When solving the control and design problems in aerospace and naval engi neering, energetics, economics, biology, etc., we need to know the state of investigated dynamic processes. The presence of inherent uncertainties in the description of these processes and of noises in measurement devices leads to the necessity to construct the estimators for corresponding dynamic systems. The estimators recover the required information about system state from mea surement data. An attempt to solve the estimation problems in an optimal way results in the formulation of different variational problems. The type and complexity of these variational problems depend on the process model, the model of uncertainties, and the estimation performance criterion. A solution of variational problem determines an optimal estimator. Howerever, there exist at least two reasons why we use nonoptimal esti mators. The first reason is that the numerical algorithms for solving the corresponding variational problems can be very difficult for numerical imple mentation. For example, the dimension of these algorithms can be very high.

This book presents a detailed examination of the estimation techniques and modeling problems. The theory is furnished with several illustrations and computer programs to promote better understanding of system modeling and parameter estimation.

Abstract: In a wide variety of research fields, dynamic modeling is employed as an instrument to learn and understand complex systems. The differential equations involved in this process are usually non-linear and depend on many parameters whose values determine the characteristics of the emergent system. The inverse problem, i.e., the inference or estimation of parameter values from observed data, is of interest from two points of view. First, the existence point of view, dealing with the question whether the system is able to reproduce the observed dynamics for any parameter values. Second, the identifiability point of view, investigating invariance of the prediction under change of parameter values, as well as the quantification of parameter uncertainty. In this paper, we present the R package dMod providing a framework for dealing with the inverse problem in dynamic systems modeled by ordinary differential equations. The uniqueness of the approach taken by dMod is to provide and propagate accurate derivatives computed from symbolic expressions wherever possible. This derivative information highly supports the convergence of optimization routines and enhances their numerical stability, a requirement for the applicability of sophisticated uncertainty analysis methods. Computational efficiency is achieved by automatic generation and execution of C code. The framework is object-oriented (S3) and provides a variety of functions to set up ordinary differential equation models, observation functions and parameter transformations for multi-conditional parameter estimation. The key elements of the framework and the methodology implemented in dMod are highlighted by an application on a three-compartment transporter model

This monograph is an exposition of a novel method for solving inverse problems, a method of parameter estimation for time series data collected from simulations of real experiments. These time series might be generated by measuring the dynamics of aircraft in flight, by the function of a hidden Markov model used in bioinformatics or speech recognition or when analyzing the dynamics of asset pricing provided by the nonlinear models of financial mathematics. Dynamic Systems Models demonstrates the use of algorithms based on polynomial approximation which have weaker requirements than already-popular iterative methods. Specifically, they do not require a first approximation of a root vector and they allow non-differentiable elements in the vector functions being approximated. The text covers all the points necessary for the understanding and use of polynomial approximation from the mathematical fundamentals, through algorithm development to the application of the method in, for instance, aeroplane flight dynamics or biological sequence analysis. The technical material is illustrated by the use of worked examples and methods for training the algorithms are included. Dynamic Systems Models provides researchers in aerospatial engineering, bioinformatics and financial mathematics (as well as computer scientists interested in any of these fields) with a reliable and effective numerical method for nonlinear estimation and solving boundary problems when carrying out control design. It will also be of interest to academic researchers studying inverse problems and their solution.