Stability of Nonlinear Systems - Department of EEgchen/pdf/C-Encyclopedia04.pdfStability of...

Stability of Nonlinear Systems

By

Guanrong Chen

Department of Electronic Engineering City University of Hong Kong

Kowloon, Hong Kong SAR, China [email protected]

In

Encyclopedia of RF and Microwave Engineering, Wiley, New York, pp. 4881-4896, December 2004

STABILITY OF NONLINEAR SYSTEMS

GUANRONG CHEN

City University of Hong KongKowloon, Hong Kong, China

1. INTRODUCTION

A nonlinear system refers to a set of nonlinear equations(algebraic, difference, differential, integral, functional, orabstract operator equations, or a combination of some ofthese) used to describe a physical device or process thatotherwise cannot be clearly defined by a set of linear equa-tions of any kind. Dynamical system is used as a synonymfor mathematical or physical system when the describingequations represent evolution of a solution with time and,sometimes, with control inputs and/or other varying pa-rameters as well.

The theory of nonlinear dynamical systems, or nonlin-ear control systems if control inputs are involved, has beengreatly advanced since the nineteenth century. Today,nonlinear control systems are used to describe a great va-riety of scientific and engineering phenomena rangingfrom social, life, and physical sciences to engineeringand technology. This theory has been applied to a broadspectrum of problems in physics, chemistry, mathematics,biology, medicine, economics, and various engineering dis-ciplines.

Stability theory plays a central role in system engineer-ing, especially in the field of control systems and automa-tion, with regard to both dynamics and control. Stability ofa dynamical system, with or without control and distur-bance inputs, is a fundamental requirement for its prac-tical value, particularly in most real-world applications.Roughly speaking, stability means that the system out-puts and its internal signals are bounded within admissi-ble limits (the so-called bounded-input/bounded-outputstability) or, sometimes more strictly, the system outputstend to an equilibrium state of interest (the so-called as-ymptotic stability). Conceptually, there are different kindsof stabilities, among which three basic notions are themain concerns in nonlinear dynamics and control systems:the stability of a system with respect to its equilibria, theorbital stability of a system output trajectory, and thestructural stability of a system itself.

The basic concept of stability emerged from the study ofan equilibrium state of a mechanical system, dated back toas early as 1644, when E. Torricelli studied the equilibri-um of a rigid body under the natural force of gravity. Theclassical stability theorem of G. Lagrange, formulated in1788, is perhaps the best known result about stability ofconservative mechanical systems, which states that if thepotential energy of a conservative system, currently at theposition of an isolated equilibrium and perhaps subject tosome simple constraints, has a minimum, then this equi-

librium position of the system is stable [23]. The evolutionof the fundamental concepts of system and trajectory sta-bilities then went through a long history, with many fruit-ful advances and developments, until the celebrated Ph.D.thesis of A. M. Lyapunov, The General Problem of MotionStability, finished in 1892 [21]. This monograph is so fun-damental that its ideas and techniques are virtually lead-ing all kinds of basic research and applications regardingstabilities of dynamical systems today. In fact, not onlydynamical behavior analysis in modern physics but alsocontrollers design in engineering systems depend on theprinciples of Lyapunov’s stability theory. This article isdevoted to a brief description of the basic stability theory,criteria, and methodologies of Lyapunov, as well as a fewrelated important stability concepts, for nonlinear dynam-ical systems.

2. NONLINEAR SYSTEM PRELIMINARIES

2.1. Nonlinear Control Systems

A continuous-time nonlinear control system is generallydescribed by a differential equation of the form

.x¼ fðx; t;uÞ; t 2 ½t0;1Þ ð1Þ

where x¼x(t) is the state of the system belonging to a(usually bounded) region Ox Rn, u is the control inputvector belonging to another (usually bounded) region Ou

Rm (often, m n), and f is a Lipschitz or continuously dif-ferentiable nonlinear function, so that the system has aunique solution for each admissible control input and suit-able initial condition x(t0)¼x0AOx. To indicate the timeevolution and the dependence on the initial state x0, thetrajectory (or orbit) of a system state x(t) is sometimesdenoted as jt(x0).

In control system (1), the initial time used is t0 0,unless otherwise indicated. The entire space Rn, to whichthe system states belong, is called the state space. Associ-ated with the control system (1), there usually is an ob-servation or measurement equation

y¼gðx; t;uÞ ð2Þ

where y¼yðtÞ 2 R‘ is the output of the system, 1‘n,and g is a continuous or smooth nonlinear function. Whenboth n, ‘ > 1, the system is called a multiinput/multioutput(MIMO) system; while if n¼ ‘¼ 1, it is called a single-input/single-output (SISO) system. MISO and SIMOsystems are similarly defined.

In the discrete-time setting, a nonlinear control systemis described by a difference equation of the form

xkþ 1¼ fðxk;k;ukÞ

yk¼gðxk; k;ukÞ;

(k¼ 0; 1; ð3Þ

SContinued

4881

where all notations are similarly defined. This article usu-ally discusses only the control system (1), or the firstequation of (3). In this case, the system state x is alsoconsidered as the system output for simplicity.

A special case of system (1), with or without control, issaid to be autonomous if the time variable t does not ap-pear separately (independently) from the state vector inthe system function f. For example, with a state feedbackcontrol u(t)¼h(x(t)), this often is the situation. In thiscase, the system is usually written as

.x¼ fðxÞ; xðt0Þ¼x0 2 Rn

ð4Þ

Otherwise, as (1) stands, the system is said to be nonau-tonomous. The same terminology may be applied in thesame way to discrete-time systems, although they mayhave different characteristics.

An equilibrium, or fixed point, of system (4), if it exists,is a solution, x, of the algebraic equation

fðxÞ¼ 0 ð5Þ

It then follows from (4) and (5) that.x ¼ 0, which means

that an equilibrium of a system must be a constant state.For the discrete-time case, an equilibrium of system

xkþ 1¼ fðxkÞ; k¼ 0; 1; . . . ð6Þ

is a solution, if it exists, of equation

x ¼ fðxÞ ð7Þ

An equilibrium is stable if some nearby trajectories of thesystem states, starting from various initial states, ap-proach it; it is unstable if some nearby trajectories moveaway from it. The concept of system stability with respectto an equilibrium will be precisely introduced in Section 3.

A control system is deterministic if there is a uniqueconsequence to every change of the system parameters orinitial states. It is random or stochastic, if there is morethan one possible consequence for a change in its para-meters or initial states according to some probabilitydistribution [6]. This article deals only with deterministicsystems.

2.2. Hyperbolic Equilibria and Their Manifolds

Consider the autonomous system (4). The Jacobian of thissystem is defined by

JðxÞ¼@f

@xð8Þ

Clearly, this is a matrix-valued function of time. If the Ja-cobian is evaluated at a constant state, say, x or x0, then itbecomes a constant matrix determined by f and x or x0.

An equilibrium x of system (4) is said to be hyperbolicif all eigenvalues of the system Jacobian, evaluated at thisequilibrium, have nonzero real parts.

For a p-periodic solution of system (4), ~xxðtÞ, with a fun-damental period p40, let Jð ~xxðtÞÞ be its Jacobian evaluated

at ~xxðtÞ. Then this Jacobian is also p-periodic:

Jð ~xxðtþpÞÞ ¼Jð ~xxðtÞÞ for all t 2 ½t0;1Þ

In this case, there always exist a p-periodic nonsingularmatrix M(t) and a constant matrix Q such that the fun-damental solution matrix associated with the JacobianJð ~xxðtÞÞ is given by [4]

FðtÞ¼MðtÞetQ

Here, the fundamental matrix F(t) consists of, as its col-umns, n linearly independent solution vectors of the linearequation

.x¼Jð ~xxðtÞÞx, with x(t0)¼x0.

In the preceding, the eigenvalues of the constant ma-trix epQ are called the Floquet multipliers of the Jacobian.The p-periodic solution ~xxðtÞ is called a hyperbolic periodicorbit of the system if all its corresponding Floquet multi-pliers have nonzero real parts.

