Stability, Controllability, Observability, and Realizability Criteria for Linear Systems on Time Scales with an Application to Fixed Point Arithmetic

The objective of this study is to unify the results on stability, controllability, observability, and realizability criteria on real time dynamical systems with those for systems over non-uniform domains such as a dynamical system on time scales or measure chains. It is shown that the conventional control systems theory on continuous/discrete systems is generalized to corresponding systems on time scales. We implement the novel standards on a first-order time-scale dynamical system using measure chains within a one-dimensional state space, featuring both discrete and continuous filters, in order to reduce the impact of noise at the filter outputs. A collection of necessary and sufficient requirements for this dynamical system to be stable and exponentially stable has been established. We complete our discussion by considering stability analysis of digital/continuous filter implementation subject to Finite Word Length (FWL) effects based on eigenvalue sensitivity analysis.