Systems & Control Letters 34 (1998) 77–83

Polytopic control Lyapunov functions for robust stabilizationof a class of nonlinear systems 1

Marc W. McConley a, Munther A. Dahleh b;∗, Eric Feron c

aCharles Stark Draper Laboratory, 555 Technology Square MS 77, Cambridge, MA 02139, USAbDepartment of Electrical Engineering, Massachusetts Institute of Technology, Room 35-402, 77 Massachusetts Avenue,

Cambridge, MA 02139, USAcDepartment of Aeronautics and Astronautics, Massachusetts Institute of Technology, Room 33-217, 77 Massachusetts Avenue,

Cambridge, MA 02139, USA

Received 18 October 1996; received in revised form 31 October 1997

Abstract

We develop a method for computing a region in state space over which a nonlinear system is guaranteed by a givenpolytopic control Lyapunov function to be stable in closed loop under some appropriate control law. For systems which arenonlinear in only a few state variables, the procedure is computationally tractable; the computation time required to evaluatestability over each cone comprising a level set of the Lyapunov function is exponential in the number of “nonlinear states”but otherwise polynomial in the dimension of the full state space. Control constraints and robustness to bounded disturbancesare easily incorporated. c© 1998 Elsevier Science B.V. All rights reserved.

Keywords: Lyapunov methods; Nonlinear control systems; Robustness; Bounded control; Linear programming

1. Introduction

Many dynamical systems can be represented byordinary di�erential equations in the physical statesof the system in uenced by other parameters, such asdisturbance and control inputs. The focus of statefeedback control theory is to design a control law (afunction which maps measured states of the systemto control inputs) which produces a desired perfor-mance for the system. Very di�erent theories apply tothis problem depending on whether the state deriva-tives are linear or nonlinear functions of the states in

∗ Corresponding author. Tel.: +1 617 253 3892; e-mail:dahleh@lids.mit.edu.

1 This work was supported in part by Charles Stark DraperLaboratory Internal Research and Development, in part by the AirForce O�ce of Scienti�c Research under Grant AFOSR F49620-95-0219, and in part by the National Science Foundation underGrants 9157306-ECS and 9409715-ECS.

the di�erential equations de�ning the system. Manysimple, straightforward techniques for robust optimalcontrol of linear systems have been developed. Ex-tensions of these methods to nonlinear systems aresometimes possible, but the analogous procedureswhich result from this exercise cannot typically beexecuted in a computationally tractable way. As aresult, control of nonlinear systems has been a topicof intense research for some time.Progress on the nonlinear control problem is dif-

�cult because of the inherent complexity of methodswhich are general enough to apply to arbitrary nonlin-ear systems. One method which has recently come intofavor is to construct a stabilizing control law basedon a known control Lyapunov function (CLF) forthe system [1, 10, 20, 21, 23]. A function is a CLF ifa control law exists to render it a Lyapunov functionfor the closed loop system. The computation of a sta-bilizing control law is straightforward from any of a

0167-6911/98/$19.00 c© 1998 Elsevier Science B.V. All rights reservedPII S0167-6911(97)00135 -7

78 M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83

number of universal formulas [10,14,21] based on theCLF and the system dynamics, so the control synthe-sis problem is reduced to constructing a CLF for thesystem and computing the region of state space overwhich a control exists to stabilize the system based onthe given CLF.In a recent work [16] the authors developed a com-

putationally e�cient procedure for control of nonlin-ear systems which is similar to gain scheduling butwhich provides stability guarantees by scheduling overLyapunov functions rather than control gains. For cer-tain broad classes of systems and candidate Lyapunovfunctions, the problem of nonconservatively comput-ing the level set over which the Lyapunov functionguarantees stability is computationally tractable. Themain contribution of the present paper is a methodfor solving a version of the nonlinear control problem(Problem 1 below) based on a given polytopic CLF,which is de�ned precisely in Section 2. The approachpresented here has the following desirable properties:• The computations are tractable even for high di-mensional systems. Thus the method is suitable forpractical application in nonlinear control systemdesign.

