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Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems

100%

Busłowicz M. , Ruszewski A.

tom Vol. 61, nr 4

779--786

In the paper the problems of practical stability and asymptotic stability of fractional discrete-time linear systems are addressed. Necessary and sufficient conditions for practical stability and for asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix of the system. In particular, it is shown that (similarly as in the case of fractional continuous-time linear systems) in the complex plane exists such a region, that location of all eigenvalues of the state matrix in this region is necessary and sufficient for asymptotic stability. The parametric description of boundary of this region is given. Moreover, it is shown that Schur stability of the state matrix (all eigenvalues have absolute values less than 1) is not necessary nor sufficient for asymptotic stability of the fractional discrete-time system. The considerations are illustrated by numerical examples.

Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix

100%

Busłowicz M.

tom Vol. 60, nr 4

809-814

In the paper the problems of practical stability and asymptotic stability of fractional discrete-time linear systems with a diagonal state matrix are addressed. Standard and positive systems are considered. Simple necessary and sufficient analytic conditions for practical stability and for asymptotic stability are established. The considerations are illustrated by numerical examples.

Fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative

100%

Kaczorek T. , Borawski K.

tom 26

nr 3

533-541

The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.

Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems

86%

Kaczorek T.

tom Vol. 11

1--10

The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.

Transformations of linear standard systems to positive asymptotically stable linear ones

86%

Kaczorek T.

tom Vol. 34, no. 3

341--348

New approaches to transformations of linear continuous-time systems to their positive asymptotically stable canonical controllable (observable) forms are proposed. It is shown that, if the system matrix is nonsingular, then the desired transformation matrix can be chosen in block diagonal form. Procedures for the computation of the transformation matrices are proposed and illustrated with simple numerical examples.

Stability diamond for linear continuous-time systems

72%

Kurek J.

tom Vol. 45, nr 8-9

73-75

An algorithm for calculation of the robust stability region - a diamond - for state space models of continuous-time systems is presented. According to the proposed algorithm one can easily calculate the robust stability region in system-parameter space. Then, the stability region tor systems described by the differential equation is calculated. Numerical example illustrates the approach.

W pracy przedstawiono algorytm wyznaczania obszaru krzepkiej stabilności dla liniowych układów z czasem ciągłym, opisanych modelem w przestrzeni stanu. Proponowany algorytm pozwala na łatwe wyznaczenie obszaru stabilności w przestrzeni parametrów układu. Następnie pokazano jak wykorzystać proponowany algorytm dla układów opisanych zwykłym równaniem różniczkowym. Przedstawiono przykład numeryczny wyznaczania obszaru stabilności.

Eigenvalues assignment in uncontrollable linear systems

58%

Kaczorek T.

tom Vol. 70, nr 6

art. no. e141987

It is shown that in uncontrollable linear system x = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A − BK has the desired eigenvalues. The procedure is illustrated by simple example.