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We have the closed loop transfer function:

$$T(s)=\frac{L(s)}{1+L(s)}$$

So as far as I understand we check along the $j\omega$-axis on the Bode plot whether $L(s)=-1$, cause that's when $T(s)$ has pole on the axis and that leads to instability.

My question is, why don't we need to check the entire right half plane whether a gain/phase modification leads to $L(s)=-1$ there? It doesn't make sense from my standpoint, cause we don't want any poles on the right half plane.

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The test known as Nyquist Stability Criterion is that -1 is not encircled clockwise when the open loop frequency response of $L(s)$ is plotted on a complex plane. This applies to open loop systems that are stable (no poles on the right half plane) as the number of encirclements of -1 is equal to the number of poles in the right half plane for the closed loop system. For systems that have open loop poles in the right half plane or RHP, the number of encirclements will be reduced by the number of open loop poles on the RHP so we need to account for that in those cases.

This is a more robust test than the similar Bode plot test that the phase must not exceed -180 degrees (-1) when the gain crosses 0 dB and is derived from Cauchy’s Argument Principle which I have further detailed here:

Nyquist's Stability Criterion and Cauchy's Argument Principle

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  • $\begingroup$ So mathematically going along the jw-axis and taking a full encirclement is the same for creating the nyquist plot? Does this mean that the boad plot method is now always adequete? $\endgroup$
    – Kakukk777
    Jun 22, 2022 at 4:35
  • $\begingroup$ Does this mean that if -1 is exactly on the nyqusit plot line then there's a pole on the imaginaty axis? And if it's inside the encirclement then it's in the RHP? $\endgroup$
    – Kakukk777
    Jun 22, 2022 at 4:54
  • $\begingroup$ @Kakukk777 if you meant “Bode plot method is not always adequate” then I agree. The Nyquist plot is a more robust test as it can handle transfer functions of open loop systems with singularities in the right half plane. $\endgroup$ Jun 22, 2022 at 15:40
  • $\begingroup$ And see this other answer my MattL with further details on how to interpret the encirclements: dsp.stackexchange.com/a/53685/21048 $\endgroup$ Jun 22, 2022 at 15:44

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