Inner-Outer Decomposition of Strictly Proper Nonlinear Systems in Output Affine Normal Form

This paper studies the Inner-Outer decomposition problem for strictly proper nonlinear MIMO systems, motivated by the relevant role of non-minimum phase dynamics in fundamental control theory. We consider systems that admit a normal-form representation with constant vector relative degree and with internal dynamics affine in the output.Building on the Hamiltonian framework for nonlinear Inner-Outer factorizations, the self-adjoint operator is represented through an associated Hamiltonian cascade but the standard partial Legendre-transform construction of an inverse cascade does not directly apply in the strictly proper case.Our main contribution is a constructive state-space procedure that bypasses this obstruction. We explicitly derive the output-nulling (zero) dynamics of the Hamiltonian cascade in normal-form coordinates, identify an autonomous Hamiltonian subsystem on the cotangent bundle of the zero-dynamics manifold. We then derive the Hamilton-Jacobi equation whose stabilizing solution parametrizes the stable invariant manifold associated with the zero-dynamics. Using such a solution, we construct a stabilizing output trajectory and introduce a back-stepping coordinate transformation that yields the Inner-Outer decomposition. The Outer component is minimum phase (its zero dynamics are stable by construction), while the Inner component is lossless with respect to the standard L2 supply rate and its origin is asymptotically stable.An illustrative non-minimum phase example demonstrates the effectiveness of the method.