dm2gm

Get disk-based margins from disk size and skew

Since R2020a

Syntax

[GM,PM] = dm2gm(alpha)

[DGM,DPM] = dm2gm(alpha,sigma)

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.

[GM,PM] = dm2gm(alpha) returns the gain and phase variations modeled by the disk with disk-size 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.

[DGM,DPM] = dm2gm(alpha,sigma) returns the disk-based gain variation 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.

Examples

Gain and Phase Variations Corresponding to Given Disk Size

Determine disk-based gain and phase variations captured by a disk with size α = 0.5.

alpha = 0.5;
[GM,PM] = dm2gm(alpha)

GM = 
1.6667

PM = 
28.0725

When you omit sigma, the dm2gm command returns the gain and phase variations corresponding to α with zero skew. Zero skew means that the disk represents gain that can increase or decrease by the same amount. In this case, α = 0.5 models a gain that can increase or decrease by up to a factor 1.6667 of its nominal value. The phase variation corresponding to this disk-based gain variation is ±28°. Visualize this disk.

diskmarginplot(alpha,0,'disk')

The plot shows the values of F in complex plane corresponding to disk size alpha = 0.5 and sigma = 0. You can see that DGM = [1/GM,GM] for this disk.

Disk-Based Margins Corresponding to Disk Size and Skew

Determine the disk-based gain and phase variations modeled by the disk parameterized by disk size α = 0.6 and skew σ = 0.75.

alpha = 0.6;
sigma = 0.75;
[DGM,DPM] = dm2gm(alpha,sigma)

DGM = 1×2

    0.6066    2.2632

DPM = 1×2

  -34.2267   34.2267

Visualize the gain and phase variations represented by this disk.

diskmarginplot(DGM)

Because σ > 0, this disk models a gain that can increase more than it can decrease relative to the nominal value.

Effect of Skew on Modeled Gain and Phase Variations

Determine the disk-based gain and phase variations represented by disks of the same size but with different skews.

alpha = 0.75;
sigma = [-0.5;0;0.5];
[DGM,DPM] = dm2gm(alpha,sigma)

DGM = 3×2

    0.3684    1.9231
    0.4545    2.2000
    0.5200    2.7143

DPM = 3×2

  -41.7908   41.7908
  -41.1121   41.1121
  -41.7908   41.7908

The disks capture roughly similar phase variations, but the skew biases the disk toward gain decrease or increase. For the disk with zero skew, the gain variation is balanced, meaning that gain can increase or decrease by the same amount. Visualize the simultaneous range of gain and phase variations corresponding to each row in DGM.

diskmarginplot(DGM)

Input Arguments

alpha — Size of gain and phase variation

scalar | vector

Disk size, specified 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.

To obtain gain and phase variations corresponding to multiple disk sizes, specify alpha as a vector.

sigma — Skew

real scalar | vector

Skew, specified 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 additional details about the meaning of sigma, see Algorithms.

To obtain gain and phase variations corresponding to multiple disks of varying skew, specify sigma as a vector.

Output Arguments

GM — Amount of gain increase or decrease

scalar | column vector

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

• If alpha is a real scalar and you omit sigma, then dm2gm returns a scalar GM such that the disk of size alpha models a symmetric gain variation in the range [1/GM,GM] and the corresponding phase variation, [-PM,PM]. For instance, GM = 2 means that the disk models a gain that can increase or decrease by a factor of 2.

• If alpha is a vector of form [alpha1;...;alphaN] and you omit sigma, the function returns GM as a column vector of the corresponding amounts of gain increase or decrease.

PM — Amount of phase variation

scalar | column vector

Amount of phase variation in degrees, returned as a real scalar or a vector.

• If alpha is a real scalar and you omit sigma, then dm2gm returns a scalar PM such that the disk of size alpha models a symmetric gain variation in the range [1/GM,GM] and the corresponding phase variation, [-PM,PM]. For instance, PM = 20 means that the disk models a phase that can increase or decrease by 20°.

• If alpha is a vector of form [alpha1;...;alphaN] and you omit sigma, the function returns PM as a column vector of the corresponding amounts of phase variation.

DGM — Range of relative gain variation

two-element vector | two-column matrix

Range of relative gain variation, returned 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). DGM is the gain variation modeled by the disk parameterized by the input arguments alpha and sigma. It is the range in which the disk crosses the real axis. gmin can be negative for large negative values of sigma, defining a range of relative gain variation that includes a change in sign. For more information about the disk-based uncertainty model, see Algorithms.

You can use DGM to create a umargin object that represents the gain and phase uncertainty described by the disk. You can visualize the disk and the associated gain and phase variations using diskmarginplot.

If alpha and sigma are vectors, then DGM is a two-column matrix of the form [gmin1,gmax1; ...;gminN,gmaxN], where each row is the disk-based gain range corresponding to [alpha1,sigma1; ...;alphaN,sigmaN].

DPM — Disk-based phase variation

two-element vector | two-column matrix

Disk-based phase variation, returned as a two-element vector or a two-column matrix.

The vector DPM = [-pm,pm], represents the relative phase variation amount determined by the geometry of the disk described by alpha and sigma. For more information, see Algorithms.

If alpha and sigma are vectors, then DPM is a two-column matrix of the form [-pm1,pm1; ...;-pmN,pmN], where each row is the phase variation corresponding to [alpha1,sigma1; ...;alphaN,sigmaN].

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