RootContourPlot - Maple Help

DynamicSystems

 RootContourPlot
 generate a root-contour plot

 Calling Sequence RootContourPlot(sys, range, opts)

Parameters

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

Options

 • algorithm = dsolve or fsolve

Select the algorithm used to generate the root-loci. The choices are dsolve and fsolve, the default is dsolve.

The dsolve algorithm generates a differential equation in the independent variable $K$ whose solutions trace the root-loci. The branches of the root-loci are generated by passing the differential equation to dsolve/numeric with initial conditions near each of the branch points. If supplied, the named options method, optimize, stiff, abserr, relerr, initstep, maxstep, and maxfun are passed to dsolve. The plot is generated with plots[odeplot], if supplied, the named option refine is passed to it.

The fsolve algorithm uses fsolve to generate the roots of the characteristic polynomial as it increments $K$ over $\mathrm{Krange}$. It reorders the list of roots so that roots in the same slot correspond to a single branch of the root-locus.  An adaptive technique reduces the amount that $K$ is incremented as the loci approach a branch point.

 • info = name

A name assigned a record containing information about the root-locus. See the Description section for details.

 • kfrac = positive

Specify the fractional distance from a branch value or critical value of $K$.  Where the $\mathrm{Krange}$ goes through a branch or critical value of $K$, the range is split, with a hole inserted around the value.  The value of frac is the fractional distance from the critical value to the end value or next branch/critical value. The default value is $1.×{10}^{-6}$. A branch value is a value of $K$ at which the root locus branches. A critical value is a value of $K$ at which the characteristic polynomial is degenerate, that is, acquires a degree lower than usual.

 • plotasymptotes = truefalse

When $\mathrm{true}$, and the root-locus has an asymptote, plot the asymptotes as dashed gray lines; the default is $\mathrm{true}$.

 • subsystem = [ posint, posint ]

Select a subsystem of sys; the default is $\left[1,1\right]$.

Description

 • The RootContourPlot command plots the root-contour of a subsystem of sys, a System object.
 • The transfer function of the selected subsystem must be a univariate rational polynomial in $s$ ($z$ for discrete systems), with one symbolic parameter. For example, $\frac{s+1}{gs+{s}^{2}+1}$ has symbolic parameter $g$.
 • The root-contour consists of the roots of $1+H\left(K\right)$, where $H$ is the transfer function of the selected subsystem of sys and $K$ is the symbolic parameter. $K$ is swept over range.
 • To plot the root-contour, the numerator of $1+H\left(K\right)$ is transformed to the equivalent expression $KG+1$; if it cannot be so transformed, an error is generated. A new system with transfer function $G$ is then generated and its root-locus plotted with a call to DynamicSystems[RootLocusPlot].
 • The info option can be used to get information about the subsystem with transformed transfer function G. This is the same record returned by DynamicSystems[RootLocusPlot].

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 RootContourPlot command takes all standard plot,options.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{sys}≔\mathrm{NewSystem}\left(\frac{1}{{s}^{2}+\mathrm{\gamma }s+1}\right):$
 > $\mathrm{RootContourPlot}\left(\mathrm{sys},0..1\right)$
 > $\mathrm{RootContourPlot}\left(\mathrm{sys},0..100\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{RootContourPlot}\left(\mathrm{sys}\right)$