D∞ 1 | 17 | Stability Domains | Lyubomir T. Gruyitch, Jean-Pierre Rich

ABSTRACT

In order to maximize the righthand side of inequality within the con

straint we have to choose then p

has to verify

Any domain X

is convenient as an initial set and then domain X

dened by the

union of the X

such than p

veries is a possible set for practical stability

We also set X

see Fig

b second study let

s X

and we want to minimize X

then p

We have now to verify and

exp

e exp

Minimization of p

leads to the same result as in Example that is

The admissible initial set is then characterized by

This leads to the set X

Fig obtained by union of X

such that p

Figure Practical stability of with the settling time s with regard to f

g second study

Conclusions

This chapter was devoted to the vectornorms approach It appears that such a

tool addresses the dierent aspects of stability questions the stability properties

themselves including practical stability with settling time but also the related

question of the domains estimation and what was not presented here the ques

tion of constrained control that appears as a direct application of the preceding

invariance properties

Vector norms constitute a simple case of vector Lyapunov functions and the

main presented results can be enlarged to this more general class provided the con

sidered vector Lyapunov functions verify the following connectivity

property

the considered VLF p is regular radially unbounded and such that the

related following set

px cg

is connected for any positive c

which property is veried for vector norms because of the triangular inequality

Therefore the presented overvaluing comparison systems appear to be a general

and practical approach to complex systems

# On the one hand the actual remaining drawback of such an approach is the

loss of information that may appear if the overvaluing procedure is too strong

The solution of this point is linked to the improvement of the state#space

model conditioning However the loss of information is encountered in

disturbing

parameters to be taken into account

Lastly it is important to remark that this chapter was presented in the frame

work of nonlinear timeinvariant systems but the method of vector norms address

es more general classes of models for example nonlinear timevarying systems or

nonlinear timedelay systems with possible timevarying delays

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BLANK PAGE

Author index

Barbashin E A

Bellman R

Bhatia N P #

Bitsoris G

Borne P

Brayton R K

Chetaev N G ix

Chiang H D

Dini U #

Genesio R

Gentina JC

Gruji'c Lj T Gruyitch L T

Hahn W #

Halanay A

Hocking J G

Infante E F

Kalman R E

Kappel F

Knobloch H W

Koteliansky D M #

Krasovskii N N

Lagrange JL xix #

LaSalle J P ix

Laurent F

Lefschetz S ix

Lurie A I xiii #

Lyapunov A M ix xv xix xx

# #

Malkin I G

Martynyuk A A

Matrosov V M

Mawhin J

McClamroch H H

McShane E J

Michel A N

Nemytskii V V

Poincar'e H

Popov V M xix

Postnykov V N

Richard J P

Rouche N

Siljak D D

Spivak M

Stepanov V V

Szeg)o G P #

Thorp J S

Tong C H

Vanelli A

Vinograd R E

Walker J A

Wa-zewski T

Weiss L

Weissenberger S

Yoshizawa T

Young G S

Zubov V I

Subject index

X is unstable

limit point

neighbourhood xiii

limit point

Ounique boundedness

Ouniquely bounded # #

Ouniquely bounded neighbourhood

Ouniquely bounded neighbourhood of

a set

Ouniquely bounded set

# #

absolute stability #

absolute stability with nite attrac

tion time

absolutely stable

absolutely stable set

absolutely stable set on N

L with

nite attraction time

absolutely stable state

absolutely stable state onN

L with

nite attraction time

aggregation function

analytic function

approach viaOuniquely bounded sets

asymptotic stability xi #

asymptotic stability domain

asymptotic stability domain estimate

asymptotic stability domain of a set

asymptotic stability domain of a state

asymptotic stability in the whole

asymptotic stability of a set

asymptotically stable

asymptotically stable equilibrium state

asymptotically stable set

asymptotically stable set with respect

to motions

asymptotically stable state

attraction xi xii

attraction domain #

attraction domain of a set

attraction domain of the origin

attraction time xvi

attraction with nite attraction time

attractive

attractive X

attractive point

attractive set

attractive set with nite attraction time

attractive state

attractive state with nite attraction

time

backwardtime Eulerian derivative

backwardtime lower right lower left

Dini derivative of

backwardtime right left Dini deriva

tive of

backwardtime unique motion

backwardtime upper right upper left

Dini derivative of

Barbashin#Krasovskii criterion

Borne and Gentina criterion

