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.
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 Citation1976). 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 (Equation12).