Minimal positive Markov realizations

TL;DR

This paper introduces a linear programming method to find minimal positive Markov realizations of transfer functions, establishing conditions under which this minimal dimension matches the overall minimal positive realization dimension.

Contribution

It proposes a novel linear programming approach for minimal positive Markov realizations and identifies conditions where this dimension is optimal.

Findings

Linear programming can compute minimal positive Markov realizations.

Minimum Markov realization dimension bounds the overall minimal positive realization.

Equality of the two dimensions holds for certain systems.

Abstract

Finding a positive state-space realization with the minimum dimension for a given transfer function is an open problem in control theory. In this paper, we focus on positive realizations in Markov form and propose a linear programming approach that computes them with a minimum dimension. Such minimum dimension of positive Markov realizations is an upper bound of the minimal positive realization dimension. However, we show that these two dimensions are equal for certain systems.