Global Nonlinear Model Identification with Multivariate Splines

Abstract

At present, model based control systems play an essential role in many aspects of modern society. Application areas of model based control systems range from food processing to medical imaging, and from process control in oil refineries to the flight control systems of modern aircraft. Central to a model based control system is a mathematical model of the physical system or process that is being controlled. The field of science concerned with the identification of models of physical systems is called system identification. In this thesis, a new methodology is proposed for the identification of models of nonlinear systems using multivariate simplex splines. This new methodology has the potential to increase the performance of any model based control system by improving the quality of system models. Multivariate simplex splines consist of polynomial basis functions, called B-form polynomials, which are defined on geometric structures called simplices. Every simplex supports a single B-form polynomial which itself consists of a linear combination of Bernstein basis polynomials. Each individual Bernstein basis polynomial is scaled by a single coefficient called a B-coefficient. The B-coefficients have a special property in the sense that they have a unique spatial location inside their supporting simplex. This spatial structure, also known as the B-net, provides a number of unique capabilities that add to the desirability of the simplex splines as a tool for data approximation. For example, the B-net simplifies local model modification by directly relating specific model regions to subsets of B-coefficients involved in shaping the model in those regions. This particular capability has the potential to play an important role in future adaptive model based control systems. In such a control system, an on-board simplex spline model can be locally adapted in real time to reflect changes in system dynamics. The approximation power of the multivariate simplex splines can be increased by joining any number of simplices together into a geometric structure called a triangulation. Triangulations come in many shapes and sizes, ranging from configurations consisting of just two simplices to configurations containing millions of simplices. Triangulations can be optimized by locally increasing or decreasing the density of simplices to reflect local system complexity. The new methodology was applied in the identification of a complete set of aerodynamic models for the Cessna Citation II laboratory using flight data obtained during seven test flights. In total, 247 flight test maneuvers were flown which together provided a significant coverage of the flight envelope of the Citation II. The complete identification dataset consisted of millions of measurements on more than sixty flight parameters. More than 2000 prototype spline based aerodynamic models were identified using a newly developed, highly optimized software implementation of the simplex spline identification algorithm. Using the developed methods for simplex spline model validation it was proved that the models are both accurate and of guaranteed numerical stability inside the spline domain. The identification and validation results of the simplex spline models were compared with those of ordinary polynomial models identified using standard identification methods. These results showed that the multivariate simplex spline based aerodynamic models were of significantly higher quality than the aerodynamic models based on ordinary polynomials.Control & OperationsAerospace Engineerin