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Tadeusz Kaczorek

Computation of initial conditions and inputs for given outputs of fractional and positive discrete-time linear systems

The problem of computation of initial conditions and inputs for given outputs of fractional standard and positive discrete-time linear systems is formulated and solved. Necessary and sufficient conditions for existence of solution to the problem are established. It is shown that there exist the unique solutions to the problem only if the pair (A, C) of the system is observable.

Open access

Tadeusz Kaczorek

Nilpotency indices of state matrices and linear dependence on time of state variable of electrical circuits with state feedbacks

Electrical circuits with state-feedbacks are addressed. It is shown that by suitable choice of the gain matrices of state-feedbacks it is possible to obtain the closed-loop system matrices with nilpotency indices equal to two and their state variables are linear functions of time. The considerations are illustrated by linear electrical circuits.

Open access

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

Computation of Realizations Composed of Dynamic and Static Parts of Improper Transfer Matrices

The problem of computing minimal realizations of a singular system decomposed into a standard dynamical system and a static system of a given improper transfer matrix is formulated and solved. A new notion of the minimal dynamical-static realization is introduced. It is shown that there always exists a minimal dynamical-static realization of a given improper transfer matrix. A procedure for the computation of a minimal dynamical-static realization for a given improper transfer matrix is proposed and illustrated by a numerical example.

Open access

Tadeusz Kaczorek

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.

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 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.

Open access

Tadeusz Kaczorek

Singular fractional linear systems and electrical circuits

A new class of singular fractional linear systems and electrical circuits is introduced. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and the Laplace transformation, the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional system if it contains at least one mesh consisting of branches only with an ideal supercapacitor and voltage sources or at least one node with branches with supercoils.

Open access

Tadeusz Kaczorek

Checking of the positivity of descriptor linear systems with singular pencils

A method for checking of the positivity of descriptor continuous-time and discrete-time linear systems with singular pencil is proposed. The method is based on elementary row and column operations on the matrices of descriptor systems. Necessary and sufficient conditions for the positivity of the descriptor systems are established.