A novel signal flow graph based solution for calculation of first and second derivatives of dynamical nonlinear systems

In this paper, the problem of calculation of first and second derivatives in general non-linear dynamical systems is addressed and an attempt of solution by means of signal flow graph (SFG) techniques is proposed. First and full second derivatives of an output of the initial system respect with the node variables of the starting SFG are delivered through an adjoint graph derived without using Lee's theorem. Mixed second derivatives are deduced by quantities attained in adjoint graphs of the original graph or graphs related to it. A detailed theoretical demonstration of these formulations is given. Even though no adjoint graph has been derived in case of mixed derivatives, the ability of the proposed method to determine all Hessian matrix entries in a complete automatic way is highlighted.