PRESERVATION OF STABILIZABILITY OF A CONTINUOUS-TIME TIME-INVARIANT LINEAR-SYSTEM WITH CONTROLLER TIME-DELAY AFTER DISCRETIZATION

Strictly speaking, all sytems have controller time delay caused by the computation time, etc. On assuming that a continuous-time time-invariant linear system (CILS) with controller time delay is reachable or stabilizable, the problem is what is a necessary and sufficient condition that the discrete-time time-invariant linear system (DILS) resulting from the discretization by the zero-order hold for the CILS is also reachable or stabilizable. After introducing the augmented state vector in which certain delayed control vectors are included, the DILS is rewritten as a new DILS in state-space form. Then a necessary and sufficient condition for the new DILS to be reachable or stabilizable is proved to be the same as that for the corresponding DILS without controller time delay. Hence the condition stated in the first problem is the same as that for the CILS without time delay. Finally it is shown that a necessary and sufficient condition for the preservation of observability or dectability of a CILS with controller time delay is the same as that of the corresponding CILS without time delay.