Rank tests for the observability of discrete-time jump linear systems with inputs

Abstract

Abstract-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.

Vulnerability analysis of large-scale dynamical networks to coordinated attacks

“…The conditions for mean square exponential stability of any MJLS are given in [12]. Conditions to check controllability and observability for Markovian systems similar to LTI systems have been discussed in [14], [15], [16]. In this paper, we consider controllability and observability gramians for MJLS as given in [12].…”

Data-driven distributionally robust control of partially observable jump linear systems

“…In [17], simple rank conditions for the observability of autonomous jump linear systems with unknown switching sequence were given. These results were later extended to systems with a control input [18]. Although these conditions are attractive from a computational point of view, since they do not suffer from exponential growth in complexity, they require a minimal dwell-time assumption on the (unknown) mode sequence.…”

“…for all j ∈ {1, 2} and i ∈ W , with O i as defined in (18) (see Appendix), where M = (M i ) i∈W is a given (potentially mode-dependent) stabilizing state-feedback gain with respect to all P i: ∈ A t,i , i ∈ W and {q l t,i }…”

Basis transform in linear switched system models from input–output data

“…Observability plays a central role in realization and identification studies of the LSSs from input–output data or the Markov parameters. The observability and the identifiability of the LSSs was studied in Reference 28 for the homogeneous and in Reference 29 for the inhomogeneous systems. A recent overview of the observability for hybrid systems can be found in Reference 30.…”