Finite memory partial inverses

Abstract

Using a state-space approach we show that an observable causal operator on a Hilbert resolution space has a finite memory decomposition property. Regardless of its past evolution, it is always possible to relate in a linear and causal way a finite segment of output with the corresponding segment of input. The decomposition property is used to extend the concept of partial system inverses. For a given operator T we construct a bounded causal map M that not only satisfies the condition <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MT= L</tex> , over a prespecified finite dimensional subspace, but also has a finite memory characteristic. Consequently, the operator M approximates the inverse of T over the subset of finite segments of linear combinations in the given subspace. The quality of the approximation depends on the length of the segments in relation to the memory length of the partial inverse. The finite memory partial inverses can be applied to the parameter sensitivity problem with time-varying parameter changes.