Transfer Functions
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Introduction
The DynamicSystems package contains many tools for manipulating transfer functions, and visualizing their response in both the time and frequency domain.
Here, we demonstrate how to define a transfer function, generate a phase plot, and convert a transfer function to the time domain. Much more is possible.
Define a Transfer Function
restart:withDynamicSystems:
tf≔sa⋅s2+b⋅s+c:
Define a transfer function object
sysTF≔TransferFunctiontf,parameters=a=1,b=1,c=1
sysTF≔Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1⁡soutputvariable=y1⁡s
Generate a Phase Plot
PhasePlotsysTF,parameters=a=3,b=4,c=5
Convert Transfer Function to a Differential Equation
sysDE≔DiffEquationtf
sysDE≔Diff. Equationcontinuous1 output(s); 1 input(s)inputvariable=u1⁡toutputvariable=y1⁡t
sysDE:-de
ⅆⅆtx1⁡t=−a⁢x2⁡tb,ⅆⅆtx2⁡t=c⁢b⁢x1⁡ta2−b⁢x2⁡ta−b⁢u1⁡ta,y1⁡t=−x2⁡tb
Discretization and Time-Domain Response to an Input Signal
Sampling time and number of samples
Ts≔0.025:Ns≔1000:
Generate the input signal
NoisyInput≔Chirp1,0.02,0.01,hertz=true,discrete=true,samplecount=Ns,sampletime=Ts+Statistics:-SampleNormal0,0.05,Ns%T:
Discretize the transfer function
sysTFD≔ToDiscretesysTF,Ts
sysTFD≔Transfer Functiondiscrete; sampletime = .25e-11 output(s); 1 input(s)inputvariable=u1⁡zoutputvariable=y1⁡z
Simulate and plot the response
OutputResponse≔SimulatesysTFD, NoisyInput, parameters=a=1,b=1,c=1
plotseqi−1⋅Ts,OutputResponsei,i=1..Ns
Applications
Amplifier Gain
Bandpass Filter Design
Inverted Pendulum
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