To the question of the minimal number of inputs for linear differential algebraic systems

Abstract

We examine a control linear system of ordinary differential equations with an identically degenerate matrix coefficient of the derivative of the unknown vector function. We study the question of the minimal dimension of the control vector when the system could be fully controllable on any segment in the domain of definition. The problem is investigated in the cases of stationary systems and the systems with real analytic and smooth coefficients for which some structural forms can be defined.Keywords: differential algebraic equations, differential algebraic system, controllability, minimal number of inputs, structural form § 1. IntroductionWe study the control linear system of ordinary differential equationswhere A(t) and B(t) are known (n × n)-matrices; x(t) is an n-dimensional vector function to be sought;These systems are called differential algebraic equations (DAE). The unsolvability measure with respect to the derivative for some DAE is some integer r, 0 ≤ r ≤ n, that is called the index of the system. In the article we analyze the following question of great importance for the synthesis of control systems: What is the minimal number of inputs l (1 ≤ l ≤ n) when (1) is fully controllable on any segment of I. As is known, for the systems that are solved with respect to the derivative, i.e. of the form x (t) + B(t)x(t) + U (t)u(t) = 0, the minimal number of inputs ensuring controllability in some sense is equal to 1. In the stationary case the minimal number of inputs is equal to the number of nontrivial invariant polynomials of B (for instance, see [1,2]).One of the approaches to studying the solvability and the qualitative properties of solutions to DAE is based on a preliminary transformation of the system under study to the form allowing us to evaluate the structure of a general solution.The notion of a "strong standard canonical form" was introduced for the first time in [3] for studying linear DAE and its existence was proven for the systems with real analytic coefficients. This form in [4] is called the central canonical form (CCF). For studying reducibility questions, the so-called "split form" was introduced in the monograph [5] which is defined for the systems with smooth coefficients. To examine observability questions for DAE of the form (1), some analogs of a split form were used in [6]. For existence of the above forms, it is not required that the rank of the matrix coefficient of the derivative of the sought vector function is a constant. A common disadvantage is the absence of constructive algorithms for constructing these forms under sufficiently general conditions. In particular, the methods for transforming DAE in CCF are developed only for the systems whose coefficients are constants or polynomials [7].