τa (X ) τ ;A) ∆ ∆ 0 ||X || 0ρ(X ,A) a) τ ;A)∈R ∈N(∆,A) | 5

ABSTRACT

Denition a The state X of the system is absolutely stable on

L with the nite attraction time

if and only if it is globally stable with

the nite attraction time

f for every f N

b The state X of the system is absolutely stable with the nite attrac

tion time

on N

L if and only if a holds and sup

f f N

for every X

Denition a A set A is absolutely stable on N

L with the nite attraction

time

if and only if it is globally stable with the nite attraction time

f for every f N

b A set A is absolutely stable with the nite attraction time

on N

L if and

only if a holds and sup

f f N

L for every X

Comments on the practical stability concept

Introductory comments

The practical stability concept aims at determining closeness in a practical engi

neering sense at clarifying the system behaviour either in the free or in a forced

regime over a time interval that can be bounded and for initial states of practical

engineering signicance The closeness can be dened via a suitably accepted dis

Figure The state portrait of the system of Example and the sets X

and X

in the sequel for the system

fX i f

where X

and i

i is an input vector function

Since the Lyapunov stability concept cannot meet the goals described above then

Chetaev emphasised the need for a nonLyapunov stability concept La Salle

and Lefschetz were probably the rst to introduce a nonLyapunov stability

concept in the western literature

Denition of practical stability

By referring to LaSalle and Lefschetz Weiss and Infante Michel

# Gruji'c # and Martynyuk we shall accept the following prac

tical stability denition in which

and I is the family of inputs of our

interest Besides a set X

is assumed to have a nonempty interior

Denition The system is practically stable with respect to f X

Ig if and only if its motions obey XtX

i X

for every tX

Comment Let the system of Example Section be reconsidered

j jx

j X

Figure The state portrait of the system of Comment and the sets X

and X

fX X

j jx

j g

fX X

j jx

j g

and

I f g

then for the system to be practically stable with respect to fX

Ig it is both

necessary and sucient that is suciently small

If it is not satised

ie if Fig then the system is not practically stable with respect to

Ig Notice again that X is asymptotically stable

Comment Let X

and let the system be specied in the following

form

j jx

j X

Evidently I f g If

fX X

j jx

j g

and

fX X

j jx

j g

then the system is practically stable with respect to fX

Ig for any

despite X is unstable Fig The same holds if we redene X

and X

that X

fX X

j jx

j g for any

Figure The system is practically stable with respect to fX

Figure The system is practically contractive with the settling time

with respect

to fX

Denition of practical contraction with settling time

The notion of practical stability does not reect any contraction property which is

expressed by the following in which T

Denition Motions of the system are practically contractive or for

short the system is practically contractive with settling time

with respect

to fX

Ig if and only if XtX

i X

for every tX

i T

The dierence between practical stability and practical contraction with settling

time

is illustrated by Fig and Fig

Figure The system is practically stable with the settling time

with respect to

Denition of practical stability with settling time

Denitions and combined imply the following notion

Denition The system is practically contractively stable for short

practically stable with the settling time

with respect to f X

if and only if both

a XtX

i X

for every t

and

b XtX

i X

for every t T

hold for every X

i X

This denition is illustrated by Fig

BLANK PAGE

Chapter

Stability domain concepts

Introductory comments

In order to get complete information about the causality between initial states and

systems motions concepts of domains of various stability properties were intro

duced In the framework of the Lyapunov stability the notion of attraction domain

of the origin was dened by Zubov and Hahn and notions of the stabil

ity domain and the asymptotic stability domain were dened by Gruji'c

# and used by Gruji'c et al # The concept of practical stability

domains was introduced by Gruji'c

In the literature eg LaSalle and Lefschetz and Zubov the notion of

region of asymptotic stability has been used in the sense of the attraction domain

In what follows the dierence between them will be claried

Domains of Lyapunov stability properties

The notion of domain

The term domain denotes a set that can be but need not be open or closed

Domains of Lyapunov stability properties will be called for short Lyapunov

stability domains in a general sense incorporating domains of stability of attraction

and of asymptotic stability In the closer sense the notion Lyapunov stability

domain will be used for the domain of stability for short the stability domain

Lyapunov stability domains will be studied herein in the framework of time

invariant continuoustime nonlinear systems governed by

with possibly certain specic features that will be described when they are needed

In the literature eg LaSalle and Lefschetz the notion region has been

to a

property

Therefore we shall use the term domain in general rather than region In case

a domain is open and connected then we can also call it a region

Denitions of stability domains

The denitions of stability domains were introduced to comply with the denitions

of stability of a state and of a set

Denition a The state X of the system has the stability domain

denoted by D

if and only if both

i for every

there is a neighbourhood D

of X such that

kXtX

k for all t

holds provided only that X

and

ii D

b The state X of the system has the strict domain of stability the

strict stability domain D

if and only if

i it has the domain D

of stability

ii D

is the largest connected neighbourhood of X which is subset

of D

for every

iii D

Comment The state X of the system has the stability domain i

it is stable due to Denition and Denition Section Moreover

it is globally stable i its strict stability domain D

is its whole state space

The signicance of the latter is the following for arbitrary X

there is possibly suciently large

such that kXtX

k for all t

In other words system motions are bounded on

for every initial state X

Since D

can be disconnected and D

cannot then the latter rather than the

former corresponds to B

for every

In other words D

is the domain

of stability in the strict Lyapunov sense while D

is the domain of stability in a

wider more practical Lyapunov sense

In order to illustrate the preceding denition let us consider the following ex

ample

Example Let a second order system be in the following form

j jx

j X

It has the innite set S

of equilibrium states

For any

the maximal denoted by

Denition Section obeys

For any

the maximal

However

kXtX

k t

i X

where

X X

j jx

which is illustrated by Fig Evidently S

It is now obvious Fig that for

so that

In this example the stability domain is the closed set S

X X

j jx

which is showed in Fig f The strict stability domain D

equals D

Denition a A set A

of states of the system has the stability

domain denoted by D

A if and only if both

i for every

there is a neighbourhood D

A of A such that

XtX

A for all t

provided only X

and

ii D

A D

b A set A

of the states of system has the strict domain of stability

the strict stability domain denoted by D

A if and only if

i it has the stability domain D

ii D

A is the largest connected neighbourhood of A which is a subset of

A D

A for all

iii D

A D

a b

c d

e f

Figure Dependence of D

Figure The state portrait of the system

X jx

j jx

j X The

shaded area together with its boundary represents the set A

Example Let the second order system be specied by

j jx

j X

We are interested in stability of the set A

X X

j jx

The state portrait of the system is given in Fig The set S

of the equilibrium

states of the system is found as

X X

X or jx

j jx

j or jx

j jx

The graphical analysis of dependence of D

A on is presented in Fig

In this example Denition Section

However

kXtX

k t

i X

N A

N A S