Gain margin

There are two basic types of specifications that are commonly used in the frequency domain. The first type [phase margin and gain margin) specifies how near the openloop polar plot is to the critical (—1,0) point. The second 

These specifications are easy to use, as we will show with some examples in Sec. 13.4. They can be related qualitatively to time-domain specifications such as damping coefficient. 

Phase margin (PM) is defined as the angle between the negative real axis and a radial line drawn from the origin to the point where the GuB curve intersects the unit circle. See Fig. 13.7. The definition is more compact in equation form. 

Gain margin (GM) is defined as the reciprocal of the intersection of the G B polar plot on the negative real axis. 

A system must be third- or higher-order (or have deadtime) to have a meaningful gain margin. Polar plots of first- and second-order systems do not intersect the negative real axis. 

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I am seleeting the bandwidth of the elosed loop (fxo) power supply to be 6 kHz. It eould be as high as 15 to 20 kHz, but there is a double pole at one-half the switehing frequency which, if approached too closely, would ruin the phase and gain margin of the closed loop. 

Gain margin. This is the value of the gain when the phase crosses over -360 degrees. 

Fig. 6.19 Gain margin (GM) and phase margin (PM) on the Nyguist diagram.

Gain Margin (GM) The gain margin is the inerease in open-loop gain required when the open-loop phase is —180° to make the elosed-loop system just unstable. 

Gain margin = 8.15dB Phase margin = 60 Bandwidth = 2.02rad/s... 

Closed-loop peak Mp = 5.5 dB Gain margin = 13.75dB Bandwidth = 5.09 rad/s Phase margin = 30.6 ... 

The compensated and uncompensated open-loop frequency response is shown in Figure 6.41. From this Figure the compensated gain margin is 12.5 dB, and the phase margin is 48°. In equation (6.117), K does not need to be adjusted, and can be set to unity. When responding to a step input, the steady-state error is now 4.6%. 

In Example 6.4, when there was no model uneertainty, K for marginal stability was 8, and for a gain margin of 6dB, K was 4. In this example with model uneertainty, from equation (9.154) marginal stability oeeurs with K = 3.5, so this is the maximum value for robust stability. For robust performanee, equation (9.150) applies. For a speeifie step input let lV(s) = 1 /s now... 

On the negative real axis (-180°), find the "distance" of GCGP from (-1,0). This is the gain margin, GM. The formal definition is... 

For most control systems, we usually take a gain margin between 1.7 and 2, and a phase... 

The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation using the Bode plot is much easier. To obtain the gain margin, we find the value of GCGP which corresponds to a phase lag of-180°. To find the phase margin, we look up the phase lag corresponding to when GCGP is 1. 

With the straight textbook definition and explanation, the gain margin and phase margin of the... 

We should find a gain margin of 2 (6 dB) and a phase margin of 25.4°, which is definitely a bit more conservative than the 1.7 gain margin result. 

We should find a gain margin of 1.47 (3.34 dB) and a phase margin of 12.3°. Both margins are a bit small. If we do a root locus plot on each case and with the help of riocf ind () in MATLAB, we should find that the corresponding closed-loop poles of these results are indeed quite close to the imaginary axis. 

If we want to increase the margin, we either have to reduce the value ofKc or increase One possibility is to keep = 1.58 min and repeat the Bode plot calculation to find a new Kc which may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we first need to find the new ultimate gain using the PI controller ... 

MATLAB should return Kcu = 40.2 (32.1 dB). Thus following Eq. (8-24), we need to use Kc = 40.2/2 = 20.1 to meet the gain margin specification of 2. You can double check the result yourself with kc=2 o. l, taui=l. 5 8. If so, you should find that the phase margin is now 23°—a bit low but we probably can accept that. After this, we may proceed to the time domain response calculations. 

Example 7.2D. Back in the last example with a proportional controller, a gain margin of 1.7 created a system with a very small phase margin. What proportional gain should we use to achieve a phase margin of at least 45° ... 

The result confirms that the system has a phase margin of 48.6°, and a gain margin of 3.2 (10.2 dB), a much more conservative design than a gain margin of 1.7. If we choose to use Kc = 1/0.037... 

A MATLAB calculation should confirm a phase margin of 30°, and find a gain margin of... 

To find the new gain margin, we need to, in theoiy, reverse the calculation sequence. We first use the phase equation to find the new crossover frequency (Dcg. Then we use the magnitude equation to find the new GOL, and the new GM is of course 1/ G0lI However, since we now know the values of td, xp, and KcKvKpKm, we might as well use MATLAB. These are the statements ... 

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