generate a root-locus plot - Maple Help

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DynamicSystems[RootLocusPlot] - generate a root-locus plot

 Calling Sequence RootLocusPlot(sys, Krange, opts)

Parameters

 sys - System; system object Krange - (optional) realcons .. realcons; range over which $K$ is swept opts - (optional) equation(s) of the form option = value; specify options for the RootLocusPlot command

Description

 • The RootLocusPlot command plots the root-locus of a subsystem of sys, a System object. The subsystem must be numeric.
 • The root-locus consists of the roots of $1+KG\left(s\right)$, where $G\left(s\right)$ is the transfer function of the selected subsystem of sys, $s$ is the frequency variable ($z$ for discrete systems), and $K$ is a parameter.
 • The optional parameter Krange assigns the range over which $K$ is swept. It is a range of type realcons; the left hand side must be less than the right hand side. The default is $0..10$.
 • The zeros and poles of $G$ are plotted as the circles and crosses, respectively. At $K=0$ the poles of $G$ lie on the root-locus. At $K=\mathrm{\infty }$ the zeros of $G$ lie on the root-locus.
 • The option info can be used to get information about the root-locus.

Info Record Details

If the value of the keyword parameter info is a name, then that name is assigned a record containing information about the root-locus. The following paragraphs describe each of the fields of the record.

 • charpoly = polynom

The characteristic polynomial of the system, with parameter $K$. This is the polynomial whose roots make up the root-locus as $K$ varies.

 • deq = equation

The differential equation passed to dsolve. If algorithm = fsolve, the value is $\mathrm{NULL}$.

 • G = ratpoly

The transfer-function of the selected subsystem of sys.

 • Kbranches = list( realcons )

A list of the values of $K$ at which the root-locus branches.

 • Kcrit = realcons

The critical value of $K$, that is, the value at which charpoly acquires a degree less than its maximum (there can be at most one such value). If no critical value exists, the value is $\mathrm{NULL}$.

 • poles = list( complexcons )

A list of the roots of the denominator of G.

 • zeros = list( complexcons )

A list of the roots of the numerator of G.

 • The RootLocusPlot command takes all standard plot,options.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{sys}:=\mathrm{NewSystem}\left(\frac{{\left(s+1\right)}^{2}}{{s}^{3}+s+1}\right):$
 > $\mathrm{RootLocusPlot}\left(\mathrm{sys}\right)$

Using fsolve is considerably faster than dsolve on the following system (this is not always the case).  These are the commands to create the plot from the Plotting Guide.

 > $\mathrm{sys}:=\mathrm{NewSystem}\left(\frac{9.505{s}^{2}+16.81s+8.225}{{s}^{4}+5.862{s}^{3}+4.823{s}^{2}}\right):$
 > $\mathrm{RootLocusPlot}\left(\mathrm{sys},\mathrm{algorithm}=\mathrm{fsolve},\mathrm{view}=\left[-3..0,-5..5\right]\right)$

Inspect the values of $K$ at which the locus branches.

 > $\mathrm{RootLocusPlot}\left(\mathrm{sys},\mathrm{algorithm}=\mathrm{fsolve},\mathrm{info}='\mathrm{data}'\right):$
 > $\mathrm{data}:-\mathrm{Kbranches}$
 $\left[{-}{0.5836509958}{,}{0.}{,}{0.9213053493}\right]$ (1)
 > $\mathrm{sys}:=\mathrm{NewSystem}\left(⟨⟨\frac{1}{{s}^{2}},\frac{{\left(s+1.{10}^{-8}\right)}^{2}}{{\left(s-1.{10}^{-8}\right)}^{2}}⟩⟩\right):$
 > $\mathrm{RootLocusPlot}\left(\mathrm{sys},0..1000,\mathrm{subsystem}=\left[2,1\right]\right)$