Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems

For planar piecewise affine systems, this paper proposes sufficient stability conditions based on discontinuous Lyapunov functions. The monotonicity condition for discontinuous functions at switching instants is presented based on the behavior of state trajectories on the switching surfaces. First, the stability conditions are derived for a typical multiple Lyapunov function and then these conditions are formulated as a set of linear matrix inequalities for piecewise quadratic Lyapunov functions. The implementation of the proposed method is illustrated by an example.