tailieunhanh - Polynomial solution of descriptor system

The aim of article is to prove that it is possible to find state function x(t) and controllability function u(t) of the descriptor systems EX'(t) = Bx(t) + Du(t) in which E, B, D are real matrices with size equivalent to state function and controllability vector in the type of polynomials of degree | ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO. 6(127).2018 41 POLYNOMIAL SOLUTION OF DESCRIPTOR SYSTEM Le Hai Trung University of Education - The University of Danang; lhtrung@ Abstract - The aim of article is to prove that it is possible to find state function x(t) and controllability function u (t ) of the descriptor systems Ex '(t ) = Bx(t ) + Du (t ) in which E, B, D are real matrices with size equivalent to state function and controllability vector in the type of polynomials of degree 2 p + 1. The basis of the 2. Results and Survey Research Consider the following lemma (см [7]): Lemma. The equation Cu = v, u R k , v R s , equivalent to system: is theory is a method to prove the cascade splitting to transform the original system into an equivalent system in the type x 'p (t ) = B p x p (t ) + D p z p (t ). In the final step, we obtain function Qv = 0 + u = C v + Pu , x p (t ) satisfying the condition and substituting this in the previous in which Pu − is an element in ker C. step. Hence continuing this process, we can find out the functions x(t ) and u (t ) of the initial descriptor system. Apply this lemma to equation (1) when C = D , then (1) is equivalent to the system: Key words - Descriptor systems; controllability function; state function; polynomial; differential algebraic equations 1. Rationale Consider the descriptor system, also known as the differential algebraic equation, as follows: (1) Ex(t ) = Bx(t ) + Du (t ) with E, B L( n , m ), D L( l , m ), x(t ) n , ; x(t ) is the state function and u(t ) is the controllability function. The system is called controllable in the interval [0, T ] if for any a, b in n , it exists the control function u(t ) so that its root x(t ) satisfies the following condition: x(0) = a, x(T ) = b (2) The problem of descriptor system has received the attention of many mathematicians around the world, such as Amit Ailon (see [8], [9]), S. P Zubova .