Duality inH$^\infty$ Cone Optimization

Abstract

Positive real cones in the space H ∞ appear naturally in many optimization problems of control theory and signal processing. Although such problems can be solved by finite-dimensional approximations (e.g., Ritz projection), all such approximations are conservative, providing one-sided bounds for the optimal value. In order to obtain both upper and lower bounds of the optimal value, a dual problem approach is developed in this paper. A finite-dimensional approximation of the dual problem gives the opposite bound for the optimal value. Thus, by combining the primal and dual problems, a suboptimal solution to the original problem can be found with any required accuracy.

Regularization of the Limiting Optimal Controller in Robust Stabilization

“…The robust stabilizability of s−δ (s−1) 2 on the uncertainty set δ ≤ ν was solved in Ghulchak and Rantzer [2002]. The stability margin ν max was calculated and the optimal controller that achieves the optimal level of stability was designed.…”

“…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. A method based on unstable cancelations has been presented to calculate the largest stability margin and to design the optimal controller of low order.…”

“…An approximation should ideally preserve the low order of the optimal controller. Different ad hoc ideas have been used in Ghulchak and Rantzer [2002] to obtain the suboptimal controller, for example, by analyzing Bode/Nyquist plot, however, a general approach to this problem still needs to be developed. In this paper, such an approach is presented.…”