Simultaneous Stabilization for a Collection of Single-Input Nonlinear ...

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005

Simultaneous Stabilization for a Collection of Single-Input Nonlinear Systems Jenq-Lang Wu

Abstract—This paper presents a novel method for designing a controller that simultaneously stabilizes a collection of single-input nonlinear systems. The control Lyapunov function approach is used to derive necessary and sufficient conditions for the existence of time-invariant simultaneously stabilizing state feedback controllers. Additionally, a universal formula for constructing a continuous simultaneously stabilizing controller when the provided sufficient condition is satisfied is presented. For any collection of second-order (and third-order) feedback linearizable systems in canonical form, global simultaneous stabilization via a single state feedback controller is shown to be always possible. Two examples are included for illustration. Index Terms—Control Lyapunov function, nonlinear systems, simultaneous stabilization, state feedback.

I. INTRODUCTION

T

HE simultaneous stabilization problem is important in the field of robust control. This problem is concerned with determining a single controller which simultaneously stabilizes a finite collection of systems. The simultaneous stabilization problem arises frequently in practice, due to plant uncertainty, plant variation, failure modes or plants with various modes of operation. References [27] and [33] first indicated that this problem reduces to a strong stabilization problem in the case of two plants. Based on the frequency domain approach, Blondel proved that the simultaneous stabilizability of more than two linear systems is rationally undecidable [4]. In [23], a nonlinear state feedback controller, which quadratically stabilizes a set of single-input linear systems simultaneously, was developed. Reference [28] obtained a sufficient condition for the existence of a stabilizing linear state feedback controller for a collection of linear single-input systems. In [15], time-varying feedback controllers based on generalized-holds (GH) were used to provide simultaneous stability. In [5], [6], [10], [11], and [22], some necessary and/or sufficient conditions for the existence of simultaneously stabilizing state feedback and/or output feedback controllers for a collection of linear systems are derived via optimization. Recently, Miller et al. (see [20] and [21]) employed linear periodically time-varying controllers for the simultaneous stabilization and disturbance rejection for a set of linear systems. All of the previous results are derived for Manuscript received November 19, 2003; revised June 8, 2004 and November 4, 2004. Recommended by Associate Editor E. Jonckheere. This work was supported by the National Science Council of the Republic of China under Contract NSC 92-2213-E-146-003. The author is with the Electronic Engineering Department, Hwa Hsia Institute of Technology, Chung-Ho 235 Taipei, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.843877

linear systems. For nonlinear systems, the simultaneous stabilization problem is more difficult to solve. Only Ho-Mock-Qai and Dayawansa [9] have presented some relevant results for nonlinear systems. Reference [9] proved that for any countable family of stabilizable nonlinear systems, a continuous state feedback law, which simultaneously stabilizes the family (nonasymptotically), always exists. Additionally, a sufficient condition for the existence of simultaneously asymptotically stabilizing controllers of a collection of nonlinear systems was provided. However, the sufficient condition in [9] is difficult to verify as it depends firstly on finding an asymptotically stabilizing state feedback controller for each system, then deriving a corresponding Lyapunov function for each closed-loop system, and finally determining whether some infinite sequences of time-varying functions exist that meet some specified conditions. Moreover, the obtained simultaneously asymptotically stabilizing controller is not easy to implement since it is constructed as a sum of infinite time-varying functions. This paper, motivated by the control Lyapunov function (CLF) [2], [3], [7], [8], [13], [14], [16]–[19], [24]–[26], and [29]–[32], develops a new method for solving the simultaneous stabilization problem for a collection of single-input nonlinear systems. The basic idea that underlies the use of a CLF for constructing stabilizing controllers for a single nonlinear system , where is the is first to determine a Lyapunov function system state, and then to derive a feedback control law that rennegative definite. As stated in [30], with an arbitrary ders this attempt may fail, but if is a CLF, a choice of stabilizing control law can always be derived. Several methods have been introduced for constructing universal stabilizing controllers for a single nonlinear plant, as in [30] and [17]. The main contribution of this paper is to provide a necessary condition, as well as a sufficient condition, for the existence of time-invariant, simultaneously asymptotically stabilizing state feedback controllers for a set of single-input nonlinear systems. Additionally, a universal formula for constructing a simultaneously asymptotically stabilizing controller when the given sufficient condition is met is provided. To the best of the author’s knowledge, this paper is the first to provide a necessary condition for the existence of simultaneously asymptotically stabilizing state feedback controllers for a collection of nonlinear systems. The sufficient condition herein is easier to check than that in [9], since checking the former depends only on finding a CLF for each system in the family, and then verifying whether a simple algebraic condition is met. Furthermore, the simultaneously asymptotically stabilizing controller provided in this paper is easier to implement as it is time-invariant, and is a simple continuous function of the

