compute the gain-margin and phase-crossover frequency - Maple Help

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DynamicSystems[GainMargin] - compute the gain-margin and phase-crossover frequency

 Calling Sequence GainMargin( sys, opts )

Parameters

 sys - System; a system object opts - (optional) equation(s) of the form option = value; specify options for the GainMargin command

Description

 • The GainMargin command returns the gain margin and corresponding phase-crossover frequency of the linear system sys.
 • This command normally returns a Matrix of lists. Each list consists of the gain-margin and associated crossover-frequency corresponding to the transfer function of each Matrix element. However, if the Matrix of transfer functions is one by one, then the single list is returned directly.
 • A phase-crossover frequency is a frequency at which the phase of the transfer function equals $\mathrm{\pi }$ radians (180 degrees).
 • If there is more than one phase-crossover frequency, then the one with the minimum gain-margin is used.
 • If there is no phase-crossover frequency for a transfer function, then the associated list is [Float(undefined),Float(undefined)].
 • For a sampled (discrete) system, the z to s transformation z = exp(s*Ts), where Ts is the sample period, is applied to the transfer function(s). The frequency range is limited to the Nyquist frequency.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{sys}:=\mathrm{TransferFunction}\left(\frac{1}{{\left(1+s\right)}^{3}}\right):$
 > $\mathrm{GainMargin}\left(\mathrm{sys}\right)$
 $\left[{18.06179974}{,}{1.732050808}\right]$ (1)
 > $g:=\mathrm{GainMargin}\left(\mathrm{sys},\mathrm{decibels}=\mathrm{false}\right)$
 ${g}{:=}\left[{8.000000000}{,}{1.732050808}\right]$ (2)
 > $\mathrm{gain}:={g}_{1}$
 ${\mathrm{gain}}{:=}{8.000000000}$ (3)
 > $\mathrm{freq}:={g}_{2}$
 ${\mathrm{freq}}{:=}{1.732050808}$ (4)
 > $\mathrm{sys}:=\mathrm{TransferFunction}\left(⟨⟨\frac{1}{{\left(s+1\right)}^{3}},\frac{1}{{s}^{3}+4{s}^{2}+s-6}⟩⟩\right):$
 > $\mathrm{GainMargin}\left(\mathrm{sys}\right)$
 $\left[\begin{array}{c}\left[{18.06179974}{,}{1.732050808}\right]\\ \left[{20.00000000}{,}{1.}\right]\end{array}\right]$ (5)
 > $\mathrm{p1}:=\mathrm{plot}\left(\left[\left[\mathrm{freq},0\right],\left[\mathrm{freq},-\mathrm{gain}\right]\right],\mathrm{color}=\mathrm{blue},\mathrm{thickness}=2\right):$
 > $\mathrm{p2}:=\mathrm{MagnitudePlot}\left(\mathrm{sys}\right):$
 > $\mathrm{plots}[\mathrm{display}]\left(\mathrm{p1},\mathrm{p2}\right)$