Generalized Minimum Phase Property for Finite-Dimensional Continuous-Time SISO LTI Systems with Additive Disturbances

For finite-dimensional continuous-time single-input and single-output linear time-invariant systems, we introduce the concept of extended zero dynamics and generalize the concept of minimum phase, which accommodates the presence of disturbance inputs. By representing a SISO LTI system with a finite relative degree in its extended zero dynamics canonical form, we obtain its extended zero dynamics, which is simply its zero dynamics (according to [1]) driven by the noiseless output of the system and the disturbance input. We then say a system is minimum phase if its extended zero dynamics is absent or satisfies that, for any bounded admissible initial condition, any bounded noiseless output, and any bounded admissible disturbance input waveform, the zero dynamics state trajectory is bounded. The system is minimum phase (according to this extended notion) if its zero dynamics is asymptotically stable. It is proved that the converse holds under the additional condition that the system be stabilizable from the control input. For a system to be minimum phase, it is necessary that the transfer function from the control input to the output has all zeros with negative real parts. The converse holds when the system is both controllable (from the control input) and observable. It is further shown that the generalized minimum phase property is necessary for model reference control that achieves 1) perfect tracking of any bounded reference trajectories with bounded derivatives up to certain order without any disturbances and 2) the existence of bounded state trajectory for any admissible bounded initial condition, any admissible bounded disturbance input waveform, and any bounded reference trajectory with bounded derivatives up to a certain order.