NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

limitations, k 1 and k 2 are tuning parameters to properly shape the closed loop en-
ergy function, and q 3 is the actuated plant variable, . The controller dynamics are
computed from the interconnection constraint andthe controller's EL equation:
u = ;k 2 tanh(qc + b ) ; k 1 tanh( )
qc _ = ;ak2 tanh(qc + b ):
(3.17)
Thus the control takes the measurable output, , as its only input, and the dynamics
in the control produce a pseudo-derivative of to generate the necessary damping.
The mathematical model for the exible beam system was slightly di erent from the
nonlinear benchmark problem due to the extra term that resulted from the fact that
we used voltage as an input instead of torque. This made the implementation of the
saturation constraint a little tricky. In fact, the particular software/hardware con g-
uration of the exible beam system required the removal of the saturation functions
in the controller dynamics. (The actual control signal was up = ;k 2(qc + b ) ; k 1 .)
To implement the control was simple: nd appropriate values for a b k 1 and k 2.
The constants a = 250, b = 2:7, k 1 = :29 and k 2 = 2:5 were chosen experimentally
to give the best possible dynamic performance. This control was relatively easy to
implement, and its great advantage is that it only uses measurable outputs of the
system. This strength can also be seen as a weakness, though the controller does
not utilize all of the available information { only one of the two truly measurable
states is used to generate the control signal. From an informational stand-point, the
ideal controller should use all of the available plant output signals. However, the
passivity-based control can clearly do more than a feedback system consisting of a
simple gain matrix { PBC systems implement a dynamic system in the feedback loop
that, in our case, provides information about _ as well as .
3.4 Backstepping
Another recent approach to the problem of robustly stabilizing non-linear sys-
tems is that of integrator backstepping. This control strategy is an attempt to con-
struct for a given non-linear system a control that will render the closed-loop system's
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