NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

disturbances, whereas the hardware results will indicate the controls' performance in
the presence of modeling errors, sensor noise, and other unmodelled disturbances.
Plots showing how each of the six control laws compares to each of the others
are included and discussed, but it is vital to have a more quantitative measure of
performance. To this end a couple of simple methods were chosen to quantify the
performance of the algorithms. First, a simple integrator was added that returns the
total energy of the rst state (the beam's de ection) from t =2:5 to t = 10 seconds.
This quanti es how quickly the oscillations died out which is the primary objective
of the control laws. This sum also includes a measure of the steady state error of the
controls, because most of the control laws achieved their steady states before t = 8
seconds. Second, an integrator returned the sum of the squared input signal from
t = 2:5 to t = 10 seconds. This number represents the amount of energy used to
implement the control: it provides an estimate of the control e ort, and it may thus
be used to evaluate the e ciency of the control laws (i.e. if two control designs had
comparable damping, which used less control e ort to achieve this damping?) These
results are compared at the end of this section in tabular form.
All of the simulations were conducted on the same SIMULINK model of the
plant. Each of the control techniques was tuned extensively using the information
from the simulations to yield the best possible performance. It should be noted that
this process of tuning, of choosing the parameters in order to implement the controls
at an optimal level, is a highly subjective activity. In fact there is no way to be sure
that the parameters nally used yielded the best possible performance for the given
implementation of the control strategy, though the utmost care was taken to try and
achieve the best possible performance with each of the design strategies.
4.2 Evaluation of the Plots in Simulation
4.2.1 Linearized Optimal H 2
The standard linear quadratic regulation method successfully regulated the
system, a fact which perhaps argues against this physical system as a benchmark
for non-linear systems: the whole purpose of nonlinear control theory is to provide
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