Itô–Volterra Optimal State Estimation With Continuous, Multirate, Randomly Sampled, and Delayed Measurements

The optimal filter for continuous, linear, stochastic, time-varying systems described by the Itocirc-Volterra equations with discontinuous measure is derived. With an appropriately selected measure, the result is applicable to a wide range of observation processes, including the hybrid case of observations formed by an arbitrary combination of continuous and discrete measurements, which may be sampled with a priori unknown, changing, and, possibly, random rates and delays. The simultaneous presence of continuous and sampled measurements causes impulsive discontinuity in the inputs of the optimal filter equations, which leads to a discontinuous change in state estimates every time a sampled measurement becomes available. Using the theory of vibrosolutions, the explicit and unique expressions for the jumps in state estimates and estimation error covariance are derived. Several examples illustrate the procedure of modeling hybrid measurement systems by selecting an appropriate discontinuous measure. We further show that the Itocirc-Volterra model and the main result of the paper can be specialized to several important cases, including state space systems, for which we recover several known state estimation results, and derive a novel optimal filter for continuous LTV systems with an arbitrary combination of continuous and delayed sampled measurements. This optimal filter updates the state estimates for incoming measurements as soon as they become available and does not require prior knowledge of sampling instants and delays, which makes it applicable when deterministic and random changes in sampling and delays are present. Several computational examples demonstrate the implementation of the developed filter and compare its performance to the traditional alternatives using Monte-Carlo simulations