Special Case When G(s)h(s) Involves Poles And/or Zeros On The Jw Axis

Special case when G(s)H(s) involves poles and/or zeros on the jωaxis

 

    

 

Fig: 1 closed contours in the s plane avoiding poles and zeros at the origin         Fig 2:s-Plane contour and the G(s)H(s) locus in the GH plane

 

We assumed that the open-loop transfer function G(s)H(s) has neither poles nor zeros at the origin. We now consider the case where G(s)H(s) involves poles and/or zeros on the jω axis.

Since the Nyquist path must not pass through poles or zeros of G(s)H(s), if the function G(s)H(s) has poles or zeros at the origin (or on the jω axis at points other than the origin), the contour in the s plane must be modified. The usual way of modifying the contour near the origin is to use a semicircle with the infinitesimal radius ε, as shown in Figure 1.A representative point s moves along the negative jω axis from – j∞ to j0-.

From s = j0- to s = j0 , the point moves along the semicircle of radius ε (where ε << 1) and then moves along the positive jω axis from j0 to j∞. From s = j∞, the contour follows a semicircle with infinite radius, and the representative point moves back to the starting point. The area that the modified closed contour avoids is very small and approaches zero as the radius ε approaches zero. Therefore, all the poles and zeros, if any, in the right-half s plane are enclosed by this contour.

Consider, for example, a closed-loop system whose open-loop transfer function is given by

 

 

The points corresponding to s = j0 and s = j0- on the locus of G(s)H(s) in the G(s)H(s) plane are – j∞ and j∞, respectively. On the semicircular path with radius ε (where ε << 1), the complex variable s can be written

 

 

The value K/ε approaches infinity as ε approaches zero, and -Φ varies from 90° to -90° as a representative point s moves along the semicircle. Thus, the points G(j0-) H(j0-) = j∞ and G(j0 )H(j0 ) = - j∞ are joined by a semicircle of infinite radius in the right-half GH plane. The infinitesimal semicircular detour around the origin maps into the GH plane as a semicircle of infinite radius. Figure 2 shows the s-plane contour and the G(s)H(s) locus in the GH plane. Points A, B, and C on the s-plane contour map into the respective points A', B ', and C' on the G(s)H(s) locus.