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What are disadvantages if I increase the gain Kd for a PD(Proportional Plus Derivative) controller, someone told me larger gain for derivative error for will reduce the stability margin, while the text books said increase the Kd could increase the damping ratio and high damping ratio means more stability margin.

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  • $\begingroup$ How did you conclude that high damping ratio means high stability margin? What is the system being controlled? Take an example (a fourth order system would be good) and increase KD; you will notice that the stability margins will reduce with increase in KD (after some point) $\endgroup$
    – AJN
    Dec 23, 2021 at 12:26
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    $\begingroup$ at least for second order system, increasing kd does increase the damping. There is app at SimulationOfFeedbackControlSystemWithControllerAndSecondOrde which you can use to try and see. $\endgroup$
    – Nasser
    Dec 23, 2021 at 14:06
  • $\begingroup$ @Nasser - cool app! But wouldn't Kd would improve phase margin at first, then after some point, push the crossover frequency high enough that additional real-world poles would become significant, thus degrading it again? $\endgroup$
    – Pete W
    Dec 23, 2021 at 14:30

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Depending on the plant, a reduction in phase margin could happen. Here is a theoretical example. Consider a plant with a double integrator and an additional pole:

$$\frac{K_1}{s^2(1+s/\omega_P)}$$

Now let's have a PD controller, which could be written either $[K_P(1+{K_D}s)]$ , or as $[K_P + {K_D}s]$ . For convenience, both of these could be re-written in the form

$$K_0(1+s/\omega_Z)$$

The loop's stability can then be analyzed by looking at the product of the gains going around the loop -- i.e. controller and plant. We have two poles at the origin, a zero at $\omega_Z$ and a pole at $\omega_P$.

An hand-drawn Bode plot is below, followed by an Octave plot of essentially the same thing. Each graph shows two different values of $\omega_Z$, corresponding to two different values of $K_D$.

hand drawn auto-loopgain-bode

For completeness, here are the corresponding closed loop bode plots and closed loop step responses

auto-cl-bode auto-cl-step

Hopefully it's possible to see from first bode plots (the ones looking at around-the-loop), that reducing $\omega_Z$ (i.e. increased $K_D$), by 10x in this example, moves the first breakpoint left by 10x. This then lifts the section of the line with the -20dB/decade slope up by 20dB, causing it to intersect the horizontal axis about 10x further to the right, meaning the loop has about a 10x higher bandwidth.

The movement of the gain crossover frequency also results in a different phase margin. In the example shown, it reduces phase margin from something like 80 degrees for line (1), to 50 degrees for line (2).

However, if the overall gain were smaller, the opposite could also be the case -- if the gain crossover started out to the left of the "peak" in the phase plot, then the phase margin would increase at first, but then decrease again after the gaincrossover frequency passes the peak in the phase plot.

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