uncertain T–S fuzzy-model-based
control system in (1), with the uncertainty forms given by (14), is robustly locally controllable if the following conditions hold simultaneously:
αiβ1iβ2i< 1 (15)
where
αi=Fi(t), β1i=Si−1UiH, β2i=Vi[In2, 0n2×n(p−1)]T
and (16), as shown at the bottom of this page, for i = 1, 2, . . . , N . The matrices Si, Ui, and Vi(i = 1, 2, . . . , N ) are,
respectively, defined in (6) and (7), and In2 denotes the n2× n2
identity matrix.
B. Robustly Global Controllability
The resulting T–S fuzzy-model-based control system, with parametric uncertainties inferred from (1), is represented as
˙x(t) =
N
i= 1
hi(z) ((Ai+ ∆Ai(t))x(t) + (Bi+ ∆Bi(t))u(t))
(17) where z = [z1, z2, . . . , zg]Tdenotes the g-dimensional premise
vector, hi(z) = wi(z)/
N
i= 1wi(z), wi(z) =
g
j = 1Mij(zj),
and Mij(zj) are the grades of membership of zj in the fuzzy
sets Mij(i = 1, 2, . . . , N , and j = 1, 2, . . . , g). It can be seen
that for all t, hi(z)≥ 0, and
N
i= 1hi(z) = 1.
From Lemma 1, it is known that the resulting uncertain T–S fuzzy-model-based control system in (17) is robustly globally controllable if and only if the n2× n(n + p − 1) matrix
˜ Q = N i= 1 hi(z)Qi+ N i= 1 m k = 1 hi(z)εik(t)Eik = N i= 1 hi(z)( ¯Q + Di) + N i= 1 m k = 1 hi(z)εik(t)Eik = ¯Q + N i= 1 hi(z)Di+ N i= 1 m k = 1 hi(z)εik(t)Eik (18)
has full row rank n2, where ¯Q is any given n2× n(n + p − 1)
constant matrix that has full row rank, Qiand Eikare given in
(5) and (6), and Di = Qi− ¯Q.
Let the singular-value decomposition of ¯Q be
¯
Q = ¯U S¯ 0n2×n(p−1)
¯
VH (19)
where ¯U ∈ Rn2×n2 and ¯V ∈ Rn (n + p−1)×n(n+p−1)are the uni-tary matrices, ¯VH denotes the complex-conjugate transpose of matrix ¯V , ¯S = diag[¯σ1, . . . , ¯σn2], and ¯σ1≥ ¯σ2 ≥ · · · ≥ ¯σn2 >
0 are the singular values of ¯Q. In the following, we present a
sufficient criterion to ensure that the resulting uncertain T–S fuzzy-model-based control system in (17) is robustly globally controllable.
Theorem 2: The resulting uncertain T–S fuzzy-model-based
control system in (17) is robustly globally controllable if the following condition holds:
N i= 1 µ(−¯Λi) + N i= 1 m k = 1 εik(t) ¯φik< 1 (20) Fi(t) = 0 0 • • • 0 0 • • • 0 MBi∆(t)NBi −MAi∆(t)NAi 0 • • • 0 0 • • • MBi∆(t)NBi 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • 0 0 • • • 0 0 0 0 • • • −MAi∆(t)NAi MBi∆(t)NBi • • • 0 0 (16)
Let V (x(t)) = xT(t)P x(t) be a quadratic Lyapunov function candidate for the system in (A4); then, we have
˙ V (x(t)) = xT(t) N i= 1 N j = 1 hi(z)hj(z) b l= 1 αij l(t)Uij lT P x(t) + x(t)TP N i= 1 N j = 1 hi(z)hj(z) b l= 1 αij l(t)Uij l x(t) = N i= 1 N j = 1 b l= 1 hi(z)hj(z)αij l(t)xT(t) Uij lTP +P Uij l x(t). (A5) It is obvious that ˙V (x(t)) < 0∀x(t) = 0 if, for the specified
fuzzy PDC controller in (27), there exists a symmetric positive-definite matrix P such that
Uij lTP + P Uij l< 0 (A6)
for i, j = 1, 2, . . . , N, and l = 1, 2, . . . , 22m.
