Pre-filtering in gain-scheduled and robust control

We revisit the issue of gain-scheduled versus robust control with a focus on matrix inequalities. It has been established that for uncertain continuous-time linear systems that depend affinely on the uncertainty, gain-scheduled stabilizability implies robust stabilizability. That is, as far as stabilizability is concerned, using a more complex gain-scheduled controller brings no advantage. In the case of performance and discrete-time systems, counter-examples exist that show that gain-scheduling can indeed be advantageous. These proof are unfortunately not constructive, and the associated necessary and sufficient conditions are hard to verify even in low dimensions. In practice, conditions based on Linear Matrix Inequalities (LMIs) are widely used to design robust and gain scheduled controllers at the expense of some conservatism. The main goal of the present paper is to explore to what extent solvability of certain LMIs for gain-scheduled control also implies solvability of the corresponding robust control inequalities. One issue investigated in detail is that of using pre-filters to handle uncertainty appearing in the input matrix. Our results show that this technique, which has been used since the 80s is rarely productive in the sense that solvability of certain gain-scheduled control design problems for the original system augmented with a pre-filter often implies existence of a robust control for the original system, which we calculate explicitly using a projection. One exception seem to be the LMIs based on the condition of Daafouz and Bernussou (2001) for discrete-time systems. A series of examples illustrate the results.