Feedback Liniarization Method for Problem of Control of a Part of Variables in Uncontrolled Disturbances

Abstract

The paper studies a problem of guaranteed transfer within a finite amount of time of a nonlinear dynamical system subjected to uncontrolled disturbances to a state where a given part of the variables equals zero. The bounded controls are offered to be generated by means of a feedback in form of nonlinear functions of phase variables of a given nonlinear controlled system of differential equations. The method of exact feedback linearization of the nonlinear system is used. As a result, the solution of the original nonlinear problem is narrowed down to solve the linear game-theoretic antagonistic control problem. Sufficient conditions are obtained with ensure that the problem has a guaranteed solution for the given domain of initial conditions. As an example, problem of the space turn of an asymmetric rigid bode (spacecraft) is considered within the framework of the method. Three reaction wheels are employed to produce necessary torque in the axes of the spacecraft. External uncontrolled disturbances, that have no statistical description, are taken into consideration in the process of reorientation. In this case the initial nonlinear controlled systems consists of dynamic Euler equations and Rodriges – Hamilton kinematic equations based on the quaternion parameterization of attitude kinematics. Two problems of the space turn of the spacecraft are considered. 1) The rest - to - rest reorientation problem. 2) The space turn from a stationary state to a given angular position; it is not assumed that the turn takes the spacecraft to a stationary state. The proposed approach allows common positions to give some already well-known solutions of these problems. A new solution of the reorientation problem is also given. For this new solution an estimation of the admissible domain of uncontrolled disturbances is found. Results of a numerical calculations are considered.