Asymptotic stability of linear dynamic time-delay systems

with parameter uncertainty of interval type

Svetlana P.Sokolova, Ruslan S.Ivlev

Institute of Informatics and Control Problems,

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

(sokolova_sv@mail.ru,    ivlevruslan@newmail.ru)

 

This paper is a development of the paper [1] for the class of interval-specified systems the perturbed motion of which is given in the state space by differential time-delay inclusion having the following vector-matrix view:

,                  ,           ,           (1)

where   is an independent variable (time);  is an initial time moment;  is a delay;  is the states vector,  are continuous functions at , ; ;  are interval matrices, ; , ;   is the set of all real intervals [9,10], ;  is an initial vector-function whose components belong to the space of continuous functions at .

Throughout this paper a mathematical model of the system written out as (1) will be understood as the family of mathematical models of the systems

            ,                   ,           ,           (2)

for which  , ,  in the formal view all just said above will be written down as

.   (3)

Definition 1. The interval-specified system (1) is said to possess the property of the asymptotic stability, if any system (2), where  and  is asymptotic stable, i.e. for any matrices ,  and for any  one can give such , that at any  and at any initial functions  given at the segment  and satisfying the condition , the solution  of the system (2) satisfies the conditions

           

and

            ,

where .

We need to obtain the conditions under which the interval-specified time-delay system (1) possesses the asymptotic stability property in the sense indicated above.

The direct method of Lyapunov and the idea by N.N. Krasovsky[2] who proposed to use functionals possessing similar properties instead of Lyapunov functions are in the base of the approach offered in the paper to solve the task of investigating the asymptotic stability of interval-specified time-delay systems (1).

Definition 2. [15] The functional  is said to be a positive definite, if there exists some continuous functions  such, that when  and

           

A negative definite functional is defined similarly.

Definition 3. An interval square matrix , , ,  is said to be positive definite and written down , if any matrix  is positive definite, i.e.  the square-law form    , .

Definition 4. A set of matrices

           

where inequality sign is understood componentwise, is said to be a symmetric interval matrix and written down .

Definition 5.  A set of matrices  , defined according to the expression

,                    (4)

is called a tolerable solution set of Lyapunov interval matrix equation [1]

            .                                                                                             (5)

The following theorem is proved in the paper.

Valid the following theorem is.

Theorem 2. Let for the given interval matrix A and some interval positive definite symmetric matrix  the tolerable solution set (12) be not empty, i.e. , some symmetric matrix  is positive definite, and there exist such constants , , that the following interval matrix

           

is negative definite, where , then the interval-specified time-delay system (1) is asymptotic stable.

            An application of the functional of Lyapunov-Krasovsky for solving the task of investigating the asymptotic stability of interval-specified time-delay systems (1) is demonstrated with a numeric example in the paper.

 

References

1. Ivlev R.S. Asymptotic stability of linear interval-specified systems// Problems of Nonlinear Analysis in Engineer Systems, № 14, V. 7, 2001, Kazan, p. 55-63. (in Russian)

2. Krasovsky N.N. Some tasks of motion control theory. M.:Physmathgis. 1959. 211 p. (in Russian)