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I have a fluid mechanics model that is heavy to simulate. Its main output is a scalar $q$ related to some energy transfer, and its input is a pressure $P$. This model, noted system 2, is coupled with a system 1 which is easier to simulate. System 1 takes $q$ as input and gives another output $w$. Numerical results show that the transfer function $q/P$ has a bode diagram which looks like that of an under-damped second-order system: it has a constant gain at low frequencies, a small resonance peak and then a constant-slope damping at higher frequencies. Here is a view of the coupled gain $(w/q).(q/P)$ of the coupled system (1+2): enter image description here

I would like to create an ad-hoc simple continuous time system to replicate this behaviour. The aim is to simulate the transient response of this simplified system to various inputs using conventional ODE integration techniques.

My requirements are:

  • have a gain of 1 at low frequencies
  • have a resonance at a given frequency $\omega_r$ which is high (typically on the order of $10^4$ rad/s)
  • have a peak gain $G_p$ which is low (typically $<10$)
  • the order of the damping slope at frequencies higher than $\omega_r$ is not really important, but it should be at least one or two. Actually, the transfer function is supposed to be proportional to $\dfrac{1}{\sqrt{\omega}}$, but if the simplified model behaves like $1/\omega$ or $1/\omega^2$, it should be ok for my applications.

Here is what the gain $q/P$ of system 2 should look like: enter image description here

Note that I don't discuss phase, as it is not that important in my current application. I may come back to this point and try to also impose a certain phase evolution, but let's forget about that for the moment.

I tried to see how to match most of these requirements with a second-order model: $$\ddot{q} + a\dot{q} + b_1 q = b_2 P$$ Now, the transfer function of such a system is usually written as: $$\dfrac{K\omega_0}{s^2 + 2\xi \omega_0 s + \omega_0^2}$$ with $s$ the Laplace variable, and $\omega_0$, $\xi$ which can be simply related to $a$ and $b_1$, and $K=\omega_0 b_2/b_1$.

Having the $b_1=b_2$ is necessary to obtain a gain of 1 at low frequencies, since I want $q$ to converge to $P$. Thus, $K=\omega_0$. The requirement of having a resonance leads to $\xi \in [0, 1/\sqrt{2}]$.

I now must find $\xi$ and $\omega_0$ which satisfy my resonance peak requirements (frequency and peak gain). The peak gain $G_p$ is attained at the resonant frequency $\omega_r$, they can be expressed as: $$G_p=\dfrac{\sqrt{\omega_0}}{2\xi\sqrt{1-\xi^2}}$$ $$\omega_r=\omega_0\sqrt{1-2\xi^2}$$

The issue is that my target $G_p$ is on the order of 1 to 10. A quick numerical study of the terms in $\xi$ in $G_p$ shows that $G_p > 1.6\sqrt{\omega_0}$. Thus, there is no combination of $(\xi,\omega_0)$ which can satisfy $G_p\approx 1$ and $\omega_0 \approx 10^4$. For instance, for that frequency, the minimum gain achievable is on the order of 250.

My question is therefore the following: what other simple system can I envision that could remedy these issues ? I have limited experience with higher-order systems, hence my request for assistance. Maybe adding a term $a_2 \dot{P}$ in my system could be useful ? (computing temporal derivatives of $P$ is slightly cumbersome in my model, but doable)

Thank you in advance for your advice !

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  • $\begingroup$ It would help immensely if you could edit your question with an actual Bode plot with the important bits labelled. $\endgroup$
    – TimWescott
    Oct 20, 2023 at 3:58
  • $\begingroup$ @TimWescott I have updated the question with more details $\endgroup$
    – Laurent90
    Oct 20, 2023 at 8:04
  • $\begingroup$ I'm not sure where you're drawing your equations from (I should have asked you to edit your question to cite those, too). Be that as it may, the minimum achievable max gain vs. gain at $\omega = 0$ is one, not 250. In general, for damping ratio $\zeta >= \sqrt(1/2)$ the gain never peaks -- it's maximally flat at $\zeta = \sqrt(1/2)$ (it's a 2nd-order Butterworth) and falls off even faster for higher damping ratios. $\endgroup$
    – TimWescott
    Oct 21, 2023 at 18:21

1 Answer 1

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A possible solution could be to add a third order term to your differential equation, which will allow you to shape the system's response at both low and high frequencies. Where a third-order system has the following form:

$$\dddot{q} + a\ddot{q} + b\dot{q} + cq = dP$$

While this will certainly make the system more complex to analyze (not too complex if you understand linear systems well), it will give you more flexibility in shaping the system's response to match your requirements. However, finding the appropriate values for the coefficients $a$, $b$, $c$, and $d$ will be a nontrivial task that will require a good understanding of the system you're trying to model (which you seem to have).

However, personally, instead of trying to find a single system that fits all your requirements, I would use system identification techniques to find a model of your system based on the input-output data. This could involve fitting a parametric model (like a transfer function or a state-space model) to the data, or using a nonparametric method like frequency response estimation to estimate the system's response at different frequencies. This would allow you to obtain a model that closely matches the behavior of your system, but it would require a good amount of data and potentially complex computational methods. Perhaps we could help you better if you could share the bode plot...

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  • $\begingroup$ Thank you for your answer. I have added more details in the question and a gain plot. $\endgroup$
    – Laurent90
    Oct 20, 2023 at 8:04

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