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Get disk-based margins from disk size and skew

`umargin`

and
`diskmargin`

model gain and phase variation as a multiplicative factor
*F*(*s*) taking values in a disk centered on the real
axis. The disk is described by two parameters: *ɑ*, which sets the size of
the variation, and *σ*, or skew, which biases the gain variation toward
increase or decrease. (See Algorithms for more details about
this model.) The disk can alternatively be described by its real-axis intercepts
`DGM = [gmin,gmax]`

, which represent the relative amount of gain
variation around the nominal value *F* = 1. Use `gm2dm`

and `dm2gm`

to convert between the
*ɑ*,*σ* values and the disk-based gain margin
`DGM = [gmin,gmax]`

that describe the same disk.

`[`

returns the gain and phase variations modeled by the disk with disk-size
`GM`

,`PM`

] = dm2gm(`alpha`

)`alpha`

and zero skew. The disk represents a gain that can vary between
`1/`

`GM`

and `GM`

times the nominal
value, and a phase that can vary by ±`PM`

degrees. If
`alpha`

is a vector, the function returns `GM`

and
`PM`

for each entry in the vector.

`[`

returns the disk-based gain variation `DGM`

,`DPM`

] = dm2gm(`alpha`

,`sigma`

)`DGM`

and disk-based phase
variation `DPM`

corresponding to the disk parameterized by
`alpha`

and `sigma`

. `DPM`

is a
vector of the form `[gmin,gmax]`

, and `DPM`

is a vector
of the form `[-pm,pm]`

corresponding to the disk size
`alpha`

and skew `sigma`

. If
`alpha`

and `sigma`

are vectors, then the function
returns the ranges for the pairs `alpha1,sigma1;...;alphaN,sigmaN`

.

`umargin`

and
`diskmargin`

model gain and phase variations in an individual feedback channel as a frequency-dependent
multiplicative factor *F*(*s*) multiplying the nominal
open-loop response *L*(*s*), such that the perturbed
response is
*L*(*s*)*F*(*s*).
The factor *F*(*s*) is parameterized by:

$$F\left(s\right)=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta \left(s\right)}{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta \left(s\right)}.$$

In this model,

*δ*(*s*) is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (||*δ*||_{∞}< 1).*ɑ*sets the amount of gain and phase variation modeled by*F*. For fixed*σ*, the parameter*ɑ*controls the size of the disk. For*ɑ*= 0, the multiplicative factor is 1, corresponding to the nominal*L*.*σ*, called the*skew*, biases the modeled uncertainty toward gain increase or gain decrease.

The factor *F* takes values in a disk centered on the real axis and
containing the nominal value *F* = 1. The disk is characterized by its
intercept `DGM = [gmin,gmax]`

with the real axis. `gmin`

< 1 and `gmin`

> 1 are the minimum and maximum relative changes in
gain modeled by *F*, at nominal phase. The phase uncertainty modeled by
*F* is the range `DPM = [-pm,pm]`

of phase values at
the nominal gain (|*F*| = 1). For instance, in the following plot, the
right side shows the disk *F* that intersects the real axis in the interval
[0.71,1.4]. The left side shows that this disk models a gain variation of ±3 dB and a phase
variation of ±19°.

```
DGM = [0.71,1.4]
F = umargin('F',DGM)
plot(F)
```

`gm2dm`

and `gm2dm`

converts between these two ways
of specifying a disk of multiplicative gain and phase uncertainty: a gain-variation range of
the form `DGM = [gmin,gmax]`

, and the
*ɑ*,*σ* parameterization of the corresponding disk.

For further details about the uncertainty model for gain and phase variations, see Stability Analysis Using Disk Margins.

`diskmargin`

| `diskmarginplot`

| `gm2dm`

| `umargin`

| `wcdiskmargin`