Quantitative measure of observability for linear stochastic systems

In this study we define a new observability measure for stochastic systems: the mutual information between the state sequence and the corresponding measurement sequence for a given time horizon. Although the definition is given for a general system representation, the paper focuses on the linear time invariant Gaussian case. Some basic analytical results are derived for this special case. The measure is extended to the observability of a subspace of the state space, specifically an individual state and/or the modes of the system. A single measurement system represented in the observable canonical form is examined in detail. A recursive form of the observability measure for a finite time horizon is derived. The possibility of using this form for designing a sensor selection algorithm is demonstrated by two examples.