• Robustness to bounded exogenous input signals ishandled with only minor modi�cations to the con-cept and without severely increasing the computa-tional complexity.

• Stability analysis under bounded control is handledwith minimal e�ect on computational complexity.

• The method is applicable to a useful class of non-linear systems.Our objective is to design a control law which sta-

bilizes a nonlinear system in the sense de�ned below.

De�nition 1. Consider a system x(t)=f(x(t); w(t))with w∈C 0(R→W⊆Rl). Assume that a solutionx(t) exists for the di�erential equation de�ned usingthe upper Dini derivative of x with respect to t, asfollows.

x(t) :=D+x(t):= lim sup

�→0+

x(t + �)− x(t)�

:

Given a positively invariant set X⊆Rn, and a com-pact subset ⊂X, the system is robustly uniformlyasymptotically stable over X with respect to , orRUAS(X; ), if it is uniformly asymptotically stablewith respect to (see [13]) whenever x(0)∈X. Wecall the set X a region of stability, or RS, for thesystem.

Note that in [22] it was shown that the property ofuniformity is implied by global asymptotic stability.In many applications, the engineer knows that only

a few states a�ect the system dynamics in a nonlinearway. In this paper, we consider the following controlsynthesis problem, where the system dynamics are as-sumed to depend nonlinearly on only the �rst k states.

Problem 1. Consider a continuous time, time invari-ant, nonlinear system in uenced by a control u(t) ina compact, convex subset U⊂Rm and a disturbancew(t) in a polytopeW⊂Rl. The state vector x(t)∈Rnis partitioned into “nonlinear states” xN∈Rk and “lin-ear states” xL∈Rn−k . The system has the form[xNxL

]=f(xN) + A(xN)xL + gw(xN)w + gu(xN)u

=[fN(xN)fL(xN)

]+[AN(xN)AL(xN)

]xL+

[gwN(xN)gwL(xN)

]w

+[guN(xN)guL(xN)

]u; (1)

where all functions of xN are C1. Use a given poly-topic CLF to construct a piecewise continuous staticstate feedback control law � :Rn→Rm and sets⊂X⊆Rn containing the origin such that the closedloop system with u= �(x) is RUAS(X; ).

Since the feedback is allowed to be discontinuous,we must be careful about the meaning of existence ofsolutions to the di�erential equation (1). This point isclari�ed in the discussion of Section 3, in which wedemonstrate that Problem 1 can be solved by deter-mining the region of stability guaranteed by a givenLyapunov function.The signi�cance of the problem formulation as it

is written here lies in the fact that the computationalcomplexity of implementing the solution presented inthis paper on each cone associated with some levelset of the polytopic CLF is exponential only in thenumber of disturbances and “nonlinear states” (k +l). Stability analysis procedures for generic nonlinearsystems typically require computation times which areexponential in the full state and disturbance dimension(n+ l). Note that Problem 1 includes the problem ofanalyzing robust stability for an autonomous systemwithout control, since this is just the case U= {0}.Obviously, we would like X to be as large as pos-

sible and as small as possible. WhenW= {0} andthe system is locally asymptotically stabilizable, localstabilization theory yields a setX such that the system

M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83 79

is RUAS(X; {0}) [25, 10]. To compute the RS gener-ally requires computation times which are exponentialin the state dimension n; for the system (1), however,the computations required to �nd X are tractable. Theapproach of this paper uses this fact to develop a con-trol design which expands the RS for the system (1).Systems of the form (1) are also considered in [2],

where the output feedback stabilization problem (withoutput xN) is solved based on an output control Lya-punov function, assuming that the solution to the statefeedback stabilization problem is already available andthe output CLF can be constructed. This paper presentsa solution to Problem 1 to determine the largest RSthat can be achieved with the available control basedon a given polytopic function which is used as a robustcontrol Lyapunov function (RCLF) for the system.The use of polytopic RCLFs for linear uncertain

