Optimal Linearization of Nonlinear Control Models

Abstract

Most of the existing linearization methods for nonlinear flight dynamics are based on the first order approximation of the model in the neighborhood of a given reference or equilibrium point. The resulting linear model approximates the nonlinear flight within relatively small variations of the state and the control around the reference point. However, the range of the domain within which the linearized model is valid is not known precisely. Besides, such an approach only applies for differentiable functions. The present paper deals with approximating a nonlinear model by a linear one in the state and control domains. Such a linearization problem is termed as optimal linearization. Existing approaches about optimal linearization deal with cases for which the functions underlying the nonlinear models are continuous, and require multidimensional integral computation. Therefore, these approaches may not be efficient for high dimension nonlinear dynamical systems. The proposed method leads to a linear regression problem based on a suitable sampling of the state and control domains, and its solution is found through a straightforward matrix inversion procedure. The method is applied to a nonlinear UAV model with much better results than using classical linearization.