Borne and Richard criterion

boundary of xiv

boundedness

boundedness of motions

centre w of unique boundedness

centre of unique boundedness

centre of uniquely bounded neighbour

hood

centre of uniquely bounded neighbour

hoodness of a set

closeness xix

closure of xiv

commutative group

compact set

comparison function

comparison function of the class

comparison functions

comparison scalar system

comparison system

complete global asymptotic stability

of sets

completely asymptotically stable in

the large

completely globally asymptotically

stable

completely globally asymptotically

stable set

completely globally asymptotically

stable state

completely globally stable set with

state with

nite attraction time

constructive algorithm

constructive computer Lyapunov func

tion generation

continuity of motions

continuous dependence

controllable

decomposition

decoupled

decoupled overvaluing system

desired output of the system

desired output vector xvi

dierential inequalities

Dini derivative

direct method of Lyapunov

disjoint

distance xix

distance function xvi

disturbance

domain xii

domain of asymptotic stability

domain of asymptotic stability of the

asymptotically stable set

domain of asymptotic stability with

nite attraction time

domain of attraction

domain of attraction of X

domain of attraction of a set

domain of exponential stability of a

set with respect to

domain of exponential stability of a

state with respect to

domain of practical contractive sta

bility with settling time of a

set with respect to

domain of practical contractive sta

bility with settling time of a

system with respect to

domain of practical contraction of a

state

settling time of a set with re

spect to

domain of practical contraction with

settling time of a system with

respect to

domain of practical stability of a set

with respect to

domain of practical stability of a sys

tem with respect to

domain of stability

domain of stability of a set

domain of system practical stability

with settling time

domain of the corresponding stability

property xx

domains of Lyapunov stability prop

erties

domains of practical contraction with

settling time

domains of practical stability

domains of practical stability with

settling time

domains of stability properties

dynamic behaviour

dynamical system #

dynamical systems

ellipsoid closed set xii

empty set xvi

equilibrium

equilibrium point #

equilibrium regime

equilibrium state

estimate E

of D

estimate domain xii

estimate of D

estimate of a domain

estimate of practical stability domain

estimate of the strict domain of ex

ponential stability of a set

estimate of the asymptotic stability

domain of a set

estimate of the asymptotic stability

domains

estimate of the domain of asymptotic

stability of X

estimate of the domain of asymp

totic stability with nite at

traction time

estimate of the domain of practical

contraction with settling

time of the system motions

with respect to

estimate of the domain of practical

stability of the system with

respect to

estimate of the domain of practical

stability with settling time

estimate of the exponential stability

domain

estimate of the exponential stability

domain of X

estimate of the exponential stability

domain of a set

estimate of the practical stability do

main

estimate of the stability domain

estimate of the stability domain of a

set

estimate of the strict asymptotic sta

bility domain #

estimate of the strict attraction do

main

estimate of the strict domain of

attraction of a set

estimate of the strict domain of sta

bility #

estimate of the strict domain of sta

bility of a set

estimate of the system practical sta

bility

bility with settling time with

respect to

Euclidean norm xvii

Eulerian derivative

existence

existence and uniqueness of motions

existence of generalised motions

existence of motion

exponential estimate

exponential stability

exponential stability domain estimate

exponential stability domain with re

spect to

exponential stability domains

exponential stability of X

exponentially stable

exponentially stable X

exponentially stable set

exponentially stable state

family of all Lurie functions xiii

nal time xvi

nite reachability time

nite time

nite time interval

forced regime

forwardtime

forwardtime Eulerian derivative

forwardtime lower right lower left

Dini derivative of

forwardtime right left Dini deriva

tive of

forwardtime unique generalised mo

tion

forwardtime unique motion

forwardtime upper right upper left

Dini derivative of

free regime

frequency matrix function

functional family #

general oneshot approach

generalised dynamical system

# #

generalised solution

generating function xiv #

