Nyquist stability criterion

In summary, the discrete Nyquist stability criterion is:

–Determine the number P  of unstable poles of the open loop transfer function F_OL(z ) = F_c (z ) R (z ).

–Plot F_OL(z ) for the unit circle, z = e ^iω∆t and 0 ≤ ω∆t  ≤ 2π. This is a counter-clockwise path around the unit circle.

–Set N  equal to the net number of counter-clockwise (same direction) encirclements of the point -1 on the plot.

–Compute Z = P – N. The system is stable if and only if Z = 0.

According to Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, Addison-Wesley, 1990

Important notice:

Nyquist stability criterion describes unconstrained systems. There may be physical constraints on either or both the controlled and the control variables. As a result of this fact, no in all cases the Nyquist stability analysis corresponds with real behavior. Despite this fact, Nyquist stability criterion is very subtle tool.