Absolute stability of nonlinear dynamic systems with parameter uncertainty of interval type

(part II)

R.S.Ivlev

Institute of Informatics and Control Problems,

125 Pushkin Str., Almaty, Republic of Kazakhstan, 480100

ivlevruslan@newmail.ru

 

In the second part of the paper the further investigation of the absolute stability of the nonlinear dynamic interval specified system of Lurye is fulfilled from the point of view of searching for new conditions of existing Lyapunov function enabling to obtain simple in calculation aspect algebraic criterion of the absolute stability.

The perturbed motion of the nonlinear dynamic system under investigation is described with the mathematical model having the following view in the states space:

љљљљљљљљљљљ ,љљљљљљљљљљљљљљљљљљљљљљљљљљ ,љљљљљ ,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (1)

where љ- states vector, љ- control action defined according to the relation

љљљљљљљљљљљ ,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

where љ- a continuous differentiable function, , satisfying the sector type restriction

,љљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)

or

љљљљљљљљљљљ ,

where , , and at љnecessarily ; the value љis determined according to the expression

љљљљљљљљљљљ ,љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (4)

where љ- a vector of dimension .

Definition 1. Any absolute continuous function љis said to be a solution ofљ (1) - (4), if for some љand љit satisfies the following nonlinear system of differential equations

љљљљљљљљљљљ ,љљљљљљљљљљљљљљљљљљ ,љљљљљљљљљљљљљљљљљ .љљљљљљљљљљљљљљљљљ (5)

Definition 2. The system of the nonlinear differential inclusions (1) - (4) is said to possess some property , if any systemљ (5) for љand љpossesses this property.

In the second part of the paper on the base of entered concepts of Lurye interval resolving system

where - an interval symmetric positive definite matrix, љ- a nonnegative number, and the generalized solution set of Lurye interval resolving system

the following theorem is proved.

Theorem 1. Let for the given interval matrix , interval vector , vector љand some interval symmetric positive definite matrix љthe following conditions be true:

1)      the generalized solution set (12) of Lurye interval resolving system (13) is not empty, i.e. ;

2)      the matrix is symmetric positive definite;

3)      the nonlinear algebraic equation: љrelative to љdoes not have real nonzero solutions;

4)      the scalar is nonnegative,

then the equilibrium position љof the system (1) - (4) under investigation is absolute stable in the chosen class of nonlinearities.

Some questions concerning the investigation of the nonemptiness of the generalized solution set of Lurye interval resolving system are considered in this part of the paper.