Input Observability of Time-Discrete Systems and Some Properties of Linear Dynamical Systems

Abstract

The measuring systems must be expressed by differential or difference equations and treated as dynamical systems, when the frequency ranges of the measuring systems cannot cover enough those of the measured values.
This paper shows first the necessary conditions and the necessary and sufficient conditions for the input observability of the linear time-discrete systems.
Comparing the conditions of controllability and obsrvability, the analogy between the time-continuous systems and the time-discrete systems are pointed out, concerning state controllability, state observability and output controllability. The principle of duality holds about the state values. It does not hold in general about the input and the output values. But the systems with one input or one output keep the principle of duality.
The time-discrete systems do not provide the sufficient informations with controllability and observability without the some numbers (n) of sampling and observation. But the time-continuous systems contain the same informations for only a little interval.
When we discretize the time-continuous systems, the controllability and observability of the time-continuous systems are the necessary conditions for those of the time-discrete systems. If we do not choose the proper sampling period when discretizing, we loose the controllability and observability. The rank of the linear coefficient matrix combining the time-continuous system with the time-discrete systems must be proper.
The condition which is very similar to the necessary and sufficient condition for the state observability is merely the necessary condition for the input observability.