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

Abstract

The problem of the existence and determination of the set of Metzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples.

Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers

“…In [23] it was shown that the matrix A 2 M 3 has real negative zero s 1 = -α 1 and two complex conjugate zeros (41) is not satisfied. In a similar way it can be shown that the fractional system with (41) and 1 0 < < α is asymptotically stable for and it is unstable.…”

A modified state variable diagram method for determination of positive realizations of linear continuous-time systems with delays

“…A method based on a similarity transformation of the standard realization to the discrete positive one was proposed by Kaczorek (2011c). Conditions for the existence of a positive stable realization with a system Metzler matrix for transfer function were established by Kaczorek (2011a), who also formulated and solved the problem of determination of the set of Metzler matrices for given stable polynomials (Kaczorek, 2012a).…”

“…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;2008;2011d;2012a;2012b;Shaker and Dixon, 1977). 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 given denominator of the transfer function.…”

A new approach to the realization problem for fractional discrete-time linear systems

“…Determination of the state space equations for given transfer matrices is a classical problem, called the realization problem, which has been addressed in many papers and books [1][2][3][4][5][6][7][8]. An overview of the positive realization problem is given in [1,2,6,9].…”