The optimal filter for continuous, linear, stochastic, time-varying systems described by the Ito-Volterra (IV) equations with discontinuous measure is derived [1]. 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.

The developed result is applicable to a broad range of systems, including systems which depend on past state trajectory, commonly known as systems with memory, or hereditary systems. The renewal equation, non-Newtonian (e.g., polymer) flows, viscoelastic problems, and heat conduction in materials with memory (such as heat conduction in tissues with properties that depend on past temperature evolution) are several examples of systems with memory. The original IV application studied by Volterra is population dynamics, which is now used to describe population balances in crystallization, polymer synthesis, and other areas. The convolution-type models, including Green's function description of distributed parameter systems, and some integral transformations can also be cast in IV form. The integral model of measurements, used in our development, is needed to improve the interpretation of noninvasive (e.g., X-ray, MRI, and ultrasound) measurements and remote sensing.

The derived result for Ito-Volterra systems an be specialized to several important cases, including state space systems, for which we 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.

References

1. H. Zhang, M. Basin and M. Skliar, “Ito-Volterra Optimal State Estimation with Continuous, Multirate, Randomly Sampled, and Delayed Measurements,'' IEEE Trans. Automat. Control, 52:401-416, 2007.