parameters = set(name = complexcons) or list(name = complexcons)

Specifies numeric values for parameters in sys. These values override those specified by the parameters field of the system object, which in turn override the settings in in SystemOptions(parameters). The numeric value on the right-hand side of each equation is substituted for the name on the left-hand side in the expressions that define the model. No checking is done during the substitution to determine whether the substituted value is valid. For example, a complex value can be substituted for the coefficient of a polynomial. If the complex value had been originally assigned to the model at creation, a warning would be generated.

radians = truefalse

Specifies the implied units of the phase margin. If true the units are radians, otherwise they are degrees. The default is the value of radians in DynamicSystems[SystemOptions].

hertz = truefalse

Specifies the implied units of the gain-crossover frequency. If true the units are hertz, otherwise they are radians/sec. The default is the value of hertz in DynamicSystems[SystemOptions].

The PhaseMargin command returns the phase margin and corresponding gain-crossover frequency of the linear system sys.

This command normally returns a Matrix of lists consisting of the phase 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.

If there is more than one gain-crossover frequency for a particular transfer function, the one with the minimum phase margin is used.

If there is no gain-crossover frequency for a transfer function then its 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.

By default, the phase margin is returned in degrees. It is computed as 180-argument(H(I*omega)), where omega is a gain-crossover frequency.

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{sys}≔\mathrm{TransferFunction}\left(\frac{2}{1+s}\right)\:$

$\mathrm{PhaseMargin}\left(\mathrm{sys}\right)$

$\left[120.0000000\,1.732050808\right]$

$\mathrm{PhaseMargin}\left(\mathrm{sys}\,\mathrm{radians}\right)$

$\left[2.094395103\,1.732050808\right]$

$g≔\mathrm{PhaseMargin}\left(\mathrm{sys}\right)$

$g≔\left[120.0000000\,1.732050808\right]$

$\mathrm{gain}≔g\left[1\right]$

$\mathrm{gain}≔120.0000000$

$\mathrm{freq}≔g\left[2\right]$

$\mathrm{freq}≔1.732050808$

$\mathrm{p1}≔\mathrm{plot}\left(\left[\left[\mathrm{freq}\,-180\right]\,\left[\mathrm{freq}\,\mathrm{gain}-180\right]\right]\,\mathrm{color}=\mathrm{blue}\,\mathrm{thickness}=2\right)\:$

$\mathrm{p2}≔\mathrm{PhasePlot}\left(\mathrm{sys}\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{p1}\,\mathrm{p2}\right)$