Next, let D be a neighborhood of an equilibrium, x, ofthe autonomous system (4). A local stable and localunstable manifold of x is defined by

Wslocðx

Þ¼ fx 2 D jjtðxÞ 2 D 8 tt0 and

jtðxÞ ! x as t!1gð9Þ

and

Wulocðx

Þ¼ fx 2 D jjtðxÞ 2 D 8 tt0 and

jtðxÞ ! x as t!1gð10Þ

respectively. Furthermore, a stable and unstable manifoldof x is defined by

WsðxÞ¼ fx 2 D jjtðxÞ \Wslocðx

ÞOfg ð11Þ

and

WuðxÞ¼ fx 2 D jjtðxÞ \Wulocðx

ÞOfg ð12Þ

respectively, where f denotes the empty set. For example,the autonomous system

.x¼ y.y¼ xð1 x2Þ

(

has a hyperbolic equilibrium (x, y)¼ (0, 0). The local sta-ble and unstable manifolds of this equilibrium are illus-trated by Fig. 1a, and the corresponding stable andunstable manifolds are visualized by Fig. 1b.

A hyperbolic equilibrium only has stable and/or unsta-ble manifolds since its associated Jacobian has only stableand/or unstable eigenvalues. The dynamics of an autono-mous system in a neighborhood of a hyperbolic equilibri-um is quite simple—it has either stable (convergent) orunstable (divergent) properties. Therefore, complex dy-namical behaviors such as chaos are seldom associatedwith isolated hyperbolic equilibria or isolated hyperbolic

4882 STABILITY OF NONLINEAR SYSTEMS

periodic orbits [7,11,15] (see Theorem 16, below); theygenerally are confined within the center manifold,Wc(x), where dim(Ws) þdim(Wc) þdim(Wu)¼n.

2.3. Open-Loop and Closed-Loop Systems

Let S be an MIMO system, which can be linear or nonlin-ear, continuous-time or discrete-time, deterministic or sto-chastic, or any well-defined input–output map. Let U andY be the sets (sometimes, spaces) of the admissible inputand corresponding output signals, respectively, both de-fined on the time domainD¼ ½a; b; 1aob1 (for con-trol systems, usually a¼ t0¼ 0 and b¼N). This simplerelation is described by an open-loop map

S : u! y or yðtÞ¼SðuðtÞÞ ð13Þ

and its block diagram is shown in Fig. 2. Actually, everycontrol system described by a differential or differenceequation can be viewed as a map in this form. But, in sucha situation, the map S can be implicitly defined only viathe equation and initial conditions.

In the control system (1) or (3), if the control inputs arefunctions of the state vectors, u¼h(x; t), then the controlsystem can be implemented via a closed-loop configura-tion. A typical closed-loop system is shown in Fig. 3, whereusually S1 is the plant (described by f) and S2 is the con-troller (described by h); yet they can be reversed.

2.4. Norms of Functions and Operators

This article deals only with finite-dimensional systems.For an n-dimensional vector-valued functions,xðtÞ¼ ½x1ðtÞ . . . xnðtÞ

T where superscript ‘‘T’’ denotes trans-pose, let || || and || ||p denote its Euclidean normand Lp norm, defined respectively by the ‘‘length’’

jjxðtÞjj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1ðtÞ þ þ x2nðtÞ

qand

jjxjjp¼

Z b

a

jjxðtÞjjp dt

1=p

; 1po1

jjxjj1 ¼ ess supatb1in

jxiðtÞj

Here, a few remarks are in order:

1. The term ‘‘ess sup’’ means ‘‘essential suprimum’’(i.e., the suprimum except perhaps over a set of

measure zero). For a piecewise continuous functionf(t), actually

ess supatb

jf ðtÞj ¼ supatb

jf ðtÞj

2. The main difference between the ‘‘sup’’ and the‘‘max’’ is that max|f(t)| is attainable but sup|f(t)|may not. For example, max0to1 j sinðtÞj ¼ 1 butsup0to1 j1 etj ¼ 1.

3. The difference between the Euclidean norm and theLp norms is that the former is a function of time butthe latters are all constants.

4. For a finite-dimensional vector x(t), with no N, allthe Lp norms are equivalent in the sense that for anyp, qA[1, N], there exist two positive constants a andb such that

ajjxjjpjjxjjqbjjxjjp

For the input–output map in Eq. (13), the so-called oper-ator norm of the map S is defined to be the maximum gainfrom all possible inputs over the domain of the map totheir corresponding outputs. More precisely, the operatornorm of the map S in (13) is defined by

jjjSjjj ¼ supu1 ;u22Uu1Ou2

jjy1 y2jjY

jju1 u2jjUð14Þ

where yi¼S(ui)AY, i¼ 1, 2, and || ||U and || ||Y arethe norms of the functions defined on the input–outputsets (or spaces) U and Y, respectively.

3. LYAPUNOV, ORBITAL, AND STRUCTURAL STABILITIES

Three different types of stabilities, namely, the Lyapunovstability of a system with respect to its equilibria, the or-bital stability of a system output trajectory, and the struc-tural stability of a system itself, are of fundamentalimportance in the studies of nonlinear dynamics and con-trol systems.

uS

y

Figure 2. The block diagram of an open-loop system.

u1 e1S1

S2

S2(e2)

S1(e1)

e2

+

_

+ +u2

Figure 3. A typical closed-loop control system.

y y

xx

W s(0,0) = W u(0,0)

W uloc(0,0)

W sloc(0,0)

(a) (b)

Figure 1. Stable and unstable manifolds.

STABILITY OF NONLINEAR SYSTEMS 4883

Roughly speaking, the Lyapunov stability of a systemwith respect to its equilibrium of interest is about the be-havior of the system outputs toward the equilibriumstate—wandering nearby and around the equilibrium(stability in the sense of Lyapunov) or gradually approach-es it (asymptotic stability); the orbital stability of a systemoutput is the resistance of the trajectory to small pertur-bations; the structural stability of a system is the resis-tance of the system structure against small perturbations[3,13,16,17,19,20,23,25,26,29,35]. These three basic typesof stabilities are introduced in this section, for dynamicalsystems without explicitly involving control inputs.

Consider the general nonautonomous system

.x¼ fðx; tÞ; xðt0Þ¼x0 2 Rn

ð15Þ

where the control input u(t)¼h(x(t), t), if it exists [seesystem (1)], has been combined into the system function ffor simplicity of discussion. Without loss of generality, as-sume that the origin x¼ 0 is the system equilibrium ofinterest. Lyapunov stability theory concerns various sta-bilities of the system orbits with respect to this equilibri-um. When another equilibrium is discussed, the newequilibrium is first shifted to zero by a change of vari-ables, and then the transformed system is studied in thesame way.

3.1. Stability in the Sense of Lyapunov

System (15) is said to be stable in the sense of Lyapunovwith respect to the equilibrium x¼0, if for any e > 0and any initial time t00, there exists a constant, d¼dðe; t0Þ > 0, such that

jjxðt0Þjjod) jjxðtÞjjoe for all tt0 ð16Þ

This stability is illustrated by Fig. 4.It should be emphasized that the constant d generally

depends on both e and t0. It is particularly important topoint out that, unlike autonomous systems, one cannotsimply fix the initial time t0¼ 0 for a nonautonomous sys-tem in a general discussion of its stability. For example,consider the following linear time-varying system with adiscontinuous coefficient:

.xðtÞ¼

1

1 txðtÞ; xðt0Þ¼ x0

It has an explicit solution

xðtÞ¼ x01 t0

1 t; 0t0to1

which is stable in the sense of Lyapunov about the equi-librium x¼ 0 over the entire time domain [0,N) if andonly if t0¼ 1. This shows that the initial time, t0, does playan important role in the stability of a nonautonomoussystem.

The above-defined stability, in the sense of Lyapunov, issaid to be uniform with respect to the initial time, if theexisting constant d¼ d(e) is indeed independent of t0 overthe entire time interval [0, N). According to the discussionabove, uniform stability is defined only for nonautono-mous systems since it is not needed for autonomous sys-tems (for which it is always uniform with respect to theinitial time).

3.2. Asymptotic and Exponential Stabilities

System (15) is said to be asymptotically stable about itsequilibrium x¼ 0, if it is stable in the sense of Lyapunovand, furthermore, there exists a constant, d¼ d(t0)40,such that

jjxðt0Þjjod) jjxðtÞjj ! 0 as t!1 ð17Þ

This stability is visualized by Fig. 5.The asymptotic stability is said to be uniform if the ex-

isting constant d is independent of t0 over [0, N), and issaid to be global if the convergence, ||x||-0, is inde-pendent of the initial state x(t0) over the entire spatialdomain on which the system is defined (e.g., when d¼N).If, furthermore

jjxðt0Þjjod) jjxðtÞjjcest ð18Þ

for two positive constants c and s, then the equilibrium issaid to be exponentially stable. The exponential stability isvisualized by Fig. 6.

