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1

A sufficient condition for existence of constantly nondominated trajectories in linear time-invariant systems

Domka M.

Automatyka / Automatics

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2018

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Vol. 22, no. 1

7--22

EN

This paper considers the multi-criteria optimal control problem in linear time-invariant systems with a single bounded input. It has been proven that, under certain assumptions, a state trajectory is non-dominated throughout the whole control process if and only if it is yielded by a control signal that belongs to a particular class of bang-bang functions. Furthermore, an explicit formula relating hyperplanes tangent to an attainable set and control switching times is presented.

PL

W artykule rozważono problem wielokryterialnego sterowania optymalnego w liniowym stacjonarnym systemie dynamicznym z jednym wejściem przyjmującym wartości z ograniczonego przedziału. W szczególności udowodniono, że przy spełnieniu określonych założeń trajektoria stanu jest niezdominowana w trakcie całego procesu sterowania wtedy i tylko wtedy, gdy jest generowana przez sterowanie należące do pewnej klasy sygnałów typu bang-bang. Ponadto został przedstawiony wzór pozwalający wyrazić hiperpłaszczyznę styczną do zbioru osiągalnego w funkcji czasów przełączeń sygnału sterującego.

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

Reachable sets concept - a general abstract analysis

Terlikowski T.

Control and Cybernetics

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2003

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Vol. 32, no 4

803-817

EN

A control problem over the infinite time horizon for periodical system is considered. Our aim is to analyze a special concept of solving problems of this type, based on the known idea of reachable sets. We try to consider this concept in a more general manner than it was done in earlier works and to find what is really essential in it. The algorithm corresponding to the proposed general description is presented.