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5.6 Stability test and controller setting with the frequency response of theopen control loop As illustrated in figure 5-23, the control loop will be opened and excitedwith as the input variable of the controller. In the settled condition the out-put variable of the controlled system will be xa(t) = Xa · cos(ωt + ϕ) . xa = −GS · GR · xe = −G0(jω) · xe = −|G0(jω)| · ejϕ0(ω) · xe and considering ejπ(2n+1) = −1 ϕ = ϕ0(ω) + (2n + 1)π . This makes the amplitude and phase relation of the output variabledependent on the frequency of the input variable. This frequency willnow be selected such that the phase angle becomes ϕ = 0 or a wholenumber multiple of 2π . The following applies for the frequency ω =ωπ selected in this manner: ϕ0(ωπ ) = −(2n + 1)π . Controller setting and stability of control loops The amplitude of the output variable can now be larger than, equal toor smaller than that of the input variable. If it should (accidentally) beequal to the amplitude of the input variable, then xe(t) = xa(t) and we can, with the switch represented in figure5-23, connect the output variable of the controlled system to the inputof the controller without changing the variables themselves. Followingthe changeover, we end up with a closed control loop which carries outcontinuous oscillations and is therefore at the limit of stability.
If for the previously defined frequency ωπ was the controller would, after changing the switch in figure 5-23, receive alarger input variable than before. This larger input variable (in additionto the switch-on procedure) would lead to a larger output variable of thecontrolled system which, as the input variable of the controller, wouldlead to further increase of the output variable. Such a closed controlloop which carries out increasing oscillations is called unstable.
If, in the opposite case, with ω = ωπ , because of then, following the changeover of the switch, the controller would re-ceive a smaller input variable than before, and the oscillation initiated 5.6 Stability test and controller setting with the frequency response of theopen control loop by the external signal would vanish and the closed control loop wouldbe stable.
From the results of these considerations we can conclude that a closedcontrol loop is stable if the amplitude of the open control loop’s fre-quency response with a frequency (or frequencies) ωπ is less than one;in which case ωπ is determined by the fact that for this frequency orthese frequencies the phase angle of the frequency response of the opencontrol loop ϕ0(ωπ ) = −(2n + 1. This can be converted to a rulewhich states that the Nyquist plot of G0(jω) may only intersect thenegative real axis to the right of point 1.
Unfortunately, this train of thought can lead to wrong conclusions ina few cases. One of these cases is illustrated in figure 5-22 where, forKR = 120, the Nyquist plot of G0 intersects the real axis, among others,at 1, 8 but the associated closed control loop is stable.
Contrary to the algebraic criteria, the Nyquist criterion applies with-out any limitation to control loops with delay elements. The (relativelyexpensive) proof will not be given here.
Gain and phase margins
From the Nyquist plot of the open control loop in the vicinity of thecritical point 1 we can obtain information regarding the quality of theclosed control loop’s stability. It is clear that control loops, for whichthe Nyquist plot of the open control loop runs very close to the criticalpoint (but still intersects the negative real axis at values > −1), are closeto the limit of stability. Such control loops appear to be poorly dampedand can, as a result of minor changes in the dynamic behaviour of theirelements, become unstable.
Contrary to this, control loops whose Nyquist plots of the open loopfrequency response run at a great distance from the critical point aremostly very sluggish because this shape of the Nyquist plot is achievede.g. by very small controller gains. Between the sketched extremes liesa central, technically usable area which will be defined by design rules.
The distance of the Nyquist plot from the critical point will be describedby means of the so-called gain margin AR and the phase margin αR; seefigure 5-24.
Controller setting and stability of control loops The gain margin is the factor by which the static gain of the open controlloop must be multiplied so that the associated closed control loop is atthe limit of stability.
ϕ0(ωπ ) = −(2n + 1)π . The gain margin is also called amplitude margin or amplitude distance.
The phase margin is the angle which a phasor drawn to the point wherethe Nyquist plot intersects a circle with a radius of one forms with thenegative real axis αR = ϕ0(ωd) − ϕ0(ωπ ) If the Nyquist plot intersects the real axis several times, the least gainmargin or the least phase margin is the crucial factor.

Source: http://www.irt.rwth-aachen.de/fileadmin/IRT/Download/Lehre/MRT/De/ACM_Aufl_8_Kap5_6_S155_bis_158.pdf

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