\({\mathcal H}^\infty\) sensitivity minimization for delay systems (Q1092852)

The paper announces results on the single input/single output \(H^{\infty}\) weighted sensitivity minimization control problem with transfer functions of the form \(P(s)=e^{-s\Delta}P_ 0(s)\), where \(P_ 0(s)\) is a minimum-phase and stable rational function, and \(\Delta >0\). The problem is to minimize \(\| W(s)(I+P(s)C(s))^{-1}\|\), where W(s) is the weighting function and C(s) ranges over all proper compensators for which the corresponding feedback system is internally stable. It is assumed that \(W(s)=(s+1)/(s+\beta)\), \(\beta >0\). Formulas for the infimal sensitivity are given, as well as the formulas for the optimal feedback compensator (when \(\beta <1)\). This compensator turns out to be an infinite dimensional system; approximation by rational proper feedback compensators is given.