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Tadeusz Kaczorek

The Choice of the Forms of Lyapunov Functions for a Positive 2D Roesser Model

The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix AT PA - P is negative definite. The theoretical deliberations will be illustrated by numerical examples.

Open access

Tadeusz Kaczorek

New stability conditions for positive continuous-discrete 2D linear systems

New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.

Open access

Tadeusz Kaczorek

Positivity of descriptor linear systems with regular pencils

The positivity of descriptor continuous-time and discrete-time linear systems with regular pencils are addressed. Such systems can be reduced to standard linear systems and can be decomposed into dynamical and static parts. Two definitions of the positive systems are proposed. It is shown that the definitions are not equivalent. Conditions for the positivity of the systems and the relationship between two classes of positive systems are established. The considerations are illustrated by examples of electrical circuits and numerical examples.

Open access

Tadeusz Kaczorek

Positive Partial Realization Problem for Linear Discrete-Time Systems

A partial realization problem for positive linear discrete-time systems is addressed. Sufficient conditions for the existence of its solution are established. A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed. The procedure is illustrated by four numerical examples.

Open access

Tadeusz Kaczorek

Fractional Positive Continuous-Time Linear Systems and Their Reachability

A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.

Open access

Tadeusz Kaczorek

Decomposition of the pairs (A, B) and (A, C) of positive discrete-time linear systems

A new test for checking the reachability (observability) of positive discrete-time linear systems is proposed. Conditions are established under which the unreachable pair (A, B) and the unobservable pair (A, C) of positive discrete-time system can be decomposed into reachable and unreachable parts and observable and unobservable parts, respectively. It is shown that the transfer matrix of the positive system is equal to the transfer matrix of its reachable (observable) part.

Open access

Tadeusz Kaczorek

Positive stable realizations with system Metzler matrices

Conditions 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.

Open access

Tadeusz Kaczorek

Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm

Necessary and sufficient conditions for the positivity of descriptor continuous-time and discrete-time linear systems are established. The shuffle algorithm is applied to transform the state equations of the descriptor systems to their equivalent form for which necessary and sufficient conditions for their positivity have been derived. A procedure for checking the positivity of the descriptor systems is proposed and illustrated by numerical examples.

Open access

Tadeusz Kaczorek

Inverse systems of linear systems

The concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.

Open access

Tadeusz Kaczorek

Positive electrical circuits and their reachability

Conditions for the positivity of linear electrical circuits composed of resistances, coils, capacitors and voltage (current) sources are established. It is shown that: 1) the electrical circuit composed of resistors, coils and voltage source is positive for any values of their resistances, inductances and source voltages if and only if the number of coils is less or equal to the number of its linearly independent meshes, 2) the electrical circuit is not positive for any values of its resistances, capacitances and source voltages if each its branch contains resistor, capacitor and voltage source, 3) the positive n-meshes electrical circuit with only one inductance in each linearly independent mesh is reachable if all resistances of branches belonging to two linearly independent meshes are zero.