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Is it common to take equal values for the proportional and derivative gains of a PID controller? if so what does it mean?

Mohammad
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The transfer function of a PID controller is given by

$$ C(s) = K_p + \frac{K_i}{s} K_d\,s = \frac{K_d\,s^2 + K_p\,s + K_i}{s}. $$

When $K=K_p=K_d$ and $K_i$ is relatively small compared to $K$, so $K_i=\varepsilon\,K$ with $|\varepsilon|<1$ this can also be written as

$$ C(s) = K\frac{s^2 + s + \varepsilon}{s}, $$

which has a zero close to $s=-1$. This means that $C(s)$ has an asymptote of slope zero below a frequency of one rad/s (0.16 Hertz) and a slope of plus one above that frequency. With slope I am referring how many times the magnitude in the Bode of $C(s)$ increases by 20 dB per decade. Usually this transition frequency is placed close to or below the bandwidth of your closed loop system (assuming your plant acts like a second order system at that frequency range).

So the assumption that $K=K_p=K_d$ might imply that the closed loop bandwidth is roughly one rad/s or higher.

fibonatic
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  • Alternatively, if means you scaled the problem so the typical time constant in the system model is about 1 unit. That is a perfectly sensible thing to do numerically, if the real-world time constant is microseconds or days, not seconds. – alephzero Feb 28 '19 at 10:19