Linearization of Nonlinear Affi ne Control Systems with Non-Involutive Distributions by the Introduction of Linearizing Controls

Abstract

In the article an algorithm for finding linear equivalents (exact linearization) for noninvolutive distributions of control vector fields is considered. In contrast to the common approach to solving this problem — the use of dynamic linearization (the introduction of integrators), which leads to an expansion of the state space — an algorithm for obtaining involutive distributions and ensuring local controllability based on linearizing controls is proposed. The essence of the algorithm: choose such a control and find an explicit expression for it that a controlled vector field associated with this control, when attached to a drift vector field, will provide local controllability and involution of the corresponding distributions. To check the involution of distributions and find the decomposition functions of vector fields on the basis of the current distribution, and on them directly the conditions imposed on the linearization controls, the author has developed an algorithm and a program in the Maple package for finding these functions. For the convenience of presentation and maximum clarity of the proposed approach, in the article is using notation not generally accepted in applied differential geometry. This applies primarily to the representation of vector fields in coordinate form or in the form of differential operators, which is often not specified, but it is assumed that the shape of the vector field is determined from the context. In the article, these forms are clearly separated and their specific use is shown. An example is considered — a nonlinear affine control system of the fifth order with three controls, in which all stages of synthesis are reflected in detail.