NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

for its true robustness, and its only drawback is the care with which one must choose
appropriate basis functions for the approximation and an appropriate region of sta-
bility, .
5.5 Tabulated Results
Table 5.1 summarizes the performance of the control strategies implemented
on the FBS. The rst row compares the e ective exponential decay rates { these
values were computed by doing a least-square t of the rst three local maxima of
the response curves. The PBC control shows an unusual perturbation in its response
envelope, and its decay number is therefore not meaningful. The second row shows
the integral of the total energy in the linear position state, y, from the time the
control is turned on till the steady-state is reached at about 6 seconds. The third row
compares the control e ort by giving the integral of the control signal from 2 to 6
seconds. (In the hardware tests steady-state error is non-existent due to the natural
damping of the system, therefore the integrator is only run to t =6seconds.)
Table 5.1: Tabular Comparison of Experimental Results
Linear Optimal Linear H1 Passivity HJB HJI
: e ; t 1.88 1.88 1.89 1.90 1.91
R y(t) 2 dt 1.71 1.71 1.63 1.41 1.1
R u(t) 2 dt .23 .33 1.35 .3 .44
Clearly the best control is the non-linear H1 (SGA/HJI) solution. It outperforms
the other control laws without using as much control e ort as the PBC control. The
next best control is the non-linear H 2 (SGA/HJB) solution. It does not attenuate the
disturbance as fast as the HJI solution, but it also uses less control e ort. These two
SGA-based controllers produced nonlinear feedback control laws that were noticeably
superior to the linearized controllers. PBC performs better than both of the linearized
controls, but it uses an unusual amount of control e ort. An advantage of the PBC
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