Forced synchronization of rigid bodies angular velocities

Abstract

The problem of synchronization on incomplete information on a state of system is considered. In control theory, one of the ways to solve the problem of incompleteness of the measured information is to obtain a vector estimate state by the values of outputs with the help of an observer -- a special dynamic system, the state of which approaches the initial trajectory. The main problem in constructing an observer is therefore, to provide a exponential dynamics of observation error reduction. Assume that a solution in the form of feedback $ u (x) $ is found for the problem of synchronization of trajectories and estimate $ \hat x $ is obtained with the help of an observer. The question arises whether thus obtained control law in the form of feedback $ u (\hat x) $ solve the original problem. For linear stationary systems, the answer to this question is positive (the separation principle): if for of a linear stationary system an exponential observer is constructed and a linear feedback is found, globally asymptotically stabilizing a given equilibrium position at a known state vector -- then with the appropriate feedback on the estimate state vector global asymptotic stability of the equilibrium position stored. For nonlinear systems in the general case the answer to this question is negative: there are examples of nonlinear systems to which the separation principle is unsuitable. The reason for this is possible phenomenon of unlimited growth of system solutions with control $ u(\hat x) $ for a finite time before the observer estimates error of the state will be reduced to zero. To construct the laws of synchronization, in contrast to the general approach, we use the method of invariant relations developed in analytical mechanics, which is designed to find partial solutions (dependences between variables) in problems of dynamics of a rigid body with a fixed point. Modification of this method to the problems of control theory allows to synthesize a manifold in the space of an extended system, which avoids possible unlimited growth of solutions and provides controlled dynamics for trajectory deviation.