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Determining stability of min phase system using GM and PM

  1. Jul 17, 2009 #1
    Alright, there's something I don't understand. The book says that the gain margin and phase margin must be non-negative for a minimum phase system to be stable. But the definition of a minimum phase system is one without right half plane poles or zeroes. Doesn't that inherently make it stable? Looking for a good explanation.
  2. jcsd
  3. Jul 18, 2009 #2


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    Your definition of minimum phase system is not correct:

    "[A] system is said to be minimum-phase if the system and its inverse are causal and stable."


    So the definition is more restrictive than just stable. The definition also doesn't care how you get the stability.

    Also, I am not 100% sure about this but I think phase margin and gain margin only apply to systems with feedback. However there is nothing in the definition of a minimum phase system that requires feedback (i.e. the system could be an open amplifier).
  4. Jul 18, 2009 #3
    Okay, maybe not the definition, but one of the properties as a result of what you just said. But that still doesn't answer the question. Even in your definition, it says the system is stable. How is there instability resulting from gain margins and phase margins in my stable system?
  5. Jul 19, 2009 #4


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    I guess I don't understand the question.
    Here is what I thought you were asking:

    System with such and such gain and phase margin = stable
    Minimum phase system = stable

    As both systems are stable, what makes them different?
  6. Jul 20, 2009 #5
    Alright, I figured out what the answer was. But my original question was this.

    The book states that, given that a system is minimum phase, if the gain margin or phase margin are negative, the system is unstable. My issue was that a minimum phase system is already stable. Turns out, the book was referring to the open loop transfer function of the system, which makes a lot more sense. If the closed loop transfer function was minimum phase, the system would definitely already be stable. The open loop transfer function being minimum phase does not guarantee stability of the closed loop transfer function.

    The stability for a closed loop system can be found using the open loop transfer function with the nyquist stability theorem which is where gain and phase margin come into play.
    Last edited: Jul 20, 2009
  7. Jul 21, 2009 #6
    Negative gain margin? What is that anyway? You mean less than 1?
  8. Jul 21, 2009 #7
    Gain margin is how much gain (in decibels) you can add to the open loop transfer function of a feedback system before it becomes unstable. Essentially, if the transfer function of the open loop is minimum phase, you want the magnitude of the open loop transfer function to be less than 1 (which, in decibels, results in a negative dB value) when the phase is 180. This criteria relates to the Nyquist stability theorem. So if the gain at 180 degrees phase has a negative dB value, the gain margin would be positive. This indicates how much gain in dB you can add before your system becomes unstable. If the magnitude is greater than 1, and your open loop transfer function is minimum phase, your system is going to be unstable. The gain in dB at 180 degrees will be some positive number which means the gain margin would be a negative value.
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