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Are unstable system really possible?

  1. Oct 18, 2012 #1
    Ok, I know it sounds ridiculous. But when reading about what unstable system is and stabilty conditions of transfer function, i got confused. If unstable system means infinite o/p for finite i/p. Does not that kind of violate physical laws? How's that even possible?

    Can anyone give me a example of physical system which is unstable?
  2. jcsd
  3. Oct 18, 2012 #2

    Filip Larsen

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    If the transfer function models a physical system where the response carries energy then infinite responses are clearly not physically possible.

    I'm a bit rusty on the control theory, but I would explain it like this: Infinite responses to finite inputs in such models are usually an "unintentional artifact" of simplifying the system as a linear system with poles which has no problem with infinite energy. Or in other words, when you model a physical system as linear with a transfer function, the state space in this model implicitly has a domain of validity outside which you cannot expect the behavior of the model to correspond to behavior in the physical system. Often this domain of validity is not explicitly stated, but its there nonetheless. If you want a "complete" model of a physical system valid for all states you can resort to tricks like piecewise linearization, where multiple transfer functions are used for different parts of the state space, or to switch over to non-linear modeling.

    Perhaps others here can provide a better explanation.
  4. Oct 18, 2012 #3
    Of course you can have unstable systems, we use them all the time.

    The trick is to exert control over them.

    I expect you are studying feedback and have started with sytems containing a single feedback loop and a linear response/transfer function.

    One way to limit the output is to employ a suitable non linear (self limiting) transfer function.

    Another way is to use more than one feedback loop. Most circuits designed for ac and dc bias work like this.

    Yet another way is to use what is known as feedforward.

    You should look up

    Automatic gain control
    Error feedforward
    Current Dumping
    Wein oscillator - oscillation critieron and stabilistation methods (there are several)
  5. Oct 18, 2012 #4


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    The transfer function is not the physical system. It is only an approximate mathematical model of how the system behaves.

    To keep the math simple, many systems are modeled assuming they are linear. That assumption isn't true for "large" inputs and outputs (whatever "large" means, for a particular system). The behaviour of the real-world system won't violate any physics laws, but a simple linear transfer function will not be an accurate description of the real world system.
  6. Oct 18, 2012 #5

    In what context are you using the word "transfer function"? Transfer functions can never be nonlinear, but you can limit their outputs after the fact that their outputs have reached the physical system's linear region limits (and no longer are modeled accurately by the transfer function). In that case you are modeling with a transfer function AND an extra set of information or conditions that is outside of the definition of a transfer function.
  7. Oct 18, 2012 #6
    Why do you say this?

    What is the transfer function of a nand gate or a comparator?
  8. Oct 18, 2012 #7
    I'm thinking in terms of systems and controls theory with the laplace operator. I always have been under the impression that this is the definition of the transfer function.

    This fact is the inherent limitation to transfer functions in nonlinear systems and is at the heart of the OP's question. I would not define a nand gate or comparator with a transfer function, but either a set of nonlinear equations or with a truth table.
  9. Oct 18, 2012 #8
    But by your definition how can you have a transfer function in a non linear system to refer to?

    My definition of a transfer function is

    Output = F (input) : O = F(I)

    Where the transfer function is F.

    Note that F may include suitable constants if O and I are not in the same units eg transconductance. Many devices do not display a linear transconductance function.
  10. Oct 18, 2012 #9
    Well, I think you are using a definition outside of OP's context. That's why I tried to clarify because it would be confusing to someone learning LTI theory. You cannot write down a non-linear transfer function that self limits in terms of a single function: O = F(I). To describe a non-linear IO system requires a set of nonlinear functions to do that (a diode for example). Edit: I'm talking about nonlinear as linearly dependent equations.

    Anyway, to answer OP's question, systems with positive feedback are often unstable. If you write down a stable transfer function, and give it enough positive feedback or too much negative feedback gain, and then use the transfer function rules to reduce it back down to a single transfer function, the poles can be moved to the right half (positive real) of the s-plane and so the system will become unstable.

