Variance-Constrained Multi-Objective Stochastic Control and Filtering

Chapter 3Robust Mixed H₂/H_∞ Filtering

The filtering problem has been playing an important role in signal processing and control engineering. Among various filtering schemes, the celebrated Kalman filtering (also known as H₂ filtering) approach minimizes the H₂ norm of the estimation error, under the assumptions that an exact model is available and the noise processes have exactly known statistical properties. However, this is seldom the case in practical applications. As an alternative, the H_∞ filtering method has been proposed, which provides an upper bound for the worst-case estimation error without the need for knowledge of noise statistics. The H_∞ filter has also been proven to be more robust than the traditional Kalman filter, when model uncertainties exist in the system. However, there is no provision in H_∞ filtering to ensure that the variance of the state estimation error lies within acceptable bounds. In this respect, it is natural to consider combining the performance requirements of the Kalman filter and the H_∞ filter into a mixed H₂/H_∞ filtering problem.

For deterministic systems, the mixed H₂/H_∞ filtering problems have been extensively studied. Various methods have been proposed to solve the mixed H₂/H_∞ filtering problems, such as the algebraic equation approach, time-domain Nash game approach, and convex optimization approach. It should be pointed out that, results of H₂/H_∞ filtering for nonlinear systems are relatively few. On the other hand, as for the stochastic ...