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# margin

Gain margin, phase margin, and crossover frequencies

## Syntax

[Gm,Pm,Wgm,Wpm] = margin(sys)
[Gm,Pm,Wgm,Wpm] = margin(mag,phase,w)
margin(sys)

## Description

margin calculates the minimum gain margin, Gm, phase margin, Pm, and associated frequencies Wgm and Wpm of SISO open-loop models. The gain and phase margin of a system sys indicates the relative stability of the closed-loop system formed by applying unit negative feedback to sys, as in the following illustration.

The gain margin is the amount of gain increase or decrease required to make the loop gain unity at the frequency Wgm where the phase angle is –180° (modulo 360°). In other words, the gain margin is 1/g if g is the gain at the –180° phase frequency. Similarly, the phase margin is the difference between the phase of the response and –180° when the loop gain is 1.0. The frequency Wpm at which the magnitude is 1.0 is called the unity-gain frequency or gain crossover frequency. It is generally found that gain margins of three or more combined with phase margins between 30 and 60 degrees result in reasonable trade-offs between bandwidth and stability.

[Gm,Pm,Wgm,Wpm] = margin(sys) computes the gain margin Gm, the phase margin Pm, and the corresponding frequencies Wgm and Wpm, given the SISO open-loop dynamic system model sys. Wgm is the frequency where the gain margin is measured, which is a –180 degree phase crossing frequency. Wpm is the frequency where the phase margin is measured, which is a 0dB gain crossing frequency. These frequencies are expressed in radians/TimeUnit, where TimeUnit is the unit specified in the TimeUnit property of sys. When sys has several crossovers, margin returns the smallest gain and phase margins and corresponding frequencies.

The phase margin Pm is in degrees. The gain margin Gm is an absolute magnitude. You can compute the gain margin in dB by

```Gm_dB = 20*log10(Gm)
```

[Gm,Pm,Wgm,Wpm] = margin(mag,phase,w) derives the gain and phase margins from Bode frequency response data (magnitude, phase, and frequency vector). margin interpolates between the frequency points to estimate the margin values. Provide the gain data mag in absolute units, and phase data phase in degrees. You can provide the frequency vector w in any units; margin returns Wgm and Wpm in the same units.

 Note:   When you use margin(mag,phase,w), margin relies on interpolation to approximate the margins, which generally produces less accurate results. For example, if there is no 0 dB crossing within the w range, margin returns a phase margin of Inf. Therefore, if you have an analytical model sys, using [Gm,Pm,Wgm,Wpm] = margin(sys) is the most robust way to obtain the margins.

margin(sys), without output arguments, plots the Bode response of sys on the screen and indicates the gain and phase margins on the plot. By default, gain margins are expressed in dB on the plot.

## Examples

Gain and Phase Margins of Open-Loop Transfer Function

Create an open-loop discrete-time transfer function.

```hd = tf([0.04798 0.0464],[1 -1.81 0.9048],0.1)
```
```hd =

0.04798 z + 0.0464
---------------------
z^2 - 1.81 z + 0.9048

Sample time: 0.1 seconds
Discrete-time transfer function.
```

Compute the gain and phase margins.

```[Gm,Pm,Wgm,Wpm] = margin(hd)
```
```Gm =

2.0517

Pm =

13.5711

Wgm =

5.4374

Wpm =

4.3544
```

Display the gain and phase margins graphically.

```margin(hd)
```

Solid vertical lines mark the gain margin and phase margin. The dashed vertical lines indicate the locations of Wpm, the frequency where the phase margin is measured, and Wgm, the frequency where the gain margin is measured.

The phase margin is computed using H theory, and the gain margin by solving $H\left(j\omega \right)=\overline{H\left(j\omega \right)}$ for the frequency ω.