Spectral Theory of H 1 Filters & Smoothers

This paper probes the spectral theory of the L 1 l-tering/ smoothing problem in the 2 block format. The solution for the lter/smoother which satisses a L 1 performance bound, is presented. Using spectral properties of the problem, it is proved that extending our scope of acceptable compensators from lters to smoothers, we cannot improve the L 1 performance. 1. Introduction and Statement of the problem In the problem of ltering, we attempt to estimate the states of a system using causal transforms of the measurements. The problem of smoothing involves estimating the states of a system using both future and past measurements. The paper by Nagpal and Khargonekar 1] presents necessary and suucient conditions for the existence of lters/smoothers and follows a time domain approach to the problem. In this paper we tackle the same problem using spectral techniques and investigate the tolerance properties of a lter/smoother in the SISO case. We present a result which says that we cannot get an improved performance by allowing the use of smoothers as opposed to lters in the original problem. This means that a lter will give the best performance for the problem set up below. This result was touched upon by Limebeer 2] using Riccati equation techniques. We present it using the spectral properties of the problem. The problem is to estimate the signal z corrupted by colored noise V v (Figure 1). We assume that the signal z can be generated from white noise w through a causal lter W. We shall use a lter/smoother Q for this purpose. We assume that w and v are white and uncorrelated relative to each other. The error term is given as e = (I ? Q)Ww ? QV v. The Filtering problem is to choose Q as a stable causal transfer function. However it might be practical to allow for non-causal components in Q, making problem that of ltering and smoothing.