Editing Step response (section)

====Control of overshoot====
How overshoot may be controlled by appropriate parameter choices is discussed next.

Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. The result for maximum step response ''S''<sub>max</sub> is:<ref name=Kuo2>{{cite book |author=Benjamin C Kuo & Golnaraghi F| title=p. 259| year=2003| publisher=Wiley| isbn=0-471-13476-7 | url=http://worldcat.org/isbn/0-471-13476-7}}</ref>

:<math>S_\max= 1 + \exp \left( - \pi \frac { \rho }{ \mu } \right). </math>

The final value of the step response is 1, so the exponential is the actual overshoot itself. It is clear the overshoot is zero if ''μ'' = 0, which is the condition:

:<math> \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2. </math>

This quadratic is solved for the ratio of time constants by setting ''x'' = (''τ''<sub>1</sub> / ''τ''<sub>2</sub>)<sup>1/2</sup> with the result

:<math>x = \sqrt{ \beta A_0 } + \sqrt { \beta A_0 +1 }. </math>

Because β ''A''<sub>0</sub> ≫ 1, the 1 in the square root can be dropped, and the result is

:<math> \frac { \tau_1} { \tau_2} = 4 \beta A_0. </math>

In words, the first time constant must be much larger than the second. To be more adventurous than a design allowing for no overshoot we can introduce a factor ''α'' in the above relation:

:<math> \frac { \tau_1} { \tau_2} = \alpha \beta A_0, </math>

and let α be set by the amount of overshoot that is acceptable.

Figure 4 illustrates the procedure. Comparing the top panel (α = 4) with the lower panel (α&nbsp;=&nbsp;0.5) shows lower values for α  increase the rate of response, but increase overshoot. The case α&nbsp;=&nbsp;2 (center panel) is the [[Butterworth filter#Maximal flatness|''maximally flat'']] design that shows no peaking in the [[Bode plot|Bode gain vs. frequency plot]]. That design has the [[rule of thumb]] built-in safety margin to deal with non-ideal realities like multiple poles (or zeros), nonlinearity (signal amplitude dependence) and manufacturing variations, any of which can lead to too much overshoot. The adjustment of the pole separation (that is, setting&nbsp;α) is the subject of [[frequency compensation]], and one such method is [[pole splitting]].