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EN
The global stability of electrical circuits composed of positive linear parts and nonlinear static element with given characteristic and positive gain feedbacks is investigated. New sufficient conditions for the global stability of this class of nonlinear positive electrical circuits are established. These new stability conditions are demonstrated on simples examples of positive nonlinear electrical circuits.
EN
Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems are established. It is shown that if the fractional system is pointwise complete in one step (q = 1), then it is also pointwise complete for q = 2, 3…
EN
The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.
EN
The global stability of discrete-time nonlinear systems with descriptor positive linear parts and positive scalar feedbacks is addressed. Sufficient conditions for the global stability of standard and fractional nonlinear systems are established. The effectiveness of these conditions is illustrated on numerical examples.
EN
The invariant properties of the stability, reachability, and transfer matrices of positive linear electrical circuits with integer and fractional orders are investigated. It is shown that the stability, reachability and transfer matrix of positive linear systems are invariant under their integer and fractional orders.
EN
The stability of positive linear continuous-time and discrete-time systems is analyzed by the use of the decomposition of the state matrices into symmetrical and antisymmetrical parts. It is shown that: 1) The state Metzler matrix of positive continuous-time linear system is Hurwitz if and only if its symmetrical part is Hurwitz; 2) The state matrix of positive linear discrete-time system is Schur if and only if its symmetrical part is Hurwitz. These results are extended to inverse matrices of the state matrices of the positive linear systems.
EN
The positivity and absolute stability of a class of nonlinear continuous-time and discrete-time systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.
EN
The positivity and absolute stability of a class of nonlinear continuous-time and discrete-time systems with nonpositive linear part are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.
9
Content available remote Normal positive linear systems and electrical circuits
EN
The notion of normal positive electrical circuits is introduced and some their specific properties are investigated. New state matrices of positive linear systems and electrical circuits are proposed and their properties are analyzed. It is shown that positive electrical circuits with diagonal state matrices are normal for all values of resistances, inductances and capacitances.
PL
W artykule zaproponowano pojęcie dodatniego obwodu elektrycznego oraz przeanalizowano specjalne własności dodatnich układów i obwodów elektrycznych. Wykazano, że dodatnie obwody elektryczne z diagonalnymi macierzami stanu są zawsze normalne dla wszystkich wartości rezystancji, indukcyjności i pojemności.
10
EN
The notion of normal positive electrical circuits is introduced and some their specific properties are investigated. New state matrices of positive linear systems and electrical circuits are proposed and their properties are analyzed. It is shown that positive electrical circuits with diagonal state matrices are normal for all values of resistances, inductances and capacitances.
PL
W artykule zaproponowano pojęcie dodatniego obwodu elektrycznego oraz przeanalizowano specjalne własności dodatnich układów i obwodów elektrycznych. Wykazano, że dodatnie obwody elektryczne z diagonalnymi macierzami stanu są zawsze normalne dla wszystkich wartości rezystancji, indukcyjności i pojemności.
11
EN
The positive linear continuous-time systems by the use of the Caputo and conformable fractional calculus are analyzed. The conditions for internally positive systems are presented. For the established electrical circuit, a solution fractional order state-space equation was developed for system with Caputo and conformable fractional derivative (CFD) definition.
EN
It is shown that the convex linear combination of the Hurwitz polynomials of positive linear systems is also the Hurwitz polynomial. The Kharitonov theorem is extended to the positive interval linear systems. It is also shown that the interval positive linear system described by state equation x ̇ = Ax, A ϵ ℜn×n, A1 ≥ A ≤ A2 is asymptotically stable if and only if the matrices Ak = 1, 2 are Hurwitz Metzler matrices.
EN
A new method for determination of positive realizations of given transfer matrices of linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are presented. A procedure for computation of the positive realizations is proposed and illustrated by an example.
EN
A new method for computation of positive realizations of given transfer matrices of descriptor linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by examples.
PL
W pracy podano nową metodę wyznaczania dodatnich realizacji dla zadanych macierzy transmitacji operatorowych deskryptowych( singularnych) ciągłych układów liniowych. Sformułowano warunki konieczne i wystarczające istnienia dodatnich realizacji dla tej klasy deskryptowych ciągłych układów liniowych. Podano procedurę wyznaczania tych realizacji. Procedura ta została zlustrowana przykładami liczbowymi.
EN
The notions of monomial generalized Frobenius matrices is proposed and the reachability and observability of positive discrete-time linear systems with positive and negative integer powers of the state matrices is addressed. Necessary and sufficient conditions for the reachability of the positive systems are established.
EN
The minimum energy control problem for the positive descriptor discrete-time linear systems with bounded inputs by the use of Weierstrass-Kronecker decomposition is formulated and solved. Necessary and sufficient conditions for the positivity and reachability of descriptor discrete-time linear systems are given. Conditions for the existence of solution and procedure for computation of optimal input and the minimal value of the performance index is proposed and illustrated by a numerical example.
EN
A new method for computation of positive realizations of given transfer matrices of fractional linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by examples.
18
Content available remote Positive linear systems and electrical circuits with inverse state matrices
EN
The positive linear systems and electrical circuits with inverse state matrices are addressed. It is shown that: 1) If the eigenvalues of asymptotically stable matrix are located outside the unit circle then the eigenvalues of asymptotically stable matrix positive and located inside the unit circle; 2) If the eigenvalues of asymptotically stable matrix are positive and located inside the unit circle then the eigenvalues of the matrix Mn are located in the left half plane outside the unit circle. A new problem concerning the discretization of positive linear systems and electrical circuits is formulated and solved.
PL
W pracy przedstawiono dodatnie układy liniowe i obwody elektryczne z odwrotnością macierzy stanu. Zostało pokazane że: 1) Jeżeli wartości własne asymptotycznie stabilnej macierzy AMn są ulokowane poza obszarem koła jednostkowego wtedy wartości własne asymptotycznie stabilnej macierzy są dodatnie i leżą wewnątrz koła jednostkowego; 2) Jeżeli wartości własne asymptotycznie stabilnej macierzy są dodatnie i leżą wewnątrz koła jednostkowego wtedy wartości własne macierzy leżą w lewej półpłaszczyźnie poza obszarem koła jednostkowego. Sformułowany zostanie nowy problem dotyczący dyskretyzacji dodatnich układów liniowych oraz obwodów elektrycznych.
EN
The concept of superstability of positive linear electrical circuits is introduced and its properties are characterized. The superstabilization of positive and nonpositive electrical circuits by state-feedbacks is analyzed.
EN
Positive descriptor fractional discrete-time linear systems with fractional different orders are addressed in the paper. The decomposition of the regular pencil is used to extend necessary and sufficient conditions for positivity of the descriptor fractional discrete-time linear system with different fractional orders. A method for finding the decentralized controller for the class of positive systems is proposed and its effectiveness is demonstrated on a numerical example.
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