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Number of results: 49
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Abstract

AbstractThe 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.
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Authors and Affiliations

T. Kaczorek
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Abstract

The notion of the normal transfer matrix and the notion of the structure decomposition of normal transfer matrix for 2D general model are introduced. Necessary and sufficient conditions for the existence of the structure decomposition of normal transfer matrix are established. A procedure for computation of the structure decomposition is proposed and illustrated by the numerical example. It is shown that the impulse response matrix of the normal model is independent of the polynomial part of its structure decomposition.

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Authors and Affiliations

T. Kaczorek
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Abstract

The positive (minimal) realization problem for a class of singular discrete-time linear single-input, single-output systems with delays in state and delays in control is addressed. Solvability conditions for the positive (minimal) realization problem are established. It is shown that there exists a positive (minimal) realization of an improper transfer function T(z) = n(z) / d(z) if the coefficients of polynomial n(z) are non-negative and of the polynomial d(z) are non-positive except the leading one, which should be positive. A procedure for computation of the positive (minimal) realization of the transfer function is proposed and illustrated by an example.

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Authors and Affiliations

T. Kaczorek
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Abstract

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.

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Authors and Affiliations

T. Kaczorek
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Abstract

It is shown that 2(n + 1) is the upper bound for the reachability index of the n-order positive 2D general models.

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Authors and Affiliations

T. Kaczorek
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Abstract

A new notion of a realization of transfer matrix of (P;Q; V)-cone-system for discrete-time linear systems is proposed. Necessary and sufficient conditions for the existence of the realizations are established. A procedure is proposed for computation of a realization of a given proper transfer matrix T(z) of (P;Q; V)-cone-system. It is shown that there exists a realization of T(z) of (P;Q; V)-cone-system if and only if there exists a positive realization of T(z) = V T(z)Q!1, where V;Q and P are non-singular matrices generating the cones V;Q and P respectively.

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Authors and Affiliations

T. Kaczorek
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Abstract

The concept of strong stability is extended for positive and compartmental linear systems. It is shown that: 1) the asymptotically stable positive and compartmental systems are strongly stable if the eigenvalues of the system matrix are distinct, 2) electrical circuits consisting of resistances, capacitances (inductances) and source voltages are strongly stable.

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Authors and Affiliations

T. Kaczorek
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Abstract

New tests (criterions) for checking the reachability and the observability of positive linear-discrete-time systems are proposed. The tests do not need checking of rank conditions of the reachability and observability matrices of the systems. Simple sufficient conditions for the unreachability and unobservability of the systems are also established.

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Authors and Affiliations

T. Kaczorek
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Abstract

The global (absolute) stability of nonlinear systems with negative feedbacks and positive descriptor linear parts is addressed. Transfer matrices of positive descriptor linear systems are analyzed. The characteristics u = f(e) of the nonlinear parts satisfy the condition
ke  ≤ f(e) ≤ ke for some positive k, k. It is shown that the nonlinear feedback systems are globally asymptotically stable if the Nyquist plots of the positive descriptor linear parts are located in the right-hand side of the circles (–¹/k₁,  –¹/k₂).

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Authors and Affiliations

T. Kaczorek
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Abstract

A new class of positive fractional 2D hybrid linear systems is introduced. The solution of the hybrid system is derived. The classical Cayley-Hamilton theorem is extended for fractional 2D hybrid systems. Necessary and sufficient conditions for the positivity are established.

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Authors and Affiliations

T. Kaczorek
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Abstract

A new concept (notion) of the practical stability of positive fractional discrete-time linear systems is introduced. Necessary and sufficient conditions for the practical stability of the positive fractional systems are established. It is shown that the positive fractional systems are practically unstable if corresponding standard positive fractional systems are asymptotically unstable.

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Authors and Affiliations

T. Kaczorek
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Abstract

A new class of positive hybrid linear systems is introduced. The solution of the hybrid system is derived and necessary and sufficient condition for the positvity of the class of hybrid systems are established. The classical Cayley-Hamilton theorem is extended for the hybrid systems. The reachability of the hybrid system is considered and sufficient conditions for the reachability are established. The considerations are illustrated by a numerical example.

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Authors and Affiliations

T. Kaczorek

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