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Responses of positive standard and fractional linear systems and electrical circuits with derivatives of their inputs

100%

Kaczorek T.

tom Vol. 66, nr 4

419--425

Responses of positive standard and fractional continuous-time and discrete-time linear systems with derivatives of their inputs are presented herein. It is shown that the formulae for state vectors and outputs are also valid for their derivatives if the inputs and outputs and their derivatives of suitable order are zero for t = 0. Similar results are also shown for positive standard and fractional discrete-time linear systems.

Descriptor fractional linear systems with regular pencils

100%

Kaczorek T.

tom Vol. 23, no. 2

309--315

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems

100%

Kaczorek T.

tom Vol. 23, no. 3

501--506

Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.

Global stability of nonlinear feedback systems with fractional positive linear parts

84%

Kaczorek T.

tom Vol. 30, no. 3

493--499

The global (absolute) stability of nonlinear systems with fractional positive and not necessarily asymptotically stable linear parts and feedbacks is addressed. The characteristics u = f(e) of the nonlinear parts satisfy the condition k1e ≤ f(e) ≤ k2e for some positive k1 and k2. It is shown that the fractional nonlinear systems are globally asymptotically stable if the Nyquist plots of the fractional positive linear parts are located on the right-hand side of the circles (−1/k1,−1/k2).

Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation

84%

Sierociuk D. , Dzieliński A.

tom Vol. 16, no 1

129-140

This paper presents a generalization of the Kalman filter for linear and nonlinear fractional order discrete state-space systems. Linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified kalman filter for the linear case is called the fractional Kalman filter and its nonlinear extension is named the extended fractional Kalman filter. The background and motivations for using such techniques are given, and some algorithms are discussed. The paper also shows a simple numerical example of linear state estimation. Finally, as an example of nonlinear estimation, the paper discusses the possibility of using these algorithms for parameters and fractional order estimation for fractional order systems. Numerical examples of the use of these algorithms in a general nonlinear case are presented.

Minimum energy control of descriptor fractional discrete-time linear systems with two different fractional orders

84%

Sajewski Ł.

2017

tom Vol. 27, no. 1

33--41

Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass–Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.

Divisibility of the second-order minors of the nominators by minimal denominators of transfer matrices of cyclic fractional linear systems

67%

Kaczorek T.

tom Vol. 31, no. 4

627--633

The divisibility of the second-order minors of the numerators of transfer matrices by their minimal denominators for cyclic fractional linear systems is analyzed. It is shown that all nonzero second-order minors of the numerators of the transfer matrices are divisible by their minimal denominators if and only if the system matrices of fractional standard and descriptor linear systems are cyclic. The theorems are illustrated by examples of fractional standard and descriptor linear systems.