  A robust controller is specified, and the stability bounds of the uncertain
closed-loop system are determined using the small gain, circle, positive real,
and Popov criteria. A graphical approach is employed in order to demonstrate
the ease with which the above robustness tests can be carried out on a problem
of practical interest. A significant improvement in stability bounds is
observed as the analysis moves from the small gain test to the circle, positive
real, and Popov tests. In particular, small gain analysis results in the most
conservative robust stability bounds, while Popov analysis yields significantly
less conservative bounds. This is because traditional small gain type tests
allow the uncertainty to be arbitrarily time-varying, whereas Popov analysis
restricts the uncertainty to be constant, real parametric uncertainty.
Therefore, the results reported here indicate the conservatism associated with
small gain analysis, and the effectiveness of Popov analysis, in gauging robust
stability in the presence of constant, real parametric uncertainty.
