Singular Control-Invariance PDEs for Nonlinear Systems

Abstract

In the present study a new method is proposed that allows the derivation of control laws capable of enforcing the desirable dynamics on an invariant manifold in state space. The problem of interest naturally surfaces in broad classes of physical and chemical systems whose dynamic behavior needs to be controlled and favorably shaped by external driving forces. In particular, the formulation of the problem under consideration is mathematically realized through a system of first-order quasi-linear singular invariance partial differential equations (PDEs), and a rather general set of conditions is derived that ensures the existence and uniqueness of a solution. The solution to the above system of singular PDEs is proven to be locally analytic, thus allowing the development of a series solution method that is easily programmable with the aid of a symbolic software package. Furthermore, through the solution to the above system of singular PDEs, an analytic manifold and a nonlinear control law are computed that render the manifold invariant for the nonlinear dynamical system considered. In particular, the restriction of the system dynamics on the invariant manifold is shown to represent exactly the desirable target dynamics of the controlled system. Finally, an illustrative case study of molecular dissociation is considered, and the proposed method is evaluated through simulation studies.