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1

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.

2

Stability of positive linear discrete-time systems

James G., Rumchev V.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2005

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Vol. 53, nr 1

1-8

EN

The main focus of the paper is on the asymptotic behaviour of linear discrete-time positive systems. Emphasis is on highlighting the relationship between asymptotic stability and the structure of the system, and to expose the relationship between null-controllability and asymptotic stability. Results are presented for both time-invariant and time-variant systems.

3

Stability of positive linear discrete-time systems

James G., Rumchev V.

Systems Science

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2004

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Vol. 30, no 3

51-67

EN

The main focus of the paper is on the asymptotic behaviour of linear discrete-time positive systems. Emphasis is on highlighting the relationship between asymptotic stability and the structure of the system, and to expose the relationship between null-controllability and asymptotic stability. Results are presented for both time-invariant and time-variant systems.

4

A fractional-flow model of serial manufacturing systems with rework and its reachability and controllability properties

James G., Rumchev V.

Systems Science

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2001

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Vol. 27, nr 2

49-59

EN

A dynamic fractional-flow model of a serial manufacturing system incorporating rework is considered. Using some recent results on reachability and controllability of positive linear systems the ability of serial manufacturing systems with rework to "move in space", that is their reachability and controllability properties, are studied. These properties are important not only for optimising the performance of the manufacturing system, possibly off-line, but also to improve its functioning by using feedback control on-line.

5

Reachability and controllability of compartmental systems.

James G., Rumchev V.

Systems Science

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2000

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Vol. 26, nr 1

5-13

EN

Compartmental models are frequently used in biology and medicine to analyze the behaviour of complex physiological systems and some of their properties (equilibrium points, stability, predicting the impact of different inflows) are well studied. Attempts have also been made to understand reachability and controllability properties of such models by employing the tests for linear systems. Linear compartmental systems as positive systems belong to the class of constrained linear systems, which are actually non-linear, and hence the reason why the general reachability and controllability tests developed for linear systems are fallible. In the paper, reachability and controllability properties of compartmental systems are studied by using some results in controllability theory for positive linear system. In particular, positive reachability and controllability criteria are in a digraph form, most suitable to the commonly used digraph representation of compartmental systems. Some specific compartmental models are also considered. The results obtained in the paper increase not only our understanding of the properties of compartmental models but they are also important for implementing feedback and optimal control strategies.