Optimal Static Output Feedback Simultaneous
Regional Pole Placement
Jenq-Lang Wu and Tsu-Tian Lee, Fellow, IEEE
Abstract—The problem of optimal simultaneous regional pole placement for a collection of linear time-invariant systems via a single static output feedback controller is considered. The cost function to be minimized is a weighted sum of quadratic perfor-mance indices of the systems. The constrained region for each system can be the intersection of several open half-planes and/or open disks. This problem cannot be solved by the linear matrix inequality (LMI) approach since it is a nonconvex optimization problem. Based on the barrier method, we instead solve an auxil-iary minimization problem to obtain an approximate solution to the original constrained optimization problem. Moreover, solution algorithms are provided for finding the optimal solution. Fur-thermore, a necessary and sufficient condition for the existence of admissible solutions to the simultaneous regional pole placement problem is derived. Finally, two examples are given for illustration. Index Terms—Barrier method, constrained optimization, re-gional pole placement, simultaneous stabilization.
I. INTRODUCTION
T
HE problem of simultaneous stabilization for a collection of linear systems via a single controller is an important issue in robust control theory (see [1], [2], [4], [13], and [25]). This problem concerned with the determination of a single con-troller which will simultaneously stabilize a finite collection of systems. The simultaneous stabilization problem arises fre-quently in practice, due to plant uncertainty, plant variation, failure modes, plants with several modes of operation, or non-linear plants non-linearized at several different equilibria. In [24], a nonlinear state feedback controller which simultaneously stabi-lizes a collection of single input systems is presented. In [10], a necessary and sufficient condition, embedded in the solvability of a constrained optimization problem, for the existence of con-trollers to simultaneously stabilize a collection of single input systems is obtained. In [11], [18], and [23], the optimal simul-taneous stabilizing state feedback controllers are found via nu-merically solving a minimization problem. The cost function to be minimized is a weighted sum of the quadratic perfor-mance indices of the systems. In [6] and [7], necessary and suffi-cient conditions for simultaneous stabilizability of a collection of multi-input multi-output (MIMO) systems via static output Manuscript received July 23, 2003; revised April 29, 2004 and February 28, 2005. This work was supported by the National Science Council of the Re-public of China under Contract NSC 91-2213-E-146-004. This paper was rec-ommended by Associate Editor S. Phoha.J.-L. Wu is with the Electronic Engineering Department, Hwa Hsia Insti-tute of Technology, Chung-Ho 235, Taipei, Taiwan, R.O.C. (e-mail: wujl@ cc.hwh.edu.tw).
T.-T. Lee is with the Electrical and Control Engineering Department, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: ttlee@cn.nctu.edu.tw).
Digital Object Identifier 10.1109/TSMCB.2005.846650
feedback and state feedback are obtained in the form of coupled algebraic Riccati inequalities. Moreover, in [20] and [21], linear periodically time-varying controllers are used for simultaneous stabilization and performance or disturbance rejection.
Although many researches have focused on the simultaneous stabilization problem in recent years, the optimal simultaneous regional pole placement problem has not been considered yet. The minimization of quadratic cost functions can indeed im-prove the systems’ static responses (see [5] and [17]). How-ever, it cannot guarantee that the closed-loop systems have good transient responses. The systems’ transient responses are deter-mined mainly by the locations of the systems’ poles. If we can assign the systems’ poles to some specified regions, then good transient responses can be guaranteed. For the single system case, in [8], [14]–[16], and [26], the authors determined a feed-back controller for a system such that the closed-loop poles lie within a specified region. Moreover, a quadratic cost function being minimized by the resultant controller is found. Never-theless, for a given cost function, how to find the optimal con-troller subject to the regional pole’s constraint has not been dis-cussed. In [9], the authors solved a modified Lyapunov equa-tion to obtain a controller which minimizes an auxiliary cost and guarantees that the resultant closed-loop poles lie in a de-sired region. This auxiliary cost provides a guaranteed upper bound on the original quadratic cost function. However, how to find the optimal controller to minimize the actual cost subject to the regional pole’s constraint is still unsolved. Up until now, the existing results about the (optimal) regional pole placement problem are focused on single system case. The optimal simul-taneous regional pole placement problem for a collection of sys-tems has yet to be addressed.
