Quadratic Surface Lyapunov Functions in the Analysis of Feedback Systems with Double Integrators and Saturations

Many systems like servo systems, satellites, harddisks, and CD players, can be modeled as linear systems with a single integrator and a saturation. Many times, such systems are controlled with a PI controller resulting in a feedback interconnection with a double integrator and a saturation. In this paper, we propose a loop transformation that results in bounded operators so that classical analysis tools like mu􀀀-analysis or IQCs can be applied. In order to show boundedness of all operators, we use quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has L2 􀀀-gain less than gamma > 0 􀀀. We show that for many of 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 .