Krzysztof Rogowski

Fractional Linear Systems and Electrical Circuits

This monograph covers some selected problems of positive and fractional electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. The book consists of 8 chapters, 4 appendices and a list of references. Chapter 1 is devoted to fractional standard and positive continuous-time and discrete-time linear systems without and with delays. In chapter 2 the standard and positive fractional electrical circuits are considered and the fractional electrical circuits in transient states are analyzed. Descriptor linear electrical circuits and their properties are investigated in chapter 3, while chapter 4 is devoted to the stability of fractional standard and positive linear electrical circuits. The reachability, observability and reconstructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. The fractional linear electrical circuits with feedbacks are considered in chapter 6. In chapter 7 solutions of minimum energy control for standard and fractional systems with and without bounded inputs is presented. In chapter 8 the fractional continuous-time 2D linear systems described by the Roesser type models are investigated.

Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Positivity and stabilization of fractional 2D linear systems described by the Roesser model A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.

Analysis of Fractional Electrical Circuit Using Caputo and Conformable Derivative Definitions

The paper presents an analysis of the fractional electrical circuit in the transient state described by the fractional-order state-space equations. General solutions to the fractional state-space equations containing two types of definitions of fractional derivative: Caputo fractional order derivative and the Conformable Fractional Derivative (CFD) definitions. The solutions are given in three cases: (1) nonzero initial conditions and zero input signal, (2) the input signal in the form of a step function and zero initial conditions; (3) the general case where the input and initial conditions are nonzero. The results for voltages across the capacitors for derivatives of fractional orders equal \(\alpha =0.7; 0.8; 0.9\) are shown. The comparison of those two definitions is presented and discussed.

Positivity Analysis of Continuous 2D Fornasini-Marchesini Fractional Model

In the chapter continuous Fornasini-Marchesini type model containing partial fractional-order derivatives described by the Caputo definition will be considered. General solution formula to the state-space equations of the model will be given. Using this solution formula the positivity of such system will be analyzed and the conditions under which the system is internally positive will be derived. Considerations will be illustrated by numerical simulations.

Reachability of Standard and Fractional Continuous-Time Systems with Piecewise Constant Inputs

The reachability of standard and fractional-order continuous-time systems with piecewise constant inputs is addressed. Necessary and sufficient conditions for the existence of such piecewise constant inputs that steers the system from zero initial conditions to the given final state in desired time are derived and proved. As examples of such systems the electrical circuits with DC switched voltage sources are presented.

Minimum Energy Control of Electrical Circuits

The minimum energy control problem for standard linear systems was formulated and solved by J. Klamka in [117, 118, 119, 120, 121, 122, 123]. The Klamka’s method has been extended to new classes of linear systems [12, 13, 71, 82, 83, 98, 84, 85, 86, 101, 94, 96, 97, 99, 100, 124]