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WU: SIMULTANEOUS STABILIZATION FOR A COLLECTION OF SINGLE-INPUT NONLINEAR SYSTEMS

system state. The perspective and approach of this work differ fundamentally from those in [9]. The method addressed herein is very simple and seems to be more appropriate for practical applications than that provided in [9]. It should be noted that finding a control Lyapunov function for an arbitrary nonlinear system is not always easy or possible. Hence, the condition provided in this paper for the simultaneous stabilizability of a collection of arbitrary nonlinear systems is not always easy or possible to check. Nevertheless, the control Lyapunov functions for nonlinear systems in some particular forms, like the strict feedback form and the feedback linearizable form, can be constructed by systematic methods. The condition presented in this paper can provide a direction for choosing the control Lyapunov functions to ensure simultaneous stabilization. Moreover, we reveal that control Lyapunov functions satisfying the provided condition always exist for any collection of second-order (and third-order) feedback linearizable systems in canonical form. Therefore, for any collection of second-order (and third-order) feedback linearizable systems in canonical form, simultaneous stabilization via a single state feedback controller is always possible. This paper is organized as follows. Section II proposes some mathematical preliminaries and formulates the problem of interest. Section III presents the main results—the necessary and sufficient conditions for the existence of simultaneously asymptotically stabilizing state feedback controllers for a collection of nonlinear systems. Moreover, this section introduces a universal formula for establishing simultaneously stabilizing controllers. Section IV provides some results concerning the simultaneous asymptotic stabilization of second-order (and third-order) feedback linearizable systems in canonical form. Section V presents two illustrative examples. Finally, Section VI draws some conclusions. II. PROBLEM FORMULATION AND PRELIMINARIES

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B. Control Lyapunov Function This subsection addresses the following nonlinear system to demonstrate the basic idea of CLF: (2) is the state and is the control input. where Suppose functions and are smooth and . Definition 1[30]: A smooth, positive–definite, and radially is a CLF for system (2) if, for any unbounded function

or, equivalently

If a continuous state feedback controller exists for (2) such becomes a globally asymptotically stable that the point equilibrium of the closed-loop system, then by standard converse Lyapunov theorems, a CLF exists [29]. of system (2) satisfies the Definition 2[30]: A CLF there is a such small control property if for each satisfies , then there is some with that, if such that

The existence of a CLF with this property is necessary if there is any smooth stabilizer continuous at zero [30]. Let

A. Problem Formulation Consider a collection of nonlinear time-invariant systems (1) is the state; is the control input, and where , and are known smooth for each functions. such The design objective is to find a continuous function that the state feedback controller

can asymptotically stabilize the collection of systems (1) simultaneously; that is, for each the closed-loop system

is globally asymptotically stable.

Reference [30] showed that if a CLF , satisfying the small control property exists for system (2), then if if is a globally asymptotically stabilizing controller. Additionally, defined previously is smooth on and the function . continuous at III. MAIN RESULTS Consider the collection of systems (1). Clearly, for each , if there is a continuous state feedback controller such that the point becomes a globally for system

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asymptotically stable equilibrium, then by the converse Lyapunov theorem, a smooth, positive definite, and radially unmust exist, such that bounded function

are globally asymptotically stable, so by converse Lyapunov theorem, there exist smooth, positive–definite, and radially un, , such that for each bounded functions

for all (3) is a CLF for the th system. However, the existence In fact, of CLFs , , for the collection of systems , , does not guarantee the existence of simultaneously asymptotically stabilizing controllers for these systems. must be globally asymptotiObviously, the pair cally stabilizable for each to ensure the exfor systems , . Now, istence of CLFs is a CLF of system . Let suppose that

for all Therefore, for each for all Then, for all for and for each

(4) and let

This implies that for all

Define

The proof is completed. Let (as motivated by Sontag’s universal formula [30]) and and and and

where

if if As shown in [30], the function is smooth on and if satisfies the small control property. continuous at For convenience, let

