The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.

Positive system problems have been developed in applications areas such as biological models, production systems, and economic applications. The realization problem for positive system has extensively been considered in many research papers as [

We will discuss the relationship between phase-type representation and positive realization by using the Perron-Frobenius theorem introduced in [

The connection between phase-type and positive realization has restrictively been proved in irreducible representation cases by remarking that it can be easily simplified to an irreducible case by discarding some states [

We will discuss the properties and characteristics, such as irreducibility, excitability, transparency, and stabilization introduced in [

An outline of the paper is as follows. In Section

Before proceeding, we introduce some basic notations. An

A matrix

Basic definitions and results of cone theory may be needed within this paper. A set

We discuss the phase-type distribution for a random variable

The probability density function (PDF), cumulative distribution function (CDF), and Laplace-Stieltjes transform (LST) of the PDF, respectively, are defined by

We note that these representations of phase-type distributions are equivalent to the state space realizations of linear systems in control. Let us consider single-input, single output linear time-invariant systems of the form

An integral function of

The augmented realization

Using a generalized version of the Perron-Frobenius theorem of nonnegative matrix theory, we derived a transformation from positive realization into phase-type realization under a constraint. The Perron-Frobenius theorem asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components [

Let

The solvability problem of the matrix equation

Assume that

There is a

Consider a continuous-time system with a positive realization

For the continuous time with a positive realization

We can choose a sufficiently large

Set

We will show that a positive realization of continuous-time positive system can be transformed into a phase-type representation normalized by a positive number. Under the irreducible assumption, it was proven that the positive realization can be transformed into phase-type representation [

Consider the continuous-time positive system with the positive realization

First, let us define an augmented realization

An important consequence of the above theorem is that an excitable positive realization can be transformed into the form of phase-type representation. Therefore, it is remarked that he concept of positive realizations is a superset of phase-type representations.

We discuss the properties and characteristics, such as stability, irreducibility, excitability, and transparency, in positive systems and phase-type distributions. A positive system with a positive realization

The irreducibility of a positive system can be defined in a similar manner. A positive system is irreducible if each state variable influences and is influenced by another variable [

The properties and characteristics of excitability and transparency are closely related to the reachability and observability of positive linear systems [

Let a transfer function

A positive system

Assume that a positive realization

Because

We discussed the method to remove unnecessary states in the unexcitable case. When the transposed realization is given by

A positive state-space realization

A discrete phase-type (DPT) distribution is the distribution of the time until one absorbing state in a discrete-state discrete-time Markov chain (DTMC) with

We can also discuss the realization between the DPH distributions and discrete-time positive systems in a similar manner. The discrete-time linear system is represented by

Assume that a realization

Because

We can easily deploy the properties and characteristics, such as irreducibility, excitability, transparency, and order reduction, in the discrete domain in a similar manner as in the continuous case. We omit the detailed exploration for the discrete case in this paper.

We considered the relation between the positive realization and the phase-type representation in continuous time and discrete time, respectively. Using the Perron-Frobenius theorem, it was shown that a phase-type representation is a special case with excitable constraint of the positive realization. We discussed their common properties and characteristics, such as irreducibility, excitability, transparency, stabilization, and order reduction. The connection between the phase-type renewal process and the feedback control of positive system was discussed. A lot of open problems related to positive system still remain and should be addressed in future research. The communities of control and probability theory can work together on solving the remaining same open problems.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by research fund of Chungnam National University.