systems is considered in [4]. However, such linear un-certain systems di�er from the system (1) in three im-portant ways. In [4], the dynamics of xN are unknown,xN is unmeasured and is not available for feedback,and xN is treated as a time varying parameter ratherthan a state to be controlled. Results on control syn-thesis using polytopic Lyapunov functions for con-strained linear systems appear in [8,24] for continuoustime systems and in [11,12] for discrete time systems.In [3], results on constrained regulation of nonlineardiscrete time systems on polyhedral sets are obtainedusing the comparison principle.We do not discuss how to generate polytopic RCLFs

for the system (1). Such results can be found, for ex-ample, in [5]. Results on the construction of polytopicLyapunov functions for stability analysis appear in[7, 17, 18].

2. Mathematical preliminaries

In this paper we propose a control design methodbased on a robust control Lyapunov function which isvalid on level sets about an equilibrium point of a sys-tem of the form (1). Before proceeding to develop thealgorithm, we de�ne some relevant terms pertainingto a system of the general form given below, whereall functions of x are C1.

x=f(x) + gw(xN)w + gu(xN)u: (2)

De�nition 2. A level set of a proper, positive de�-nite function V :Rn→R is de�ned by real numbersc2¿c1¿0 via V−1[c1; c2]

:= {x∈Rn | c16V (x)6c2}.

De�nition 3 (Sontag [20]). Given a function V :Rn→R and a state trajectory x :R→Rn, we de�ne theLie derivative of V by

V (x(t)) :=D+(V ◦ x)(t)= lim sup

�→0+

V (x(t + �))− V (x(t))�

:

If V is locally Lipschitz and the solution to x=f(x; w)exists, then by [4] we have

V (x(t))=LfV (x):=lim sup

�→0+

V (x + �f(x; w))− V (x)�

:

Moreover, if V (x) is di�erentiable at a point x∈Rn,then LfV (x)= [@V (x)=@x]f(x).

De�nition 4 (Freeman and Kokotovi�c [10] andSontag [20, 21]). Consider a polytope W⊂Rl, acompact, convex subset U⊂Rm, a positive de�-nite function W (x), and real numbers c2¿c1¿0. Alocally Lipschitz, proper, positive de�nite functionV (x) is a robust control Lyapunov function (RCLF)with stability margin W (x) with controls in U overV−1[c1; c2] for the system (2) if there exists a function� :Rn→U such that

supx∈V−1[c1 ; c2]

maxw∈W

LfV (x)

+LgwV (x)w + LguV (x)�(x) +W (x)60; (3)

in other words, the Lyapunov function V (x) decreasesover time at the rate W (x).

Throughout this paper we consider polytopic Lya-punov function candidates, whose level sets are scaledversions of a given polytope in the state space. In par-ticular, we deal with functions satisfying the follow-ing.

De�nition 5. A polytopic function is a proper positivede�nite function of the form

V (x)= max16s6M

‘Ts x; (4)

where the ‘s are distinct.

To each ‘s in Eq. (4), there corresponds a con-vex cone Ns

:= {x∈Rn |V (x)= ‘Ts x} with the prop-erty that int(Ns)∩ int(Nr)= ∅ for all r 6= s. Hence,V (x) is linear in x on each such cone. This is an im-portant property which we shall use throughout thispaper.

80 M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83

3. Main stability analysis procedure

Given a function V (x) of the form (4), our objectiveis to compute real numbers c2¿c1¿0 such that V (x)is an RCLF with stability margin W (x) with controlsin U over V−1[c1; c2]. To do this, we need to checkwhether condition (3) holds over each of the conesNs

comprising a given level set V−1[c1; c2]. We begin bychecking the condition at states having a given valueof xN∈Rk . This problem can be solved e�ciently dueto the special structure of the system (1), and we canuse this result to solve the robust stability analysisproblem. To show this, we de�ne I(M) to be thecollection of all subsets of the index set {1; : : : ; M}.De�ningNI

:=⋂s∈I Ns for each I ∈I(M), we intro-

duce the following de�nitions:

