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Nyquist Criterion

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Suppose we have a linear time invariant system with a feedback loop. We label the open loop system function G(s)G(s)G(s). This applet allows the feedback to have a constant gain kkk and delay aaa. Thus, in the frequency domain, the feedback is represented by the function K(s)=ke−asK(s) = ke^{-as}K(s)=ke−as. By Black's formula, the closed loop system function is given by GCL(s)=K(s)G(s)1+K(s)G(s)=ke−asG(s)1+ke−asG(s). G_{CL}(s) = \frac{K(s)G(s)}{1+K(s)G(s)} = \frac{ke^{-as}G(s)}{1+ke^{-as}G(s)}.GCL​(s)=1+K(s)G(s)K(s)G(s)​=1+ke−asG(s)ke−asG(s)​. The Nyquist plot is the trajectory of K(iω)G(iω)=ke−iaωG(iω)K(i\omega) G(i\omega) = ke^{-ia\omega}G(i\omega)K(iω)G(iω)=ke−iaωG(iω) , where iωi\omegaiω traverses the imaginary axis. If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. This applet restricts G(s)G(s)G(s) to be a rational function

Suppose we have a linear time invariant system with a feedback loop. We label the open loop system function \(G(s)\). This applet allows the feedback to have a constant gain \(k\) and delay \(a\). Thus, in the frequency domain, the feedback is represented by the function \(K(s) = ke^{-as}\). By Black's formula, the closed loop system function is given by \[ G_{CL}(s) = \frac{K(s)G(s)}{1+K(s)G(s)} = \frac{ke^{-as}G(s)}{1+ke^{-as}G(s)}.\] The Nyquist plot is the trajectory of \(K(i\omega) G(i\omega) = ke^{-ia\omega}G(i\omega)\) , where \(i\omega\) traverses the imaginary axis. If the number…

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