In this paper, we provide a new method to solve output feed-back optimal simultaneous regional pole placement problem for a collection of systems. The considered cost function is a weighted sum of quadratic performance indices of the systems; and the constrained region for each system can be the intersec-tion of several open half-planes and/or open disks. This is a con-strained optimization problem and its minimum point may not exist. It often happens that its infimum point lies on the boundary of the admissible solution set, and it is not a stationary point. Therefore, the Lagrange multiplier method cannot be employed to derive the necessary conditions for optimum for this problem. To solve this problem analytically is quite difficult. Moreover, this problem cannot be solved via the linear matrix inequality (LMI) approach since the admissible solution set may be non-convex. In general, static output feedback control problems are very difficult to solve [28]. It has been shown in [3] that simul-taneous stabilization by static output feedback is NP-hard. In 1083-4419/$20.00 © 2005 IEEE
this paper, based on the barrier method (see [22]), we instead solve an auxiliary minimization problem to obtain an approxi-mate solution of the original problem. The new cost function is the sum of the actual cost function of the original problem and a weighted “barrier function.” Necessary and sufficient conditions for the existence of admissible solutions are given. We prove that the minimal solution of the auxiliary minimization problem exists if the admissible solution set is nonempty. Moreover, it is a stationary point. Then the Lagrange multiplier method can be used to derive the necessary conditions for optimum of the auxiliary minimization problem. In fact, the minimal solution of the auxiliary minimization problem converges to the infimal solution of the original problem if the weighting factor of the barrier function approaches zero. Unlike the approaches pre-sented in [8], [14]–[16], and [26] for the single-system case, in our approach, we can get a solution very close to a local in-fimal solution of the considered problem. When the poles’ con-straint is relaxed, a necessary and sufficient condition for the existence of the simultaneous stabilizing static output feedback controller is found in form of coupled matrix equalities. Finally, two numerical examples are provided for illustration. Based on the gradient method, numerical algorithms are provided in Ex-ample 1 to demonstrate how to solve the auxiliary minimization problem.
A. Notations
expected value;
spectrum of the matrix ;
Tr trace of the matrix ;
(conjugate) transpose of the matrix ; , the spectral norm of the matrix ;
matrix is positive (semi)definite; complex conjugate of ;
approaches ; order of.
II. PROBLEMFORMULATION ANDPRELIMINARIES Consider a collection of linear time-invariant systems
(1) where is the state of the th system, is the control input of the th system, and is the output of the th system; , , and are constant matrices of appropriate dimensions. Suppose that is controllable, is observable and has full row rank for all . Let , ,
, and . Define Re
Note that denotes an open half-plane and is an open disk with radius and centered at . The region
is the open left half-plane.
The design goal is to find a static output feedback gain such that the controllers
(2) achieve the infimum of the cost function
(3) subject to the constraints that
where , , are weighting factors and
is defined as
Suppose , with
being observable, and the constrained region is represented by
Re and
Each can be the intersection of several open half-planes and open disks. Note that the region must be symmetric with respect to the real axis in order to obtain a real feedback gain. The selection of weighting factors , , depends on requirements of practical applications. If we want the -th system has better LQ performance, then we can choose larger . In contrast, if the LQ performance of the -th system is less important comparing to the other systems, then we can choose smaller .
Let and let
and
The set is the collection of all matrices such that the th closed-loop system is stable; the set is the collection of all matrices such that all the closed-loop systems are stable; the set is the collection of all matrices
such that all the closed-loop poles of the th system lie in the region ; and the set is the collection of all matrices
such that all the closed-loop poles of the systems are located in the desired regions.
It is shown in [17] that the objective function is equiv-alent to
Tr if
otherwise
where and is the unique
solution of
Therefore, the cost function becomes
Tr if
otherwise.
(5)
Suppose that , , are positive definite. Two useful lemmas are introduced in the following.
Lemma 1 [12]: All the eigenvalues of the matrix lie in the region if, and only if, for any given matrix
, the equation
has a unique solution .
Lemma 2 [8]: All the eigenvalues of the matrix lie in the region if, and only if, for any given matrix
, the equation
has a unique solution .
