Robustness optimization of linear feedback systems

The robustness of a linear, finite-dimensional, time-invariant, single-input single-output feedback system with respect to plant variations is analyzed and expressed in terms of the sensitivity function of the feedback system and its complement. The analysis leads to defining optimally robust control systems as control systems for which the supremum over all frequencies of a weighted combination of the sensitivity function and its complement is minimal. Besides accomodating robustness considerations, the approach allows handling specifications on the closed-loop bandwidth and controller pole excess. A complete solution of the problem is given. Specialization of the general formulation to the minimization of the supremum of the sensitivity function or its complement alone affords determining the achievable performance of a single-input single-output plant. This performance depends on the plant pole-zero configuration and is characterized by the so-called regulability number of the plant. The ideas are illustrated with an example involving a double pendulum system in four different equilibrium positions.