Clearly, exponential stability implies asymptotic stabil-ity, and asymptotic stability implies the stability in thesense of Lyapunov, but the reverse need not be true. Forillustration, if a system has output trajectory x1(t)¼x0 sin(t), then it is stable in the sense of Lyapunov about0, but is not asymptotically stable; a system with output

x (t0)

t

x (t)

Figure 4. Geometric meaning of stability in the sense of Lya-punov.

x (t0)

tx(t)

Figure 5. Geometric meaning of the asymptotic stability.


trajectory x2(t)¼ x0(1þ t t0)1 is asymptotically stable

(so also is stable in the sense of Lyapunov) if t0o1 but isnot exponentially stable about 0; however, a system withx3(t)¼ x0e t is exponentially stable (hence, is both asymp-totically stable and stable in the sense of Lyapunov).

3.3. Orbital Stability

The orbital stability differs from the Lyapunov stabilitiesin that it concerns with the stability of a system output (orstate) trajectory under small external perturbations.

Let jt(x0) be a p-periodic solution, p40, of the auton-omous system

.xðtÞ¼ fðxÞ; xðt0Þ¼x0 2 Rn

ð19Þ

and let G represent the closed orbit of jt(x0) in the statespace, namely,

G¼fy jy¼jtðx0Þ; 0topg

If, for any e40, there exits a constant d¼ d(e)40 such thatfor any x0 satisfying

dðx0;GÞ : ¼ infy2Gjjx0 yjjod

the solution of the system, jt(x0), satisfies

dðjtðx0Þ;GÞoe; for all tt0

then this p-periodic solution trajectory, jt(x0), is said to beorbitally stable.

Orbital stability is visualized by Fig. 7. For a simpleexample, a stable periodic solution, particularly a stableequilibrium of a system is orbitally stable. This is becauseall nearby trajectories approach it and, as such, it becomesa nearby orbit after a small perturbation and so will moveback to its original position (or stay nearby). On the con-trary, unstable and semistable (saddle-type of) periodicorbits are orbitally unstable.

A more precise concept of orbital stability is given inthe sense of Zhukovskij [37].

A solution jt(x0) of system (19) is said to be stable in thesense of Zhukovskij, if for any e > 0, there exists a d¼d(e)40 such that for any y0ABd(x0), a ball of radius d cen-tered at x0, there exist two functions, t1¼ t1(t) and t2¼

t2(t), satisfying

jjxt1 ðx0Þ xt2 ðx0Þjjoe

for all tt0, where t1 and t2 are homeomorphisms (i.e., acontinuous map whose inverse exists and is also continu-ous) from [0, N) to [0, N) with t1(0)¼ t2(0)¼ 0.

Furthermore, a Zhukovskij stable solution jt(x0) of sys-tem (19) is said to be asymptotically stable in the sense ofZhukovskij, if for any e40, there exists a d¼ d(e)40 suchthat for any y0ABd(x0), a ball of radius d centered at x0,

there exist two functions, t1¼ t1(t) and t2¼t2(t), satisfying

jjxt1ðx0Þ xt2

ðx0Þjj ! 0

as t-N, where t1 and t2 are homeomorphisms from[0, N) to [0, N) with t1(0)¼ t2(0)¼ 0.

It can be verified that (asymptotic) Zhukovskij stabilityimplies (asymptotic) Lyapunov stability about an equilib-rium. However, the converse may not be true. Moreover,these two types of stabilities are equivalent if the orbitjt(x0) is an equilibrium of the system.

3.4. Structural Stability

Two systems are said to be topologically orbitally equiva-lent, if there exists a homeomorphism that transforms thefamily of trajectories of the first system to that of the sec-ond while preserving their motion directions. Roughly,this means that the geometrical pictures of the orbit fam-ilies of the two systems are similar (no one has extraknots, sharp corners, bifurcating branches, etc.). For in-stance, systems

.x¼ x and

.x¼ 2x are topologically orbitally

equivalent, but are not so between.x¼ x and

.x¼

ffiffiffixp

. Thesethree system trajectories are shown in Fig. 8.

Return to the autonomous system (19). If the dynamicsof the system in the state space changes radically, for ex-ample by the appearance of a new equilibrium or a newperiodic orbit, due to small external perturbations, thenthe system is considered to be structurally unstable.

x (t0)x (t )

t

Figure 6. Geometric meaning of the exponential stability.

d (x0, Γ) <

d (t (x0), Γ) <

Figure 7. Geometric meaning of the orbital stability.

x

21

t0

2

0t t

xx

0

(a) (b) (c)

Figure 8. Trajectories of three systems for comparison: (a).x¼ x;

(b).x¼2x; (c)

.x¼

ffiffiffixp

.


To be more precise, consider the following set of func-tions:

S¼ gðxÞ jjgðxÞjjo1 ;

@gðxÞ

@x

o1 for all x 2 Rn

If, for any g 2 S, there exists an e > 0 such that the orbitsof the two systems

.x¼ fðxÞ and

.x¼ fðxÞþ egðxÞ

are topologically orbitally equivalent, then the autono-mous system (19), namely, the first (unperturbed) systemabove, is said to be structurally stable.

For example,.x¼ x is structurally stable but

.x¼ x2 is

not, in a neighborhood of the origin. This is because whenthe second system is slightly perturbed, to become, say,.x¼ x2þ e, where e > 0, then the resulting system has twoequilibria, x1¼

ffiffiep

and x2¼ ffiffiep

, which has more num-bers of equilibria than the original system that possessesonly one, x¼ 0.

4. VARIOUS STABILITY THEOREMS

Consider the general nonautonomous system


ð20Þ

where f :D ½0;1Þ ! Rn is continuously differentiable ina neighborhood of the origin, D Rn, with a given initialstate x0 2 D. Again, without loss of generality, assumethat x¼ 0 is a system equilibrium of interest.

4.1. Lyapunov Stability Theorems

First, for the autonomous system (19), an important spe-cial case of (20), with a continuously differentiable f: D-Rn, the following criterion of stability, called the first (orindirect) method of Lyapunov, is very convenient to use.

Theorem 1 (First Method of Lyapunov: For Continuous-TimeAutonomous Systems). Let J¼ ½@f=@xx¼x ¼0 be the systemJacobian evaluated at the zero equilibrium of system (19).If all the eigenvalues of J have a negative real part, thenthe system is asymptotically stable about x¼ 0.

First, note that this and the following Lyapunov theo-rems apply to linear systems as well, for linear systemsare merely a special case of nonlinear systems. When f(x)¼Ax, the linear time-invariant system

.x¼Ax has the

only equilibrium x¼ 0. If A has all eigenvalues with neg-ative real parts, Theorem 1 implies that the system is as-ymptotically stable about its equilibrium since the systemJacobian is simply J¼A. This is consistent with thefamiliar linear stability results.

Note also that the region of asymptotic stability givenby Theorem 1 is local, which can be quite large for somenonlinear systems but may be very small for some others.However, there is no general criterion for determining the

boundaries of such local stability regions when this andthe following Lyapunov methods are applied.

Moreover, it is important to note that this theoremcannot be applied to a general nonautonomous system,since for general nonautonomous systems this theorem isneither necessary nor sufficient [36]. A simple counterex-ample is the following linear time-varying system [17,34]:

.xðtÞ¼

1þ 1:5 cos2ðtÞ 1 1:5 sinðtÞ cosðtÞ

1 1:5 sinðtÞ cosðtÞ 1þ 1:5 sin2ðtÞ

" #xðtÞ

This system has eigenvalues l1;2¼ 0:25 j0:25ffiffiffi7p

, bothhaving negative real parts and being independent of thetime variable t. If this theorem is used to judge the system,the conclusion would be that the system is asymptoticallystable about its equilibrium 0. However, the solution ofthis system is

xðtÞ¼e0:5t cosðtÞ et sinðtÞ

e0:5t sinðtÞ et cosðtÞ

" #x1ðt0Þ

x2ðt0Þ

" #

which is unstable, for any initial conditions with a bound-ed and nonzero value of x1(t0), no matter how small thisinitial value is. This example shows that by using theLyapunov first method alone to determine the stability of ageneral time-varying system, the conclusion can be wrong.

