Editing Minimum phase (section)

== Non-minimum phase ==

Systems that are causal and stable whose inverses are causal and unstable are known as ''non-minimum-phase'' systems. A given non-minimum phase system will have a greater phase contribution than the minimum-phase system with the equivalent magnitude response.

=== Maximum phase ===
{{unreferenced section|date=September 2014}}
A ''maximum-phase'' system is the opposite of a minimum phase system. A causal and stable LTI system is a ''maximum-phase'' system if its inverse is causal and unstable.{{dubious|date=September 2014}} That is,
* The zeros of the discrete-time system are outside the [[unit circle]].
* The zeros of the continuous-time system are in the right-hand side of the [[complex plane]].
Such a system is called a ''maximum-phase system'' because it has the maximum [[group delay]] of the set of systems that have the same magnitude response. In this set of equal-magnitude-response systems, the maximum phase system will have maximum energy delay.

For example, the two continuous-time LTI systems described by the transfer functions
<math display="block">\frac{s + 10}{s + 5} \qquad \text{and} \qquad \frac{s - 10}{s + 5}</math>

have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift. Hence, in this set, the second system is the maximum-phase system and the first system is the minimum-phase system. These systems are also famously known as nonminimum-phase systems that raise many stability concerns in control. One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method.<ref>{{Cite book|title=Analytical Statistical Study of Linear Parallel Feedforward Compensators for Nonminimum Phase Systems|last=Noury|first=K. |date=2019|doi = 10.1115/DSCC2019-9126 |chapter = Analytical Statistical Study of Linear Parallel Feedforward Compensators for Nonminimum-Phase Systems|isbn = 978-0-7918-5914-8|s2cid=214446227 }}</ref>

=== 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.

=== Linear phase ===

A [[linear phase|linear-phase]] system has constant [[group delay]]. Non-trivial linear phase or nearly linear phase systems are also mixed phase.