Impulse and Step Responses DC motor example
Impulse response Transfer function
If we apply a unit impulse δ(t) as input, output is called the impulse response.
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Laplace transform of δ(t) is 1. So,
Transfer function G = Laplace transform of impulse response g!
Step responses
Impulses are infinitely large for an infinitely short time. Not often encountered in control.
In control, for time-domain analysis often of more interest to study the step response, i.e. the output when the input is a unit step
Reveals how system behaves when reference suddenly changes from one constant to anothe∫r
u(t)=1, t >0U(s) = 1/s Y(s)=G(s)/s𝑦𝑦 𝑡𝑡 = 𝑡𝑡𝑔𝑔 𝜏𝜏 𝑑𝑑𝜏𝜏
I.e., impulse response = derivative of step response
Example (DC motor)
Determine the impulse response of the motor for the input being the supply voltage and the output being the angle of the shaft.
Determine the step response for the case when the output is the angular velocity of the shaft and the input is the supply voltage.
Example (DC motor)
Assume zero initial conditions:
Graphical interpretation
Exercise: redraw this diagram so that it represents the block diagram where each block is given by its transfer function.
Direct calculations yield
We want to rewrite the transfer function as
Let us calculate . We multiply by s:
and obtain by setting , which yields
Compute and in the same way.
Example (summary)
Transfer function from to
Example (impulse response)
Example (angular velocity step response)
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