Convert disk-based gain margin to disk size and skew
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
dm2gm to convert between the
ɑ,σ values and the disk-based gain margin
DGM = [gmin,gmax] that describe the same disk.
Compute the disk size α of the disk that represents a gain variation of ±6 dB, that is, gain that can increase or decrease by about a factor of 2.
GM = db2mag(6); [alpha,sigma] = gm2dm(GM)
alpha = 0.6646
sigma = 0
For symmetric gain variations, the skew
sigma is 0. Examine the disk corresponding to this gain variation.
The disk that captures gain variations of a factor of two in either direction also models phase variations of ±37°.
Determine the disk size and skew needed to capture gain variations between 80% and 150% of nominal and phase variation between –20 and +40 degrees. First, use
getDGM to find
DGM = [gmin,gmax] that describes a disk that captures these target ranges.
DGM = getDGM([0.8,1.5],[-20,40],'tight')
DGM = 1×2 0.2031 1.5000
gm2dm to convert that disk-based gain variation into the α,σ parameterization of the disk.
[alpha,sigma] = gm2dm(DGM)
alpha = 0.6145
sigma = -1.7451
For the modeled gain and phase variations, the skew is less than zero because the disk-based gain range
DGM = [0.2 1.5] includes more gain decrease than increase.
Determine the disk size and skew of the disks which capture gain ranges [0.2,1.3], [0.5,2] and [0.8,3].
GainRange1 = [0.2,1.3]; GainRange2 = [0.5,2]; GainRange3 = [0.8,3];
For the gain ranges above, compute the disk-based gain margin.
[alpha,sigma] = gm2dm([GainRange1;GainRange2;GainRange3])
alpha = 3×1 0.4364 0.6667 0.3636
sigma = 3×1 -3.0833 0 3.5000
For the vector
sigma, the first entry is negative because first entry of
DGM has bias toward gain decrease. Similarly, second entry is zero because of balanced gain variation and third entry is positive because of bias toward gain increase. The plot shows the disks corresponding to range of gain variations specified above.
DGM— Range of relative gain variation
Range of relative gain variation, specified as a two-element vector of the form
gmin < 1 and
gmax > 1. For instance,
DGM = [0.8 1.5]
represents a gain that can vary between 80% and 150% of its nominal value (that is,
change by a factor between 0.8 and 1.5).
gmin can be negative,
defining a range of relative gain variation that includes a change in sign.
[gmin,gmax] describes a disk of gain and phase
uncertainty where the gain can vary by
[gmin,gmax] and the phase can
vary by an amount determined by the disk geometry. The
command finds the disk size
alpha and skew
sigma that parameterize this disk. For more information about the
disk-based uncertainty model, see Algorithms.
You can obtain
DGM from desired gain and phase variations (or
GainMargin field of the output structures of the
diskmargin command is also a disk-based gain range of this form.
sigma corresponding to
multiple gain variation ranges at once, specify
DGM as a two-column
matrix of the form
[gmin1,gmax1; ...;gminN,gmaxN], where each row is
a corresponding disk-based gain range.
GM— Amount of gain increase or decrease
Amount of gain increase or decrease in absolute units, specified as a real scalar or a vector.
GM is a real scalar, then
returns the disk size
alpha corresponding to the symmetric gain
variation in the range
[1/GM,GM]. For instance,
GM = 2 specifies a gain that can increase or decrease by a
factor of 2. For such symmetric gain variation, the skew
GM is a vector of form
[GM1;...;GMN], the function returns
a column vector of the corresponding disk sizes.
alpha— Disk size
Disk size of the uncertainty corresponding to the input gain range, returned as a
scalar or vector. Disk-based gain-margin analysis represents gain and phase variation as
a multiplicative uncertainty F, which is a disk of values containing
F = 1, corresponding to the nominal value of the system. The disk
is parameterized by
alpha, which sets the size of the disk, and
sigma, which biases the gain variation toward gain increase or
decrease. See Algorithms for details about
the meaning of
Skew of the modeled uncertainty disk, returned as a scalar or vector. The skew biases the modeled gain variation toward gain increase or decrease.
sigma = 0 for a balanced gain range
gmin = 1/gmax.
sigma is positive for a varying gain that can increase more
than it can decrease,
gmax > 1/gmin.
sigma is negative for a varying gain that can decrease more
than it can increase,
gmin < 1/gmax.
The more the gain range is biased, the larger the absolute value of
sigma. For a scalar gain variation input
sigma is always zero. For additional details about the
sigma, see Algorithms.
DGM is a two-column matrix, then
is a vector of the form
[sigma1;...;sigmaN] of the corresponding skew
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
The factor F(s) is parameterized by:
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
DGM = [gmin,gmax] with the real axis.
< 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 converts between these two ways
of specifying a disk of multiplicative gain and phase uncertainty: a gain-variation range of
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.