A robust semi-definite optimization based solution to the robust order reduction problem for parametric uncertain dissipative linear systems

In this paper we address the problem of reducing the order of a linear system affected by uncertainties from the robust dissipative perspective introduced in Barb. Firstly, we show that all major balanced truncation techniques developed and reported in the literature of the last two decades see Enns, Glover, Moore, Harshavardhana, Jonckheere, Opdenacker, Phillips) can be treated in a uniform fashion within the framework of dissipative systems. Accordingly, we shall generalize these results to uncertain dissipative systems. The key role is played by balancing two positive definite robust solutions to the uncertain dissipativity LMIs associated with the linear system in question and its dual. Determining the maximal level of uncertainty for which such two solutions exists and computing them efficiently is well known to be NP-hard. Our method is based on determining robust tractable approximations of these NP-hard entities by following the novel method known as Matrix-Cube Theory. The proven results are accompanied by a numerical example.