gm2dm

Convert disk-based gain margin to disk size and skew

Description

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.

example

[alpha,sigma] = gm2dm(DGM) returns the disk size alpha and skew sigma corresponding to the disk-based gain margin DGM. The gain margin DGM is a vector of the form [gmin,gmax].

example

[alpha,sigma] = gm2dm(GM) is the same as gm2dm([1/GM,GM]). This syntax returns the disk size for gain that can increase or decrease by a factor GM. This syntax always returns sigma = 0.

Examples

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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.

diskmarginplot(alpha,sigma,'disk')

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

Now use 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.

diskmarginplot(alpha,sigma,'disk')

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.

diskmarginplot(alpha,sigma,'disk')

Input Arguments

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Range of relative gain variation, specified as a two-element vector of the form [gmin,gmax], where 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.

The range [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 gm2dm 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 margins) using getDGM. The GainMargin field of the output structures of the diskmargin command is also a disk-based gain range of this form. .

To get alpha and 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.

Amount of gain increase or decrease in absolute units, specified as a real scalar or a vector.

• If GM is a real scalar, then gm2dm 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 sigma is zero.

• If GM is a vector of form [GM1;...;GMN], the function returns alpha as a column vector of the corresponding disk sizes.

Output Arguments

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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 alpha.

If DGM is a two-column matrix or GM is a column vector, then alpha is a vector of the form [alpha1;...;alphaN] of the corresponding disk sizes.

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,gmax], with 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 GM, the skew sigma is always zero. For additional details about the meaning of sigma, see Algorithms.

If DGM is a two-column matrix, then sigma is a vector of the form [sigma1;...;sigmaN] of the corresponding skew values.

Algorithms

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.

Version History

Introduced in R2020a