Rank Tests for the Observability of Discrete-Time Jump Linear Systems with Inputs

We analyze the observability of the continuous and discrete states of discrete-time jump linear systems (JLSs) with deterministic inputs. We consider several definitions of observability for JLSs depending on whether some or all inputs are considered. Unfortunately, checking these definitions can involve an exponential number of rank tests on the parameters of the JLS when the discrete state sequence is arbitrary. Our key contribution is to demonstrate that when there is a minimum separation between consecutive switches, one can verify observability by checking a number of rank tests that is only quadratic in the number of discrete states. The observability conditions we derive are natural generalizations of the well known rank test for linear systems. Moreover, they can be related to the Markov parameters of the individual linear systems.