Almost input-decoupled filtering under white noise input

Abstract

Chapter 7 considers exact input-decoupled (EID) filtering problems. In that chapter, we seek perfect performance; that is, we try to make the impact of the unknown input on the asymptotic error absolutely zero irrespective of what is the input, whether it is persistent or not. Such a severe performance measure demands that the transfer function or transfer matrix from the input to the estimation error be identically zero. Of course, as discussed in Chapter 7, the EID filtering problem is not always solvable. It is natural then to think of methods of relaxing the performance requirements so that the solvability conditions can possibly be weakened and thus allowing us to deal with a larger class of systems. There are a number of ways by which the performance requirements can be weakened. As we said in the introduction to this book, we plan to relax the performance requirements progressively layer by layer to form a hierarchy of problems. In this chapter, we introduce the first layer of relaxing the performance requirements. We seek here to make the impact of the unknown input on the asymptotic error to be almost zero or equivalently arbitrarily small or as small as desired instead of being identically zero. To be more precise, we try to find a family of filters parameterized by some positive ε such that when applied to the system, the asymptotic error converges to zero as ε ↓ 0. Thus, in this chapter, we deal with what can be termed as almost-input-decoupled (AID) filtering problems, or for short AID filtering problems. The filters that solve the AID filtering problems are termed not surprisingly as AID filters.