Some applications of projections in nonlinear control and estimation
The present thesis uses an approach which regards nonlinear systems as a pair (z(·),zf) trajectory-final state, for the case of controlability or (zO,z(·)) initial state-trajectory, for the case of state estimation, in a space M which is the cross product 1x Z or Z x F between the space of trajectories F and the state space Z. In this setting some mappings F:M + M are constructed using projections P onto specific subsets S of M (i.e. p2 = P and Rep} = S). The solution of the problems of nonlinear controllability, state estimation and state and parameter estimation are obtained via the fixed points of such F's. Fixed points theorems have been used in [8, 9, 25, 35 and 15] to provide global controllability, state estimation ,and joint state and parameter estimation. Some theoretical results are presented here, which show that it is possible to eliminate some of the assumptions which restricted the systems treated in these papers and at the same time to , obtain mappings with fixed points which contain all the possible solutions for the problem of nonlinear controllability, state estimation and the joint state and parameter estimation. Among the relaxations allowed now are, for example, in the problem of control, the possibility of a set of admissible controls Uad different from the set U of all input controls of the system. In order to obtain continuous projections P, S must be closed in M. This will occur naturally in the case of state estimation however, for the control case, in general, it will be necessary to adjust the topologies of the spaces U and/or M in order to achieve this. A comprehensive theory which shows that this adjustment is always possible as well as a complete procedure for obtaining the adjusted spaces, U and M are presented here.