Lyapunov design of stabilizing controllers for cascaded systems

The design of a state feedback law for an affine nonlinear system to render a (as small as possible) compact neighborhood of the equilibrium of interest globally attractive is discussed. Following Z. Artstein´s theorem (1983), the problem can be solved by designing a so-called control Lyapunov function. For systems which are in a cascade form, a Lyapunov function meeting Artstein´s conditions is designed, assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem. In particular, for planar systems, this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized