Digital aspects of nonlinear synthesis problems.

Abstract

This dissertation treats two aspects of control synthesis problems for discrete-time and sampled-data nonlinear systems. First, the effects of time-sampling on the solvability conditions of a "popular" method for designing observers for a class of nonlinear systems, the so-called observer error linearization method, are investigated. It is shown that the class of systems for which this design method can be applied for an open set of sampling times is quite small. The fact that the methodology is not robust with respect to time-sampling implies that digital implementation of this methodology will have to be approximate, either by implementing a continuous-time design through rapid sampling, or by seeking an approximate solution to the discrete-time version of the problem. In response to this, approximate solutions to observer error linearization, both in the state variables and in the sampling interval, are sought. Unfortunately, it is found that the conditions for achieving approximate observer error linearization for an open set of sampling-times are essentially as restrictive as the conditions for the exact solutions. One possible conclusion of all these investigations is that linearization methods in designing a nonlinear observer, whether they are exact or approximate, are not very natural, at least in the context of sampling. Second, controllability distributions are developed for a class of discrete-time nonlinear systems with invertible transition maps, and as an application of them, the problem of discrete-time block noninteracting control with internal exponential stability is treated. A characterization of local controllability distributions is provided by an algorithm. Such controllability distributions are important for solving the problem of discrete-time block noninteracting control with internal exponential stability, where they are essential for the identification of certain fixed modes common to every noninteractive closed loop. From the fixed modes, a necessary condition for achieving an exponentially stable solution to the problem is obtained; through a special decomposition of the given system, a sufficient condition is provided. It is noticed that due to peculiarities intrinsic to discrete-time nonlinear systems, the development of the solution is not completely parallel to that of the solution for the continuous-time systems previously treated in the literature. The treatment of the problem of discrete-time noninteracting control with internal stability will serve as a basis for studying the effects of time-sampling on the ability to achieve noninteracting control with internal stability.