Interpret disk gain and phase margins

dmplot dmplot(diskgm)

`[dgm,dpm] = dmplot`

`dmplot`

plots disk gain
margin (`dgm`

) and disk phase margin (`dpm`

).
Both margins are derived from the largest disk that

Contains the critical point (–1,0)

Does not intersect the Nyquist plot of the open-loop response

*L*

`diskgm`

is the radius of this disk and a lower
bound on the classical gain margin.

`dmplot(diskgm)`

plots the
maximum allowable phase variation as a function of the actual gain
variation for a given disk gain margin `diskgm`

(the
maximum gain variation being `diskgm`

). The closed-loop
system is guaranteed to remain stable for all combined gain/phase
variations inside the plotted ellipse.

`[dgm,dpm] = dmplot`

returns
the data used to plot the gain/phase variation ellipse.

When you call `dmplot`

(without an argument),
the resulting plot shows a comparison of a disk margin analysis with
the classical notations of gain and phase margins. The Nyquist plot
is of the loop transfer function L(s)

$$L(s)=\frac{\frac{s}{30}+1}{(s+1)({s}^{2}+1.6s+16)}$$

dmplot

This figure shows a comparison of a disk margin analysis with the classical notations of gain and phase margins. The Nyquist plot is of the loop transfer function L = 4(s/30 + 1)/((s+1)*(s^2 + 1.6s + 16)) - The Nyquist plot of L corresponds to the blue line - The unit disk corresponds to the dotted red line - GM and PM indicate the location of the classical gain and phase margins for the system L. - DGM and DPM correspond to the disk gain and phase margins. The disk margins provide a lower bound on classical gain and phase margins. - The disk margin circle corresponds to the dashed black line. The disk margin corresponds to the largest disk centered at (GMD + 1/GMD)/2 that just touches the loop transfer function L. This location is indicated by the red dot.

The Nyquist plot of

*L*corresponds to the blue line.The unit disk corresponds to the dotted red line.

GM and PM indicate the location of the classical gain and phase margins for the system

*L*.DGM and DPM correspond to the disk gain and phase margins, respectively. The disk margins provide a lower bound on classical gain and phase margins.

The disk margin circle, represented by the dashed black line, corresponds to the largest disk centered at

`(DGM + 1/DGM)/2`

that just touches the loop transfer function*L*. This location is indicated by the red dot.

The *x*-axis corresponds to the gain variation,
in dB, and the *y*-axis corresponds to the phase
variation allowable, in degrees. For a disk gain margin corresponding
to 3 dB (1.414), the closed-loop system is stable for all phase and
gain variations inside the blue ellipse. For example, the closed-loop
system can simultaneously tolerate +/– 2 dB gain variation
and +/– 14 deg phase variations.

dmplot(1.414)

Barrett, M.F., Conservatism with robustness tests for linear feedback control systems, Ph.D. Thesis. Control Science and Dynamical Systems, University of Minnesota, 1980.

Blight, J.D., R.L. Dailey, and Gangsass, D., "Practical
control law design for aircraft using multivariable techniques," *International
Journal of Control*, Vol. 59, No. 1, 1994, 93-137.

Bates, D., and I. Postlethwaite, Robust Multivariable Control of Aerospace Systems, Delft University Press, Delft, The Netherlands, ISBN: 90-407-2317-6, 2002.

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