Modern Control System Theory and Design, 2nd Edition

6.16.  ROOT-LOCUS METHOD FOR POSITIVE-FEEDBACK SYSTEMS

The rules presented in the previous section for constructing the root locus were directed toward negative-feedback systems. For positive-feedback systems, several of these rules must be modified. The purpose of this section is to indicate the changes to the rules and apply them to the second problem considered in Section 6.14 for positive, instead of negative, feedback.

For positive feedback, Eq. (6.130) becomes

or

where n = 0, ±1, ±2, ±3, . . .. Equation (6.184) specifies two conditions that must be satisfied for the existence of a closed-loop pole in positive-feedback systems.

1. The angle of G(s)H(s) must be an even multiple of π:

   

   where n = 0, ±1, ±2, ±3, . . ..

2. The magnitude of G(s)H(s) must be unity:

   

Based on Eqs. (6.185) and (6.186), it is necessary to modify Rules 4, 5, 7, and 9 given in Section 6.14 for construction of the root locus with negative feedback as follows:

Rule 4. This rule is modified for positive feedback so that sections of the real axis are part of the root locus if the number of poles and zeros to the right ...