This type of counterexamples can be easily found [35].On the one hand, this demonstrates the necessity of othergeneral criteria for asymptotic stability of nonautonomoussystems. On the other hand, however, a word of caution isthat this types of counterexamples do not completely ruleout the possibility of applying the first method of Lyapu-nov to some special nonautonomous systems in case stud-ies. The reason is that there is no theorem saying that ‘‘theLyapunov first method cannot be applied to all nonauton-omous systems.’’ Because of the complexity of nonlineardynamical systems, they often have to be studied class byclass, or even case by case. It has been widely experiencedthat the first method of Lyapunov does work for some,perhaps not too many, specific nonautonomous systems incase studies (e.g., in the study of some chaotic systems [7];see also Theorem 18, below). The point is that one has tobe very careful when this method is applied to a particularnonautonomous system; the stability conclusion must beverified by some other means at the same time.

Here, it is emphasized that a rigorous approach for as-ymptotic stability analysis of general nonautonomous sys-tems is provided by the second method of Lyapunov, forwhich the following set of classK functions are useful:

K¼fgðtÞ : gðt0Þ¼ 0; gðtÞ > 0 if t > t0;

gðtÞ is continuous and nondecreasing on ½t0;1Þg

Theorem 2 (Second Method of Lyapunov: For Continuous-Time Nonautonomous Systems). The system (20) is globally(over the entire domain D), uniformly (with respect to theinitial time over the entire time interval [t0, N)), andasymptotically stable about its zero equilibrium, if there


exist a scalar-valued function, V(x, t), defined on D[t0, N), and three functions a( . ), b( . ), g( . )AK, such that

(a) V(0, t0)¼0.(b) V(x, t)40 for all xa0 in D and all tt0.(c) a(||x||) V(x, t) b(||x||) for all tt0.(d)

.Vðx; tÞ gðjjxjjÞo0 for all tt0.

In Theorem 2, the function V is called a Lyapunov func-tion. The method of constructing a Lyapunov function forstability determination is called the second (or direct)method of Lyapunov.

The geometric meaning of a Lyapunov function used fordetermining the system stability about the zero equilibri-um may be illustrated by Fig. 9. In this figure, assumingthat a Lyapunov function, V(x), has been found, which hasa bowl shape as shown based on conditions (a) and (b).Then, condition (d) is

.VðxÞ¼

@V

@x

.xo0 ð21Þ

where [@V/@x] is the gradient of V along the trajectory x. Itis known, from calculus, that if the inner product of thisgradient and the tangent vector

.x is constantly negative,

as guaranteed by condition (21), then the angle betweenthese two vectors is larger than 901, so that the surface ofV(x) is monotonically decreasing to zero (this is visualizedin Fig. 9). Consequently, the system trajectory x, the pro-jection on the domain as shown in the figure, converges tozero as time evolves.

As an example, consider the following nonautonomoussystem

.x¼Axþgðx; tÞ

where A is a stable constant matrix and g is a nonlinearfunction satisfying g(0, t)¼ 0 and ||g(x, t)|| c||x|| fora constant c40 for all tA[t0, N). Since A is stable, the fol-lowing Lyapunov equation

PAþATPþ I¼ 0

has a unique positive definite and symmetric matrixsolution, P. Using the Lyapunov function V(x, t)¼xTPx,it can be easily verified that

.Vðx; tÞ¼xT½PAþATPxþ 2xTPgðx; tÞ

xTxþ 2lmaxðPÞcjjxjj2

where lmax(P) is the largest eigenvalue of P. Therefore, ifthe constant c o 1/(2lmax(P)) and if the classK functions

aðzÞ¼ lminðPÞz2; bðzÞ¼ lmaxðPÞz

2;

gðzÞ¼ ½1 2clmaxðPÞz2

are used, then conditions (c) and (d) of Theorem 2 are sat-isfied. As a result, the above system is globally, uniformly,and asymptotically stable about its zero equilibrium. Thisexample shows that the linear part of a weakly nonlinearnonautonomous system can indeed dominate the stability.

Note that in Theorem 2, the uniform stability is guar-anteed by the classK functions a, b, g stated in conditions(c) and (d), which is necessary since the solution of a non-autonomous system may sensitively depend on the initialtime, as seen from the numerical example discussed inSection 3.1. For autonomous systems, these classK func-tions [hence, condition (c)] are not needed. In this case,Theorem 2 reduces to the following simple form.

Theorem 3 (Second Method of Lyapunov: For Continuous-Time Autonomous Systems). The autonomous system (19)is globally (over the entire domain D) and asymptoticallystable about its zero equilibrium, if there exists a scalar-valued function V(x), defined on D, such that

(a) V(0)¼ 0.(b) V(x)40 for all xa0 in D.(c)

.VðxÞo0 for all xa0 in D.

Note that if condition (c) in Theorem 3 is replaced by (d).VðxÞ0 for all xAD, then the resulting stability is only inthe sense of Lyapunov but may not be asymptotic. For ex-ample, consider a simple model of an undamped pendulumof length ‘ described by

.x¼

g

‘sinðyÞ

.y¼ x

8<:

where x¼ y is the angular variable defined on po yo p,with the vertical axis as its reference, and g is the gravityconstant. Since the system Jacobian at the zero equilibri-um has one pair of purely imaginary eigenvalues,l1;2¼

ffiffiffiffiffiffiffiffiffiffiffig=‘

p, Theorem 1 is not applicable. However, if

one uses the Lyapunov function

V ¼g

‘ð1 cosðyÞÞ þ

1

2x2

then it can be easily verified that.

V¼ 0 over the entiredomain. Thus, the conclusion is that the undamped

x0

x.

V (x)

Vx

Figure 9. Geometric meaning of the Lyapunov function.


pendulum is stable in the sense of Lyapunov but may notbe asymptotically, consistent with the physics of the un-damped pendulum.

Theorem 4 (Krasovskii Theorem: For Continuous-TimeAutonomous Systems). For the autonomous system (19),let J(x)¼ [qf/qx] be its Jacobian evaluated at x(t). A suf-ficient condition for the system to be asymptotically stableabout its zero equilibrium is that there exist two real pos-itive definite and symmetric constant matrices, P and Q,such that the matrix

JTðxÞPþPJðxÞþQ

is seminegative definite for all xa0 in a neighborhood Dof the origin. For this case, a Lyapunov function is given by

VðxÞ¼ fTðxÞPfðxÞ

Furthermore, ifD¼Rn and V(x)-N as ||x||-N, thenthis asymptotic stability is also global.

Similar stability criteria can be established for discrete-time systems [10]. Two main results are summarized asfollows.

Theorem 5 (First Method of Lyapunov: For Discrete-Time‘‘Autonomous’’ Systems). Let x¼ 0 be an equilibrium ofthe discrete-time ‘‘autonomous’’ system

xkþ 1¼ fðxkÞ ð22Þ

where f : D-Rn is continuously differentiable in a neigh-borhood of the origin, DDRn, and let J¼ ½@f=@xkxk ¼x¼ 0

be the Jacobian of the system evaluated at this equilibri-um. If all the eigenvalues of J are strictly less than one inabsolute value, then the system is asymptotically stableabout its zero equilibrium.

Theorem 6 (Second Method of Lyapunov: For Discrete-Time‘‘Nonautonomous’’ Systems). Let x¼ 0 be an equilibriumof the ‘‘nonautonomous’’ system

xkþ 1¼ fkðxkÞ ð23Þ

where fk :D! Rn is continuously differentiable in aneighborhood of the origin, DDRn. Then, system (22) isglobally (over the entire domain D) and asymptoticallystable about its zero equilibrium, if there exists a scalar-valued function, V(xk, k), defined on D and continuous inxk, such that

(a) V(0, k)¼ 0 for all kk0.(b) V(xk, k)40 for all xka0 in D and for all kk0.(c) DV(xk, k) :¼V(xk, k)V(xk1, k 1) o 0 for all

xka0 in D and all kk0þ 1.(d) 0o W(||xk||)o V(xk, k) for all kk0þ1, where

W(t) is a positive continuous function defined onD,satisfying Wðjjxk0

jjÞ ¼ 0 and limt-N W(t)¼N

monotonically.

As a special case, for discrete-time ‘‘autonomous’’ sys-tems, Theorem 6 reduces to the following simple form.

Theorem 7 (Second Method of Lyapunov: For Discrete-Time‘‘Autonomous’’ Systems). Let x¼ 0 be an equilibrium forthe ‘‘autonomous’’ system (22). Then the system is globally(over the entire domain D) and asymptotically stableabout this zero equilibrium if there exists a scalar-valuedfunction, V(xk), defined on D and continuous in xk, suchthat

(a) V(0)¼ 0.(b) V(xk)40 for all xka0 in D.(c) DV(xk):¼V(xk)V(xk 1)o 0 for all xka0 in D.(d) V(x)-N as ||x||-N.

