Monotonic relaxations for robust control: new characterizations

Parameterized linear matrix inequalities (PLMIs), that is LMIs depending on a parameter confined to a compact set frequently arise in both analysis and synthesis problems of robust control. As a major difficulty, PLMIs are equivalent to an infinite family of LMI constraints and consequently are very hard to solve numerically. Known approaches to find solutions exploit relaxations inferred from convexity arguments. These relaxations involve a finite family of LMIs the number of which grows exponentially with the number of scalar parameters. In this note, we propose a novel approach based on monotonicity concept which allows us to solve PLMIs via a finite and of polynomial order family of LMIs. The effectiveness and viability of our approach are demonstrated by numerical examples such as robust stability analysis and linear parameter varying (LPV) synthesis for which we clearly show that no additional conservatism is entailed as compared to earlier techniques.