Input-output stability for optimal estimation problems

Optimal estimation problems for a class of dynamic systems described by nonlinear differential equations are considered under the effect of disturbances. The estimator is a Luenberger observer that depends on an innovation function to be suitably chosen. The optimality criterion is taken as the norm of the estimation error in a function space and is expressed by means of a cost functional dependent on the innovation function. The well-definiteness of such a functional can be guaranteed via a Lyapunov approach and in terms of input-output stability of mappings between function spaces, where the disturbances are the input and the estimation error is the output. In particular, Lp and Sobolev optimality criteria are adopted. In these cases, relationships between internal (asymptotic and exponential) stability and input-output stability are studied and upper bounds on the estimation error are given. The bounds are illustrated by an example and a converse result is presented. In summary, the paper provides conditions for the well-definiteness of a class of optimal estimation problems and represents a departure point to develop efficient solution methodologies.