To this end, it is important to emphasize that all theLyapunov theorems stated above only offer sufficient con-ditions for asymptotic stability. On the other hand, usuallymore than one Lyapunov function may be constructed forthe same system. For a given system, one choice of aLyapunov function may yield a less conservative result(e.g., with a larger stability region) than other choices.However, no conclusion regarding stability may be drawnif, for technical reasons, a satisfactory Lyapunov functioncannot be found. Nevertheless, there is a necessary con-dition in theory about the existence of a Lyapunov func-tion [15], as follows.

Theorem 8 (Massera Inverse Theorem). Suppose that theautonomous system (19) is asymptotically stable about itsequilibrium x and f is continuously differentiable withrespect to x for all tA[t0, N). Then a Lyapunov functionexists for this system.

4.2. Some Instability Theorems

Once again, consider a general autonomous system

.x¼ fðxÞ; xðt0Þ¼x0 2 Rn

ð24Þ

with an equilibrium x¼ 0. To disprove the stability, thefollowing instability theorems may be used.

Theorem 9 (A Linear Instability Theorem). In system (24),let J¼ ½@f=@xx¼x ¼ 0 be the system Jacobian evaluated atx¼ 0. If at lease one of the eigenvalues of J has a positivereal part, then x¼ 0 is unstable.

For discrete-time systems, there is a similar result. Adiscrete-time ‘‘autonomous’’ system

xkþ 1¼ fðxkÞ; k¼ 0;1; 2; ;

is unstable about its equilibrium x¼ 0 if at least one ofthe eigenvalues of the system Jacobian is larger than 1 inabsolute value.

The following two negative theorems can be easily ex-tended to nonautonomous systems in an obvious way.


Theorem 10 (A General Instability Theorem). For system(24), let V(x) be a positive and continuously differentiablefunction defined on a neighborhood D of the origin, satis-fying V(0)¼ 0. Assume that in any subset, containing theorigin, of D, there is an ~xx such that V( ~xx)40. If, moreover

d

dtVðxÞ > 0 for all xO0 inD

then the system is unstable about the equilibrium x¼ 0.

One example is the system

.x¼ yþ xðx2þ y4Þ

.y¼ xþ yðx2þ y4Þ

(

which has equilibrium (x, y)¼ (0, 0). The system Jacobi-an at the equilibrium has a pair of imaginary eigenvalues,l1;2¼

ffiffiffiffiffiffiffi1p

, so Theorem 1 is not applicable. On the con-trary, the Lyapunov function

V ¼1

2ðx2þ y2Þ

leads to.

V¼ ðx2þ y2Þðx2þ y4Þ > 0 for all (x, y)a(0, 0). There-fore, the conclusion is that this system is unstable aboutits zero equilibrium.

Theorem 11 (Chetaev Instability Theorem). For system (24),let V(x) be a positive and continuously differentiable func-tion defined on D, and let O be a subset, containing theorigin, of D, (i.e., 0AD-O). If (a) V(x)40 and

.VðxÞ > 0 for

all xa0 in D and (b) V(x)¼ 0 for all x on the boundaryof O, then the system is unstable about the equilibriumx¼0.

This instability theorem is illustrated by Fig. 10, whichgraphically shows that if the theorem conditions are sat-isfied, then there is a gap within any neighborhood of theorigin, so that a system trajectory can escape from theneighborhood of the origin along a path in this gap [23].

As an example, consider the system

.x¼ x2þ 2y5

.y¼ xy2

(

with the Lyapunov function

V ¼ x2 y4

which is positive inside the region defined by

x¼ y2 and x¼ y2

Let D be the right half-plane and O be the shaded areashown in Fig. 11. Clearly, V¼0 on the boundary of O, andV40 and

.V¼ 2x3 > 0 for all (x, y)A D. According to the

Chetaev theorem, this system is unstable about its zeroequilibrium.

4.3. LaSalle Invariance Principle

Consider again the autonomous system (24) with anequilibrium x¼ 0. Let V(x) be a Lyapunov functiondefined on a neighborhood D of the origin. Let also jt(x0)be a bounded solution orbit of the system, with the initialstate x0 and all its limit states being confined in D. More-over, let

E¼fx 2 D j.

VðxÞ¼ 0g ð25Þ

and M E be the largest invariant subset of E in the sensethat if the initial state x0AM, then the entire orbitjt(x0)M for all tt0.

Theorem 12 (LaSalle Invariance Principle). Under the as-sumptions above, for any initial state x0AD, the solutionorbit satisfies jtðx0Þ !M as t!1.

This invariance principle is consistent with the Lyapu-nov theorems when they are applicable to a problem[11,19]. Sometimes, when

.V¼ 0 over a subset of the

domain of V, a Lyapunov theorem is not easy to directlyapply, but the above LaSalle invariance principle may beconvenient to use. For instance, consider the system

.x¼ xþ

1

3x3þ y

.y¼ x

8<:

The Lyapunov function V¼ x2þ y2 yields

.V¼

1

2x2 1

3x2 1

D

0

V (x) = 0

V (x) =

0

V (x ) > 0.

Ω

Figure 10. Illustration of the Chetaev theorem.

y

0D

x

Ω

Figure 11. The defining region of a Lyapunov function.


which is negative for x2o 3 but is zero for x¼0 and x2¼ 3,

regardless of variable y. Thus, Lyapunov theorems do notseem to be applicable, at least not directly. However, observethat the set E defined above has only three straight lines:x¼

ffiffiffi3p

, x¼ 0, and x¼ffiffiffi3p

, and that all trajectories thatintersect the line x¼ 0 will remain on the line only if y¼ 0.This means that the largest invariant subset M containingthe points with x¼ 0 is the only point (0,0). It then followsfrom the LaSalle invariance principle that starting from anyinitial state located in a neighborhood of the origin boundedwithin the two stripes x¼

ffiffiffi3p

, say, located inside the disk

D¼fðx; yÞ j x2þ y2o3g

the solution orbit will always be attracted to the point (0, 0).This means that the system is (locally) asymptotically stableabout its zero equilibrium.

4.4. Comparison Principle and Vector Lyapunov Functions

For large-scale and interconnected nonlinear (control) sys-tems, or systems described by differential inequalitiesrather than differential equations, the stability criteriaabove may not be directly applicable. In such cases,the comparison principle and vector Lyapunov functionmethods turn out to be advantageous [18,24,32].

To introduce the comparison principle, consider thegeneral nonautonomous system

.x¼ fðx; tÞ; xðt0Þ¼x0 ð26Þ

where f(0, t)¼ 0 is continuous on a neighborhood D of theorigin, t0to1.

In this case, since f is only continuous (but not neces-sarily satisfying the Lipschitz condition), this differentialequation may have more than one solution [12]. Letxmax(t) and xmin(t) be its maximum and minimum solu-tions, respectively, in the sense that

xminðtÞxðtÞxmaxðtÞ componentwise; for all t 2 ½t0;1Þ

where x(t) is any solution of the equation, and xmin(t0)¼x(t0)¼xmax(t0)¼x0.

Theorem 13 (The Comparison Principle). Let y (t) be asolution of the following differential inequality:

.yðtÞfðy; tÞ with yðt0Þx0 componentwise

If xmax(t) is the maximum solution of system (26), then

yðtÞxmaxðtÞ componentwise for all t 2 ½t0;1Þ

The next theorem is established based on this compari-son principle. First, recall that a vector-valued function,gðx; tÞ¼ ½g1ðx; tÞ gnðx; tÞ

T is said to be quasimonotonic, if

xi¼ ~xxi and xj ~xxj ðjOiÞ

) giðx; tÞ gið ~xx; tÞ; i¼ 1; ;n

Theorem 14 (Vector Lyapunov Function Theorem). Letv (x, t) be a vector Lyapunov function associated with the

nonautonomous system (26), vðx; tÞ¼ ½V1ðx; tÞ Vnðx; tÞT

in which each Vi is a continuous Lyapunov function for thesystem, i¼ 1; . . . ;n; satisfying ||v(x, t)||40 for xa0.Assume that

.vðx; tÞgðvðx; tÞ; tÞ componentwise

for a continuous and quasimonotonic function g defined onD. Then

(a) If the system.yðtÞ¼gðy; tÞ

is stable in the sense of Lyapunov (or asymptotical-ly stable) about its zero equilibrium y*¼ 0, then sois the nonautonomous system (26).

