NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

Chapter 1
Introduction
1.1 Motivation and Problem Description
The reliability and e ectiveness of feedback control systems has been recog-
nized for centuries: they provide a simple and robust way of regulating a large class
of physical systems, and there exists a rich body of mathematical results that enable
the systematic design of feedback control laws. A signi cant limitation of most of the
classic results is that they apply to a rather narrow class of systems { those which can
be modeled by systems of linear di erential equations. Despite the fact that many
physical plants can be linearized, and thus an adequate small-signal control can be
realized for the linearized system, the fact remains that almost all physical systems in-
volve dynamics that cannot be modeled completely by a linear system. Such nonlinear
systems have been the subject of much research in the past two decades researchers
have sought to extend the results obtained for linear plants to non-linear plants.
A good example of this is the search for the nonlinear optimal feedback con-
troller. It is well known that the solution of the optimal linear control problem
depends upon the solution of an algebraic Riccati equation. For the non-linear case,
this result generalizes nicely: the optimal non-linear controller requires the solution of
the Hamilton-Jacobi-Bellman (HJB) equation (which was actually discovered before
the Riccati equations). This partial di erential equation is very di cult to solve,
however, and the quest for a reliable and accurate approximation of its solution is an
open problem. The Successive Galerkin Approximation (SGA) is one such method of
approximation, and the details of its implementation will be described in Chapter 3.
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