denotes an empty set. Clearly

if if

and

if if Additionally, sets , , , and are disjoint, and , , and are also disjoint. Functions , sets , are smooth so the origin belongs to . The following theorem presents a necessary condition for the existence of simultaneously asymptotically stabilizing state feedback controllers for a collection of single-input nonlinear systems. Theorem 1: Consider the collection of systems (1). If there such that for exists a continuous feedback law , the closed-loop system each is globally asymptotically stable, then there exists a collection of smooth, positive–definite, and radially unbounded , , satisfying (3), such that for all functions , the following inequality: (5) holds, where and are as defined in (4). Proof: The closed-loop systems (6)

and undefined

if elsewhere.

Surprisingly, with only a little modification, the necessary condition provided in the aforementioned theorem is also a sufficient condition for the existence of simultaneously asymptotically stabilizing state feedback controllers for the collection of systems (1). The following theorem presents the main results of this paper. Theorem 2: Consider the collection of systems (1). If there exist a collection of smooth, positive–definite, and radially un, , satisfying the small bounded functions control property and (3), such that for all (7) where and are as defined in (4), then there exists such that for each a continuous feedback law , the closed-loop system


is globally asymptotically stable. Moreover, if condition (7) is satisfied, then the feedback law if if if if

331

For each

,

so

That is, if

, then

(8)

is continuous in and it can globally asymptotically stabilize the collection of systems (1) simultaneously, where

if undefined

if if

undefined

if

for all Case 3: For In this case

.

and if elsewhere.

undefined

which is positive in this region. For all can be easily shown:

, the following

Proof: We will prove that if condition (7) holds, then the feedback law defined in (8) is continuous in , and the following inequality: for each for all

For each

,

That is, if

, then

so

(9)

holds and so all of the closed-loop systems are globally asymptotically stable. The asymptotic stability of the closed-loop systems is first proven, and then the continuity of the feedback law defined in (8) is established. A. Asymptotic Stability of the Collection of Closed-Loop Systems Case 1: For Equations (3) and (4) imply that if

, then

for all

for all In this case,

Case 4: For Note that, for all

, so if

.

for all Case 2: For In this case

.

Then, in this region for all

which is negative in this region. Then, for all following can be easily shown:

, the

From the definition of can be easily verified:

, for all

and all , the following

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Therefore, in this region

1) for all

Let to a point

be a sequence of vectors that converges on the boundary of ; that is, for all . Then

and

(10) for all

Notably, also for all

2) 3)

which implies that if for all

is continuous on the Therefore, the function and . boundary between Similarly, the function is continuous on the and . boundary between be a sequence of vectors that converges Let on the boundary of ; that is, to a point for all . Then

The discussions of Cases 1–4 indicate that if condition (7) holds, then (9) will hold so all of the closed-loop systems will be globis continuous. ally asymptotically stable, provided that B. Continuity of Function The continuity of the provided feedback law (8) in must now be proven. Notably, . The defined in (8) is claim is initially made that the function , , and . continuous in the interiors of regions is smooth on and continuous at so Since the only possible points of discontinuity of are those points on the boundary between the set and the set , for each . is continuous in the inteFirst, note that the function rior of . Then, since is smooth in and contin, the function is continuous in the interior uous at , and the function is continuous in the interior of of . Now, the continuity of in the interior of must be proven. Let . Let be a sequence of vectors that converges to a point on ; that is, and (but the boundary of ). Note that other Then

as exists such that

(11)

4)

Hence, the function is continuous on the and . boundary between Let be a sequence of vectors that converges on the boundary of ; that is, to a point for all . Notably, and , so is finite as as Accordingly, as

. Hence

for sufficiently large , so . Since , so some for sufficiently large .

