Rules for Making Complementary Root Locus Plot

Rules for drawing the Complementary Root Locus (K≤0), are similar to those for the Standard Root Locus (K≥0).  Derivations are left to the reader, but follow closely the rules for the Standard Root Locus

The table below gives the rules for both the Standard Root Locus as well as the Complementary Root Locus.  Where the rules are identical, they are only stated once.

Rule Name Standard Root Locus Complementary Root Locus
Definitions
• The loop gain is KG(s)H(s) which can be rewritten as KN(s)/D(s).
• N(s), the numerator, is an mth order polynomial; D(s) is nth order.
• N(s) has zeros at zi (i=1..m);  D(s) has them at pi (i=1..n).
• The difference between n and m is q, so q=n-m.
Symmetry The locus is symmetric about real axis (i.e., complex poles appear as conjugate pairs).
Number of Branches There are n branches of the locus, one for each pole of the loop gain.
Starting and Ending Points The locus starts (when K=0) at poles of the loop gain, and ends (when K→∞) at the zeros. Note: there are q zeros of the loop gain as |s|→∞.
Locus on Real Axis The locus exists on real axis to the left of an odd number of poles and zeros. The locus exists on real axis to the left of an even number of poles and zeros.
Asymptotes as |s|→∞ If q>0 there are asymptotes of the root locus that intersect the real axis at $\sigma = {{\sum\limits_{i = 1}^n {{p_i}} - \sum\limits_{i = 1}^m {{z_i}} } \over q}$, and radiate out with angles $\theta = \pm r{{180} \over q}$, where r=1, 3, 5… If q>0 there are asymptotes of the root locus that intersect the real axis at $\sigma = {{\sum\limits_{i = 1}^n {{p_i}} - \sum\limits_{i = 1}^m {{z_i}} } \over q}$, and radiate out with angles $\theta = \pm p{{180} \over q}$, where p=0, 2, 4…
Break-Away and -In Points on Real Axis There are break-away or -in points of the locus on the axis wherever $N(s)D'(s)-N'(s)D(s)=0$.
Angle of Departure from Complex Pole Angle of departure from pole pj is
$${\theta _{depart,{p_j}}} = 180^\circ + \sum\limits_{i = 1}^m {\angle \left( {{p_j} - {z_i}} \right) - } \sum\limits_{i = 1,\;i \ne j}^n {\angle \left( {{p_j} - {p_i}} \right)}$$
Angle of departure from pole pj is $${\theta _{depart,{p_j}}} = \sum\limits_{i = 1}^m {\angle \left( {{p_j} - {z_i}} \right) - } \sum\limits_{i = 1,\;i \ne j}^n {\angle \left( {{p_j} - {p_i}} \right)}$$
Angle of Arrival at Complex Zero Angle of arrival at zero zj is $${\theta _{arrive,{z_j}}} = 180^\circ - \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j} - {z_i}} \right) + } \sum\limits_{i = 1}^n {\angle \left( {{z_j} - {p_i}} \right)}$$ Angle of arrival at zero zj is
$${\theta _{arrive,{z_j}}} = \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j} - {z_i}} \right) - } \sum\limits_{i = 1}^n {\angle \left( {{z_j} - {p_i}} \right)}$$
Locus Crosses Imaginary Axis Use Routh-Horwitz to determine where the locus crosses the imaginary axis.
Determine Location of Poles, Given Gain "K" Rewrite characteristic equation as D(s)+KN(s)=0. Put value of K into equation, and find roots of characteristic equation.  (This may require a computer)
Determine Value of "K", Given Pole Locations  Rewrite characteristic equation as $K = - {{D(s)} \over {N(s)}}$, replace "s" by the desired pole location and solve for K.  Note: if "s" is not exactly on locus, K may be complex, but the imaginary part should be small. Take the real part of K for your answer.

Note the derivation of the rules for drawing the Complementary Root Locus (K≤0), are similar to those for the Standard Root Locus (K≥0),  except the angle condition becomes

\eqalign{ \angle \left( {K{{N(s)} \over {D(s)}}} \right) &= \angle \left( { - 1} \right) \cr \angle \left( K \right) + \angle \left( {{{N(s)} \over {D(s)}}} \right) &= \angle \left( { - 1} \right) \cr 180^\circ + \angle \left( {{{N(s)} \over {D(s)}}} \right) &= \pm r180^\circ ,\quad r = 1,\;3,\;5,\; \ldots \cr \angle \left( {{{N(s)} \over {D(s)}}} \right) &= \pm p180^\circ ,\quad p = 0,\;2,\;4,\; \ldots \cr}

References