5.6.4 Examples of the Application of the Nyquist Stability CriteriaExample 5.7Consider the open-loop transfer function L\left( s \right) = \frac{5}{1 - s} = - \frac{5}{s - 1}. Let us analyze the stability of the closed-loop control system.The system has a pole in the right half-plane, thus P = 1 . The Nyquist diagram is shown in Fig. 5.23. As the Nyquist diagram does not encircle the point - 1 {\,+\,} 0j , R = 0 , thus the closed-loop is unstable.The system can be stabilized if a so called compensation element of a constant gain by A = - 1 is connected into the forward path. This element changes the sign of the points of the Nyquist diagram reflecting it about the origin (dashed-dotted curve). Thus the number of windings around - 1 {\,+\,} j0 will be R = P = 1 . \hskip 2pc{\blacksquare } Example 5.8Let us consider for example the case when the open-loop is an L\left( s \right) = K_{{\rm I}} /s integrator, whose pole is at the origin. The closed curve is created by getting around the pole from the right. By mapping this curve according to L\left( s \right) the complete Nyquist diagram shown in Fig. 5.24a is obtained. The case involving getting around the pole from the left is demonstrated in Fig. 5.24b. In the s-plane, the points denoted by 1, 2 and 3 on the small circle surrounding the pole are mapped into the points 1′, 2′ and 3′ in the L\left( s \right) -plane. In case (a) P = 0 and R = 0 , in case (b) P = 1 and R = 1 , thus in both cases the stable behavior of the system can be established. \hskip 22.9pc{\blacksquare } Example 5.9Let the transfer function of an open-loop be a proportional element with three time lags, L\left( s \right) = K/\left[ {\left( {1 {\,+\,} sT_{1} } \right)\left( {1 {\,+\,} sT_{2} } \right)\left( {1 {\,+\,} sT_{3} } \right)} \right] . The poles p_{1} = - 1/T_{1} , p_{2} = - 1/T_{2} , p_{3} = - 1/T_{3} are all in the left half-plane, thus P = 0 . Let us apply the generalized Nyquist stability criterion. The complete Nyquist diagram obtained by mapping of the curve given in Fig. 5.17 is shown in Fig. 5.25. If the Nyquist diagram goes through - 1 {\,+\,} j0 , the system is at the stability limit. If the Nyquist diagram does not include the point - 1 {\,+\,} j0 ( K_{1} loop gain), R = P = 0 , thus the control system is stable. If the Nyquist diagram includes the point - 1 {\,+\,} 0j ( K_{2} loop gain), R \ne P , thus the control system is unstable. To determine the number of windings R , let us put the spike of an imaginary compass on the point - 1 {\,+\,} 0j , and with the other end of the compass pass through the Nyquist diagram from \upomega = - \infty to {\,+\,} \infty . The number of windings is R = - 2 (clockwise). The characteristic equation has two roots in the right half-plane, so, Z = 2 , and as R = - 2 = P - Z = 0 - Z , in this case the system is unstable. \hskip 6pc{\blacksquare } In the case of a stable open-loop, it is sufficient to use the simple Nyquist stability criterion. In the stable case - 1 {\,+\,} 0j lies to the left of the Nyquist diagram drawn for positive frequencies, whereas in the unstable case it is to the right of that curve. The simplified stability investigation can be applied also to the cases when the open-loop contains integrators, and thus there are poles at the origin.