ℒ2-Gain of Double Integrators with Saturation Nonlinearity

This paper uses quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has ℒ2-gain less than γ > 0. We show that for many such systems, the ℒ2-gain is non-conservative in the sense that this is approximately equal to the lower bound obtained by replacing the saturation with a constant gain of 1. These results allow the us of classical analysis tools like μ-analysis or IQCs to analyze systems with double integrators and saturations, including servo systems like some mechanical systems, satellites, hard-disks, CD players, etc.