Finite spectrum assignment of multi-input systems with non-commensurate delays

Abstract

We present a new framework for finite spectrum assignment for multi-input systems with non-commensurate delays using an algebraic approach over multidimensional polynomial matrices. By focusing on the solvability of a Bezout equation over multidimensional polynomial matrices, we derive a necessary and sufficient condition for finite spectrum assignability under which a finite number of spectra can be assigned by a control law using a ring of entire functions, i.e. Laplace transforms of all exponential time functions with compact support. Furthermore, using a solution to the Bezout equation, we present a design method for a controller that achieves finite spectrum assignment.

Keywords:

• finite spectrum assignment
• non-commensurate delays
• multi-input system
• Bezout equation

Notes

Notes

1.  is a first-order exponential entire function in s. Thus, if it has no zeros, it must be of the form e ^qs+r from Hadamard separation theorem (Titchmarsh 1976). This contradicts Proposition 2.5. Thus, it has zeros.

2. For brevity, we write as z (s _k ) in the proof of Proposition 3.1.

3. To take a polynomial part, before the division manipulation, for the terms of degrees not less than ν in s of the (i, j)-element of , we reduce the degrees in s, if necessary, by using the relationship , which can be easily derived from (12).