An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions

We consider a nonlinear state transformation that allows us to work with non-quadratic polynomial Lyapunov functions. We use these polynomials to form Lyapunov functions to demonstrate simultaneous stability for a finite collection of linear systems. Under a weak definiteness condition, our main result, Theorem 3, shows that the minimum degree polynomial Lyapunov function that demonstrates simultaneous stability for a collection of linear systems can be written as a homogeneous polynomial.