Editing Minimum phase (section)

=== Mixed phase ===

A ''mixed-phase'' system has some of its [[Zero (complex analysis)|zero]]s inside the [[unit circle]] and has others outside the [[unit circle]]. Thus, its [[group delay]] is neither minimum or maximum but somewhere between the [[group delay]] of the minimum and maximum phase equivalent system.

For example, the continuous-time LTI system described by transfer function
<math display="block">\frac{ (s + 1)(s - 5)(s + 10) }{ (s+2)(s+4)(s+6) }</math>
is stable and causal; however, it has zeros on both the left- and right-hand sides of the [[complex plane]]. Hence, it is a ''mixed-phase'' system. To control the transfer functions that include these systems some methods such as internal model controller (IMC),<ref>{{Cite book |title=Robust process control|author =Morari, Manfred |date=2002| publisher=PTR Prentice Hall|isbn=0137821530|oclc=263718708}}</ref> generalized Smith's predictor (GSP)<ref>{{Cite journal|last1=Ramanathan|first1=S. |last2=Curl|first2=R. L.| last3=Kravaris|first3=C.|date=1989 | title=Dynamics and control of quasirational systems |journal=AIChE Journal |language=en |volume=35 |issue=6 |pages=1017–1028 |doi=10.1002/aic.690350615 |bibcode=1989AIChE..35.1017R |issn=1547-5905 |hdl=2027.42/37408 |s2cid=20116797|hdl-access=free}}</ref> and parallel feedforward control with derivative (PFCD)<ref>{{Cite book|title=Class of Stabilizing Parallel Feedforward Compensators for Nonminimum Phase Systems |last=Noury|first=K. |date=2019|doi = 10.1115/DSCC2019-9240|chapter = Class of Stabilizing Parallel Feedforward Compensators for Nonminimum-Phase Systems |isbn = 978-0-7918-5914-8|s2cid=214440404 }}</ref> are proposed.