Synthesis Of Input Signals In Parameter Estimation Problems

Synthesis of Input Signals in Parameter Estimation Problems

Optimal Input Signals for Parameter Estimation

The aim of this book is to provide methods and algorithms for the optimization of input signals so as to estimate parameters in systems described by PDE’s as accurate as possible under given constraints. The optimality conditions have their background in the optimal experiment design theory for regression functions and in simple but useful results on the dependence of eigenvalues of partial differential operators on their parameters. Examples are provided that reveal sometimes intriguing geometry of spatiotemporal input signals and responses to them. An introduction to optimal experimental design for parameter estimation of regression functions is provided. The emphasis is on functions having a tensor product (Kronecker) structure that is compatible with eigenfunctions of many partial differential operators. New optimality conditions in the time domain and computational algorithms are derived for D-optimal input signals when parameters of ordinary differential equations are estimated. They are used as building blocks for constructing D-optimal spatio-temporal inputs for systems described by linear partial differential equations of the parabolic and hyperbolic types with constant parameters. Optimality conditions for spatially distributed signals are also obtained for equations of elliptic type in those cases where their eigenfunctions do not depend on unknown constant parameters. These conditions and the resulting algorithms are interesting in their own right and, moreover, they are second building blocks for optimality of spatio-temporal signals. A discussion of the generalizability and possible applications of the results obtained is presented.

Synthesis of Input Signals in Parameter Estimation Problems

Design of input signals to enhance the estimation on unknown parameters in discrete time dynamics is considered. The system identification problem can be considered as the initial phase of a stochastic control problem of a dynamic system. In such a situation it is desirable to estimate the parameters as rapidly as possible without disturbing the normal operation of the process. The problem is represented in a Baysian framework. The optimality criterion for the synthesis problem is defined in terms of the statistical distance measure between distributions characterized by distinct parameter values. In this dissertation we define this quantity in terms of the pairwise Bhattacharyya distance. The distance measure has the desirable property of ordering the Bayes' probability of error as a function of two experimental design programs. The statistical distance measures have been successfully used in studying taxonomy and characterization of racial distributions. It is shown that such an approach is also feasible in applications to more general dynamic situation.