The maximal controllable set for open-loop unstable systems under input saturation

An optimization technique is presented for approximating the controllable set by an ellipsoid for a linear time-invariant open-loop unstable system subject to input saturation. A technique and algorithms for maximizing the controllable set are also presented. Instead of starting from a positive definite right-hand side matrix Q of the Lyapunov equation as done in almost all applications of the Lyapunov functions, we start from a positive definite Hessian matrix P for the Lyapunov function so that the resulting Lyapunov function can be convex.