# Benefits of higher order low pass in a negative feedback loop

Let $G(s) = \frac{Y(s)}{R(s)}$ be the loop gain of a negative feedback loop system.

For example a system like this, where Y(s) and R(s) are the laplace transformed functions y(t) and r(t):

At the loop bandwith $\omega_0$ $|G(i\omega_0)|$ becomes unity. Why is it now benefical to have a roll-off (for $\omega>\omega_0$) as steep as possible? For example in a PLL you often find several low pass filters in the loop filter:

• The question doesn't seem to make sense because it's certainly not the case that the frequency response should be as steep as possible past the natural resonant frequency. Please clarify your question. – Andy aka Feb 16 '18 at 9:55
• Could you explain why? – user2224350 Feb 16 '18 at 10:22
• It's called phase margin - too steep means instability when the loop is closed. – Andy aka Feb 16 '18 at 10:36
• Sure. You want to have a phase margin above 45° or so when crossing unity. But afterwards you want a steep roll-off right? In my question I was refering to this region – user2224350 Feb 16 '18 at 12:41
• No, I don't see the general case at all. If it rolls off at 60 dB/dec beyond $\omega_0$ or carries on at a lower rate there is no general problem. The PLL comparison is a flawed argument. – Andy aka Feb 16 '18 at 13:06

Do not confuse a hardware feedback loop such as a PLL, which needs a very fast correction factor (intergral), with a servo-loop that controls a slow mechanical device, such as a motor.

A PLL needs no derivative feedback, while a motor needs it for sudden changes in load or speed, then the integral dominates during study speed or study loads.

High-order integrals are best used in frequency-hopping communications gear, which allows for FM and FSK transmissions. There is a maximum time to change frequency bands before the loop tries to stabilize the frequency.

Other than that type of use high-order LPF are seldom used, as they cause too much delay in the correction time, such as a single frequency PLL.

• Correct reference to frequency hopping. These applications use higher loop bandwidth to assure short settling time. This makes them prone to PLL-reference spurs. Third order filter is used to better suppress these spurs. – Andreas Feb 17 '18 at 12:44

Here's one reason: in a real system, some features of the plant such as mechanical resonances of actuators may be present at frequencies above the unity gain frequency defined by the control law, which should not be driven by the feedback. For example, some types of mechanical resonance (e.g. from piezoelectric transducers) have high quality factor, which focus oscillation in narrow frequency bands when energy is provided to drive the oscillation. While with a low pass filter the majority of the plant's deviation from the set point would be suppressed below the controller's unity gain frequency (as defined by the low pass filter in this case), sharp mechanical resonances must not be driven to the extent that they resonate with a magnitude that again crosses the unity gain frequency. In such a case, if the phase at the resonant frequency is close to an odd multiple of 180°, the system can oscillate and become unstable.

A single low pass filter provides suppression above the corner frequency inverse to frequency. Using a higher order low pass filter allows the magnitude of the feedback to be further suppressed at higher frequencies (for a second order filter, by inverse frequency squared, and so on). Higher order low pass filters therefore help to prevent "ringing up" resonant features that can create instabilities.