A Study of the Observability of Multidimensional Hybrid Linear Systems

A class of multidimensional hybrid linear systems is presented, with the time vector composed by q continuous-time real components and by r discrete-time integer ones, q, r ≥ 1. The state equation is of multidimensional partial differential-difference type. A generalized variation-of-parameters formula is provided and it is used to obtain the state and the general response of the system. The fundamental concept of observability is studied for these systems. An observability Gramian is introduced, which is a generalization of the Gramians corresponding to the classical 1D continuous-time and 1D discrete-time systems. In the case of completely observable systems this Gramian is used to obtain a formula which provides the initial state of the system for any input-output pair. A list of observability criteria is given for time-invariant systems and the duality between the concepts of observability and reachability is emphasized. Key–Words: multidimensional hybrid systems, observability, observability Gramian, time-varying systems, timeinvariant systems.