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dm2gm

Get disk-based margins from disk size and skew

Since R2020a

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Syntax

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

Description

umarginanddiskmarginmodel gain and phase variation as a multiplicative factorF(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, andσ, or skew, which biases the gain variation toward increase or decrease. (SeeAlgorithmsfor more details about this model.) The disk can alternatively be described by its real-axis interceptsDGM = [gmin,gmax], which represent the relative amount of gain variation around the nominal valueF= 1. Usegm2dmanddm2gmto convert between theɑ,σvalues and the disk-based gain marginDGM = [gmin,gmax]that describe the same disk.

example

[GM,PM] = dm2gm(alpha)returns the gain and phase variations modeled by the disk with disk-sizealphaand zero skew. The disk represents a gain that can vary between1/GMandGMtimes the nominal value, and a phase that can vary by ±PMdegrees. Ifalphais a vector, the function returnsGMandPMfor each entry in the vector.

example

[DGM,DPM] = dm2gm(alpha,sigma)returns the disk-based gain variationDGMand disk-based phase variationDPMcorresponding to the disk parameterized byalphaandsigma.DPMis a vector of the form[gmin,gmax], andDPMis a vector of the form[-pm,pm]corresponding to the disk sizealphaand skewsigma. Ifalphaandsigmaare vectors, then the function returns the ranges for the pairsalpha1,sigma1;...;alphaN,sigmaN.

Examples

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Open Live Script

确定基于磁盘的丐帮n and phase variations captured by a disk with sizeα= 0.5.

alpha = 0.5; [GM,PM] = dm2gm(alpha)
GM = 1.6667
PM = 28.0725

When you omitsigma,dm2gmcommand returns the gain and phase variations corresponding toαwith 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')

图包含一个坐标轴对象。坐标轴对象with title Values of multiplicative factor F contains 8 objects of type patch, line, text. This object represents alpha = 0.5, skew = 0.

The plot shows the values of F in complex plane corresponding to disk sizealpha= 0.5 andsigma= 0. You can see thatDGM=[1/GM,GM]for this disk.

Open Live Script

Determine the disk-based gain and phase variations modeled by the disk parameterized by disk sizeα= 0.6 and skewσ= 0.75.

alpha = 0.6; sigma = 0.75; [DGM,DPM] = dm2gm(alpha,sigma)
DGM =1×20.6066 2.2632
DPM =1×2-34.2267 34.2267

Visualize the gain and phase variations represented by this disk.

diskmarginplot(DGM)

图包含一个坐标轴对象。坐标轴对象with title Range of gain and phase variations, xlabel Gain margin (dB), ylabel Phase margin (degrees) contains 5 objects of type patch, text, line.

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

Open Live Script

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 =3×20.3684 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, and meaning that gain can increase or decrease by the same amount. To Visualize the simultaneous range of gain and phase variations corresponding to each row inDGM.

diskmarginplot(DGM)

图包含一个坐标轴对象。坐标轴对象with title Range of gain and phase variations, xlabel Gain margin (dB), ylabel Phase margin (degrees) contains 3 objects of type patch. These objects represent DGM = [0.368,1.92], DPM = 41.8, DGM = [0.455,2.2], DPM = 41.1, DGM = [0.52,2.71], DPM = 41.8.

Input Arguments

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Disk size, specified as a scalar or vector. Disk-based gain-margin analysis represents gain and phase variation as a multiplicative uncertaintyF, which is a disk of values containingF= 1, corresponding to the nominal value of the system. The disk is parameterized byalpha, which sets the size of the disk, andsigma, which biases the gain variation toward gain increase or decrease. SeeAlgorithmsfor details about the meaning ofalpha.

To obtain gain and phase variations corresponding to multiple disk sizes, specifyalphaas a 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], withgmin = 1/gmax.

  • sigmais positive for a varying gain that can increase more than it can decrease,gmax > 1/gmin.

  • sigmais 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 ofsigma. For additional details about the meaning ofsigma, seeAlgorithms.

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

Output Arguments

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

  • Ifalphais a real scalar and you omitsigma,ndm2gmreturns a scalarGMsuch that the disk of sizealphamodels 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.

  • Ifalphais a vector of form[alpha1;...;alphaN]and you omitsigma,function returnsGMas a column vector of the corresponding amounts of gain increase or decrease.

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

  • Ifalphais a real scalar and you omitsigma,ndm2gmreturns a scalarPMsuch that the disk of sizealphamodels 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°.

  • Ifalphais a vector of form[alpha1;...;alphaN]and you omitsigma,function returnsPMas a column vector of the corresponding amounts of phase variation.

Range of relative gain variation, returned as a two-element vector of the form[gmin,gmax], wheregmin< 1 andgmax> 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).DGMis the gain variation modeled by the disk parameterized by the input argumentsalphaandsigma. It is the range in which the disk crosses the real axis.gmincan be negative for large negative values ofsigma, defining a range of relative gain variation that includes a change in sign. For more information about the disk-based uncertainty model, seeAlgorithms.

You can useDGMto create aumarginobject that represents the gain and phase uncertainty described by the disk. You can visualize the disk and the associated gain and phase variations usingdiskmarginplot.

Ifalphaandsigmaare vectors, thenDGMis 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].

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

The vectorDPM = [-pm,pm], represents the relative phase variation amount determined by the geometry of the disk described byalphaandsigma. For more information, seeAlgorithms.

Ifalphaandsigmaare vectors, thenDPMis 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

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

F ( s ) = 1 + α [ ( 1 − σ ) / 2 ] δ ( s ) 1 − α [ ( 1 + σ ) / 2 ] δ ( s ) .

In this model,

  • δ(s) is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (||δ||∞< 1).

  • ɑsets the amount of gain and phase variation modeled byF. For fixedσ,parameterɑcontrols the size of the disk. Forɑ= 0, the multiplicative factor is 1, corresponding to the nominalL.

  • σ, called theskew, biases the modeled uncertainty toward gain increase or gain decrease.

The factorFtakes values in a disk centered on the real axis and containing the nominal valueF= 1. The disk is characterized by its interceptDGM = [gmin,gmax]with the real axis.gmin< 1 andgmin> 1 are the minimum and maximum relative changes in gain modeled byF, at nominal phase. The phase uncertainty modeled byFis the rangeDPM = [-pm,pm]of phase values at the nominal gain (|F| = 1). For instance, in the following plot, the right side shows the diskFthat 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)

Multiplicative disk and range of gain and phase variations for umargin block modeling gain variation of plus or minus 3 dB and phase variation of plus or minus 19 degrees.

gm2dmandgm2dmconverts between these two ways of specifying a disk of multiplicative gain and phase uncertainty: a gain-variation range of the formDGM = [gmin,gmax], and theɑ,σparameterization of the corresponding disk.

为进一步细节abo血型ut the uncertainty model for gain and phase variations, seeStability Analysis Using Disk Margins.

Version History

Introduced in R2020a

See Also

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