AI (c1; c2; xN):= {xL∈Rn−k | x∈V−1[c1; c2]∩NI};

YI (c1; c2):= {xN∈Rk |AI (c1; c2; xN) 6= ∅};

�s(x; w):= ‘Ts f(x) + ‘

Ts gw(xN)w +W (x);

�I (c1; c2; xN):=maxs∈I

maxxL∈AI (c1 ; c2 ; xN)

maxw∈W

�s(x; w):

The utility of the above de�nitions becomes appar-ent in Section 3.1, where we cast the stability anal-ysis problem as a family of linear programs in the“linear states” xL, parameterized over the “nonlinearstates” xN. Note that AI (c1; c2; xN) is compact for allxN∈Rk since V (x) is proper. Also, YI (c1; c2) is com-pact. The quantities de�ned above can be used to an-alyze robust stability for any system of the form (2)as follows.

Theorem 1. If W (x) is continuous on every coneNs; then V (x) is an RCLF with stability marginW (x) with controls in U over V−1[c1; c2] if andonly if

�I (c1; c2; xN) + minu∈U

maxr∈I

‘Tr gu(xN)u60 for all

xN∈YI (c1; c2) for all I ∈Iadm ; (5)

where

Iadm:= {I ∈I(M) |V−1[c1; c2]∩N1 6= ∅}:

Proof. First note that⋃Ms=1Ns=Rn, so that V (x) is

an RCLF by De�nition 4 if and only if there exists

� :Rn→U such that, for every I ∈I(M),

supx∈V−1[c1 ; c2]∩NI

maxw∈W

LfV (x) + LgwV (x)w

+LguV (x)�(x) +W (x)60:

To prove su�ciency, note that for every I ∈Iadm andevery (x; w; u) such that x∈V−1[c1; c2]∩NI for alls∈ I , there exists r∈ I satisfying

LfV (x) + LgwV (x)w + LguV (x)u+W (x)

= ‘Tr f(x) + ‘Tr gw(xN)w + ‘

Tr gu(xN)u+W (x)

6maxs∈I

�s(x; w) + maxr∈I

‘Tr gu(xN)u:

This also holds trivially for I 6∈Iadm since V−1[c1; c2]∩NI = ∅. Therefore, V (x) is an RCLF if

supx∈V−1[c1 ; c2]∩NI

minu∈U

maxw∈W[

maxs∈I

�s(x; w) + maxr∈I

‘Tr gu(xN)u]

60 for all I ∈Iadm : (6)

However, since �s(x; w) is continuous in (x; w), it isclear that Eq. (6) is equivalent to Eq. (5).To prove necessity, note that if V (x) is an RCLF,

then there exists � :Rn→U such that, for everys∈{1; : : : ; M}, we have

‘Ts [f(x) + gw(xN)w + gu(xN)�(x)] +W (x)

= �s(x; w) + ‘Ts gu(xN)�(x)60

for every x∈V−1[c1; c2]∩Ns and w∈W. Therefore;since �s(x; w) is continuous in x on V−1[c1; c2]∩Ns,we obtain

�s(c1; c2; xN) + maxxL∈As(c1 ; c2 ; xN)

‘Ts gu(xN)�(x)60

for all xN∈Ys(c1; c2), which completes the proof.

Given a polytopic RCLF, the computation ofa piecewise continuous stabilizing control law isstraightforward under the regularity assumptions ofthe system described in Problem 1. To see this, notethat, for each s∈{1; : : : ; M}, a control law whichsatis�es the Lie derivative requirements and whichis continuous over int(Ns) can be constructed basedon the RCLF V (x)= ‘Ts x. (See [10, 14, 21], for ex-ample.) For any state on one of the cone boundaries(x∈NI ), the corresponding control u= �(x) must beconstructed from one of the RCLFs V (x)= ‘Tr x, r∈ I ,

M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83 81

in such a way that the Lie derivative requirements aresatis�ed relative to these RCLFs. In other words, uneeds to satisfy

�I (c1; c2; xN) + maxr∈ I

‘Tr gu(xN)u60: (7)