# #

generating function of a uniquely

bounded set

generating function of an Ouniquely

bounded set

generation of a function

generation of a system Lyapunov func

tion

global asymptotic stability

global positive deniteness

global stability in the whole in the

large

globally asymptotically stable

globally asymptotically stable stable

in the whole state

globally asymptotically stable set

globally attractive set attractive in

the whole in the large

globally attractive set with nite at

traction time

globally attractive state attractive in

the whole in the large

globally attractive state with nite at

traction time

globally exponentially stable

globally exponentially stable set

exponentially stable in the

whole in the large

globally exponentially stable state

exponentially stable in the

whole in the large

globally Lipschitzian

globally negative denite

globally negative denite derivative

globally positive denite function

globally positive denite function with

respect to

globally positive denite radially un

bounded function

globally positive semidenite with re

spect to a set

globally stable stable in the whole

stable in the large

globally stable set with nite attrac

tion time

globally stable state with nite attrac

tion time

greatest smallest partial limit

group property

hermitian part

Holders norm

homogenous function

hyperbolic

hyperstability concept xix

ideal relay function xiv

importance eigenvalue xvi

importance value

importance vector xiv

increasing with respect to the diago

nal elements

independent scalar xiv

innite time interval

initial moment xiv

integral curve xv

interior of xiv

invariance

invariance features

invariance principle

invariance properties of limit sets

invariance properties of sets

invariance with respect to system mo

tions

invariant

points

invariant set relative to the system

inverse function

irreducible

Jacobian

Jacobian matrix

Knobloch#Kappel theorem

Koteliansky conditions #

Krasovskii criterion

Kronecker delta xvi

Lstability concept xix

Lagrange stability xix

Lagrange stable point relative to

Lagrange stable set relative to

LaSalle principle

leading principal minors

left lower semicontinuous

limit cycle

limit points

limit sets

linear part of the system

Lipschitz condition

Lipschitz constant

Lipschitz continuous

Lipschitz function

Lipschitzian

local overvaluingmatrix

local overvaluing pair

local overvaluing system

Lurie form

Lurie matrix xiii

Lurie matrix equations

Lurie nonlinearities

Lurie system #

Lyapunov closeness

Lyapunov function

Lyapunov function construction

Lyapunov functional family #

Lyapunov method

Lyapunov sense

Lyapunov stability

Lyapunov stability concept xix xx

Lyapunov stability criteria

Lyapunov stability domains

Lyapunov stability theory

Lyapunovs conditions

Lyapunovs denition of stability

Lyapunovs methodology for nonlin

ear systems

Lyapunovs methodology for timeinva

riant linear systems

Lyapunovs original denition

Lyapunovs original methodology

Mmatrix

matrix Popov criterion

matrix transfer function

max norm

maximal eigenvalue xvi

maximal Lyapunov function

measure of matrix

minimal eigenvalue xvi

motion xv # #

multiplicative semigroup

natural form

negative denite # #

negative denite function with respect

to a set on a set

negative denite matrix

negative deniteness #

negative limit point

negative limit set xiii

negative semidenite matrix

negatively invariant

negatively invariant set

negatively Lagrange stable

negatively precompact

negatively stable point

new methodology

nominal desired pair with respect

nominal desired regime with respect

nominal motion with respect to

nonattractive

nondenite

nonLyapunov stability concept

nonradially unbounded function

nonsemidenite

nonsingular

norm of matrix

novel development of the Lyapunov

method

observable

oneshot approach

oneshot construction

open neighbourhood xiii

open set

output vector xvi

overvaluation lemma

overvaluing comparison systems

overvaluing input vector

overvaluing matrix

overvaluing pair

overvaluing system

Pmatrix

parallelepipedic closed set xii

partial dierential equation

period

periodic regime

persistency ix xix

perturbed motions

polyhedral neighbourhood

Popov criterion

Popov frequency approach

positive denite

# #

positive denite function with respect

positive denite matrix

positive denite on

positive denite with respect to

positive denite with respect to a set

positive deniteness

positive deniteness criterion

positive deniteness with respect

positive function

positive importance vector

positive invariance

positive invariant

positive limit

positive limit point

positive limit set xiii

positive lower limit

positive semidenite matrix

