Canonical parameterization of the dual problem in robust optimization: Non-rational case

Abstract

In this paper, we consider the problem of robust optimization for a system with uncertainty of rank one. The main result is the canonical parameterization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case has been previously stated in terms of unstable zero-pole cancellations.In this paper the result is extended to the class of non-rational systems with continuous nominal factors. For non-rational systems the situation with the common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. The result has certain similarities with the parameterization of the classical Nehari problem [9]. To illustrate the duality principle, the result is applied to a system with delay. The dual problem can be interpreted as the shortest distance from the nominal plant to all non-stabilizable plants in some metric that has a strong connection to and may be considered as a generalization of the υ-gap metric.