    For your example, just consider the classic guitar amp squeel when the output is fed back into the input to make the new output even louder. If you just pluck a string softly (finite input) it starts as a quiet hum and gets louder and louder unbounded (infinite output), without you continuing to apply input by plucking the string, until it no longer acts linearly; once the amplifier cannot apply higher voltage than its power supply gives it to make the sound any louder, it behaves nonlinearly and does not obey the transfer function, which prevents it from violating the laws of physics.
    Last edited: Oct 18, 2012
  11. Oct 18, 2012 #10


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    This is probably just an argument about terminology, but LTI systems are precisely the subset of all systems where the transfer functions are linear.

    There is no reason why you should limit the idea of "transfer functions" to simple algebraiic equations. In real life engieering, often you just MEASURE them, and use the measured data directly, without caring whether it is linear or not. (Of course you might have to measure them at several different amplitudes, etc).

    IMO the OP's basic problem is not realizing that the simple maths of LTI systems is not an "exact" description of anything in the real world, even though the maths of LTI systems is a good, simple, and useful approximation in many real world situations.
  12. Oct 18, 2012 #11

    D H

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    You never get infinite output, of course. The system instead falls over far and breaks into little pieces. Or it tears itself apart. Or it goes veering out of control, only to have the system safety officer push the big red button. Or it spins itself into an uncontrollable tizzy right before it becomes shrapnel. You get some kind of nasty failure long before the system builds up to an infinite output.

    That's why it's so important to do stability analysis, and so important to have some kind of FDIR in case things go awry even with all that analysis.
  13. Oct 18, 2012 #12
    I give the OP credit for realising the impossibility of infinite output or any other parameter in the real world.

    He/she has obviously been given that definition for an unstable system as one that leads to some sort of runaway condition with (potentially) infinite output.

    The question is how is this possible ?

    Clearly we use such systems all the time. People place microphones in front of loudpeakers and pay the howling penalty.The output stages of amplifiers and nuclear power stations are subject to thermal runaway (thank goodness the latter is not infinite).

    But mostly we can live with these systems and control them and the OP appears to be asking how.
  14. Oct 18, 2012 #13

    jim hardy

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    recall what happens to transfer function when you add feedback:

    forward G, feedback H

    withouot feedback gain (transfer function) is G
    with feedback it's G/(1+GH)

    if GH = -1 you have division by zero or infinite GAIN (not infinite output)

    which means you can have output with no input

    and that's unstable

    for a real example consider that amplifier somebody mentioned earlier,

    or look up "Dihedral" in airplane wings
    and observe downswept (or forward swept) wings make the craft unstable , and unfliable in the days before computers.

    Mother Nature loves a balance. Learn to operate feedback systems in your head it'll help you in conrtol courses.

    See also "Thermostat Hypothesis" in climate feedback at "watts up with that"

    gotta go now library closing

    old jim
  15. Oct 19, 2012 #14
    I think unstable not just means having infinite output or but also means a system that doesn't settle on a fixed value but keeps on oscillating.
    The later type do exist and isn't hard to believe.
    The former type however can't exist in reality.
    Such types of system results because of incomplete modelling of a real physical system.
    For example consider a system about trying to balance a pencil on its tip on a table.
    For very small Displacement around the vertical position it can be assumed that the top of the pencil follows linear motion. Assuming the system is started at equilibrium with the pencil perfectly balanced, if we give a small displacement to the tip of the pencil on either side, the (approximate) mathematical model we developed may dictate that the tip of the pencil will keep on moving in that direction for infinite distance, proving that the system is unstable. But this has clearly resulted because of our incomplete modelling. If a complete modelling was done then the system would be stable, and the stable position would be that the pencil will lie flat on the table (after some rollings.)
    Just my two cents.
  16. Oct 19, 2012 #15
    As an example - Read up on the X-29 Aircraft.
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