Filtering theory

Abstract

The stochastic state space model introduced in Section 2.4 is an internal model: its states x _k are not observed directly but do contribute to the observed outputs y _k as specified by the observation equation in (2.4.3). It is natural then to consider the problem of forming ‘best estimates’ of the state x _k give the available data (y _O, y ₁,…,y _k ). This procedure is known as filtering. There are at least three situations in which filtering is required. Firstly, it may be an end in itself: this is the case when, as often happens, the state variables xⁱ _k represent important physical quantities in a system which we need to know as accurately as possible even though they cannot be measured directly. Secondly, if we wish to control systems described by state space models then the natural class of controls to consider is that of state feedback controls where the control variable u_k takes the form u _k = u(k, x _k ). If x _k is not ‘known’ then in some circumstances it can be replaced by a best estimate x _k produced by filtering; this topic is described at length in Chapter 6. Finally, filtering is relevant when we wish to replace the state space model by an ‘equivalent’ external model; see section 3.4 below.