The robustness of the stability property of multivariable feedback
control systems with respect to model uncertainty is studied and
discussed. By introducing a topological notion of arcwise connectivity,
existing and new robust stability tests are combined and unified under a
common framework. The new switching-type robust stability test is easy
to apply, and does not require the nominal and perturbed plants to share
the same number of closed right half-plane poles, or zeros, or both. It
also highlights the importance of both the sensitivity matrix and the
complementary sensitivity matrix in determining the robust stability of a
feedback system. More specifically, it is shown that at those
frequencies where there is a possibility of an uncertain pole crossing
the jw-axis, robust stability is "maximized" by minimizing the maximum
singular value of the sensitivity matrix. At frequencies where there is
a likelihood of uncertain zeros crossing the imaginary axis, it is then
desirable to minimize the maximum singular value of the complementary
sensitivity matrix.
A robustness optimization problem is posed as a non-square
H∞-optimization problem. All solutions to the optimization problem are
derived, and parameterized by the solutions to an "equivalent" two-parameter
interpolation problem. Motivated by improvements in
disturbance rejection and robust stability, additional optimization
objectives are introduced to arrive at the 'best' solution.</p