NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

equation. The Matlab commands lqr and hinf automate the entire procedure. The
Galerkin approximations are also fairly easy to implement, requiring only the e ort to
master the software package. All of the work is then done by the algorithm. Choosing
an appropriate required several attempts: a balance must be struck between having
the largest possible region of stabilityandhaving improved performance on the region
of operation.
The backstepping control is by far the most di cult to implement, as it requires
a good deal of e ort to do all of the variable transformations and to create the stable
control Lyapunov functions. Tuning for this algorithm is also di cult due to the
complexity of the feedback law: it is not clear how changing the weighting coe cients
will a ect the control signal (i.e. in the LQR and SGA designs Q and R weight the
cost of the states, there's no such physical intuition here.)
The passivity based control is also relatively di cult to implement, and its
results are the poorest of the test group. It is also di cult to tune this control its
performance is very sensitive to changes in the feedback loop parameters, a b k 1
and k 2, and it took the longest time to adjust this control in order to provide an
acceptable response. Perhaps, its poor performance is due to an inappropriate choice
of these parameters, but tuning this control is unclear at best { raising the gains does
not translate into more instability and better performance. In fact, the nal design
is the result of lowering the values of k 1 and k 2.
4.5 Robustness Analysis
The true test of any given algorithm's robustness lies in how it will perform in
hardware: imperfect sensors, actuators, and unmodelled disturbances and dynamics
will always degrade control e orts. So in many ways, the best measure of robustness
for the six control designs studied is to examine how their simulated results compare
with their results as tested in hardware. Another way to gauge robustness, especially
with respect to modeling errors, is to design for a speci c set of system parameters, run
the control on a system with di erent parameter values, and measure the degradation
in the performance. This was easily accomplished with the TORA system, because
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