Max-plus based computation of nonlinear ℒ2-gain performance bounds using a piecewise affine-quadratic basis

Nonlinear ℒ₂-gain is a generalization of the well-known finite ℒ₂-gain robust stability property for nonlinear systems. Computation of tight performance bounds associated with this nonlinear ℒ₂-gain property is key to avoiding conservatism in its application, for example in small-gain based design. In previous work, a number of max-plus eigenvector methods have been proposed to facilitate this computation. Those methods have each employed quadratic basis functions, which have been shown to lead to a specific computational issue concerning continuity of the associated Hamiltonian. In this paper, an alternative piecewise affine-quadratic basis is proposed that allows the development of a refined max-plus eigenvector method that avoids this computational issue.