Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations

A piecewise quadratic Lyapunov function is developed for the analysis of the global and regional performances for systems with saturation/deadzone in a general feedback configuration with an algebraic loop. This piecewise quadratic Lyapunov function effectively incorporates the structure of the saturation/deadzone nonlinearity. Several sector-like conditions are derived to describe the complex nonlinear algebraic loop. These conditions transform several performance analysis problems into optimization problems with linear (or bilinear) matrix inequalities. The effectiveness of the results is demonstrated with numerical examples.