(b) If, moreover, ||v(x, t)|| is monotonically decreas-ing with respect to t and the preceding stability (orasymptotic stability) is uniform, then so is the non-autonomous system (26).

(c) If, furthermore, ||v(x, t)|| c||x||s for two pos-itive constants c and s, and the proceeding stability(or asymptotic stability) is exponential, then so isthe nonautonomous system (26).

A simple and frequently used comparison function is

gðy; tÞ¼Ayþhðy; tÞ; limjjyjj!0

jjhðy; tÞjj

jjyjj¼ 0

where A is a stable M matrix (Metzler matrix). Here,A¼ [aij] is an M matrix if

aiio0 and aii0 ðiOjÞ; i; j¼ 1; . . . ;n

4.5. Orbital and Structural Stability Theorems

Theorem 15 (Orbital Stability Theorem). Let ~xxðtÞ be a p-pe-riodic solution of an autonomous system. Suppose that thesystem has Floquet multipliers li, with l1¼ 0 and |li|o 1for i¼ 2; . . . ;n. Then this periodic solution ~xxðtÞ is orbitallystable.

Theorem 16 (Peixoto Structural Stability Theorem). Consid-er a two-dimensional autonomous system. Suppose that fis twice differentiable on a compact and connected subsetD bounded by a simple closed curve G, with an outwardnormal vector ~nn. Assume that f ~nnO0 on G. Then the sys-tem is structurally stable on D if and only if

(a) All equilibria are hyperbolic.(b) All periodic orbits are hyperbolic.(c) If x and y are hyperbolic saddles (probably x¼ y),

then W s(x) and Wu(y) are transversal.

5. LINEAR STABILITY OF NONLINEAR SYSTEMS

The first method of Lyapunov provides a linear stabilityanalysis for nonlinear autonomous systems. In this sec-tion, the following general nonautonomous system is


considered:


ð27Þ

which is assumed to have an equilibrium x¼ 0.

5.1. Linear Stability of Nonautonomous Systems

In system (27), Taylor-expanding the function f aboutx¼0 gives

.x¼ fðx; tÞ¼JðtÞxþgðx; tÞ ð28Þ

where JðtÞ¼ ½@f=@xx¼ 0 is the Jacobian and g(x, t) is theresidual of the expansion, which is assumed to satisfy

jjgðx; tÞjjajjxjj2 for all t 2 ½t0;1Þ

where a40 is a constant. It is known from the theory ofelementary ordinary differential equations [12] that thesolution of equation (28) is given by

xðtÞ¼Fðt; t0Þx0þ

Z t

t0

Fðt; tÞgðxðtÞ; tÞdt ð29Þ

where F(t, t) is the fundamental matrix associated withmatrix J(t).

Theorem 17 (A General Linear Stability Theorem). For thenonlinear nonautonomous system (28), if there are twopositive constants c and s such that

jjFðt; tÞjjcesðttÞ for all t0tto1

and if

limjjxjj!0

jjgðx; tÞjj

jjxjj¼0

uniformly with respect to tA[t0, N), then there are twopositive constants, g and d, such that

jjxðtÞjjcjjx0jjegðtt0Þ

for all jjx0jj d and all tA[t0, N).

This result implies that under the theorem conditions,the system is locally, uniformly, and exponentially stableabout its equilibrium x¼ 0.

In particular, if the system matrix J(t)¼J is a stableconstant matrix, then the following simple criterion isconvenient to use.

Theorem 18 (A Special Linear Stability Theorem). Supposethat in system (28), the matrix J(t)¼J is a stable constantmatrix (all its eigenvalues have a negative real part), andg(0, t)¼ 0. Let P be a positive definite and symmetricmatrix solution of the Lyapunov equation

PJþJTPþQ¼ 0

where Q is a positive definite and symmetric constantmatrix. If

jjgðx; tÞjjajjxjj

for a constant ao12 lmaxðPÞ uniformly on [t0, N), where

lmax(P) is the maximum eigenvalue of P, then system (28)is globally, uniformly, and asymptotically stable about itsequilibrium x¼ 0.

This actually is the example for illustration of Theo-rem 2 discussed in Section 4.1.

Again consider system (27) with equilibrium x¼ 0.

Theorem 19 (The Lyapunov Converse Theorem). Supposethat f is continuously differentiable in a neighborhood ofthe origin, and its Jacobian J(t) is bounded and Lipschitzin the neighborhood, uniformly in t. Then, the system isexponentially stable about its equilibrium if and only if itslinearized system

.x¼JðtÞx is exponentially stable about

the origin.

5.2. Linear Stability of Nonlinear Systems withPeriodic Linearity

Consider a nonlinear nonautonomous system of the form

.x¼ fðx; tÞ¼JðtÞxþgðx; tÞ; xðt0Þ¼x0 2 Rn

ð30Þ

where g(0, t)¼ 0 and J(t) is a p-periodic matrix (p40):

JðtþpÞ¼JðtÞ for all t 2 ½t0;1Þ

Theorem 20 (Floquet Theorem). In system (30), assumethat g(x, t) and qg(x, t)/qx are both continuous in a bound-ed regionD containing the origin. Assume, moreover, that

limjjxjj!0

jjgðx; tÞjj

jjxjj¼ 0

uniformly on [t0, N). If the system Floquet multiplierssatisfy

jlijo1; i¼ 1; ;n; for all t 2 ½t0;1Þ ð31Þ

then system (30) is globally, uniformly, and asymptoticallystable about its equilibrium x¼ 0.

Note that if g(x, t)¼ 0 in system (28) or (30), all thelinear stability results presented above still hold and areconsistent with the familiar results from the linear sys-tems theory.

6. TOTAL STABILITY: STABILITY UNDERPERSISTENT PERTURBANCES

Consider a nonautonomous system of the form

.x¼ fðx; tÞþhðx; tÞ; xðt0Þ¼x0 2 Rn

ð32Þ


where f is continuously differentiable, with f(0, t)¼ 0, andh is a persistent perturbation in the sense that for anye > 0, there are two positive constants, d1 and d2, such thatif jjhð ~xx; tÞjjod1 for all tA[t0, N) and if jj ~xxðt0Þjjod2, thenjj ~xxðtÞjjoe.

The equilibrium x¼ 0 of the unperturbed system [sys-tem (32) with h¼ 0 therein] is said to be totally stable, ifthe persistently perturbed system (32) remains to be sta-ble in the sense of Lyapunov.

As the next theorem states, all uniformly and asymp-totically stable systems with persistent perturbations aretotally stable, namely, a stable orbit starting from a neigh-borhood of another orbit will stay nearby [13,15].

Theorem 21 (Malkin Theorem). If the unperturbed system(32), that is, with h¼ 0 therein, is uniformly and asymp-totically stable about its equilibrium x¼ 0, then it is to-tally stable, namely, the persistently perturbed system(32) remains to be stable in the sense of Lyapunov.

Next, consider an autonomous system with persistentperturbations:

.x¼ fðxÞþhðx; tÞ; x 2 Rn

ð33Þ

Theorem 22 (Perturbed Orbital Stability Theorem). If jt(x0)is an orbitally stable solution of the unperturbed autono-mous system (33) (with h¼ 0 therein), then it is totallystable, that is, the perturbed system remains to be orbi-tally stable under persistent perturbations.

7. ABSOLUTE STABILITY AND FREQUENCY-DOMAIN CRITERIA

Consider a feedback system in the Lur’e form

.x¼AxþBhðyÞ

y¼Cx

(ð34Þ

where A, B, C are constant matrices, in which A is non-singular but B and C are not necessarily square (yet,probably, B¼C¼ I), and h is a vector-valued nonlinearfunction. By taking the Laplace transform with zero ini-tial conditions, and denoting the transform by xx¼Lfxg,the state vector is obtained as

xx¼ ½sI A1BLfhðyÞg ð35Þ

so that the output is given by

yy¼CGðsÞLfhðyÞg ð36Þ

with the system transfer matrix

GðsÞ¼ ½sI A1B ð37Þ

This can be implemented via the block diagram shown inFig. 12, where, for notational convenience, both time- andfrequency-domain symbols are mixed.

