Phase margin is defined as the amount of change in open-loop phase needed to make a closed-loop system unstable. 2.25, meaning a positive gain margin, then the closed-loop system is stable. If the gain | G H ( j ω ) | at the frequency of ∠ G H ( j ω ) = − 180 ° is greater than 0 dB as shown in the left of Fig. The gain margin is the difference between 0 dB and the gain at the phase cross-over frequency that gives a phase of −180°.

Gain margin is defined as the amount of change in open-loop gain needed to make a closed-loop system unstable. The MATLAB commands for obtaining frequency response plots (which work for both continuous-time and discrete-time systems) areįigure 2.25. It is sometimes convenient to plot the frequency response using the Bode plot, but the availability of frequency response plotting commands in MATLAB reduces the necessity for such plots. In some cases, the intercept with the real axis can be obtained as the value where z = −1 provided that the system has no pole at −1 (i.e., the frequency response has no discontinuity at the folding frequency ω s/2). However, because only approximate values are needed in practice, it is easier to use MATLAB to obtain both margins. The phase margin can be obtained by equating the magnitude of the frequency response to unity and solving for angle, and then adding 180°. The gain margin can be obtained analytically by equating the imaginary part of the frequency response to zero and solving for the real part. Nyquist plot with negative gain margin (dBs) and phase margin. The two distances are given in the following definitions.įigure 4.13. However, it is more convenient to define relative stability in terms of two distances: a magnitude distance and an angular distance. Typically, the distance between a set of points and the single point (−1, 0) is defined as the minimum distance over the set of points. For open-loop stable systems that are nominally closed-loop stable, the distance from instability can be measured by the distance between the set of points of the Nyquist plot and the point (−1, 0).

To keep our discussion simple, we restrict it to open-loop stable systems where zero encirclements guarantee stability. This degree of stability is known as relative stability. We therefore need to determine how far the system is from instability. This is because of the approximate nature of mathematical models and our imperfect knowledge of the system parameters.

In practice, the stability of a mathematical model is not sufficient to guarantee acceptable system performance or even to guarantee the stability of the physical system that the model represents. Sami Fadali, Antonio Visioli, in Digital Control Engineering, 2009 4.6.1 Phase Margin and Gain Margin

Phase shift controller setup