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On polynomial solutions of the linear stationary control system

Research paper by S. P. Zubov, E. V. Raetskaya, Le Hai Trung

Indexed on: 23 Nov '08Published on: 23 Nov '08Published in: Automation and Remote Control



Abstract

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x0 to an arbitrary finite state x(T) = xT by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p − 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.