Sterowalność liniowych układów dynamicznych z opóźnienieniami w sterowniu przy ograniczeniach na sterowanie

Artykuł zawiera definicje oraz podstawowe twierdzenia dotyczące względnej oraz absolutnej sterowalności przy ograniczeniach na sterownie liniowych, ciągłych, skończenie-wymiarowych układów dynamicznych z opóźnieniami w sterowaniu.

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In this article we consider linear, continuous, finite dimensional dynamical with delays in control described by state equation of the form (1). We give some definitions of different kind relative and absolute controllability for dynamical system(1) with constrained controls. We formulate some theorems which institute the criteria of relative and absolute controllability with delays of systems with contrained controls. In presented examples we illustrate mutual dependences between controllability with constrained controls of system without delays and relative controllability with constrained controls of system with delays in control (Example 1,2 and 3) as well as implication between absolute and relative controllability (Example 4). We show that constraining the controls reduces set of initial states from which we can reach required final state and, on the other hand, also may reduce the attainability domain of dynamical sytem. But let us notice that certain advantages can arise through introducing the delays in control in system without delays. As presents in Example 2, owing to properly selected delays we are able instance, increase the initial states set. It is worth also to reflect in which way even a little changes of constrains putting on controls influence the set initial states from which we can reach demanded final state or attainability domain of the given system.