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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.

• 

Added NicholsPlot to the context menu.

 

Example

withDynamicSystems:

Assign a differential system with derivatives in the input.

deq:=3ⅆ2ⅆt2yt2yt=ut+ⅆⅆtut:

sys:=DiffEquationdeq,u,y:

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

plots[display]MagnitudePlotsys,MagnitudePlotsys,frequencies=0.1,1,10,style=point,symbolsize=20,symbol=circle

 

 

 

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:=1ss+1s2+1:

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).

NicholsPlotTransferFunctionG,gainrange=20..20,frequencies=0.8

 

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

CL:=TransferFunctionG1+G:

MagnitudePlotCL,range=0.1..10

 

See Also

DynamicSystems


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