III. AUXILIARYMINIMIZATIONPROBLEM
The considered problem is a constrained optimization problem. To solve this problem analytically is difficult since its minimal solution may not exist. In fact, its infimal solution may lie on the boundary of the set ; and furthermore, it may not be a stationary point. In this paper, motivated by the barrier method (Luenberger [22]), we instead solve an auxiliary
minimization problem to obtain an approximate solution of
the original problem. The auxiliary cost function is the sum of the actual cost function and an additional weighted barrier function . The auxiliary minimization
problem is formulated as: Find , over , to minimize the
auxiliary cost function
where the term is defined in (3), is the weighting factor
Tr Tr if
otherwise (6) and matrices and are the solutions of
(7) and
(8)
respectively, with and .
Let denote the Kronecker product, vec denote the op-erator of stacking the column vectors of a matrix to a 1 nm column vector, and vec be the inverse operator of vec (see [9]). As shown in [22], a barrier function must satisfy: 1) it is continuous, 2) it is non-negative over the set , and 3) it will approach infinity as approaches the boundary of the set . Now we will show that the function satisfies these three conditions.
Lemma 3: The function defined in (6) satisfies the following.
1) is continuous in the set .
2) over the set .
3) approaches infinity as approaches the boundary of the set .
Proof:
1) We first show that Tr is continuous in the set for fixed and . Using vec operator in (7) yields
vec vec
where
If , then is nonsingular and
Tr Tr vec vec
(9) The right-hand side of (9) is smooth in .
Note that the solution of discrete Lyapunov equation (8) can be expressed as (10), shown at the bottom of the page, which is a rational function of the matrix . So, Tr is smooth in the set (see [19]). From the definitions of and , it follows that
is continuous in the set .
2) As stated in Lemmas 1 and 2, and are positive definite in the set . Therefore, in the set .
3) Let be an infinite sequence of gain ap-proaching the boundary of from the interior. Then there exists an eigenvalue
approaching the boundary of as .
Sup-pose and are the solutions of (7)
and (8), respectively, with being replaced by . We first show that if the sequence is
such that Re , then
Tr . Suppose is the normalized eigenvector corresponding to . Premultiplying and postmultiplying (7) by and , respectively, and after some manipulations, we have
Since
we have as
. Note that since is positive
definite, as
. Similarly, we can prove if the se-quence is such that
, then Tr . These show that
Tr Tr if the
sequence approaches the boundary of . From the definitions of and , we can conclude that
if approaches the boundary of .
As stated in [22], although the auxiliary minimization problem is, from a formal viewpoint, a minimization problem with inequality constraints; for a computational viewpoint it is unconstrained. The advantage of the auxiliary minimization problem is that it can be solved by unconstrained search tech-niques.
Remark 1: It is shown in [22] that the optimal solution of
the auxiliary minimization problem converges to the solution of the original problem as the weighting factor . This suggests a way to approximate the infimal solution of the orig-inal problem in our approach. It should be noted that even for the single system case, the optimal solutions obtained by the approaches presented in [9], [27], and [29] might be far away from the infimal solutions of the original constrained optimiza-tion problems.
Next, we will prove that if the set is nonempty, then the auxiliary cost function has a minimum point in the set
.
Lemma 4: If the admissible set is nonempty, then the aux-iliary cost function has a minimum point in the interior of the set .
Proof: From (4), we have
Tr Tr
For matrices and , Tr Tr (see
[29]). Therefore
Tr Tr (11)
Since has full rank, , and , then the right hand side of (11) is as . This means Tr
as . Moreover, since is by assumption positive
definite, then Tr as . This implies
that as . As a result, the level set
is bounded for
any . Moreover, since is continuous in the set and as approaches the boundary of the set from the interior, the set is closed and then is compact. From the Weiestrass theorem (see [19]), there exists a
such that
This implies that
and completes the proof.
Since the minimum point of the auxiliary cost function lies in the interior of the admissible solution set, it must be a stationary point. The Lagrange multiplier method can be employed to derive the necessary conditions for local optimum of cost function .
Theorem 1: Let minimize . Then there exist
, , , , , and ( , , and ) satisfying vec vec vec vec (12) (13) vec vec vec vec (14) (15) vec vec vec vec (16) and (17)
where we have the first equation at the bottom of the next page, and
vec vec
such that the optimal feedback gain is given by
vec vec (18)
Proof: The Lagragian is defined as the second equation at the bottom of the next page. The necessary
condi-tions for local optimum are , ,
, and . After some ma-nipulations, we have (19)–(25), shown at the bottom of the page. From (19), we can derive (18). By substituting (18) into (19)–(25), (12)–(17) can be obtained.
