Dimensional Unfolding and Controllability of Non-Linear Dynamical Systems

In linear dynamical systems, one has an elegant way to study dynamics and control using the state space formulation. However, in non-linear systems, there is no dynamics matrix to begin with. In this talk, we discuss how a system of non-linear ODEs can be dimensionally unfolded such that non-linear terms in the vector field can be re-expressed using auxiliary dynamical variables. This results in an unfolded dynamical system with only polynomial non-linearities. It turns out that this works for a large class of non-linear ODEs. For these, there exists an equivalent unfolded dynamical system with only polynomial non-linearities. Expressing generic non-linear vector fields using polynomials has some advantages. Methods of algebraic geometry can now be used to find attractors by organizing the system in Gröbner basis. More importantly, once we have a polynomial vector field, the system can straightforwardly be expressed in state space form, yielding a state-dependent dynamics matrix. Additionally, we will discuss a graphical representation of this matrix as a network with non-linear edge weights, which generalizes the usual notion of networks with dyadic edges to networks with compounded edges. A non-linear state space formulation of dynamical systems suggests a way to formalize stability and controllability criteria of non-linear dynamical systems. More specifically, with this we derive a non-linear generalization of the Kalman controllability matrix. Another consequence of expressing systems in generalized state space form is that this admits an eigenvalue flow in the spectrum of the system. This suggests a way to perform eigenvalue assignments for non-linear control. We demonstrate these applications for simple systems.