# Physical interpretation of a Bode plot

I have a few questions regarding the physical interpretation of the information presented on a Bode plot. I believe it is one of those cases where I "don't know what I don't know", so please bear with me if I meander.

1) The magnitude plot of a Bode diagram represents the ratio of a system's output and input amplitudes, in dB. Given any system, is there a threshold dB which represents "too much", or is it a system-by-system assessment which must be made by the designer? (Note: I believe I know the answer to be the latter, but want to be sure). Could you provide a contrasting example, such as a filter design vs. something with vastly different technical requirements?

2) What is the physical interpretation of a magnitude plot having infinite gain at $$0$$ rad/s (for example, the bode plot of a pure integrator)? When an undamped second order system hits resonance, you get a theoretically infinite-dB magnitude response, which is obviously bad. So why does this logic not apply to very low frequencies? Is the infinite gain of a pure integrator simply a mathematical construct?

3) In controller design, what is the physical interpretation of phase margin and gain margin? I understand the underlying mathematics which brings about those particular properties, but not what they represent. For example, the gain margin is the amount of gain which can be added to the system prior to it going unstable, but why would you want more gain? Is it simply that more controller gain allows it to better track the input signal?

4) How does the gain of the system as represented by the magnitude bode plot relate to the gains of the controller? I feel as though much of my confusion stems from this distinction, where large bode plot magnitudes are not necessarily desirable but large controller gains always seem to be so.

Thank you very much in advance.

## 1 Answer

I apologize in advance for the length of the text but as you can understand these concepts can't be easily defined in few words. I will do some research about the second question and come back to update the answer.

1. The magnitude bode plot of a system indeed represents the ratio you mentio. However, there is not something like a general "good" requirement for the gain ratio of any given system. Each system represents a different procedure which includes different characteristics. The gain ratio may come from a design specification. Although, the output of any system is very often needed to track some reference either time dependent (like a trajectory or a simple sinusoidal) or constant (a step input of any amplitude). This means you want your gain to be equal to $$1 \ (0dB)$$ ideally in order to have zero steady state error. Consider the following first order lowpass filters with transfer functions (they are randomly chosen for demonstration purposes):

$$L(s) = \frac{1}{s+15}$$ $$F(s) = \frac{15}{s+15}$$

The first one has DC gain $$K_1 = 0.0667$$ which means there is much loss of the input. The second one has DC gain $$K_2 = 1$$ which is the ideal in general and also what you want by a filter (at least very often). The bode plots of these two filters are the following:

Notice that while the pase plots are identical the magnitude plots have significant differences. The plot for $$F$$ is the ideal case where the gain at low frequencies is $$0dB$$ while the gain of $$L$$ at low frequencies is very much undesirable. For a more complex example I place here the bode plot of a closed loop system which corresponds to my diploma thesis and I have designed. The closed loop system consists of the following parts:

$$T(s) = \frac{b_0}{s^2+a_1s+a_2} \rightarrow Plant \ Dynamics$$

$$C(s) = K_p + K_d\frac{15s}{s+15} \rightarrow PD-Controller$$

$$F(s) = \frac{15}{s+15} \rightarrow Lowpass \ Filter$$

See that at low frequencies the gain is about ($$0dB$$) with too litle variation and approaches the ideal case. Generally the engineer is responsible for tuning the system in order to achieve certain behaviour, which in many cases dictates that the DC gain should be $$0$$.

3. The purpose of control systems is to be able to control real mechanical systems and in general procedures which take place in the real world. In order to be able to do so, it is almost always great to have a mathematical model which represents the procedure you are trying to control and force certain behaviours. However, since the real world is not perfect you will never (and believe me never) be able to obtain a mathematical model which represents perfectly the real world procedure. First of all, it is almost always prefered to have a linear model (because we have a bunch of design tools for linear models and they are much simpler than non-linear) although the real world is full of non-linearities which means you only approach the real procedure. You will also need sensors to measure data that correspond to the system's behaviour and measurements have noise. Most importantly though is that there are always disturbances and the so called unmodeled dynamics (dynamics that can't be modeled but we know they exist) that influence your system's behaviour and have not been taken into account when designing your controllers and perfoming simulations since the simulations perfect. All these things (linearization, disturbances, noise, unmodeled dynamics) influence your system and add phase and gain to your system which you are not able to take, as said before, into account when designing your controllers. This means that the more gain and phase margin the more robust your systems to these variations is. For example, if you have a gain margin of $$5dB$$ and there are some variations that may add up to $$4.5dB$$ gain, your system will remain stable regardless of these variations but if these variations add up to $$6dB$$ gain your system most probably will go unstable. The same goes for phase margin as well. That does not mean you make your system "infinite" robust because you pay a cost for more robustness but I will leave it to you to earch for robust control (very interesting subject). I suggest you also to look for the so-called disk margin which can tell you many great things about your system including phase and gain margin.

4. The issue regarding large controller gains is very much debatable. Generally, the larger the controller gains are the more aggresive and fast your system is. For example, if you have a closed loop system with third poles, you want your third pole to be like $$10$$ times larger than the two dominant poles in order not to affect the behaviour which is dictated by the dominant poles. This means that the gains of the controller will surely be very large in order to achieve this. However, when controlling a real system you need actuators (dc motors, servo motors, stepper motors etc) which feed the control signal produced by your controller to the real system. These actuators always come with certain limitations. They can produce a certain amount of power, force or torque which are generated by the controller. The larger the controller gains the more torque, for example, is required by the actuator yet the actuator can only provide a certain amount of torque. You need to be very careful because if you do not take into consideration these limitations you can damage the actuator or even the system itself. You always need to saturate the control signal and the more saturation the control signal undergoes the worst your produced results regarding the system's behaviour may be because there are phenomena like integral windup that may come up.

• Hi Teo, That was beautifully answered, so thank you for your explanations. With respect to Question 2), I actually resolved that myself: the pure integrator goes to infinity in the open loop state, but the Bode plot of an integrator with unity feedback actually has 0 DC gain, which, as you pointed out above, is often desirable. Again, thank you for the response. Apr 26 '20 at 1:31
• You are welcome :) Don’t forget to upvote the answer as well ! Apr 26 '20 at 1:33