NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

to the Galerkin approximation technique, where again an optimal set of coe cients
is sought that will regulate the system optimally in some region of the state space,
. The di erence is that the linearized controls are only guaranteed to locally stabi-
lize the system, whereas the Galerkin approximations take into account more global
information. Also, the SGA technique creates a truly nonlinear control signal, and
this translates into better and more robust performance when implemented on the
exible beam set-up.
On the other hand, the backstepping approach uses a Lyapunov approach {
building a control Lyapunov function step by step, then implementing a very complex
and non-linear control that is a composite of the control signals at the various steps.
The passivity-based strategy seeks to exploit the structure and ow of the
energy in the system by adding a dynamic in the feedback loop.This control dynamic
is chosen to add damping arti cially to the closed-loop system and to shape the
potential energy function for closed-loop stability andperformance.
6.2 Extensions to this Research
Some interesting things have come to light throughout the testing of these
controls: there does not seem to be any advantage of adding higher order terms to
the Galerkin approximations. As higher power terms are added to the basis functions
there was a slight degradation in performance. This would be an excellent extension
to this research: to discover why adding higher power terms to the approximation
does not improve the control. Also, it would be valuable to experiment with non-
polynomial basis functions, to see if other functions might provide better control
signals perhaps exponential, logarithmic, Bessel, or other functions would generate
useful nonlinearities in the feedback signal.
The selection of an appropriate and appropriate basis functions is also an
area where more research should be conducted. It was observed that changing the
region of stability, , could signi cantly improve or degrade the control. Future
research might explore ways of mathematically determining the optimal size of for
a given demand of robustness and performance within a given region of the state
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