The Lur’e system shown in Fig. 12 is a closed-loop con-figuration, where the feedback loop is usually consideredas a ‘‘controller.’’ Thus, this system is sometimes written inthe following equivalent form:

.xþAxþBu

y¼Cx

u¼hðyÞ

8>><>>: ð38Þ

7.1. SISO Lur’e Systems

First, single-input/single-output Lur’e systems are dis-cussed, where u¼h(y) and y¼cTx are both scalar-valuedfunctions:

.x¼Axþbu

y¼ cTx

u¼hðyÞ

8>><>>: ð39Þ

Assume that h(0)¼ 0, so that x¼ 0 is an equilibrium ofthe system.

7.1.1. The Sector Condition. The Lur’e system (39) issaid to satisfy the local (global) sector condition on thenonlinear function h( ), if there exist two constants, ao b,such that

1. Local sector condition:

ay2ðtÞyðtÞhðyðtÞÞby2ðtÞ for all

ayðtÞb and t 2 ½t0;1Þð40Þ

2. Global sector condition:

ay2ðtÞyðtÞhðyðtÞÞby2ðtÞ for all

1oyðtÞo1 and t 2 ½t0;1Þð41Þ

Here, [a, b] is called the sector for the nonlinear functionh( ). Moreover, the system (39) is said to be (globally)absolutely stable within the sector [a, b] if the systemis (globally) asymptotically stable about its equilibriumx¼ 0 for any nonlinear function h( ) satisfying the

+

−G (s ) C

yx

h (y )

Figure 12. Configuration of the Lur’e system.


(global) sector condition. These local and global sector con-ditions are visualized by Figs. 13a and 13b, respectively.

7.1.2. The Popov Criterion.

Theorem 23 (Popov Criterion). Suppose that the SISOLure’s system (39) satisfied the following conditions:

(a) A is stable and A, b is controllable.(b) The system satisfies the global sector condition

with a¼ 0 therein.(c) For any e > 0, there is a constant g40 such that

Refð1þ jgoÞGðjoÞgþ1

be for all o0 ð42Þ

where G(s) is the transfer function defined by (37),and Re denotes the real part of a complex num-ber (or function). Then, the system is globally as-ymptotically stable about its equilibrium x¼ 0within the sector.

The Popov criterion has the following geometric mean-ing. Separate the complex function G(s) into its real andimaginary parts, namely

GðjoÞ¼GrðoÞþ jGiðoÞ

and rewrite condition (C) as

1

b> GrðoÞþ goGiðoÞ for all o0

Then the graphical situation of the Popov criterion shownin Fig. 14 implies the global asymptotic stability of thesystem about its zero equilibrium.

7.1.3. The Circle Criterion. The Popov criterion has anatural connection to the linear Nyquist criterion

[17,26,27,29]. A more direct generalization of the Nyquistcriterion to nonlinear systems is the following.

Theorem 24 (Circle Criterion). Suppose that the SISOLur’e system (39) satisfies the following conditions:

(a) A has no purely imaginary eigenvalues, and has keigenvalues with positive real parts.

(b) The system satisfies the global sector condition.(c) One of the following situation holds:

(1) 0o ao b—the Nyquist plot of G(jo) encircles thedisk D[ (1/a), (1/b)] counterclockwise k timesbut does not enter it.

(2) 0¼ ao b—the Nyquist plot of G(jo) stays withinthe open half-plane Res4 (1/b).

(3) ao 0o b—the Nyquist plot of G(jo) stays withinthe open disk D[ (1/b), (1/a)].

(4) aobo0—the Nyquist plot of G(jo) encirclesthe disk D[(1/a), (1/b)] counterclockwise k timesbut does not enter it.

Then, the system is globally asymptotically stable aboutits equilibrium x¼ 0.

Here, the disk D[ (1/a), (1/b)], for the case of0oaob, is shown in Fig. 15.

7.2. MIMO Lur’e systems

Consider a multiinput/multioutput Lur’e system, asshown in Fig. 12, described by

xðsÞ¼GðsÞuðsÞ

uðtÞ ¼ hðyðtÞÞ

(ð43Þ

with G(s) as defined in (37). If this system satisfies thefollowing Popov inequality

Z t1

t0

yTðtÞxðtÞdt g for all t1t0 ð44Þ

for a constant g0 independent of t, then it is said to behyperstable.

The linear part of this MIMO system is described by thetransfer matrix G(s), which is said to be positive real if

1. There are no poles of G(s) located inside the openhalf-plane Res40.

2. Poles of G(s) on the imaginary axis are simple, andthe residues form a semipositive definite matrix.

a0 b y (t ) y (t )

y (t ) y (t )

y (t )y (t )

h (y (t ))

h (y (t ))

0

(a) (b)

Figure 13. Local and global sector conditions.

1

0

−Gr() + Gi ()

Figure 14. Geometric meaning of the Popov criterion.

1

1

0

Figure 15. The disk D[ (1/a), (1/b)].


3. The matrix [G(jo)þGT(jo)] is a semipositive defi-nite matrix for all real values of o that are not polesof G(s).

Theorem 25 (Hyperstability Theorem). The MIMO Lur’esystem (43) is hyperstable if and only if its transfermatrix G(s) is positive real.

7.3. Describing Function Method

Return to the SISO Lur’e system (39) and consider its pe-riodic output y(t). Assume that the nonlinear function h( )therein is a time-invariant odd function, and satisfies theproperty that for y(t)¼ a sin(ot), with real constants o andaa0, only the first-order harmonic of h(y) in its Fourierseries expansion is significant. Under this setup, the spe-cially defined function

CðaÞ¼ 2oap

Z p=o

0hða sinðotÞÞ sinðotÞdt ð45Þ

is called the describing function of the nonlinearity h( ),or of the system [1,26,27].

Theorem 26 (First-Order Harmonic Balance Approxima-tion). Under the conditions stated above, if the first-orderharmonic balance equations

GrðjoÞCðaÞ¼ 1 and Gið joÞ¼ 0

have solutions o and aa0, then

yh1iðtÞ¼ja2

ejot ja2

ejot

is the first-order approximation of a possible periodic orbitof the output of system (39); but if the preceding harmonicbalance equations have no solution, then the system isunlikely to have any periodic output.

When solving the equation Gr( jo)C(a)¼ 1 graphically,one can sketch two curves in the complex plane: Gr( jo)and 1/C(a) by increasing gradually o and a, respec-tively, to find their crossing points:

1. If the two curves are (almost) tangent, as illustratedby Fig. 16a, then a conclusion drawn from the de-scribing function method will not be satisfactory ingeneral.

2. If the two curves are (almost) transversal, as illus-trated by Fig. 16b, then a conclusion drawn from thedescribing function analysis will generally be reliable.

Theorem 27 (Graphical Stability Criterion for a PeriodicOrbit). Each intersection point of the two curves above,Gr(jo) and 1/C(a), corresponds to a periodic orbit, yh1iðtÞ,of the output of system (39). If the points, near the inter-section and on one side of the curve 1/C(a) where a isincreasing, are not encircled by the curve Gr(jo), then thecorresponding periodic output is stable; otherwise, it isunstable.

8. BIBO STABILITY

A relatively simple, and also relatively weak, notion ofstability is discussed in this section. This is the bounded-input/bounded-output (BIBO) stability, which refers tothe property of a system that any bounded input to thesystem produces a bounded output throughout the system[8,9,17].

It can be verified that the linear system.x¼AxþBu is

BIBO stable if the matrix A is asymptotically stable.Here, the focus is on the input-output map (13), with its

configuration shown in Fig. 2.The system S is said to be BIBO stable from the input

set U to the output set Y, if for each admissible input uAUand the corresponding output yAY, there exist two non-negative constants, bi and bo, such that

jjujjUbi ) jjyjjYbo ð46Þ

Since all norms are equivalent for a finite-dimensionalvector, it is generally insignificant to distinguish underwhat kind of norms for the input and output signals theBIBO stability is defined and achieved. Moreover, it is im-portant to note that in the definition above, even if bi issmall and bo is large, the system is still considered to beBIBO-stable. Therefore, this stability may not be verypractical for some systems in certain applications.

−1/Ψ()

−1/Ψ()

G ( j) G ( j)

0Re Re

lm lm

0

(a) (b)Figure 16. Graphical depiction of describing func-tion analysis.


