Analysis of systems with saturation/deadzone via piecewise-quadratic Lyapunov functions

We study the global and regional stability and performance of saturated systems in a general feedback configuration. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of disturbances with bounded energy and the nonlinear L₂ gain. Motivated by the Lure-Postnikov type Lyapunov function, we develop a piecewise quadratic function which effectively incorporates the structure of the saturation/deadzone nonlinearity. The global and regional analysis are established through an effective treatment of the algebraic loop containing exogenous inputs. The corresponding conditions for global stability and performance are derived as Linear Matrix Inequalities (LMIs), and the conditions for regional ones yield bilinear matrix inequalities (BMIs). The BMI conditions cover the corresponding LMI conditions when the bounded energy of the disturbance goes to infinity. Numerical examples demonstrates the effectiveness of this approach and the great potential of the piecewise quadratic Lyapunov function.