Absolute stability of nonlinear dynamic systems
with parameter uncertainty of interval type
(part I)
R.S.Ivlev
Institute
of Informatics and Control Problems,
125
Pushkin Str., Almaty, Republic of Kazakhstan, 480100
1.
Introduction
In the
paper on the base of direct method of Lyapunov the absolute stability property
of the nonlinear dynamic parametric uncertain system is investigated. A
mathematical model of the system under investigation is given in the states
space as the nonlinear differential inclusions of the interval type:
ššššššššššš 
,šššššššššššššššššššššššššš 
,ššššš 
,ššššššššššššššššššššššššššššš (1)
where
š- states vector,š 
šare continuous
functions at 
, i.e. 
, 
, 
š- a constant interval
matrix of dimension 
, 
, 
š- the set of all real
intervals, 
, 
š- an interval vector
of dimension 
,
, 
, 
, 
š- control
action that satisfies the relation
ššššššššššš 
,ššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (2)
whereš 
š- a continuous differentiable function, 
, such, that true the following
inequality is
ššššššššššš 
,ššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (3)
where 
š- a real quadratic form of the variables 
šand 
; the value 
šis determined according to the expression
ššššššššššš 
,ššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (4)
where 
š- a vector of dimension 
.
Definition
1. Any absolute continuous function 
šis said to be a solution ofš
(1) - (4), if for some 
šand 
šit satisfies the following nonlinear system of differential equations
ššššššššššš 
,šššššššššššššššššš 
,ššššššššššššššššš 
.ššššššššššššššššš (5)
Definition
2. The system of the nonlinear differential
inclusions (1) - (4) is said to possess some property 
, if any systemš (5) for 
šand 
špossesses this property.
After choosing Lyapunov function in the form of
Lurye-Postnikov
ššššššššššš 
,šššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (6)
where 
, 
- a symmetric positive definite matrix, 
š- a
nonnegative number, and applying S-procedure the S-form has the view:
,šššššššššššššššššššššššššššššššššššššššš (7)
where 
š- a
nonnegative number, the vector 
šand the matrix
šis defined as
.
Definition
3. An interval square matrix 
, 
, 
, 
šis said to be positive definite and to write down 
, if any matrix 
šis positive definite, i.e. 
šthe quadratic form 
š
, 
.
Definition
4.š The set of
thin matrices
ššššššššššš 
,
where
the inequality sign is understood componentwise, is said to be a symmetric
interval matrix and write down 
.
Definition
5. The set of square thin matrices
ššššššššššššššššššššššš (16)
is
called the tolerable solution set of the interval matrix equation of Lyapunov
ššššššššššš 
.ššššššššššššššššššššššššššššššššššš šššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (17)
Definition
6. The interval
ššššššššššš 
is called a determinant of
an interval square matrix 
.
Theorem. Let
for the given interval matrix 
šand some interval symmetric positive definite matrix 
šthe tolerable solution set (16) of the interval matrix equation of
Lyapunov (17) be not empty, i.e. 
, some symmetric matrix belong to this
set 
, and there exist such nonnegative
numbers 
šand 
, that for the given interval vector 
šthe determinant of the following interval matrix
ššššššššššš 
šššššššššššššššššššššššššššššššššššššššš (18)
satisfies
the condition
ššššššššššš 
,šššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššššš (19)
then
the equilibrium position 
šof the interval specified system (1) - (4) is
absolute stable in the chosen class of nonlinearties.
In the
paper an algorithm for investigating the absolute stability of the system under
consideration in the chosen class of nonlinearities is shown.
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