M. Zeitz [2] discusses the inconsistent definitions of the minimum-phase property in a German paper. The abstract in English states the following [2]:

Originally, the minimum phase property has been defined by H. W. Bode [3] in order to characterize the unique relationship between gain and phase of the frequency response. With regard to the design of digital filters, another definition of minimum phase is used and a filter is said to be minimum phase if both the filter and its inverse are asymptotically stable. Finally, systems with asymptotically stable zero dynamics are named as minimum phase by C. I. Byrnes and A. Isidori [1]. Due to the inconsistent definitions, avoiding the minimum phase property for control purposes is advocated and the well-established criteria of Hurwitz or Ljapunow to describe the stability of filters and zero dynamics are recommended.

Inputs

sys

LTI system.

Optional tolerance. tol must be a real-valued, non-negative scalar. Default value is 0.

Outputs

bool

True if the system is minimum-phase and false otherwise.

   real (z) < -tol*(1 + abs (z))    continuous-time
   abs (z) < 1 - tol                discrete-time
 

References
[1] Byrnes, C.I. and Isidori, A. A Frequency Domain Philosophy for Nonlinear Systems. IEEE Conf. Dec. Contr. 23, pp. 1569–1573, 1984.

[2] Zeitz, M. Minimum phase – no relevant property of automatic control!. at – Automatisierungstechnik. Volume 62, Issue 1, pp. 3–10, 2014.

[3] Bode, H.W. Network Analysis and Feedback Amplifier Design. D. Van Nostrand Company, pp. 312-318, 1945. pp. 341-351, 1992.

Source Code: @lti/isminimumphase