4. Composite Quadratic Lyapunov Functions

Because of their simplicity, quadratic Lyapunov functions are most commonly used Lyapunov functions in stability analysis of linear systems subject to actuator saturation. In particular, in Chapter 2, quadratic Lyapunov functions were utilized for regional stability analysis of a linear system with different types of saturation nonlinearities, and contractively invariant ellipsoids, level sets of quadratic Lyapunov functions, have been used as estimates of the domain of attraction. Moreover, in Chapter 3, we discussed the maximal contractively invariant ellipsoid, and developed an algebraic computational approach to determining the maximal contractively invariant ellipsoid. In spite of their simplicity, quadratic Lyapunov functions lead to conservative results in estimating the domain of attraction for a linear system subject to actuator saturation.