NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

Chapter 3
Overview of Control Strategies Implemented on the Flexible
Beam System
This Chapter will provide a review of all of the control strategies that were
tested on the NLBP. The focus of this thesis is the comparison of the SGA method
of control with a broad sample of current robust nonlinear control methodologies.
Thus the following nonlinear design methodologies are included mainly to highlight
the uniqueness of the SGA design method. This chapter will explain the mathemat-
ical derivation of two linearized controls, a passivity based approach, abackstepping
controller, and the SGA algorithm. Additionally, the details of the implementation
of these controllers will be explained.
3.1 Linear Quadratic Regulation
As a starting point of comparison, an H 2 optimal control was rst designed
for the linearized system. With optimal control, the objective is to minimize a cost
functional of the following form:
V (x 0)=
Z 1
(x
0
T (t)Qx(t)+u T Ru)dt (3.1)
where V (x 0) represents the cost associated with moving from a given state, x 0, under
a control signal u to the origin. Q and R are matrices that weight the cost of the
states and the control respectively. The goal is to nd the stabilizing control signal
u (x) that will minimize the cost V (x). Linear quadratic regulation has a well known
solution, where the optimal stabilizing control u (x) is a simple linear combination of
the states: u (x) =;Kx. K is a matrix that depends upon the input matrix B, the
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