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Suppose there are two linear devices (containing some multi-physics but the details don't matter here), and these devices are given to us as two black boxes that can be studied experimentally. Black box #1 takes signal X as input and produces signal Y as output. Black box #2 takes signal Y as input and produces signal X as output. It is conceptually similar to a microphone converting an acoustic signal to an electric signal, and a loudspeaker converting an electric signal to an acoustic signal. If these two black boxes get connected to each other then an instability can probably arise. The question is what measurements on the two black boxes would produce all necessary and sufficient information to assess whether the coupled system would go unstable, and at what frequency. Would the right approach be to study the response of each black box to a sinusoidal input? Or the response to a delta-pulse input? What is the criterion for instability for the coupled system, and what measurements and calculations would be needed for assessing it?

Maxim Umansky
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  • Is this a homework question? – MBaz Apr 25 '21 at 17:04
  • No, the question is from my work in plasma fusion physics. The actual context of what I mean by the two black boxes would be hard to explain here but conceptually it is like a microphone and a loudspeaker. I am guessing this kind of system is very basic and standard in EE or signal processing field, but my background is far away from those. By searching online a little bit I learned about those transfer functions, and that seems to be the right direction. But I want to check with people who know this area, to make sure I understand right the procedure of dealing with this kind of problem. – Maxim Umansky Apr 25 '21 at 17:11
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    First, if you were to preface your question with the note that what you're working on isn't actually acoustics, but rather plasma fusion, that would help us help you. Second, the field of dynamic systems applies to - well - everything. Electronics engineers are quite accustomed to having analog and/or digital electronics plopped into the middle of some feedback loop with just about any arbitrary system outside of it. Air, sewage, motors and gears -- all can be dealt with. You can trust us not to get discombobulated if you just tell us what you really want to know. – TimWescott Apr 25 '21 at 17:35
  • Fair enough, I reworded the question to make it clear that it is not from the acoustics field. The microphone and loudspeaker analogy is just my toy model to make it easier to think about it. What actually I have in mind is two parts of a plasma fusion system, and I have independent simulation models for each of them. I suspect that there is an instability there if the system was coupled. But coupling these two simulation models into an integrated simulation would be a large effort, and before doing it I'd like to do a scoping study using the linear response of these two models separately. – Maxim Umansky Apr 25 '21 at 17:46
  • @MaximUmansky Are the two systems connected in series? Is there a feedback loop anywhere in the system? – MBaz Apr 25 '21 at 18:09
  • @Mbaz In real life there is an interface between two parts of the physical system. One part (box 1) produces power flux through the interface, as a function of temperature on the interface. The other part (box 2) controls the temperature at the interface, as a function of the power flux across the interface. Would you call this connected in series? On the feedback loop, I would say the two parts of my physical system provide feedback on each other. There is no external stabilization in my model but if the results of this study show instability then such external feedback must be provided. – Maxim Umansky Apr 25 '21 at 18:19
  • It sounds like a complex interaction between two non-linear systems. Quite far from my own area. Let's see if an expert chips in; in the meantime Tim's answer is a good place to start. – MBaz Apr 25 '21 at 18:39
  • @MBaz Yes, the actual system is nonlinear indeed, and it can be viewed as a combination of two parts that interact through an interface between them. However there is an equilibrium point for the coupled system, and at the equilibrium one can use a linearized model for each subsystem. The question is whether this equilibrium is stable or not. I believe by studying linear response of each of the two subsystems one can assess the presence of instability there. – Maxim Umansky Apr 25 '21 at 18:44
  • Sounds like a potentially metastable system; there may be good references here: https://en.wikipedia.org/wiki/Metastability Unfortunately I don't know much about this subject. – MBaz Apr 25 '21 at 19:01
  • It is actually much simpler than metastable. Think about a ball on a top of a hill. It is in equilibrium (i.e. in force balance), but this equilibrium is unstable (a small perturbation will grow exponentially). But this discussion is very useful, thanks a lot. – Maxim Umansky Apr 25 '21 at 19:17

2 Answers2

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Assuming both black boxes are LTI systems, that's reasonably easy to do.

Let's call the transfer function of the first box $H_A(\omega)$ and the second one $H_B(\omega)$. The open look transfer function is simply the product of the, i.e. $H_O(\omega) = H_A(\omega) \cdot H_B(\omega)$

A sufficient stability condition is $|H_O(\omega)| < 1$. For an actual acoustic feedback loop that's really all there is to it since there is always plenty of group delay in the system and the phase is spinning so fast that any type of phase criteria is pointless.

So measure the transfer functions (fully calibrated !), multiply them and check whether the max gain is smaller than unity.