The above theorem provides not only a necessary conditions for optimum but a method to calculate the gradient direction of at a given point as well. The gradient of at a fixed point is shown in the equation at the bottom of the next
page, where , , , , , and ( ,
, and ) are the solution of
(20)–(25). In the solution algorithms, this gradient direction is used as the searching direction.
Note that if , then the solution of (4) is positive definite. Based on the Theorem 1, a necessary and sufficient con-dition for the existence of admissible solutions to the simulta-neous static output feedback regional pole placement problem is given in the following.
Corollary 1: The set is nonempty if, and only if, for any given positive definite Hermitian matrices , , ,
, , and ( , , and Tr Tr Tr Tr Tr Tr (19) (20) (21) (22) (23) (24) (25)
), there exist positive definite Hermitian
solutions , , , , , and ( ,
, and ) to the cross-coupled (12) to (17).
Proof: The “Sufficiency” part is obvious and, therefore, is
omitted.
Necessity: Suppose the set is nonempty. From Lemma 4, has a minimum in . It has been shown in the Theorem 1 that there must exist some positive definite matrices , ,
, , , and ( , , and
) satisfying the cross-coupled (12) to (17). This completes the proof.
When the poles’ constraints are relaxed, the considered problem is reduced to the optimal simultaneous static output
feedback stabilization problem. It is obvious that is finite if , and will approach infinity if approaches the boundary of . Following the same procedure provided above, we can show that the cost function is continuous in the
set and the level set for
any is compact. Therefore, has a minimum in . Hence, the following results can be obtained.
Corollary 2: Let minimize . Then there exist
and , , satisfy vec vec vec vec vec vec vec vec (26) and vec vec vec vec (27) where
such that the optimal feedback gain is given by
vec vec (28)
Proof: The proof is similar to that of the Theorem 1 and,
therefore, is omitted.
Consequently, a necessary and sufficient condition for the ex-istence of admissible solutions to the simultaneous output
feed-back stabilization problem can be obtained.
Corollary 3: The set is nonempty if, and only if, for any given positive definite symmetry matrices and ,
, there exist positive definite symmetry solutions and , , to the cross-coupled (26) and (27). In this case, the output feedback gain given in (28) will simul-taneously stabilize the collection of systems (1).
Proof: The “Sufficiency” part is obvious and, therefore, is
omitted.
Necessity: Suppose the set is nonempty. We have shown that has a minimum in . It has been shown in Corollary 2 that there exists positive definite matrices and , , satisfying the cross-coupled (26) and (27). In this case, it is obvious that the feedback matrix given in (28) is a solution to the simultaneous output feedback stabilization problem.
Remark 2: For state feedback case, we only need to let
for all in the above results. IV. ILLUSTRATIVEEXAMPLES
Example 1: Consider the following collection of systems
and
where
and
Suppose and
The design goal is to find a static output feedback gain such that the controllers
and achieve the infimum of the cost function
subject to the constraints that and , where
and the constrained regions and are represented by
Re Re
Re Re
Let
Suppose the weighting factors and . As shown in Section II, we have
Tr Tr Tr Tr
where and are the positive definite solutions of
(29) (30) Let the infimal solution of this problem be denoted by
. Choose and
. From the discussions in Section III, we solve the following auxiliary minimization problem: Find
, over , to minimize the auxiliary cost function
Tr
where is a weighting factor to be chosen, and matrices , , , and are the positive definite solutions of
(31)
(32) (33) (34)
Suppose matrices , , , , , and are the solu-tions of (35) (36) (37) (38) (39) (40) From the Theorem 1, we know that the gradient of at a fixed point is
Based on the gradient method, an algorithm is presented in the following to solve the auxiliary minimization problem.
Main-Algorithm: Find the optimal solution of the auxil-iary minimization problem.
1) Choose a . Set .
2) Solving (29)–(40), where is
substi-tuted by , yields , , ,
, , , , , ,
, , and .
3) Let .
4) If , where is a small
positive number, then , end;
else find , via line search, such
that shall minimize
. Let , go to step 2).
