Robustness of optimally robust controllers

The largest robust stability radius γ(P₀) of a system P₀ is defined as the radius of the largest ball B_max in the gap metric centered at P₀ which can be stabilized by one single controller. Any controller which stabilizes B _max is called an optimally robust controller of P₀. Any controller, regarded as a system, should have its own largest robust stability radius also. In the paper it is shown that the largest robust stability radius of any optimally robust controller of P ₀ is larger than or equal to γ(P₀). Moreover, the variation of the closed-loop transfer matrix caused by the perturbation of the system is estimated