8.1. Small-Gain Theorem

A convenient criterion for verifying the BIBO stability of aclosed-loop control system is the small-gain theorem[8,9,17], which applies to almost all kinds of systems(linear and nonlinear, continuous-time and discrete-time,time-delayed, of any dimensions), as long as the mathe-matical setup is appropriately formulated to meet the the-orem conditions. The main disadvantage of this criterionis its overconservativity.

Return to the typical closed-loop system shown inFig. 3, where the inputs, outputs, and internal signalsare related via the following equations:

S1ðe1Þ¼ e2 u2

S2ðe2Þ¼u1 e1

(ð47Þ

It is important to note that the individual BIBO stabil-ity of S1 and S2 is not sufficient for the BIBO stability ofthe connected closed-loop system. For instance, in the dis-crete-time setting of Fig. 3, suppose that S11 andS2 1, with u1(k)1 for all k¼ 0; 1; . Then S1 and S2

are BIBO-stable individually, but it can be easily verifiedthat y1(k)¼ k-N as the discrete-time variable k evolves.Therefore, a stronger condition describing the interactionof S1 and S2 is necessary.

Theorem 28 (Small-Gain Theorem). If there exist four con-stants, L1, L2, M1, M2, with L1L2 o 1, such that

jjS1ðe1ÞjjM1þL1jje1jj

jjS2ðe2ÞjjM2þL2jje2jj

(ð48Þ

then

jje1jjð1 L1L2Þ1ðjju1jj þL2jju2jj þM2þL2M1Þ

jje2jjð1 L1L2Þ1ðjju2jj þL1jju1jj þM1þL1M2Þ

(ð49Þ

where the norms || || are defined over the spaces thatthe signals belong. Consequently, (48) and (49) togetherimply that if the system inputs (u1 and u2) are boundedthen the corresponding outputs [S1(e1) and S2(e2)] arebounded.

Note that the four constants, L1, L2, M1, M2, can besomewhat arbitrary (e.g., either L1 or L2 can be large)provided that L1L2o 1, which is the key condition for thetheorem to hold [and is used to obtain (1L1L2) 1 in thebounds (49)].

In the special case where the input–output spaces Uand Y are both the L2 space, a similar criterion based onthe system passivity property can be obtained [9,17]. Inthis case, an inner product between any two vectors in thespace is defined by

hx; Zi¼Z 1

t0

xTðtÞZðtÞdt

Theorem 29 (Passivity Stability Theorem). If there exist fourconstants, L1, L2, M1, M2, with L1þL240, such that

he1;S1ðe1ÞiL1jje1jj22þM1

he2;S2ðe2ÞiL2jjS2ðe2Þjj22þM2

(ð50Þ

then the closed-loop system (47) is BIBO-stable.

As mentioned, the main disadvantage of this criterionis its overconservativity in providing the sufficient condi-tions for the BIBO stability. One resolution is to transformthe system into the Lur’e structure, and then apply thecircle or Popov criterion under the sector condition (if itcan be satisfied), which can usually lead to less conserva-tive stability conditions.

8.2. Contraction Mapping Theorem

The small-gain theorem discussed above, by nature, is akind of contraction mapping theorem. The contractionmapping theorem can be used to determine the BIBO sta-bility property of a system described by a map in variousforms, provided that the system (or the map) is appropri-ately formulated. The following is a typical (global) con-traction mapping theorem.

Theorem 30 (Contraction Mapping Theorem). If the opera-tor norm of the input–output map S, defined on Rn, sati-sfies |||S||| o 1, then the system mapping

yðtÞ¼SðyðtÞÞ þ c

has a unique solution for any constant vector c 2 Rn. Thissolution satisfies

y ð1 jjjSjjjÞ1 ck k

In particular, the solution of the equation

ykþ 1¼SðykÞ; y0 2 Rn; k¼ 0; 1; . . .

satisfies

jjykjj ! 0 as k!1

9. CONCLUDING REMARKS

This article has offered a brief introduction to the basictheory and methodology of Lyapunov stability, orbital sta-bility, structural stability, and input–output stability fornonlinear dynamical systems. More subtle details for sta-bility analysis of general dynamical systems can be foundin the literature [3,5,9,11–13,16,17,19,20,23,26,27,29,30].In cases where control was explicitly involved, stabilityand stabilization issues have also been studied[8,17,25,28,33].

Several important classes of nonlinear (control) sys-tems have been omitted in the discussion of various sta-bility issues: some general functional systems such as


systems with time delays [14], measure ordinary differen-tial equations such as systems with impulses [2,31], andsome weakly nonlinear systems such as piecewise linearand switching (non)linear systems. Discussion of more ad-vanced nonlinear systems, such as singular nonlinear sys-tems (perhaps with time delays), infinite-dimensional(non)linear systems, spatiotemporal systems describedby nonlinear partial differential equations, and nonlinearstochastic (control) systems, are beyond the scope of thiselementary expository article.

Finally, for more recent studies of system stabilitytheories, the reader is referred to the reviews edited byMartynyuk [22].

BIBLIOGRAPHY

1. D. P. Atherton, Nonlinear Control Engineering (student ed.),Van Nostrand Reinhold, New York, 1982.

2. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect:Stability, Theory and Applications, Ellis Horwood, Chichester(UK), 1989.

3. N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical

Systems, Springer-Verlag, Berlin, 1970.

4. F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary

Differential Equations: An Introduction, 1969; Dover, NewYork, 1989.

5. L. Cesari, Asymptotic Behavior and Stability Problems in Or-dinary Differential Equations, Springer-Verlag, New York,1971.

6. G. Chen, G. Chen, and S. H. Hsu, Linear Stochastic Control

Systems, CRC Press, Boca Raton, FL, 1995.

7. G. Chen and X. Dong, From Chaos to Order: Methodologies,Perspectives and Applications, World Scientific, Singapore,1998.

8. R. J. P. de Figueiredo and G. Chen, Nonlinear Feedback Con-

trol Systems: An Operator Theory Approach, Academic Press,San Diego, 1993.

9. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–

Output Properties, Academic Press, New York, 1975.

10. H. Freeman, Discrete-Time Systems: An Introduction to the

Theory, Wiley, New York, 1965.

11. P. Glendinning, Stability, Instability and Chaos, CambridgeUniv. Press, New York, 1994.

12. R. Grimshaw, Nonlinear Ordinary Differential Equations,CRC Press, Boca Raton, FL, 1993.

13. W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967.

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15. F. C. Hoppensteadt, Analysis and Simulation of Chaotic

Systems, Springer-Verlag, New York, 1977.

16. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential

Equations, 2nd ed., Oxford Univ. Press, New York, 1987.

17. H. K. Khalil, Nonlinear Systems, 2nd ed., Prentice-Hall,Upper Saddle River, NJ, 1996.

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ear Systems, Kluwer Academic, Boston, 1991.

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20. S. Lefschetz, Differential Equations: Geometric Theory, 2nded., 1963; Dover, New York, 1977.

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(100th Anniversary), Taylor & Francis, London, 1992.

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of the 20th Century, Taylor & Francis, London, 2003.

23. D. R. Merkin, Introduction to the Theory of Stability, Spring-er-Verlag, New York, 1997.

24. A. N. Michel and R. K. Miller, Qualitative Analysis of LargeScale Dynamical Systems, Academic Press, New York, 1977.

25. N. Minorsky, Theory of Nonlinear Control Systems, New York:McGraw-Hill, 1969.

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Control, Prentice-Hall, Englewood Cliffs, NJ, 1991.

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of Differential Equations, Birkhaser, Boston, 1992.

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Equations, World Scientific, Singapore, 1995.

32. D. D. Siljack, Large Scale Dynamic Systems: Stability and

Structure, North-Holland, Amsterdam, 1978.

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Finite Dimensional Systems, 2nd ed., Springer-Verlag,New York, 1998.

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Systems, Springer-Verlag, Berlin, 1990.

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STANDING WAVE METERS ANDNETWORK ANALYZERS

BURKHARD SCHIEK

ANDREAS GRONEFELD

Ruhr-Universitat Bochum

1. MEASUREMENT OF NETWORKS ATMICROWAVE FREQUENCIES

Determining the characteristics of electrical networks(network analysis) is an important measurement toolboth for developers of electrical circuits and systems andin production testing. The measurement of network char-acteristics at microwave frequencies employs differentconcepts than are used at lower frequencies, where mea-suring port voltages and currents readily allows determi-nation of impedance (Z) or admittance (Y) parameters. Athigher frequencies, neither high-impedance voltageprobes nor low-impedance current measurements arefeasible, because parasitic capacitance and inductance of

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