# Multiple-loop System Of A Polar Plot

**Multiple-loop system of a polar plot**

*Fig: 1 Multiple-loop system*

Consider the system shown in Figure 1. This is a multiple-loop system. The inner loop has the transfer function

If G(s) is unstable, the effects of instability are to produce a pole or poles in the right half s plane. Then the characteristic equation of the inner loop, 1 G_{2}(s)H_{2}(s) = 0, has a zero or zeros in this portion of the plane. If G_{2}(S) and H_{2}(S) have P_{1} poles here, then the number Z_{1} of right-half plane zeros of 1 G_{2}(s)H_{2}(S) can be found from Z_{1} = N_{1} P_{1}, where N_{1} is the number of clockwise encirclements of the -1 j0 point by the G_{2}(s)H_{2}(S) locus.

Since the open-loop transfer function of the entire system is given by G_{1}(s)G(s)H_{1}(s), the stability of this closed-loop system can be found from the Nyquist plot of G_{1}l(s)G(s)H_{1}(s) and knowledge of the right-half plane poles of G_{1}(s)G(s)H_{1}(s).

Notice that if a feedback loop is eliminated by means of block diagram reductions there is a possibility that unstable poles are introduced; if the feedforward branch is eliminated by means of block diagram reductions, there is a possibility that right-half plane zeros are introduced. Therefore, we must note all right-half plane poles and zeros as they appear from subsidiary loop reductions. This knowledge is necessary in determining the stability of multiple-loop systems.