What is a minimum phase system?

The minimum phase system is a system with the smallest phase shift under certain amplitude-frequency characteristics; if the real parts of the poles and zeros of the open-loop transfer function of the closed-loop system are less than or equal to zero, it is called Minimum phase system. The characteristic of the minimum phase system is that the amplitude-frequency characteristics and phase-frequency characteristics are directly related. All poles and zeros are in the left half plane, and all poles and zeros are within the unit circle.

The operating environment of this tutorial: Windows 10 system, DELL G3 computer.

What is a minimum phase system

For a closed-loop system, if the real parts of its open-loop transfer function poles and zeros are less than or equal to zero, it is called a minimum phase system, if there are zeros or poles in the open-loop transfer function with a positive real part, or a delay link, the system is said to be a non-minimum phase system. Because if the delay link is approximately expressed in the form of zeros and poles (Taylor series expansion), it will be found that it has positive real part zeros.

Features: Amplitude-frequency characteristics and phase-frequency characteristics are directly related

Quality: All poles and zeros are in the left half-plane (continuous time system); all poles and zeros are in unit Inside the circle

The minimum-phase system is the system whose phase shift is the smallest under certain amplitude-frequency characteristics, also called the minimum phase shift system. Compared with the system function (also called network function or transfer function) of this system, the amplitude-frequency response characteristics of the two systems are the same, but the absolute value of the phase of the former is smaller than that of the latter. While keeping the amplitude-frequency response characteristics of the system function unchanged, the necessary and sufficient condition for minimizing its phase is: for an analog signal system, its zero point (that is, the complex frequency value at which the system function is zero) is only located on the S plane (i.e. On the left half plane or imaginary axis of the complex frequency domain plane); for discrete signal systems, the zero point is required to be located only within or on the unit circle of the Z plane (that is, the complex frequency domain plane of the discrete signal). Often used for phase correction.

For a continuous-time system, if all poles and zeros of the open-loop transfer function of the control system are located on the left half-plane of s, the system is called a minimum phase system. For a discrete-time system, all zeros and poles are located within the unit circle.

A system is called a minimum phase system if and only if the system is causally stable, has a rational form of the system function, and has a causally stable inverse function.

The minimum phase system mainly has the following two characteristics:

1. If the two systems have the same amplitude-frequency characteristics, then for any frequency greater than zero, the minimum phase The phase angle of the phase system is always smaller than that of the non-minimum phase system;

2. The amplitude-frequency characteristics of the minimum phase system are directly related to the phase-frequency characteristics. That is to say, an amplitude-frequency characteristic can only have one phase-frequency characteristic and Correspondingly, a phase-frequency characteristic can only have one amplitude-frequency characteristic corresponding to it. For a minimum phase system, the transfer function of the system can be written based on the logarithmic amplitude-frequency curve.

For more related knowledge, please visit the FAQ column!

The above is the detailed content of What is a minimum phase system?. For more information, please follow other related articles on the PHP Chinese website!