Polyhedral functions, composite quadratic functions, and equivalent conditions for stability/stabilization

Relationship between polyhedral functions and composite quadratic functions is investigated in this paper. The two composite quadratic functions considered are the pointwise maximum of quadratics and the convex hull of quadratics. It is shown that these two composite quadratic functions are universal for robust, possibly constrained, stabilization problems. In particular, a linear differential inclusion is stable (stabilizable with/without constraints) iff it admits a Lyapunov (control Lyapunov) function in these classes. Relationships between the existing stability/stabilization conditions derived from these functions are also investigated. It is shown that a well known stability condition in terms of matrix equalities is equivalent to a stability condition in terms of bilinear matrix inequalities (BMIs). Similar conclusions are made about conditions for stabilization of linear differential/difference inclusions and constrained control systems. This investigation provides insight into the relationship between two alternative approaches to various analysis and design problems, making it possible to transform some synthesis problems derived from polyhedral functions into LMI-based optimization problems.