(12)

5)

means that but . This means that the function is continuous on the , of . Similarly, that is boundary, contained in continuous on the boundary, contained in , of can also be proven. Accordingly, the function and, therefore, the function are continuous in the . This discussion indicates that the function interior of is continuous in the interior of the regions , , and . defined in (8) is proven In the following, the function , , , also continuous on the boundaries of the regions and .

and

Therefore, the function is continuous on the boundary between and . be a sequence of vectors that converges Let on the boundary of ; that is, to a point for all . Notably, and , so

where

is finite as

and

as Hence,

as . Therefore

(13)


6)

7)

Accordingly, the function is continuous on the and . boundary between It should be noted that, by definition, the elements on and belong to . the boundary between is continuous both in Since the function and , it must be continuous on the boundary and . between For a point located on the boundary be, , and , it can be concluded from tween is continuous at (10) and (11) that the function . This means that function is continuous on the boundary between , , and . Similarly, it is continuous on the can be shown that function , , and , and continuous boundary between , , and . Notably, on the boundary between by definition, the elements on the boundary between , , and belong to . Since the function is continuous both in and , it must be continuous on the boundary between , , and . Therefore, it is also contin, , , and . uous on the boundary between

These discussions and the definition of demonstrate that is continuous in , completing the proof. Remark 1: In fact, another simultaneously asymptotically stabilizing controller can be found. If in Theorem 2 is redefined as if elsewhere

undefined

then the feedback law (8) can also globally asymptotically sta, simultaneously. bilize all the systems , Remark 2: Theorem 2 provides another important result: If CLFs can be found for the collection of systems (1) such that , then (7) in Theorem 2 does not need to be checked and simultaneous stabilization can be guaranteed. The following section shows that CLFs always can be found for any collection of second-order (and third-order) feedback linearizable systems . Therefore, in canonical form, and the corresponding simultaneous stabilization can be guaranteed for any collection of systems in this form.

333

and , , are smooth where . For all , suppose functions. Let and suppose for all . Additionally, assume all functions , , have the same sign. Without loss of generality, suppose , . The systems in (14) are feedback linearizable, as introduced in [12]. This class of systems is frequently used in the study of nonlinear system theory. The dynamics of many physical systems can be represented in this form. If and is a linear function, then this form reduces to the well known second-order controllable companion form (controller form) for linear systems [1]. That is a control Lyapunov function for the th system in (14), and that it satisfies the small control property, can be easily proven. Note that

Let

and if if Since

,

and

Hence, from Theorem 2, the following result is provided without proof. Corollary 1: Consider the collection of systems (14). A conalways exists such that the tinuous function feedback law globally asymptotically stabilizes the collection of systems (14) simultaneously. Additionally

IV. CASE STUDY

if

This section will show that simultaneous stabilization can be guaranteed for any collection of second-order (and third-order) feedback linearizable systems in canonical form under standard assumptions.

if

A. Second-Order Feedback Linearizable Systems in Canonical Form Consider the following collection of second-order feedback linearizable systems:

(14)

if (15) is one such controller. B. Third-Order Feedback Linearizable Systems in Canonical Form Consider the following collection of third-order feedback linearizable systems:

(16)

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where and , , are . For all , smooth functions. Let and for each suppose . Moreover, assume all functions , , have the same sign. Without loss of generality, . That suppose

is a control Lyapunov functions for the th system of (16), and that it satisfies the small control property, can be easily proven. Note that

Lyapunov function for an arbitrary nonlinear system is not always easy or possible. Hence, the simultaneous stabilizability for a collection of arbitrary nonlinear systems is not always easy or possible to check. However, the control Lyapunov functions for nonlinear systems in some particular forms, such as the strict feedback form and the feedback linearizable form, can be constructed by systematic methods. Then, the condition presented in Theorem 2 provides a direction for choosing the control Lyapunov functions to ensure simultaneous stabilization. Moreover, Section IV reveals that control Lyapunov functions, which satisfy the condition provided in Theorem 2, always exist for any collection of second-order (and third-order) feedback linearizable systems in canonical form, so simultaneous stabilization can be guaranteed. In the linear case, any collection of second-order or third-order linear systems in controllable companion form can always be simultaneously stabilized.

Let V. EXAMPLES Example 1: Consider the following three nonlinear systems:

and if if

(17)

Since . That , and are control and , respectively, Lyapunov functions for systems , and all satisfy the small control property, can be easily shown. Notably Let

and

Accordingly, from Theorem 2, the following result is offered without proof. Corollary 2: Consider the collection of systems (16). A conalways exists such that the tinuous function globally asymptotically stabilizes the feedback law collection of systems (16) simultaneously. Furthermore

if

Let

, , , , , and . That and , , and , and can be contains only the origin. For convenience, derived. In fact, define

if if and is one such controller. Remark 3: Based on the frequency domain approach, Blondel proved that the simultaneous stabilizability of more than two linear systems is rationally undecidable [4]. The sufficient condition for the simultaneous stabilizability of nonlinear systems provided in Theorem 2 is derived based on the control Lyapunov functions of these systems. In fact, finding a control

and

Clearly, and

. Moreover, if

,

and


Fig. 1.

