Computation of transition matrices of positive linear electrical circuits

A method is proposed for calculation of transition matrices of positive electrical circuits. It is shown that if the transition matrix is presented as finite series of the Metzler matrix with real distinct eigenvalues then the coefficients of the series are nonnegative function of time. The method is applied to positive linear electrical circuits.

Metzler cyclic electric systems

In applications we meet with positive (nonnegative) systems. A system generated by a linear differential equation with a Metzler-matrix is called a nonnegative system. The paper shows the basic dynamic properties of nonnegative continuous-time systems. Considered was the example of an oscillating electrical circuit with a cyclic Metzler-matrix, enabling the correct physical interpretations. Cykliczne obwody elektryczne Metzlera Streszczenie: Układy dodatnie (nieujemne) można spotkać w wielu zastosowaniach. Układem nieujemnym jest układ generowany równaniem różniczkowym liniowym z macierzą Metzlera. W artykule przedstawiono podstawowe własności dynamiczne układów nieujemnych z czasem ciągłym. Rozważono przykład elektrycznego obwodu oscylacyjnego z cykliczną macierzą Metzlera umożliwiający odpowiednie interpretacje fizyczne. Słowa kluczowe: układy dodatnie, układy Metzlera, układy cykliczne, obwody elektryczne

Stability of fractional positive nonlinear systems

The conditions for positivity and stability of a class of fractional nonlinear continuous-time systems are established. It is assumed that the nonlinear vector function is continuous, satisfies the Lipschitz condition and the linear part is described by a Metzler matrix. The stability conditions are established by the use of an extension of the Lyapunov method to fractional positive nonlinear systems.

Positive stable realizations with system Metzler matrices

Positive stable realizations with system Metzler matricesConditions for the existence of positive stable realizations with system Metzler matrices for linear continuous-time systems are established. A procedure for finding a positive stable realization with system Metzler matrix based on similarity transformation of proper transfer matrices is proposed and demonstrated on numerical examples. It is shown that if the poles of stable transfer matrix are real then the classical Gilbert method can be used to find the positive stable realization.

On the Properties of Reachability, Observability, Controllability, and Constructibility of Discrete-Time Positive Time-Invariant Linear Systems with Aperiodic Choice of the Sampling Instants

This paper investigates the properties of reachability, observability, controllability, and constructibility of positive discrete-time linear time-invariant dynamic systems when the sampling instants are chosen aperiodically. Reachability and observability hold if and only if a relevant matrix defining each of those properties is monomial for the set of chosen sampling instants provided that the continuous-time system is positive. Controllability and constructibility hold globally only asymptotically under close conditions to the above ones guaranteeing reachability/observability provided that the matrix of dynamics of the continuous-time system, required to be a Metzler matrix for the system's positivity, is furthermore a stability matrix while they hold in finite time only for regions excluding the zero vector of the first orthant of the state space or output space, respectively. Some related properties can be deduced for continuous-time systems and for piecewise constant discrete-time ones from the above general framework.