Temporal and differential stabilizability and detectability of piecewise constant rank systems

SUMMARY In a past note we drew attention to the fact that time‐varying continuous‐time linear systems may be temporarily uncontrollable and unreconstructable and that this is vital knowledge to both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability require considering piecewise constant rank (PCR) systems and the differential Kalman decomposition. In this note for conventional as well as PCR systems measures of temporal and differential stabilizability and detectability are developed. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed‐loop system when designing a static state or dynamic output feedback controller. It is indicated how to compute the measures from the system matrices. The importance of our developments for control system design is illustrated through three numerical examples concerning LQ and LQG perturbation feedback control of a non‐linear system about an optimal control and state trajectory. Copyright © 2011 John Wiley & Sons, Ltd.