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When using step response to a feedback loop with disturbance. Why do we analyze the step response of the disturbance D and the step response of the input R separately. Instead of considering them together like with $$Y = \frac{GR}{1+HG} + \frac{GD}{1+HG}$$ given that \$Y\$ is the output.

I'm assuming we can't take the step response with both together in the function as we want the transfer function to be in terms of \$\cfrac{Y}{R}\$ or \$\cfrac{Y}{D}\$, but I just wanted to confirm that this was the case.

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  • \$\begingroup\$ I think it came directly from mathematics. Where solution to a non-homogenous differential equation is calculated as complementary solution + particular solution. \$\endgroup\$ – nidhin May 29 '14 at 19:32
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Mainly because it's easier. As long as your system is both linear and time-invariant (LTI), the principle of "superposition" allows the probelm to be decomposed this way and still get valid results.

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  • \$\begingroup\$ Dave, please apologize me for this: I've also mentioned the superposition on answer (when I started to respond, the web browser was opened too long without updating). \$\endgroup\$ – Dirceu Rodrigues Jr May 30 '14 at 12:45
  • \$\begingroup\$ @DirceuRodriguesJr: No need to apologize; there's nothing wrong with redundant answers. In the end, the community and the OP will decide which one best addresses the question. \$\endgroup\$ – Dave Tweed May 30 '14 at 13:00
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The input step response and load step response are often different when the drivers are not linear or constant impedance throughout the entire load curve.

Often power supplies are tested 10 to 90% or 50 to 100% or some other different step size due to this characteristic. The step size in fact will rarely give an identical time response because designs are rarely ideal linear current vs voltage and time invariant over the whole range.

Disturbances can come from either inputs or outputs so it best to understand how they can be different and change with step size and direction, for both inputs and outputs. This is the essence to understanding the non-linearities of circuits.

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Due to the property of superposition that holds for linear systems, the response to the reference and disturbance applied simultaneously is equal to the sum of those ones. I believe that using state-space methods (MISO - Multiple Input, Single-Output), you can create an input vector having the reference and disturbance as components. Alternatively, "Robust Control" and QFT (Quantitative Feedback Theory) are methods that emphasizes command / disturbance treatment.

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