An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index

This paper deals with singular systems of index k ≥ 1. Our main goal is to find a state-feedback such that the closed-loop system satis- fies the regularity condition and it is nonnegative and stable. In order to do that, the core-nilpotent decomposition of a square matrix is applied to the singular matrix of the system. Moreover, if the Drazin projector of this matrix is nonnegative then the previous decomposition allows us to write the core-part of the matrix in a specific block form. In addition, an algorithm to study this kind of systems via a state-feedback is designed.This paper has been partially supported by Ministry of Education of Spain [grant number DGI MTM2010-18228].Herrero Debón, A.; Francisco J. Ramírez; Thome, N. (2014). An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index. Linear and Multilinear Algebra. 1-11. https://doi.org/10.1080/03081087.2014.904559S11

Warunki stabilności oraz odpornej stabilności modelu ogólnego skalarnych liniowych układów ciągło-dyskretnych

The problems of asymptotic stability and robust stability of the general model of scalar linear dynamic continuous-discrete systems, standard and positive, are considered. Simple analytic conditions for asymptotic stability and for robust stability are given. These conditions are expressed in terms of coefficients of the model. The considerations are illustrated by numerical examples.W pracy rozpatrzono problemy stabilności oraz odpornej stabilności modelu ogólnego (1) skalarnych liniowych układów ciągło-dyskretnych, standardowych oraz dodatnich. Bazując na podanym w twierdzeniu 3 kryterium stabilności analizowanej klasy układów, wyprowadzono proste analityczne warunki asymptotycznej stabilności oraz odpornej stabilności. Warunki asymptotycznej stabilności oraz odpornej stabilności standardowego układu ciągło-dyskretnego podano w twierdzeniu 4 oraz w twierdzeniu 6, odpowiednio. Natomiast warunki asymptotycznej stabilności oraz odpornej stabilności dodatniego układu ciągło-dyskretnego podano w twierdzeniach 5 i 8, odpowiednio. Wszystkie warunki są wyrażone w terminach współczynników modelu (1) (lub wartości krańcowych przedziałów (13), z których te współczynniki mogą przyjmować swoje wartości). Rozważania zostały zilustrowane przykładami liczbowymi

Robust stability of a class of uncertain fractional order linear systems with pure delay

The paper considers the robust stability problem of uncertain continuous-time fractional order linear systems with pure delay in the following two cases: a) the state matrix is a linear convex combination of two known constant matrices, b) the state matrix is an interval matrix. It is shown that the system is robustly stable if and only if all the eigenvalues of the state matrix multiplied by delay in power equal to fractional order are located in the open stability region in the complex plane. Parametric description of boundary of this region is derived. In the case a) the necessary and sufficient computational condition for robust stability is established. This condition is given in terms of eigenvalue-loci of the state matrix, fractional order and time delay. In the case b) the method for determining the rectangle with sides parallel to the axes of the complex plane in which all the eigenvalues of interval matrix are located is given and the sufficient condition for robust stability is proposed. This condition is satisfied if the rectangle multiplied by delay in power equal to fractional order lie in the stability region. The considerations are illustrated by numerical examples

Controllability, reachability and minimum energy control of fractional discrete-time linear systems with multiple delays in state

In the paper the problems of controllability, reachability and minimum energy control of a fractional discrete-time linear system with delays in state are addressed. A general form of solution of the state equation of the system is given and necessary and sufficient conditions for controllability, reachability and minimum energy control are established. The problems are considered for systems with unbounded and bounded inputs. The considerations are illustrated by numerical examples. Influence of a value of the fractional order on an optimal value of the performance index of the minimum energy control is examined on an example

Stability conditions for linear continuous-time fractional-order state-delayed systems

The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example

Stability conditions for linear continuous-time fractional-order state-delayed systems

The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example

Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders

The stability problem of continuous-time linear systems described by the state equation consisting of n subsystems with different fractional orders of derivatives of the state variables has been considered. The methods for asymptotic stability checking have been given. The method proposed in the general case is based on the Argument Principle and it is similar to the modified Mikhailov stability criterion known from the stability theory of natural order systems. The considerations are illustrated by numerical examples

Simple conditions for robust stability of linear positive discrete-time systems with one delay

Simple new necessary and sufficient conditions for robust stability of the positive linear discrete-time systems with one delay in the general case and in the two special cases: 1) linear unity rank uncertainty structure, 2) linear uncertainty structure with non-negative perturbation matrices, are established. The conditions are based on the new simple criterion for asymptotic stability of the positive linear discrete-time systems with one delay, proved in the paper. The considerations are illustrated by numerical examples