Minimum phase system

The minimum phase system is an ambiguous term from systems theory and the related disciplines of communications engineering, control engineering and electrical engineering. Different, non-consistent definitions are used in the various specialist areas. For example, a minimum phase system can denote a linear time-invariant system whose system function only has zeros in the stable area of ​​the complex image plane or, in general (also for non-linear systems) whose zero dynamics are stable. The concept of the minimum-phase system applies to both time-continuous and time-discrete systems. Linear systems that are minimally in phase in the sense of Bode's original definition have the property of having the smallest possible group delay for a given amplitude curve . Often an "inverse response behavior" of the step response of a system is linked to the concept of minimum phase.

Linear, time-continuous systems

For time-continuous systems, the transfer function of which is determined as a Laplace transform of the impulse response , the unstable area of ​​the image plane is the right half plane with a positive real part. A time-continuous minimal-phase system only has zeros and - depending on the definition - poles in the left area of ​​the complex half-plane. In other words, a system with a rational transfer function G (s) is:

${\displaystyle G(s)={\frac {Z(s)}{U(s)}}}$

minimal phase if and only if it has no zeros to the right of the imaginary axis. Depending on the definition,

${\displaystyle \operatorname {Re} (s_{z})\leq 0}$

or

${\displaystyle \operatorname {Re} (s_{z})<0}$

required. If the position of the poles is also included in the definition of the minimum phase, this must also be done, for example

${\displaystyle \operatorname {Re} (s_{u})\leq 0}$

fulfill.

Discrete-time systems

For time-discrete systems whose transfer function is determined as the z-transform of the impulse response , the unstable area of ​​the image plane is that outside the unit circle . A time-discrete minimal-phase system has zeros only within the unit circle or (depending on the definition) exactly on the unit circle.

meaning

Some authors are of the opinion that minimum-phase systems are important, for example in the field of control engineering. Non-minimum-phase systems can always be broken down into a minimum-phase component and an all-pass , which can lead to a better consideration of the system or to simpler development of a controller. According to other sources, the name-giving, minimal course of the phase characteristic of a minimum phase system is not of interest for control-related issues.

literature

• Alan V. Oppenheim, Ronald W. Schafer, John R. Buck: Discrete-time signal processing . Pearson, ISBN 3-8273-7077-9 , chapter 5.6.

Individual evidence

1. ↑ Michael Zeitz: Minimum phase - not a relevant property for control engineering! In: at - automation technology . tape 62 , no. 1 , 2014, p. 3–10 , doi : 10.1515 / auto-2014-1067 .
2. ^ Otto Föllinger: Control engineering . 11th edition. VDE Verlag, Berlin 2013, ISBN 978-3-8007-3231-9 , pp. 141 .