Existence and determination of the set of Metzler matrices for given stable polynomials

Determination of the state space equations for a given transfer matrix is a classical problem, called the realization problem, which has been addressed in many papers and books (Farina and Rinaldi, 2000; Benvenuti and Farina, 2004; Kaczorek, 1992; 2009b; 2011d; 2012; Shaker and Dixon, 1977). An overview on the positive realization problem is given by Farina and Rinaldi (2000), Kaczorek (2002), as well as Benvenuti and Farina (2004). The realization problem for positive continuous-time and discrete-time linear systems was considered by Kaczorek (2006a; 2006b; 2011a; 2011b; 2006c; 2004; 2011c) along with the positive realization problem for discrete-time systems with delays (Kaczorek, 2006c; 2004; 2005). Fractional positive linear systems were addressed by Kaczorek (2008c; 2009a; 2011d), together with the realization problem for fractional linear systems (Kaczorek, 2008a) and for positive 2D hybrid systems (Kaczorek, 2008b). A method based on similarity transformation of the standard realization to the discrete positive one was proposed (Kaczorek, 2011c), and conditions for the existence of a positive stable realization with a system Metzler matrix for a transfer function were established (Kaczorek, 2011a). The problem of determination of the set of Metzler matrices for given stable polynomials was formulated and partly solved by Kaczorek (2012). It is well known (Farina and Rinaldi, 2000; Kaczorek, 2002; 1992) that to find a realization for a given transfer function, first we have to find a state matrix for a given denominator of the transfer function. In this paper the problem of the existence and determination of the set of Metzler matrices for a given stable polynomial will be established and solved. Necessary and sufficient conditions will be established for the existence of the set of Metzler matrices for a given stable polynomial and a procedure will be proposed for finding the desired set of Metzler matrices. The paper is organized as follows. In Section 2 some preliminaries concerning positive stable continuous-time linear systems are recalled and the problem formulation is given. The problem solution is presented in Section 3, which consists of four subsections. In Section 3.1 the problem is solved for second-order stable polynomials, and in Section 3.2 and 3.3 for thirdand fourthorder stable polynomials. The general case is addressed in Section 3.4. Concluding remarks are given in Section 4. The following notation will be used: R is the set of real numbers, Rn×m is the set of n × m real matrices,