The control law is synthesized using a universal for-mula. For example, the pointwise min-norm controllaw of [10] can be used to obtain the following whenx∈V−1[c1; c2]∩NI :

�(x)= argmin{uTu | u∈U and (7) holds}:We can select the appropriate RCLF to use in the

universal formula in the following manner. We let�s(x) denote the control input constructed from theuniversal formula applied to V (x)= ‘Ts x, s∈ I , sub-ject to the constraint (7) on the control input. Havingcomputed each of the us, we de�ne

�s(x):=maxr∈I

‘Tr gu(xN)�s(x);

s∗(x) :=argmins∈I

�s(x):

Then s∗(x) gives the index s of the appropriate control�s(x) to use.To show that the control law �(x) is piecewise con-

tinuous, note that each of the �s(x) is continuous onNs (by [10, 14, 21]). Since gu(xN) is also continu-ous, we conclude that �s(x) is continuous on eachNs.This implies that s∗(x) is a piecewise constant func-tion of x on each of theNI . De�ning s∗(x)= r for allx∈ int(Nr), for all r ∈ I , we conclude that�(x) = �s∗(x)(x)

is piecewise continuous on V−1[c1; c2].The reader will notice that, in general, the solution

to the di�erential equation (2) does not formally existon the cone boundaries, where �(x) is discontinuous.However, the state trajectory still moves in a directionof decreasing value of V (x) provided we interpret thestate dynamics using Filippov’s construction of theequivalent dynamics along the cone boundary [9]. Inthis regard, the dynamics at the cone boundary areanalogous to the dynamics along a sliding surface insliding mode control [19]. The state trajectory doesnot necessarily remain on the cone boundary for allfuture time, however.As a consequence of the above, we consider Prob-

lem 1 to be solved if we can determine the level setV−1[c1; c2] over which V (x) is an RCLF.

3.1. Stability analysis over a given level set

In the remainder of this paper we demonstrate thatthe structure of the system (1) renders the problem ofevaluating the condition in Theorem 1 a linear pro-gramming problem parameterized over xN. Therefore,the solution procedure is computationally tractablewhen dim(xN) is �xed. Since V (x)= ‘Ts x overNs, wecan represent the set AI (c1; c2; xN) de�ned above asfollows:

AI (c1; c2; xN) = {xL∈Rn−k | c16‘Ts x6c2; ‘Ts x¿‘Tr x;∀s∈ I; ∀r=1; : : : ; M}:

The polytope W can be represented in the formW= {w∈Rl | qTj w¿rj; j=1; : : : ; K}: De�ningy = [xL;w], we write AI (c1; c2; xN) × W={y∈Rn−k+l |Ey¿b}, where, for I = {s1; : : : ; sP},

E =

Es1 0...

...EsP 00 qT1...

...0 qTK

; b =

bs1...bsPr1...rK

:

Using the partition ‘s= [‘sN; ‘sL], we de�ne

Es =

‘TsL−‘TsL

(‘sL − ‘1L)T...

(‘sL − ‘ML)T

; bs =

c1 − ‘TsNxN−c2 + ‘TsNxN

−(‘sN − ‘1N)TxN...

−(‘sN − ‘MN)TxN

:

Suppose that W (x) is a�ne in xL on every cone Ns,so that W (x)= as(xN)Tx onNs. Then we obtain

�I (c1; c2; xN )= maxs∈I

maxEy¿b

cTs y + ds;

cTs = [‘Ts A(xN) + asL(xN)

T | ‘Ts gw(xN)];ds = ‘Ts f(xN) + asN(xN)

TxN; (8)

for the partition as= [asN; asL]. Note that [minu∈U

maxr∈I ‘Tr gu(xN)u] is a constant at a given xN over agivenNI . We therefore check the condition in Theo-rem 1 over each coneNs by gridding the compact setYI (c1; c2) and solving the linear programming prob-lem (8) at each grid point to determine whether

�I (c1; c2; xN)6−[minu∈U

maxr∈I

‘Tr gu(xN)u]:

