NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

all of the physical parameters except for the motor constants can be combined into
the term as described in Chapter 1 = p mc
(I+mc2 )(m+M )
. All of the controllers
were designed for the actual value of = :2. The simulations were then run with
varying values of until the controller no longer regulated the system. Thus, the
more robust controllers could tolerate greater changes in epsilon than the less robust
designs. In Table 4.2 below, the maximum and minimum values of that the control
designs could regulate, given that they were assuming a plant value of = :2, are
compared:
Table 4.2: Tabular Comparison of Simulated Robustness
LQR Lin. H1 PBC Backstep HJB HJI
max .277 .275 .245 .251 .262 .275
min .016 .015 .014 .015 .102 .093
Interestingly, the linearized controls outperformed the nonlinear designs in this
simulation as well: the linearized LQR control had the highest tolerance for raising
the value of in the system, with the linearized H1 and the SGA to the nonlinear
H1 yielding only slightly less robustness with respect to this upper bound. On the
lower bound, the system value for could be lowered to = :014 before the passivity
based control failed to regulate the system, over a 1000 percent change in the value
of ! In fact, all of the controls gave similarly robust results for the lower bound of ,
except the Galerkin approximations whose lower bounds were notably inferior.
Robustness is a property that is very di cult to measure precisely, and the
above simulated results only give a rough idea of how the controllers behave when
there are signi cant modeling errors present. It should be noted that they all exhibit
a wide range of values for , wherein the controls continue to regulate the system,
and they are all, therefore, meaningfully robust.
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