Duality relations for the optimal two-disk H∞ problem

Classical compensator design in the frequency domain amounts to chosing a linear controller which makes the sensitivity function S and the complementary sensitivity function T obey bounds on their gain, where the desired bounds in general vary with frequency and vector direction. The combination of the Youla Parameterisation and the concept of the Cartesian product of vector spaces allows the problem to be written in the form: Find a vector belonging to a subspace which minimises the distance from this vector to another given vector. The convexity of the underlying optimal synthesis problem follows immediately from the formulation given it. Function space duality theory is used to develop the duality relations for the problem. Bounds are established for the general case. For the case where the plant is both stable and minimum phase, the optimal achievable performance is explicitly determined.