Robust stabilizability and low-order controllers: duality principle in case studies

Regularization of the Limiting Optimal Controller in Robust Stabilization

“…The problem of finding h ∈ H ∞ such that condition (1) holds is considered as a primal problem. We would like to solve it for as large ν as possible, that is, for In Ghulchak [2004] a dual problem was introduced as follows:…”

“…In Ghulchak [2004], the maximal stability margin was calculated and found the optimal controller that has achieved this level of stability. The unstable pole in the numerator a = 1 + √ 2, the stability margin ν opt = √ 2 − 1 ≈ 0.414 and the optimal controller is K opt = s − a.…”

“…Doyle et al [1992] have shown that the largest achievable k opt = 4 and suggested a sixth order controller for the gain margin k = 3.5. Ghulchak [2004] has found the 2-d order optimal controller and regularized it by perturbing the Nyquist plot.…”

“…The convex duality principle has brought a new insight into robust stabilization. The dual problem for systems with uncertainties of rank one has been introduced in Ghulchak [2004]. Several examples have been studied in Ghulchak and Rantzer [2002] to illustrate the power and simplicity of the principle.…”