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1

On stabilizability-holdability problem for linear discrete time systems

Rumchev V., G., Higashiyama Y.

Systems Science

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2010

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Vol. 36, no 2

33-38

EN

Consider the following problem. Given a linear discrete-time system, find if possible a linear state-feedback control law such that under this law all system trajectories originating in the non-negative orthant remain non-negative while asymptotically converging to the origin. This problem is called feedback stabilizability-holdabiltiy problem (FSH). If, in addition, the requirement of non-negativity is imposed on the controls, the problem is a positive feedback stabilizability-holdabiltiy problem (PFSH). It is shown that the set of all linear state feedback controllers that make the open-loop system holdable and asymptotically stable is a polyhedron and the external representation of this polyhedron is obtained. Necessary and sufficient conditions for identifying when the open-loop system is not positive feedback R+n-invariant (and therefore there is no solution to the PFSH problem) are obtained in terms of the system parameters. A constructive linear programming based approach to the solution of FSH and PFSH problems is developed in the paper. This approach provides not only a simple computational procedure to find out whether the FSF, respectively the PFSH problem, has a solution or not but also to determine a linear state feedback controller (respectively, a non-negative linear state feedback controller) that endows the closed-loop (positive) system with a maximum stability margin and guarantees the fastest possible convergence to the origin.

2

Monomial Subdigraphs of Reachable and Controllable Positive Discrete-time Systems

Bru R., Cacetta L., Rumchev V. G.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 1

159-166

EN

A generic structure of reachable and controllable positive linear systems is given in terms of some characteristic components (monomial subdigraphs) of the digraph of a non-negative a pair. The properties of monomial subdigraphs are examined and used to derive reachability and controllability criteria in a digraph form for the general case when the system matrix A may contain zero columns. The graph-theoretic nature of these criteria makes them computationally more efficient than their known equivalents. The criteria identify not only the reachability and controllability properties of positive linear systems, but also their reachable and controllable parts (subsystems) when the system does not possess such properties.

3

Reachability and controllability of time-variant discrete-time positive linear systems

Rumchev V. G., Adeane J.

Control and Cybernetics

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2004

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Vol. 33, no 1

85-94

EN

In this paper necessary and sufficient conditions (and criteria) for null-controllability, reachability and controllability of time-variant discrete-time positive linear systems are established. These properties appear to be entirely structural properties, that is, they do depend on the zero-nonzero pattern of the pair (A(k), B[k)) > 0 and do not depend on the values of its entries. An interesting phenomenon has been revealed namely the time needed to reach the origin for a null-controllable system as well as the time to reach a (non-negative) state from the origin for a reachable system can be less, equal or greater than the dimension of the system. This phenomenon has no equivalent in the case of time-invariant discrete-time positive linear systems where this time is always less or equal to the system dimension. Examples are provided.

4

Positive linear dynamic model of mobile source air pollution and problems of control

Rumchev V.G., Caccetta L., Kostova S.

Systems Science

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2003

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Vol. 29, nr 1

31-46

EN

A dynamic model of mobile source air pollution (MSAP) is developed in this paper. The model exhibits positive linear system behaviour. One of its specific features is that the system is restricted to evolve in a given polyhedral region in the state space in order to keep the total emissions from the vehicles within the allowed levels. Three problems of control namely maintainability, finite-time holdability and feedback holdability are formulated and the corresponding solution procedures developed in the paper. A policy implementing the solutions to these problems leads to sustainable levels of the total emissions. Our method can be immediately used to support the decision making process of regional and state environmental authorities.

5

Controlled balanced growth of positive linear discrete-time systems

Rumchev V.G., James D.J.G.

Systems Science

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2003

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Vol. 29, nr 1

5-14

EN

In this paper properties of M-matrices are used to develop procedures to ensure that a positive discrete-time linear system achieves a specified desired balanced growth rate. Both closed-loop and open-loop control procedures are considered, with state feedback being adopted for implementing the closed-loop control. Procedures developed are illustrated by simple examples.

6

The pole-assignment problem for positive linear discrete-time systems with scalar controls

James D.J., Rumchev V.G.

Systems Science

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2002

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Vol. 28, nr 2

15-27

EN

The paper considers the pole-assignment problem on the basis of the single-input positive linear discrete-time systems, compares it with the unrestricted case and develops procedures for finding a linear state-feedback controller that assigns the spectrum of the closed-loop to the desired spectrum

7

Cohort-type models and their reachability and controllability properties.

James D.J.G., Rumchev V.G.

Systems Science

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2000

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Vol. 26, nr 2

43-54

EN

Cohort-type models have a wealth of structure and are important in many fields of application. Being a sub-class of linear positive systems, such a model structure belongs to this class of constrained linear systems which are defined on cones and not on linear spaces. As a consequence much of the theory developed for linear systems is not directly applicable. Whilst the dynamics of cohort-type models. Are well understood their reachability and controllability properties have received little attention in the literature. Using a graph-theoretic characterisation the paper examines these fundamental properties and includes application to some specific models.

8

Asymptotic reachable sets for discrete-time positive linear systems.

Rumchev V.G., Anstee R.P.

Systems Science

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1999

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Vol. 25, nr 1

41-47

EN

Reachable sets of discrete-time positive linear systems with no additional constraints imposed on control are polyhedral cones for any finite t but the asymptotic reachable set may not be polyhedral. In this paper asymptotic reachable properties of such sets are studied and characterisations of polyhedrality/non-polyhedrality of the asymptotic reachable set are developed.