Therefore, from the results mentioned before, we can con-clude that the closed-loop uncertain T–S fuzzy-model-based dynamic system in (28) can be robustly stabilized by the spec-ified fuzzy PDC controller if, for the specspec-ified fuzzy PDC con-troller in (27), there exists a symmetric positive-definite matrix
P such that the conditions in (29) are satisfied simultaneously.
Thus, the proof is completed. Q.E.D.
ACKNOWLEDGMENT
The authors thank the reviewers and the associate editor for their constructive comments and suggestions.
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Shinn-Horng Chen received the B.S. and M.S.
de-grees in mechanical engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 1987 and 1989, respectively, and the Ph.D. degree in mechanical and mechatronic engi-neering from the National Sun Yat-Sen University, Kaohsiung City, Taiwan, in 1996.
He is currently a Professor of mechanical engi-neering with the National Kaohsiung University of Applied Sciences, Kaohsiung City, where he was a Professor and the Chairman with the Mechanical En-gineering Department from October 2006 to July 2009. His research interests include the areas of robust control, vibration control, and optimal design.
Wen-Hsien Ho received the B.S. degree in marine
engineering from the National Taiwan Ocean Uni-versity, Keelung, Taiwan, in 1991, the B.S. degree in industrial and information management from the National Cheng-Kung University, Tainan, Taiwan, in 1998, and the M.S. degree in mechanical and au-tomation engineering and the Ph.D. degree in engi-neering science and technology from the National Kaohsiung First University of Science and Tech-nology, Kaohsiung City, Taiwan, in 2002 and 2006, respectively.
From September 1991 to July 2006, he was an Engineer with the Design Department, China Shipbuilding Corporation, Taiwan. He is currently an As-sistant Professor with the Department of Medical Information Management, Kaohsiung Medical University, Kaohsiung City. His research interests include intelligent systems and control, computational intelligence and methods, robust control, and quality engineering.
Jyh-Horng Chou (M’04–SM’04) received the B.S.
and M.S. degrees in engineering science from the National Cheng-Kung University, Tainan, Taiwan, in 1981 and 1983, respectively, and the Ph.D. degree in mechatronic engineering from the National Sun Yat-Sen University, Kaohsiung City, Taiwan, in 1988. From August 1983 to July 1986, he was a Lec-turer with the Mechanical Engineering Department, National Sun Yat-Sen University. From August 1986 to July 1991, he was an Associate Professor with the Mechanical Engineering Department, National Kaohsiung University of Applied Sciences, where he was also the Director of the Center for Automation Technology. From August 1991 to July 1999, he was a Professor with the Mechanical Engineering Department, National Yun-lin University of Science and Technology, YunYun-lin, Taiwan, where he was also the Chairman. From August 1999 to September 2004, he was a Professor and the Chairman with the Mechanical and Automation Engineering Department, National Kaohsiung First University of Science and Technology, Kaohsiung City, where he was a Professor and the Dean of the Engineering College from October 2004 to December 2005 and where he is currently a Professor, the Vice President, and the Acting President. He has coauthored three books. He has also authored or coauthored more than 205 papers published in refereed journals and 190 conference papers. He holds three patents (two are in the technology area of automation, and one is in the area of computational intelligence). His cur-rent research interests include intelligent systems and control, computational intelligence and methods, automation technology, robust control, and quality engineering. He is an Editorial Member or an Associate Editor of nine interna-tional journals.
Prof. Chou received both the Research Award and the Excellent Research Award from the National Science Council of Taiwan 14 times. He also received the 2004 Excellent Project Outcome Award from the Educational Promotion Project on Integrated Manufacturing and e-Commerce Technology from the Ministry of Education, Taiwan; the 2005 and the 2008 Distinguished Research Award from the National Kaohsiung First University of Science and Technol-ogy; the 2005 Distinguished Achievement Award for alumnus graduated from the Engineering Science Department, National Cheng-Kung University; the Best Paper Award at the 2006 Conference on Artificial Intelligence and Applications, Taiwan; the 2007 Distinguished Electrical Engineering Professor Award from the Chinese Institute of Electrical Engineering, Taiwan; and the 2007 Excellent Educator Award from the Ministry of Education, Taiwan. He has been listed in Marquis Who’s Who in World and Marquis Who’s Who in Science and
En-gineering. He has also been a member of the program committees of many