82 M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83

3.2. Computational complexity analysis

The procedure presented in this paper provides theadvantage that the computational complexity of thestability analysis solution for systems which are non-linear in only a few variables is commensurate withthe actual degree of nonlinearity in the problem. Thisis in contrast to other methods in nonlinear control(for example, methods based on the solution to aHamilton–Jacobi equation) for which computationalcomplexity is not reduced even when the system dy-namics have the special form (1) with k¡n. For thearbitrary nonlinear system (2), the computation timerequired to evaluate stability is exponential in dim(x)because it is necessary to evaluate the expression inEq. (3) at grid points over the entire level set. Also, thecomplexity of this procedure is typically exponentialin dim(w) because of the maximization over w∈Win Eq. (3). For the system (1), the method proposedin this paper only requires gridding over dim(xN)dimensions.The linear program (8) has dim(xL)+dim(w) vari-

ables and up to P(M + 2) + K active constraints;therefore, the complexity of the stability analysis overeach NI is polynomial in dim(xL) and dim(w) [6],provided M and K are also polynomial in these quan-tities. (Note that P6M for all I ∈I(M).) This pro-vision may or may not hold depending on how V (x)and W are constructed. For example, the number offaces of a hypercube is linear in its dimension. If sig-ni�cantly more complex polytopes are constructed,however, the method introduced in this paper maynot yield any computational savings over the griddingmethod.We may still have a large number of setsNI corre-

sponding to the elements I ∈Iadm over which to an-alyze stability. However, this is also true in stabilityanalysis for a general nonlinear system, so the reduc-tion in computation time over eachNI still representsan improvement in computational complexity.The complexity of computing [minu∈Umaxr∈I ‘Tr

gu(xN)u] depends on the setU. IfU is a polytope, thisquantity can be computed by solving the followinglinear program:

min �;

subject to u∈U; ‘Ts1gu(xN)u6�; : : : ; ‘TsP gu(xN)u6�:

This is a linear program with m + 1 variables andMU+P constraints, whereMU is the number of conescomprising U, and P6M .

3.3. Iteration over level values

For any �xed xN; �I (c1; c2; xN) is nonincreasing inc1 and nondecreasing in c2. Therefore, simple bisec-tion algorithms can be used to �nd the smallest c1 andlargest c2 satisfying c2¿ �c¿c1¿0; for any given �c¿0;such that the condition in Theorem 1 holds. A detailedalgorithmic procedure for �nding an appropriate valueof �c is presented in [15], but such a value can also befound by trial and error with minimal di�culty. Notethat if W= {0} and the system (2) linearized aboutthe origin is stabilizable, then there exists a polytopicLyapunov function proving global stability of the lin-earized system in closed loop with a piecewise linearcontrol law [4]. In this case, we can use �c=0 pro-vided 0∈ int(U) and W (x) is su�ciently small, sincethe piecewise linear control law is bounded byU oversome level set V−1[0; c2].

4. Conclusions

The most serious hindrance to progress on the non-linear control problem is the inherent complexity ofthe class of arbitrary nonlinear systems. Therefore, itis important to identify a class of systems which is suf-�ciently restricted so that computations can be madetractable, yet which is general enough to apply to awide variety of real control systems. We have shownsome ways in which systems of the form (1) formsuch a class. When the dynamics are nonlinear in onlya few variables xN, the computation time required toanalyze robust stability over cones corresponding toa given polytopic control Lyapunov function variesexponentially in dim(xN) but polynomially in the fullstate dimension. Exponential growth in computationtime is required to analyze stability for a general non-linear system.

Acknowledgements

The authors wish to thank the anonymous reviewersfor their helpful suggestions on this paper.

References

[1] Z. Artstein, Stabilization with relaxed controls, NonlinearAnal. Theory Methods Appl. 7 (1983) 1163–1173.

[2] S. Battilotti, Stabilization via dynamic output feedback forsystems with output nonlinearities, Systems Control Lett. 23(1994) 411–419.