positive semidenite with respect

positive semidenite with respect to

a set

positive semidenite with respect to

a set in the whole

positive semidenite with respect to

a set on a set

positively negatively Lagrange

stable

positively negatively precompact

positively invariant #

positively invariant relative to

positively invariant set

positively Lagrange stable

positively precompact

positively stable point

positively stable set

practical contraction with settling

time

practical Lyapunov sense

practical stability

practical stability concept xix

practical stability criteria

practical stability domains

practical stability with settling time

with regard to

practical system stability with settling

time

practically contractively stable with

settling time with respect

practically stable with respect to

practically stable with settling time

with respect to

precompact

precompact generalised motion

precompact relative to

precompact set relative to

precompactness

principal minors

quadratic form

qualitative features of stability do

mains properties

quasiincreasing

quasiincreasing function

RVN # regular vector norm

radial unboundedness

radially increasing

radially increasing on a neighbour

hood

radially increasing on the boundary of

a set

radially increasing on the boundary of

a set to

radially increasing positive denite

function on a set

radially increasing with respect to

radially increasing with respect to set

on its neighbourhood

radially unbounded

radially unbounded globally positive

denite

radially unbounded with respect to

rational function

recursive equations

recursive relation

region of attraction

regular

relative to the system

rhomboid closed set xii

right upper right lower limit

saturation function xiv

scalar Lyapunov function xv

second method of Lyapunov

semideniteness property

set S

of the equilibrium states

set of admitted system states

set of all the equilibrium points xiv

set of permitted inputs

set of the equilibrium states

settling time xvi

sign denite function

sign function xiv

sign semidenite function

simplicial cone

smoothness property

solution motion of the system

solution function

solutions

specic smoothness property

stability xii xix

stability domain # #

stability domain D

A of the set A

stability domain estimate

stability domain of X

stability domain of a set

stability in the Lyapunov sense

stability of X

stability of a set

stability of equilibrium points

stability of motion

stability problem

stability properties of sets

stability theorems

stability with nite attraction time

stable # #

stable closed set

stable equilibrium state

stable limit cycle

stable point

stable set

stable set with nite attraction time

stable state with nite attraction time

stable unperturbed motion

stable with respect to

state portrait

state variable xv

state vector xv

stationary point

stationary regime

stationary state

steady state

strict asymptotic stability domain

strict asymptotic stability domain of

a set

strict asymptotic stability domain of

a state

strict attraction domain

strict domain of attraction of a set

strict domain of exponential stability

of a set with respect to

strict domain of exponential stability

of a respect

strict domain of stability

strict exponential stability domain

strict Lyapunov sense

strict stability domain

strong smoothness property #

surjective

symmetric matrix

symmetric part

system

system aggregation function

system is practically contractive with

settling time with respect

system is practically stable with re

spect to

system Lyapunov function

systemmotions

system regimes

system solutions

systems mathematical model

systems with continuous motions

time interval xix

timeinvariant continuoustime sys

tems

timevarying continuoustime sys

tems

timevarying systems

trajectory xv

twostage approach

uniformly continuous

uniformly nonsingular monotone

unique generalised motion

unique boundedness

unique equilibrium state

unique generalised motion

unique motion #

unique solution

uniquely bounded xiv #

uniquely bounded neighbourhood #

uniquely bounded neighbourhood of a

set

uniquely bounded set

uniqueness

uniqueness of motions

unperturbed motion

unstable

unstable equilibrium in the asymp

totic stability domain of a set

unstable unperturbed motion

Van der Pol equation

Vanelli#Vidyasagar approach

Vanelli#Vidyasagar recursive algorithm

Vanelli#Vidyasagar results

Vanelli#Vidyasagar theorem

vector Lyapunov functions

vector norm

Wa-zewski conditions

weak smoothness property #

weakly invariant #

weakly invariant set relative to the

system

with respect to a set

Yakubovich#Kalman lemma

Yoshizawa criterion

Zmatrix

Zubov theorem