Lifted Inner-Outer Decomposition for Nonlinear Systems in Affine Normal Form

Non-minimum phase dynamics play a central role in control theory, yet available analysis tools for strictly proper nonlinear systems with unstable internal dynamics remain limited. This limitation motivates the Inner-Outer decomposition developed in this paper. We focus on systems that admit a normal form representation with constant vector relative degree and affine internal dynamics. Our main contribution is a constructive state-space procedure for building the inverse of the system self-adjoint operator. The procedure bypasses the obstruction encountered with the standard partial Legendre transform, which does not apply to strictly proper nonlinear systems. Existing Inner-Outer decomposition methods also do not cover the general affine internal dynamics setting. To address this limitation, we carry out the analysis on an auxiliary system. We explicitly derive the zero dynamics of a pseudo-Hamiltonian cascade and identify an autonomous Hamiltonian subsystem on the cotangent bundle of the zero-dynamics manifold. This leads to a Hamilton-Jacobi equation whose stabilizing solution characterizes the generating function of the stable invariant manifold associated with the zero dynamics. Using this solution, we construct a stabilizing output trajectory and a coordinate transformation that yields the Inner-Outer decomposition of the auxiliary system. Cascading this decomposition with the original output map defines the proposed "lifted" Inner-Outer decomposition. A non-minimum-phase example illustrates the effectiveness of the method.