if

if

State trajectories (initial state x(0) = [2

, and . We can show that if

0 2] and

335

) and control inputs of the three systems in (17) with the same controller (18).

if , then

and

if if if if

undefined

, then

if if if

undefined and if

and

if elsewhere

undefined

, then with

That is, for all

if

, the inequality

if

and

if holds. Therefore, from Theorem 2, continuous state feedback controllers exist that asymptotically stabilize systems , and simultaneously. Let if if if if if if

undefined

elsewhere.

Then, from Theorem 2 if if if

and

(18) can globally asymptotically stabilize the systems , , and simultaneously. Fig. 1 shows the state trajectories and control inputs of these three systems with the same controller (18). The results indicate that the provided controller (18) indeed asympsimultaneously. totically stabilizes the systems , , and

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Fig. 2.


State trajectories (initial state x(0)[1 0

0 1]

) and control inputs of the four systems in (19) with the same controller (20).

Example 2: Consider the following collection of third-order feedback linearizable nonlinear systems:

and (19) where

and

Suppose

. For

, let

can globally asymptotically stabilize all the systems in (19) simultaneously. Fig. 2 depicts the state trajectories and control inputs of these four systems with the same controller (20). VI. CONCLUSION This paper presented a novel method for finding a continuous state feedback controller to globally asymptotically stabilize a collection of single-input nonlinear systems simultaneously. The control Lyapunov function approach was used to derive necessary and sufficient conditions for the existence of simultaneously asymptotically stabilizing controllers. Additionally, a universal formula for constructing a simultaneously asymptotically stabilizing state feedback controller was proposed. Any collection of second-order (and third-order) feedback linearizable systems in canonical form was shown always to be simultaneously asymptotically stabilized under standard assumptions. Another topic for further study involves the extension of these results to multiple-input nonlinear systems. The use of output feedback rather than the state feedback considered in this paper may also be considered. The provided method can also be extended to the time-varying case. Some other classes of nonlinear systems in special forms could be found, such that simultaneous stabilization can be guaranteed. ACKNOWLEDGMENT

and

The author would like to thank the Associate Editor and the anonymous reviewers for their many helpful suggestions.

if if

REFERENCES

From Corollary 2, the following controller: if if if

(20)

[1] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGrawHill, 1998. [2] Z. Artstein, “Stabilization with relaxed control,” Nonlinear Anal., Theory, Meth., Appl., vol. 7, pp. 1163–1173, 1983. [3] S. Battilotti, “Robust stabilization of nonlinear systems with pointwise norm-bounded uncertainties: a control Lyapunov function approach,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 3–17, Jan. 1999.


[4] V. Blondel, Simultaneous Stabilization of Linear Systems. New York: Springer-Verlag, 1994. [5] Y. Y. Cao and J. Lam, “A computational methods for simultaneous LQ optimal control design via piecewise constant output feedback,” IEEE Trans. Syst., Man, Cybern. B: Cybern., vol. 31, no. 5, pp. 836–842, Oct. 2001. [6] Y. Y. Cao, Y. X. Sun, and J. Lam, “Simultaneous stabilization via static output feedback and state feedback,” IEEE Trans. Autom. Control, vol. 44, no. 6, pp. 1277–1282, Jun. 1999. [7] D. V. Efimov, “A condition of CLF existence for affine systems,” in Proc. 41st IEEE Conf. Decision and Control, 2002, pp. 1882–1887. [8] K. Ezal, Z. Pan, and P. V. Kokotovic, “Locally optimal and robust backstepping design,” IEEE Trans. Autom. Control, vol. 45, no. 2, pp. 260–271, Feb. 2000. [9] B. Ho-Mock-Qai and W. P. Dayawansa, “Simultaneous stabilization of linear and nonlinear systems by means of nonlinear state feedback,” SIAM J. Control Optim., vol. 37, pp. 1701–1725, 1999. [10] G. D. Howitt and R. Luus, “Simultaneous stabilization of linear singleinput systems by linear state feedback control,” Int. J. Control, vol. 54, pp. 1015–1039, 1991. , “Control of a collection of linear systems by linear state feedback [11] control,” Int. J. Control, vol. 58, pp. 79–96, 1993. [12] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: SpringerVerlag, 1995. [13] M. Jankovic, “Control Lyapunov–Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1048–1060, Jul. 2002. [14] M. Jankovic, P. Sepulchre, and P. V. Kokotovic, “CLF based designs with robustness to dynamic input uncertainties,” Syst. Control Lett., vol. 37, pp. 45–54, 1999. [15] P. T. Kabamba and C. Yang, “Simultaneous controller design for linear time-invariant systems,” IEEE Trans. Autom. Control, vol. 36, no. 1, pp. 106–110, Jan. 1991. [16] P. P. Kokotovic and M. Arcak, “Constructive nonlinear control: a historical perspective,” Automatica, vol. 37, pp. 637–662, 2001. [17] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [18] D. Liberzon, E. D. Sontag, and Y. Wang, “Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation,” Syst. Control Lett., vol. 46, pp. 111–127, 2002. [19] Y. Lin and E. D. Sontag, “Control-Lyapunov universal formulas for restricted inputs,” Control-Theory Adv. Technol., vol. 10, pp. 1981–2004, 1995. [20] D. E. Miller and T. Chen, “Simultaneous stabilization with near-optimal performance,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 1986–1998, Dec. 2002. [21] D. E. Miller and M. Rossi, “Simultaneous stabilization with near-optimal LQR performance,” IEEE Trans. Autom. Control, vol. 46, no. 10, pp. 1543–1555, Oct, 2001.

