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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

Maintainability of positive linear discrete-time systems

James G., Rumchev V.

Systems Science

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2007

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Vol. 33, no 3

21-30

EN

The concept of maintainability for (time-invariant) positive linear discrete-time systems (PLDS) is introduced and studied in detail. A state x(t) of a PLDS is said to be maintainable if there exists an admissible control such that x(t + 1) = x(t) for t = 0, 1, 2, ... For time-invariant systems, if a given state is maintainable it is maintainable at all times. The set of all maintainable states is called a maintainable set. Maintainability and stability are different concepts - while stability is an asymptotic ("long-term") notion, maintainability is a "short-term" concept. Moreover, stability always implies maintainability but maintainability does not necessarily imply stability. If no additional constraints are imposed on the states and controls except the standard non-negativity restrictions, the maintainable sets are polyhedral cones. Their geometry is determined completely by the structural and spectral properties of nonnegative system pair (A, B) ≥ 0. Different cases are studied in the paper and relevant numerical examples are presented. PLDS with two-side bounded controls are also discussed and an interesting result is obtained namely the maintainable set of an asymptotically stable PLDS coincides with its asymptotic reachable set.

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.