In fact, the step 1) of the Main-Algorithm is not an easy task. In the following, we will provide a Pre-Algorithm to find a
. Let
for non-negative , , , and . It is clear that
and . Define
and
Note that and .
Let , , , , ,
, , and are the solutions of
(41) (42) (43) (44) (45) (46) (47) (48) Let Tr
From the discussions in Section III, we know that if approaches the
boundary of from the interior. The
gradient of at a fixed point is
Now we are ready to provide the Pre-Algorithm for finding a .
Pre-Algorithm: Find a .
1) Choose arbitrary . Find sufficient
large , , , and
such that
. Set .
2) Solving (41)–(48), where is
substi-tuted by , yields , ,
, , , ,
, and .
3) Let
.
4) Find , via line search, such that
shall minimize
5) Let . Suppose , ,
are the eigenvalues of matrix
and , , are the eigenvalues
of matrix . Choose . If Re Im let , else . If Re Im let , else . If Re , let , else . If Re , let , else . Repeat 2)-5) until , , , and . Then, .
From the Pre-Algorithm we know that
if , if
, if , and
if . The values of , ,
, and are monotonically decreasing if they are nonzero. Thus we can expect that if the admissible solution set
is nonempty and the set
is connected in the iteration, there is some finite such that
, , , , and
.
For the considered problem, we choose the weighting factor . The Pre-Algorithm is started with an
initial guess . After some iteration, a
ma-trix is obtained. Then, let
and start the Main-Algorithm
with stop condition . After
some iteration, we get the following results (the solutions of (29)–(40)):
Since all the above matrices are positive definite, this verifies the results of the Corollary 1 that the admissible solution set is nonempty. The resultant optimal feedback gain for the auxiliary minimization problem is
We have
and
as desired. The resultant optimal value of cost function is
From the discussions in Remark 1, we can expect that it is very close to the (local) infimal value since is very small.
Note that the Pre-Algorithm is not sensitive with respect to the initial guess . Even for the extreme case
, which is far away from the admissible
solution set , a matrix is
ob-tained after some iteration. Note also that since the considered constrained optimization problem is not a convex optimization problem, it may have several local infimal (minimal) solutions. Thus the result obtained via the Main-Algorithm may be a local optimal solution of the auxiliary minimization problem. However, for this example, we have started the algorithms with several different initial guesses; they all finally converge to the
solution .
For comparison, we consider the same optimization problem with the following new constrained regions:
Re Re
We find that for several different initial guesses , the Sub-Algorithm never converges. Therefore, we expect that the set
and
is empty and the considered problem is unsolvable.
Example 2: For comparison, we now consider a static state
feedback simultaneous regional pole placement problem. Con-sider the systems described in Howitt and Luus [11], Paskota et
al. [23], and Petersen [24]
and where
The design goal is to find a static state feedback gain such that the controllers
and achieve the infimum of the cost function
subject to the constraints that
and
where the constrained regions , , 2, 3, and 4, are repre-sented by Re Re Re and Re Re Re Re and Let and As shown in Section II Tr Tr
where , , , and are the positive definite solutions of (49) (50) (51) (52) Let the infimal solution of this problem be denoted by
.
Note that for this problem, ,
, , and
. Moreover, choose
and
From the discussions in Section III, we solve the following aux-iliary minimization problem: Find , over , to mini-mize the auxiliary cost function
Tr Tr
where is a weighting factor to be chosen, and matrices , , , , , , , , and are the positive def-inite solutions of (53) (54) (55) (56) (57) (58) (59) (60) (61) Suppose matrices , , , , , , , , , ,
, , and are the positive definite solutions of
(62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74)
From the Theorem 1, we know that the gradient of at a fixed point is
The solution algorithms are similar to those presented in the Example 1 and, thus, are omitted here to save space.
The following four cases are considered.
Case 1: The poles’ constraints are relaxed ( ). This is the optimal simultaneous stabilization problem considered in [23].
We start the algorithms with the initial guess
. For saving space, we only give the final positive definite solutions , , , and :
The resultant optimal feedback gain for the auxiliary minimiza-tion problem is
We have
Moreover, .
Case 2: The weighting factor . We start the algorithms with the initial guess
. For saving space, we only give the final positive definite solutions , , , and :
The resultant optimal feedback gain for the auxiliary minimiza-tion problem is
The final solutions of , , , , , , , ,
, , , , , , , , , , , , ,
and can be easily obtained by solving (53) – (74) with and thus are omitted here for the consideration of space.
