Optimal solutions to matrix-valued Nehari problems and related limit theorems

In a 1990 paper Helton and Young showed that under certain conditions the optimal solution of the Nehari problem corresponding to a finite rank Hankel operator with scalar entries can be efficiently approximated by certain functions defined in terms of finite dimensional restrictions of the Hankel operator. In this paper it is shown that these approximants appear as optimal solutions to restricted Nehari problems. The latter problems can be solved using relaxed commutant lifting theory. This observation is used to extent the Helton and Young approximation result to a matrix-valued setting. As in the Helton and Young paper the rate of convergence depends on the choice of the initial space in the approximation scheme.Comment: 22 page

Taylor approximations of operator functions

This survey on approximations of perturbed operator functions addresses recent advances and some of the successful methods.Comment: 12 page

An explicit state-space approach to the one-block super-optimal distance problem

An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115–1134, 1993) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960–982, 1993; Limebeer et al. in Int J Control 50(6):2431–2466, 1989). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control