Dynamic Systems - Maple Programming Help

 Dynamic Systems

Several improvements have been made to the DynamicSystems package, including:

 • Extended FrequencyResponse to handle differential equations with input derivatives.
 • Extended all models to accept linear, non-differential systems.
 • Added frequencies option to all frequency-based plots, which permits specifying the precise frequencies at which expressions are evaluated.
 • Extended Grammians to work with discrete systems.

Example

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

Assign a differential system with derivatives in the input.

 > $\mathrm{deq}:=3\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}y\left(t\right)\right)-2y\left(t\right)=u\left(t\right)+\frac{ⅆ}{ⅆt}u\left(t\right):$
 > $\mathrm{sys}:=\mathrm{DiffEquation}\left(\mathrm{deq},u,y\right):$

Plot the magnitude of the response vs frequency, adding circles at selected frequencies. This is done by generating and combining two plots.

 > $\mathrm{plots}[\mathrm{display}]\left(\mathrm{MagnitudePlot}\left(\mathrm{sys}\right),\mathrm{MagnitudePlot}\left(\mathrm{sys},\mathrm{frequencies}=\left[0.1,1,10\right],\mathrm{style}=\mathrm{point},\mathrm{symbolsize}=20,\mathrm{symbol}=\mathrm{circle}\right)\right)$

Example

A Nichols plot is useful for quickly estimating the closed-loop response of system with unity-feedback, given its open-loop transfer function. For example, let the open-loop transfer function be:

 > $G:=\frac{1}{s\left(s+1\right)\left(\frac{s}{2}+1\right)}:$

By default a Nichols plot includes constant-phase and constant-magnitude contour plots of a closed-loop system.  From the graph below, the peak closed-loop response is about 5 dB, at 0.8 rad/s, because that is the highest constant-magnitude contour that it touches (estimating).

 > $\mathrm{NicholsPlot}\left(\mathrm{TransferFunction}\left(G\right),\mathrm{gainrange}=-20..20,\mathrm{frequencies}=\left[0.8\right]\right)$

Here we plot the actual closed-loop response and confirm that the maximum gain is approximately 5 dB, at 0.8 rad/s.

 > $\mathrm{CL}:=\mathrm{TransferFunction}\left(\frac{G}{1+G}\right):$
 > $\mathrm{MagnitudePlot}\left(\mathrm{CL},\mathrm{range}=0.1..10\right)$