Inner-Outer Decomposition for Strictly Proper Linear Time Invariant Systems and Non-Minimum Phase Performance Limitations

In this work, we deal with an equivalent input-output description of non-minimum phase multi-input multi-output Linear Time Invariant systems and its use in the output feedback stabilisation problem. We first present a state-space inner-outer decomposition of the original system, with the original system shown to be input-output equivalent to the cascade of a right invertible minimum phase (outer) system feeding and an all-pass non-minimum phase (inner) one. By assuming the existence of a state feedback stabiliser for the outer minimum-phase system, we then show the existence of an output feedback stabiliser for the original system such that its output behaviour is arbitrarily close to the output of the controlled outer system (e.g., reducing the size of undershoot in the system response), except that in an initial time interval that can be kept arbitrarily short, provided that the dynamics of the controlled outer system are sufficiently slow.