# Choice of transfer function in a system where the controller adapts to give the intended behavior

Consider a negative feedback loop with input $r(t)$ and output $x(t)$. The difference

$$e(t) = r(t) - x(t)$$

is the input to a controller $C(s)$ and the output is $u(t)$. Then, $u(t)$ is the input to $G(s)$ and the output is $x(t)$. The transfer function for this system is

$$H(s) = \frac{G(s)C(s)}{1 + G(s)C(s)}$$

It is known that $H(s)$ has the behavior of a forced-damped system. Given a $G(s)$, $C(s)$ is chosen so that $H(s)$ will reflect a forced-damped system.

If $C(s)$ is always chosen to give the intended behavior for $H(s)$, is there any reason that one form of $G(s)$ may be better than another? For example, consider $G(s)$

• $G_{LP}(s) = \frac{1}{s}$
• $G_{HP}(s) = \frac{1}{M s^2}$

where $M$ is a constant. The corresponding $C(s)$ are

• linear low-pass filter: $C_{LP}(s) = \frac{K}{p + s}$
• linear high-pass filter: $C_{HP}(s) = \frac{Ks}{p + s}$

where $K$ and $p$ are constants. The quantities

• $G_{LP}(s)C_{LP}(s) = \frac{K}{s(p + s)}$
• $G_{HP}(s)C_{HP}(s) = \frac{K}{Ms(p + s)}$

only differ by a constant $M$, but $K$ can always be tuned for each choice of $C(s)$ so that the appearance of this extra constant $M$ does not matter.

To reiterate, would one choice of $G(s)$ be preferable from a physics/engineering perspective if the behavior of the overall system remains the same?

• I don’t understadt your question. Normally, G(s) is given as the plant model and your design task is to select an appropriate contoller with desired properties. You should edit your question, because it is not clear what you are asking for. Jul 4 '17 at 20:17
• It is not clear (at least to me) what you are exactly asking for here. In general you would (normally) select the controller structure $C(s)$ such that it is suitable for the plant / system to be controlled (i.e. $G(s)$). In that way, asking for the preferred $G(s)$ is somewhat strange but there might be an application I'm missing here. With respect to the plants you define: $G_{LP}$ corresponds to a single integrator, $G_{HP}$ corresponds to a double integrator (mechanically: an unconstrained mass $M$). Both "might" be difficult to realize in physics/engineering practice. Jul 5 '17 at 5:08

Yes, the choice of $G(s)$ and $C(s)$ is preferable from a physics/engineering perspective.
$C(s)$ stands for controller and $G(s)$ for plant. As an engineer you want these transfer functions to match the real physical system including the corresponding parameters. Otherwise you would rather write down $H(s) = \frac{K}{s(p+s)+K}=\frac{K}{s^2+ps+K}$.
So if your plant is an unconstrained mass $G(s) =\frac{1}{Ms^2}$ is preferred.
If your plant is an integrator (like a watertank) $G(s)=\frac{1}{s}$ is preferred.
If it is known that $H(s)$ has the behavior of a forced-damped system, the notation $H(s)=\frac{K\cdot\omega_n^2 }{s^2+ 2 \beta \omega_n s+\omega_n^2 }$ or $H(s)=\frac{K}{(s+\lambda )^2+\omega_d^2}$ would be preferred, since the parameters have a real physical representation.