I have been reading several manuscripts, primarily involving modified Smith Predictors to improve disturbance rejection for time-delayed systems. The overwhelming majority of these manuscripts discuss in rejecting input disturbances to the process.
In my application, I have output disturbance (disturbance coming after the process). Is it incorrect to assume that these control systems (designed and simulated for input disturbances) will perform similarly if the disturbance came after the plant. I haven't taken linear controls for the longest time, but I am certain the closed-loop transfer function is different with output disturbances vs. input disturbances.
In regards to my application. I am controlling the contact-force of an end-effector onto a moving surface. I am considering that the moving surface is output disturbance. Correct me if I am wrong.
Thank you,
In general if one could describe your system with the following block diagram:
Then in the frequency domain the output $y$ can be written as,
$$ Y = \frac{C\,P}{1 + C\,P}\, R + \frac{P}{1 + C\,P}\, W + \frac{1}{1 + C\,P}\, V. $$
So the input disturbance gets an additional "coloring" from $P$ of the noise $w$ compared to $v$. I am not sure what kind of methods you refer to with "these control systems", but if those methods specifically use the fact that the system amplifies the input disturbance with the process sensitivity (instead of the sensitivity for the output disturbance), then I would say that those methods will not perform similarly.
If you would like to find a controller, using Smith predictor, for a plant $P$ which is delayed by $T$ seconds,
$$ \hat{P} = P\,e^{-T\,s}, $$
by using a controller $C$, which was designed for plant $P$, you get,
$$ \hat{C} = \frac{C}{1 + C\,P\left(1 - e^{-T\,s}\right)}. $$
The corresponding transfer functions from $r$, $w$ and $v$ to $y$ can be found te be equal to,
$$ Y = \frac{C\,P\,e^{-Ts}}{1 + C\,P}\, R + \frac{\left(1 + C\,P\left(1 - e^{-T\,s}\right)\right) P\,e^{-T\,s}}{1 + C\,P}\, W + \frac{1 + C\,P\left(1 - e^{-T\,s}\right)}{1 + C\,P}\, V. $$
For small $s$ the term $1 - e^{-T\,s}$ can be approximated with $T\,s$, which can be shown with the finite differencing for the derivative,
$$ \frac{dx}{dt} = \lim_{h \to 0} \frac{x(t) - x(t-h)}{h}, $$
$$ s\,X(s) = \lim_{T \to 0} \frac{1 - e^{-T\,s}}{T} X(s). $$
So if you would design a high bandwidth controller $C$ for $P$, such that $C\,P$ has always a minus one slope below the bandwidth frequency, then the sensitivity function for the delayed control loop will behave approximately as the undelayed sensitivity function plus $T\,s$ for low frequencies. Therefore, for such a controller, frequencies in $v$ above approximately $T^{-1}$ rad/s will be visible in the output. Even though the undelayed sensitivity function, at those frequencies, should still be very small. At this frequency the approximation of $T\,s$ will start to break down, but the order of magnitude will still be the same, so $T^{-1}$ rad/s should still be a reasonable estimation of above which frequencies the delayed sensitivity function will start to stop suppressing frequencies in $v$.
Another option could be to design a controller, such that for frequencies near $T^{-1}$ rad/s, or lower if one knows that $v$ is limited to frequencies lower than that, the numerator of the the sensitivity goes to zero. This would be the case when,
$$ \left. C = \frac{1}{P \left(e^{-T\,s} - 1\right)}\right|_{s\approx i\,T^{-1}}. $$
This also have the advantage that the process sensitivity also goes to zero near those frequencies. If the those "control systems" you talked about also make use of this property, then you should also be able to apply it to output disturbances.