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T. Kaczorek

Stability of continuous-discrete linear systems described by the general model

New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete linear systems described by the general 2D model are established. A procedure for checking the asymptotic stability is proposed. The effectiveness of the procedure is demonstrated on examples.

Open access

T. Kaczorek

Asymptotic stability of positive fractional 2D linear systems

New necessary and sufficient conditions for the asymptotic stability of the positive fractional 2D systems are established. It is shown that the checking of the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems.

Open access

T. Kaczorek

Stability and stabilization of positive fractional linear systems by state-feedbacks

Notations of the practical stability and of the asymptotic stability of positive and cone fractional 1D and 2D linear systems are introduced. Necessary and sufficient conditions for the practical stability and the asymptotic stability of positive and cone fractional 1D and 2D linear systems are established. It is shown that the checking of the practical stability and asymptotic stability of positive 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems. Three LMI approaches are proposed for checking the stability of positive fractional linear systems. LMI approach is applied to compute gain matrices of state-feedbacks such that closed-loop systems are positive and asymptotically stable. The proposed methods are illustrated on numerical examples.

Open access

T. Kaczorek

Asymptotic stability of positive 2D linear systems with delays

New necessary and sufficient conditions for the asymptotic stability of positive 2D linear systems with delays described by the general model, Fornasini-Marchesini models and Roesser model are established. It is shown that checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to the checking of the asymptotic stability of corresponding positive 1D linear systems without delays. The efficiency of the new criterions is demonstrated on numerical examples.

Open access

T. Kaczorek

Stability of positive continuous-time linear systems with delays

The asymptotic stability of positive continuous-time linear systems with delays is addressed. It is shown that: 1) the asymptotic stability of the positive systems with delays is independent of their delays, 2) the checking of the asymptotic stability of the positive systems with delays can be reduced to checking of the asymptotic stability of positive systems without delays. Simple stability conditions for the positive systems with delays are give and illustrated by numerical examples.

Open access

T. Kaczorek

Positive linear systems with different fractional orders

A new class of positive linear systems with different fractional orders is introduced. A solution to the set of linear differential equations with different fractional orders is derived Necessary and sufficient conditions for the positivity of the fractional systems are established. It is shown that the linear electrical circuits composed of resistors, supercondensators, coils and voltage (current) sources are positive systems with different fractional orders.

Open access

T. Kaczorek

Computation of positive stable realizations for linear continuous-time systems

Conditions for the existence of positive stable realizations with system Metzler matrices for proper transfer function are established. It is shown that there exists a proper stable realization of transfer function of second order if and only if the transfer function has real negative poles. Sufficient conditions for the existence of positive stable realizations of transfer function of third order are established. A method based on the decomposition of transfer functions into the first, second and third orders transfer functions for computation of positive stable realizations is proposed. A method for computation of positive stable realizations of transfer functions with real negative poles and zeros is given.

Open access

T. Kaczorek

Positive fractional 2D continuous-discrete linear systems

A new class of positive fractional 2D continuous-discrete linear systems is introduced. The solution to the equations describing by the new class of systems is derived. Necessary and sufficient conditions for the positivity of the fractional 2D continuous-discrete linear systems are established.

Open access

T. Kaczorek

Positive fractional continuous-time linear systems with singular pencils

A method for checking the positivity and finding the solution to the positive fractional descriptor continuous-time linear systems with singular pencils is proposed. The method is based on elementary row and column operations of the fractional descriptor systems to equivalent standard systems with some algebraic constraints on state variables and inputs. Necessary and sufficient conditions for the positivity of the fractional descriptor systems are established.

Open access

T. Kaczorek

Abstract

The Caputo-Fabrizio definition of the fractional derivative is applied to minimum energy control of fractional positive continuous- time linear systems with bounded inputs. Conditions for the reachability of standard and positive fractional linear continuous-time systems are established. The minimum energy control problem for the fractional positive linear systems with bounded inputs is formulated and solved.