Nyquist Stability Criterion

Discussion in 'Math' started by jrv9090, Dec 25, 2014.

  1. jrv9090

    Thread Starter New Member

    Nov 26, 2014

    I am reading about Nyquist stability criterion, and in order to get better understanding I decided to tey a test function of my own. My test function is 1/(s+1)(s-2).

    Lets say GH = 1/(s+1)(s-2). Also, I am interested in finding the stability of GH alone and not 1+GH (which is usually done with nyquist criterion). Hence I am concerned about the encirclements only about the origin and not about -1.

    Plotting the test function in matlab, I get the attached plot. Since, my test function contains a pole in the right half plane, these should be on counter clockwise rotation about origin. But the figure shows no such rotation. Kindly explain.

    Matlab code:
    sys = tf([1],poly([-1 2]))
  2. t_n_k

    AAC Fanatic!

    Mar 6, 2009
    Your transfer function GH presents a problem from the start as it is already open loop unstable, having a pole in the right half of the complex plane.

    It is possible to get some unstable open loop systems to be stable by applying closed loop feedback. But not in your example.

    Careful consideration will reveal that a simple closing of the loop on GH will not produce a stable system in any event, as the unstable open loop condition "persists" into the closed loop form still having an unstable RHP pole - irrespective of whether non-unity feedback of any magnitude is applied. One could argue that there is no point in testing for stability since no stable condition can ever exist in this case. This is readily confirmed by use of the root locus technique. So the 'ambiguous' behavior of the Nyquist plot to which you allude isn't surprising. But it may lead one to unwittingly conclude that the plot indicates that the closed loop system is stable for a range of loop gains.

    To indicate a stable closed loop system having RHP open loop poles, the Nyquist plot must encircle the (-1+j0) point in a counter-clockwise direction N times where N=Z-P. Z is the number of zeros of the function (1+GH) in the RHP and P the number of poles of GH in the RHP. So for Z to be zero one requires |N|=P.

    It is possible to get the Nyquist plot of your function to encircle the (-1+j0) point by using greater-than-unity feedback factor >2. However the (single) encirclement is in a clockwise (CW) direction which indicates an unstable system. The test (a necessary variation of the conventional interpretation) for stability in the case where GH has one or more RHP poles, is to confirm that there is a commensurate number of counter-clockwise (CCW) encirclements of the (-1+j0) point by the Nyquist locus.

    A different unstable open loop TF will perhaps produce something more in line with your expectations.

    For instance, if GH=1/((s-0.5)*(s+1)) the Nyquist plot encircles the (-1+j0) point once in a counter-clockwise direction. In this example one would need a non-unity feedback factor of 0.5 or less to have an unstable closed loop situation. A slightly interesting comparison is to do the Nyquist plot for my suggested case and your example with a fixed loop gain factor of 4x(your)GH. The plots are very similar (in shape) except for the locus direction as angular frequency varies from -infinity to + infinity.

    Your GH:
    My GH:

    If you want to examine the case when there are two open loop poles in the RHP and where feedback can stabilize the system, look at the hypothetical open loop function:


    which produces a Nyquist plot as shown in the attachment. You will note two CCW encirclements of (-1+j0) as expected for a stable closed loop system.
    Nyquist plot.jpg
    Last edited: Dec 29, 2014