"1+L(s)" in the right half plane (which is the same as the number Since we know N and P, we can determine Z, the number of zeros of.We can factor L(s) to determine the number of poles that are in the.From the mapping we find the number N, which is the number ofĮncirclements of the -1+j0 point in "L(s)."Įquivalent to the number of encirclements of the origin in "1+L(s).".Make a mapping from the "s" domain to the "L(s)"ĭomain where the path of "s" encloses the.Start with a system whose characteristic equation is given by.To determine the stability of a system we: This is because the origin in "1+L(s)" corresponds to the "-1+j0" point in "L(s)" (if L(s)=-1, then 1+L(s)=0). Instead of mapping from "s" to "1+L(s)" and counting encirclements of the origin, we map from "s" to "L(s)" and count encirclements of the point -1+j0 in the complex plane. Mapping) and we know P (from L(s)), so we can easily determine Z.īefore continuing we make one small change. We can easily find P, the numbers of poles of "1+L(s)." This is becauseĪny pole of L(s) is also a pole of "1+L(s)." So now we know N (from the What we want, though, is Z, the number of zeros in the right half plane. If we perform a mapping (as explained on the previous page) of the function "1+L(s)" with a path in "s" that encircle the entire right half plane and we count the encirclements of the origin in the "1+L(s)" domain in the clockwise direction we get the number N=Z-P (where Z is the number of zeros, and P is the number of poles). Response of a transfer function with poles in the right half plane grows Has any zeros in the right half of the s-plane (recall that the natural Transfer function (which is the characteristic equation of the system) This is equivalent to asking whether the denominator of the We would like to be able to determine whether or not the closed loop system, The Nyquist Path (with poles on jω axis).Determining Stability using the Nyquist Plot
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