We can see that
All the closed-loop poles of the four systems are located in the desired regions. Moreover, we can expect that
will be very close to the (local) infimal value
since is very small.
Case 3: The weighting factor .
For saving space, we only give the final positive definite so-lutions , , , and :
The resultant optimal feedback gain for the auxiliary minimiza-tion problem is
We have
as desired. Moreover, .
Case 4: The weighting factor .
For saving space, we only give the final positive definite so-lutions , , , and :
The optimal feedback gain for the auxiliary minimization problem is
We have
as desired. Moreover, .
Note that our results in case 1 are almost the same as the results presented in [23]. The resultant cost
is the minimal cost for optimal simultaneous stabilization problem (without regional pole constraints). However, since the constraints on closed-loop poles are not considered, the
resultant for and 4.
The other cases show that the closed-loop poles of each system are assigned to the prespecified region as desired since the constraints on closed-loop poles are considered. The minimal value of the auxiliary minimization problem will be closer to its infimal value of the original constraint optimization problem if the weighting factor becomes smaller. No matter how small the weighting factor is, the resultant closed-loop poles of each system will still lie inside the desired regions. Note that the weighting factor in case 2 is very small, we can expect that the infimal solution (may be a local one) of the original problem is very close to
.
V. CONCLUSIONS
In this paper, a new method for approximate solving the optimal output feedback simultaneous regional pole placement problem is provided. The constraint region for each system can be the intersection of several open half-planes and/or open disks. Good transient responses of the closed-loop systems can be guaranteed since the closed loop poles are restricted to lie in some desired regions, and good static state responses of the systems are also guaranteed since a quadratic type cost function is minimized. This problem cannot be solved via LMI approach since its admissible solution set may be nonconvex. Based on the barrier method, we instead solve an auxiliary minimization problem to obtain an approximate solution to the original constrained optimization problem. We have shown that the minimum point of the auxiliary cost function does exist if the admissible solution set is nonempty. Moreover, the necessary conditions for which the optimal solution of the auxiliary minimization problem must be satisfied have been derived. Based on gradient method, numerical algorithms have been provided to find the optimal solution.
ACKNOWLEDGMENT
The authors would like to thank the Associate Editor and the anonymous reviewers for their many helpful suggestions.
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Jenq-Lang Wu was born in Yunlin, Taiwan, R.O.C.,
in 1968. He received the B.S. and Ph.D. degrees in electrical engineering from the National Taiwan In-stitute of Technology, Taipei, Taiwan, in 1991 and 1996, respectively.
Since 1998, he has been with the Department of Electronic Engineering, Hwa Hsia Institute of Tech-nology, Taipei, where he is currently an Associate Professor. His current research interests include non-linear control,H control, switched systems, and networked control 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 at 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), Taipei. In 1998, he became the Professor and Dean of the Office of Research and Devel-opment, NTUST. In 2000, he was with the Department of Electrical and Control Engineering, NCTU, where he served as a Chair Professor. Since 2004, he has been with National Taipei University of Technology (NTUT), where he is now the President. He has published more than 180 refereed journal and conference papers in the areas of automatic control, robotics, fuzzy systems, and neural net-works. His current research involves motion planning, fuzzy and neural control, optimal control theory and application, and walking machines.
Prof. 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, the TECO Sciences and Technology Award from TECO Foundation in 2003, the Academic Achievement Award in Engineering and Applied Sci-ence from the Ministry of Education, R.O.C., in 1998, and the National Endow Chair from Ministry of Education, RO.C., in 2003. He was elected IEE Fellow in 2000. He became a Fellow of New York Academy of Sciences (NYAS) in 2002. His professional activities include serving on the Advisory Board of Di-vision of Engineering and Applied Science, National Science Council, serving as the Program Director, Automatic Control Research Program, National Sci-ence Council, and serving as an Advisor of Ministry of Education, Taiwan, and numerous consulting positions. He has been actively involved in many IEEE ac-tivities, including as Member of Technical Program Committee and Member of Advisory Committee for many IEEE sponsored international conferences. He is now the Vice President for Membership, a member of the Board of Governors, and the Newsletter Editor of the IEEE Systems, Man, and Cybernetics Society.