A duality-based approach to the multiobjective H₂/H_∞ optimisation problem

Abstract

In this article, a duality approach to multiobjective H ₂/H _∞ problems is pursued in which real-rational, para-Hermitian multipliers and real-valued ones are associated to H _∞ and (as usual) H ₂ constraints, respectively. It is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective H ₂/H _∞ problem. To compute lower bounds on the latter and the corresponding approximate solutions to the original problem, the maximisation of the dual functional over linearly-parameterised, finite-dimensional classes of real-rational multipliers is shown to be equivalent to semi-definite, linear programming problems – once the optimal multipliers in such a class are obtained, the corresponding approximate solutions can be computed from an unconstrained H ₂ problem. Iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem. This is done on the basis of (locally) increasing directions for the dual functional which go beyond the finite-dimensional class of multipliers considered in a given step. Finally, a numerical example is presented to illustrate the way the presented results can lead to approximate solutions to the multiobjective H ₂/H _∞ problem together with tight estimates of the corresponding deviation from its optimal value.

Keywords:

• H ₂ optimisation
• linear systems
• H _∞ control
• Lagrangian duality
• linear matrix inequalities
• multivariable systems
• Youla parameterisation