Models for Dynamic Systems. Since their introduction by Isaac Newton in the seventeenth century, differential equations have provided concise mathematical models for many dynamic systems of importance to humans. By this device, Newton was able to model the motions of the planets in our solar system with a small number of variables and parameters. Given a ®nite number of initial conditions (the initial positions and velocities of the sun and planets will do) and these equations, one can uniquely determine the positions and velocities of the planets for all time | Kalman Filtering Theory and Practice Using MATLAB Second Edition Mohinder S. Grewal Angus P. Andrews Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-39254-5 Hardback 0-471-26638-8 Electronic 2 Linear Dynamic Systems What we experience of nature is in models and all of nature s models are so R. Buckminster Fuller 1895-1983 CHAPTER FOCUS Models for Dynamic Systems. Since their introduction by Isaac Newton in the seventeenth century differential equations have provided concise mathematical models for many dynamic systems of importance to humans. By this device Newton was able to model the motions of the planets in our solar system with a small number of variables and parameters. Given a finite number of initial conditions the initial positions and velocities of the sun and planets will do and these equations one can uniquely determine the positions and velocities of the planets for all time. The finite-dimensional representation of a problem in this example the problem of predicting the future course of the planets is the basis for the so-called state-space approach to the representation of differential equations and their solutions which is the focus of this chapter. The dependent variables of the differential equations become state variables of the dynamic system. They explicitly represent all the important characteristics of the dynamic system at any time. The whole of dynamic system theory is a subject of considerably more scope than one needs for the present undertaking Kalman filtering . This chapter will stick to just those concepts that are essential for that purpose which is the development of the statespace representation for dynamic systems described by systems of linear differential equations. These are given a somewhat heuristic treatment without the mathematical rigor often accorded the subject omitting the development and use of the transform methods of functional analysis for solving differential equations when they serve no purpose in