H

337

[22] M. Paskota, V. Sreeram, K. L. Teo, and A. I. Mees, “Optimal simultaneous stabilization of linear single-input systems via linear state feedback control,” Int. J. Control, vol. 60, pp. 483–493, 1994. [23] I. R. Petersen, “A procedure for simultaneous stabilizing a collection of single input linear systems using nonlinear state feedback control,” Automatica, vol. 23, pp. 33–40, 1987. [24] J. A. Primbs, V. Nevistic, and J. C. Doyle, “A receding horizon generalization of pointwise min-norm controllers,” IEEE Trans. Autom. Control, vol. 45, no. 5, pp. 898–909, May 2000. [25] L. Rifford, “Semiconcave control-Lyapunov functions and stabilizing feedbacks,” SIAM J. Control Optim., vol. 41, pp. 659–681, 2002. [26] M. A. Rotea and P. P. Khargonekar, “Stabilization of uncertain systems with norm bounded uncertainty—A control Lyapunov function approach,” SIAM J. Control Optim., vol. 27, pp. 1462–1476, 1989. [27] R. Saeks and J. Murray, “Fractional representation, algebraic geometry, and the simultaneous stabilization problem,” IEEE Trans. Autom. Control, vol. AC-27, no. 4, pp. 895–903, Aug. 1982. [28] W. E. Schmitendorf and C. C. Hollot, “Simultaneous stabilization via linear state feedback control,” IEEE Trans. Autom. Control, vol. 34, no. 9, pp. 1001–1005, Sep. 1989. [29] E. D. Sontag, “A Lyapunov-like characterization of asymptotic controllability,” SIAM J. Control Optim., vol. 21, pp. 462–471, 1983. , “A “universal” constructive of Artstein’s theorem on nonlinear [30] stabilization,” Syst. Control Lett., vol. 12, pp. 542–550, 1989. [31] J. Tsinias, “Asymptotic feedback stabilization: a sufficient condition for the existence of control Lyapunov functions,” Syst. Control Lett., vol. 15, pp. 441–448, 1990. [32] , “Existence of control Lyapunov functions and applications to state feedback stabilizability of nonlinear systems,” SIAM J. Control Optim., vol. 29, pp. 457–473, 1991. [33] M. Vidyasagar and N. Viswanadham, “Algebraic design techniques for reliable stabilization,” IEEE Trans. Autom. Control, vol. AC-27, no. 5, pp. 1085–1095, Oct. 1982.

Jenq-Lang Wu was born in Yunlin, Taiwan, in 1968. He received the B.S. and Ph.D. degrees in electrical engineering from the National Taiwan Institute of Technology, Taipei, Taiwan, in 1991 and 1996, respectively. Since 1998, he has been with the Department of Electronic Engineering, Hwa Hsia Institute of Technology, Taipei, Taiwan, where he is currently an Associate Professor. His current research interests include nonlinear control, control, switched systems, and networked control systems.

H