Hilmar
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  • Great, so the first thing is to look at $|H_o(\omega)| = |H_A(\omega)| |H_B(\omega)| $ - if below unity then there is no instability guaranteed? And if above unity then analyze the phase? In the actual physical system that I have in mind the phase rotation may be not so fast on the time scale of interest. – Maxim Umansky Apr 25 '21 at 20:06
  • Yep. As long as your loop gain is below unity, it's technically stable although you will see significant "ringing" behavior above 0.1 or thereabouts so in practice you should stay way clear of unity. Controlling the phase can buy you more margin but then you need to go into all the control loop details that @TimWescott talked about in his answer – Hilmar Apr 25 '21 at 20:47
  • Can you clarify, you call this "open loop transfer function" - so this is not the situation with a closed-loop transfer function described in https://en.wikipedia.org/wiki/Closed-loop_transfer_function because there is no external signal and no summation? That's why the relevant transfer function is just the product? – Maxim Umansky Apr 26 '21 at 14:35
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This is a bog-standard feedback control problem.

Rather than trying to summarize all of signals and systems and all of control theory in one itty bitty post -- the field of knowledge you're looking for is control theory. Key phrases are

  • "Signals and Systems" (I suggest any of the books by Oppenheimer et all -- Oppenheimer has teamed up with a number of people over the years; they're all good).
  • "Control Theory" or "Feedback Control Theory" or "Dynamical Systems".
    • Start with classical linear control theory (that's typically a full-year class for 3rd-year University students on that track).
    • Next up will be nonlinear control theory
    • and/or state-space control. It just goes on from there.

Expect to take years to master it all. Signals and Systems is a one-semester introductory course, but that's just a life raft to the ocean-going boat you'll need before you're done. Junior-level control theory only equips you to formally handle single-input, single-output linear systems. I suspect that before you're done you'll need to get your head wrapped around multiple-input, multiple-output nonlinear systems. If you need to actually control systems, and not just analyze their behavior, then you'll need to pay more attention yet.

Best of luck. Buy books and get cracking, or if your work is associated with a university, start taking or at least auditing classes.

TimWescott
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  • Thanks, I don't have time or interest to fully master Control Theory, I just want to learn enough to deal with my problem in hand. Do I understand it right that the first step is applying sinusoidal input to each of my black boxes to construct the Bode plot for each? Do I understand it right that from the Bode plot I can estimate zeroes and poles of the transfer functions A and B for the two black boxes, to represent them as rational functions? Do I understand it right that the transfer function of the coupled system is $A B /(1 + A B)$, and positive real part of its poles means instability? – Maxim Umansky Apr 25 '21 at 18:12
  • It would be better to work that question into your question above. The answers are all "yes, but" -- the biggest "but" being that if your systems are in the form of differential equations and you aren't investigating what happens when parameters change, then Bode plot analysis may not be the right approach. – TimWescott Apr 25 '21 at 18:16
  • Ok, thanks, I'll edit the question to make it clear what I am after. My physical system is represented by differential equations (many multidimensional nonlinear PDEs) but it is so complicated that it would not be feasible to analyze the underlying mathematical model. Rather I'd try to treat my simulation model as two black boxes that have an equilibrium point, and there is a linear response to small-amplitude perturbations of the equilibrium. From studying the linear response of each box I hope to assess the presence of a physical instability (just a linearized model but still something). – Maxim Umansky Apr 25 '21 at 18:24
  • Can you clarify what you mean saying "if your systems are in the form of differential equations and you aren't investigating what happens when parameters change, then Bode plot analysis may not be the right approach"? The systems are in the form of differential equations (deep under the hood). What do you mean by changing parameters? My nonlinear system has a point of equilibrium (which I know), and my model is linearized at this equilibrium point. I believe everything is fully defined by that equilibrium point, beyond that I am not sure if there are any parameters there that could be changed. – Maxim Umansky Apr 25 '21 at 18:52
  • If it's all ordinary differential equations, linearize around the equilibrium point, express it in state space, and check the eigenvalues of the matrix. "Varying parameters" means a knob that you might tweak, or a "constant" that may vary over time, or have been determined with some uncertainty. – TimWescott Apr 25 '21 at 21:41
  • Ok, understood, thanks. My physical system can be represented by a set of ODEs, but there would be about a million of ODEs there. Dealing with such large eigenvalue problem is not impossible but it seems to me the approach based on treating my physical systems as black boxes and measuring and analyzing the transfer functions would be easier. Thanks a lot for the discussion. – Maxim Umansky Apr 25 '21 at 22:08
  • Ah -- you didn't mention that part. As long as the phase response is smooth, then that technique should work. – TimWescott Apr 26 '21 at 00:01