Algebraic differential equations and nonlinear control systems.

摘要

This dissertation establishes a precise correspondence between realizability of operators defined by convergent generating series and the existence of high order differential equations ("i/o equations") relating derivatives of inputs and outputs.; State space models are central to modern nonlinear control theory, since they permit the application of techniques from various mathematics branches such as differential equations, dynamical systems and optimization theory. A natural question is to decide when a given i/o operator admits a representation by an initialized state space system (the operator is realizable).; To investigate the relation between i/o equations and realizability, we introduce and study the structures of observation spaces, observation algebras and observation fields. In realization theory and many other areas of nonlinear control, the concept of observation space plays a central role. One may define observation spaces in two very different ways. Roughly, one possibility is to take the functions corresponding to derivatives with respect to switching times in piecewise constant controls, and the other is to take high-order derivatives at the final time, if smooth controls are used. It turns out that the existence of algebraic i/o equations is closely related to the finiteness properties of the observation algebra and field associated with the first type of observation space, while realizability is closely related to the finiteness properties of the algebraic objects associated with the other type of observation space. One of the central technical results, given in Chapter 3, shows that the two types of spaces coincide.; Based on the results mentioned above, we get our main results: Realizability by singular polynomial systems is equivalent to existence of algebraic i/o equations. We also provide other results relating various special kinds of i/o equations to some specific classes of realizations, for instance, what are called recursive i/o equations are related to realizability by polynomial systems.; In Chapter 7, our results relating algebraic i/o equations to realizability by "rational" systems are extended to analytic i/o equations and local realization by analytic systems. By studying properties of meromorphically finitely generated field of functions, together with the application of some known facts in the literature of nonlinear realization, we conclude that the existence of analytic i/o equations implies local realizability by analytic systems.