Optimal inversions of uncertain matrices - an estimation and control perspective
K. Mike Tao · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2003
Many system and signal related problems involve matrix inversion of some kind. For example, in estimation and signal recovery applications, inversion of the channel response matrix is often required in order to estimate the source signals. In the control of multivariable systems, inverting a process gain matrix may be called for in order to deliver appropriate control actions. There are situations where these matrices should be considered as uncertain (or random): for example, when the process/channel environments vary randomly, or when significant uncertainties are involved in estimating these matrices. Based on a unified approach, this paper considers both the right inversion (for control) and the left inversion (for estimation) of random matrices. In both cases, minimizing a statistical error function leads to the determination of optimal or linear optimal inversion. Connections with related techniques, such as the total least squares (TLS), the ridge regression, the Levenberg-Marquardt algorithm and the regularization theory are discussed. A variant Kalman filtering problem with randomly varying measurement gain matrix is among the applications addressed. Monte Carlo simulation results show substantial benefits by taking process/model uncertainty into consideration.