M.W. McConley et al. / Systems & Control Letters 34 (1998) 77–83 83

[3] G. Bitsoris, E. Gravalou, Comparison principle, positiveinvariance and constrained regulation of nonlinear systems,Automatica 31 (1995) 217–222.

[4] F. Blanchini, Nonquadratic Lyapunov functions for robustcontrol, Automatica 31 (1995) 451–461.

[5] F. Blanchini, S. Miani, Constrained stabilization ofcontinuous-time linear systems, Systems Control Lett. 28(1996) 95–102.

[6] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, LinearMatrix Inequalities in System and Control Theory, Societyfor Industrial and Applied Mathematics, Philadelphia, 1994.

[7] R.K. Brayton, C.H. Tong, Constructive stability andasymptotic stability of dynamical systems, IEEE Trans.Circuits Systems 27 (1980) 1121–1130.

[8] E.B. Castelan, J.-C. Hennet, Eigenstructure assignment forstate constrained linear continuous time systems, Automatica28 (1992) 605–611.

[9] A.F. Filippov, Di�erential equations with discontinuous right-hand sides, Mathematicheskii Sbornik 51 (1960), in Russian.Translated in English, Amer. Math. Soc. Trans. 62 (1964).Cited in [19].

[10] R.A. Freeman, P.V. Kokotovi�c, Inverse optimality in robuststabilization, SIAM J. Control Optim. 34 (1996) 1365–1391.

[11] P.-O. Gutman, M. Cwikel, Admissible sets and feedbackcontrol for discrete-time linear dynamical systems withbounded controls and states, IEEE Trans. Automat. Control31 (1986) 373–376.

[12] J.B. Lasserre, Reachable, controllable sets and stabilizingcontrol of constrained linear systems, Automatica 29 (1993)531–536.

[13] Y. Lin, E. Sontag, Y. Wang, Recent results on Lyapunov-theoretic techniques for nonlinear stability, in: Proc. Amer.Control. Conf., Baltimore, 1994, pp. 1771–1775.

[14] Y. Lin, E.D. Sontag, A universal formula for stabilization withbounded controls, Systems Control Lett. 16 (1991) 393–397.

[15] M.W. McConley, A computationally e�cient Lyapunov-based procedure for control of nonlinear systems with stabilityand performance guarantees, Ph.D. Thesis, MassachusettsInstitute of Technology, Department of Aeronautics andAstronautics, Cambridge, MA, 1997.

[16] M.W. McConley, B.D. Appleby, M.A. Dahleh, E. Feron, Acontrol Lyapunov function approach to robust stabilizationof nonlinear systems, in: Proc. Amer. Control Conf.,Albuquerque, 1997, pp. 329–333.

[17] A.N. Michel, B.H. Nam, V. Vittal, Computer generatedLyapunov functions for interconnected systems: Improvedresults with applications to power systems, IEEE Trans.Circuits Systems 31 (1984) 189–198.

[18] Y. Ohta et al., Computer generated Lyapunov functions fora class of nonlinear systems, IEEE Trans. Circuits Systems— I: Fundam. Theory Appl. 40 (1993) 343–354.

[19] J.-J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cli�s, NJ, 1991.

[20] E.D. Sontag, A Lyapunov-like characterization of asymptoticcontrollability, SIAM J. Control Optim. 21 (1983) 462–471.

[21] E.D. Sontag, A ‘universal’ construction of Artstein’s theoremon nonlinear stabilization, Systems Control Lett. 13 (1989)117–123.

[22] E.D. Sontag, Y. Wang, New characterization of input-to-statestability, IEEE Trans. Automat. Control 41 (1996) 1283–1294.

[23] J. Tsinias, Versions of Sontag’s “input to state stabilitycondition” and the global stabilizability problem, SIAM J.Control Optim. 31 (1993) 928–941.

[24] M. Vassilaki, G. Bitsoris, Constrained regulation of linearcontinuous-time dynamical systems, Systems Control Lett. 13(1989) 247–252.

[25] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall,Englewood Cli�s, NJ, 1993.