Global solutions for nonlinear systems using qualitative reasoning

Abstract

This paper explores how qualitative information can be used to improve the performance of global optimization procedures. Specifically, we have constructed a nonlinear parameter estimation reasoner (NPER) for finding parameter values that match an ordinary differential equation (ODE) model to observed data. Qualitative reasoning is used within the NPER, for instance, to intelligently choose starting values for the unknown parameters and to empirically determine when the system appears to be chaotic. This enables odrpack, the nonlinear leastdsquares solver that lies at the heart of this NPER, to avoid terminating at local extrema in the regression landscape. odrpack is uniquely suited to this task because of its efficiency and stability. The NPER’s robustness is demonstrated via a Monte Carlo analysis of simulated examples drawn from across the domain of dynamics, including systems that are nonlinear, chaotic, and noisy. It is shown to locate solutions for noisy, incomplete realdworld sensor data from radiodcontrolled cars used in the University of British Columbia’s soccerdplaying robot project. The parameter estimation scheme described in this paper is a component of pret, an implemented computer program that uses a variety of artificial intelligence techniques to automate system identification – the process of inferring an internal ODE model from external observations of a system – a routine and difficult problem faced by engineers from various disciplines.