Complementary Root Locus

The complementary root locus extends the classical root locus concept to systems operating under negative gain, where the loop gain K varies from 0 to negative infinity. While the standard root locus handles positive feedback gain K from 0 to positive infinity, many practical control problems involve sign-reversed loop gains, requiring a separate locus construction with modified angle conditions.

Core Concept Explanation

In classical root locus, the characteristic equation is written as 1 + KG(s)H(s) = 0, where K is positive. The solution locus on the s-plane satisfies the angle condition: the net angle of G(s)H(s) at any point on the locus must equal an odd multiple of 180 degrees. This condition directly reflects positive feedback of the error signal through the plant.

When K is allowed to take negative values, the characteristic equation still holds, but the angle condition changes. For the complementary root locus, the angle condition becomes an even multiple of 180 degrees, that is, 0 degrees, 360 degrees, and so on. This is because with negative K, the sign inversion is absorbed into the angle, shifting the valid phase condition by exactly 180 degrees.

The magnitude condition remains unchanged between both root loci: the absolute value of KG(s)H(s) must equal 1 at any closed-loop pole location. Only the phase condition differs, which is what creates a geometrically distinct set of locus branches in the s-plane.

It is important to understand that the complementary root locus is not the mirror image of the standard root locus. The two loci are complementary in the sense that together they trace all possible closed-loop pole locations as K varies from negative infinity to positive infinity. Every segment of the real axis not covered by the standard locus is covered by the complementary locus, and vice versa.

Mathematical Expression

The general characteristic equation for a unity feedback system is 1 + KG(s) = 0. Rearranging, this gives G(s) = -1/K. For positive K, -1/K is real and negative, so the angle of G(s) must be an odd multiple of 180 degrees. For negative K, -1/K is real and positive, so the angle of G(s) must be an even multiple of 180 degrees including zero.

The two angle conditions are therefore:

• Standard Root Locus: angle of G(s)H(s) = (2q+1) x 180 degrees, q = 0, +/-1, +/-2, ...
• Complementary Root Locus: angle of G(s)H(s) = 2q x 180 degrees, q = 0, +/-1, +/-2, ...

The real axis rule for the complementary locus states that a point on the real axis lies on the complementary root locus if the total number of open-loop poles and zeros to the right of that point is even (including zero). This is the direct opposite of the standard root locus rule, which requires an odd count.

The asymptote angles for the complementary root locus are given by the formula: angle = 2q x 180 degrees divided by (n minus m), where n is the number of open-loop poles and m is the number of open-loop zeros. These asymptotes originate from the same centroid as in the standard case but point in directions shifted by 180/(n-m) degrees relative to the standard locus asymptotes.

Practical Understanding

The complementary root locus finds direct application in systems where the controller gain can be negative, such as in positive feedback loops, temperature-control systems with derivative action that might invert sign, or conditionally stable systems. In a conditionally stable system, the closed-loop system is stable only for a specific range of K, and both lobes of the complete root locus (standard plus complementary) together explain why stability is lost both at very low and very high gains.

In GATE problems, the complementary root locus is typically tested through identification of the real axis segment that belongs to it, determination of asymptote angles, and checking angle condition at a given test point. Students frequently confuse the even-multiple rule with the odd-multiple rule, which is the most common error.

Another practical significance is in pole-zero cancellation analysis. When a controller zero is placed near a plant pole to cancel it, the sensitivity of this cancellation to gain sign becomes visible through the complementary locus. If the cancelled pole moves under negative gain perturbations, the complementary locus shows where it migrates.

Given:
Open-loop transfer function G(s) = K / [s(s+4)]
Test point: s = -2 (on real axis)
Determine: Does s = -2 lie on standard or complementary root locus?

Why this formula applies:
Real axis rule — count poles and zeros to the right of test point.
Poles at s=0 and s=-4. Zeros: none.
Poles to the right of s=-2: only s=0 → count = 1 (odd)

Formula:
Standard locus rule: odd count → lies on standard root locus
Complementary locus rule: even count → lies on complementary root locus

Substitution:
Poles to the right of s = -2: {s=0} → 1 pole
Count = 1 (odd)

Calculation:
Odd count → s = -2 belongs to standard root locus (K > 0)
For complementary locus, need even count
Example: s = -5 has poles {s=0, s=-4} to its right → count = 2 (even)

Final Answer:
s = -2 lies on standard root locus (K > 0)
s = -5 lies on complementary root locus (K < 0)

Exam Tip: In GATE, when asked to find the real axis segment of the complementary root locus, apply the even-count rule for poles plus zeros to the right. The segment between two poles (or two zeros) with an even count belongs to the complementary locus, not the standard locus.

Mechanism: Construction Rules Summary

• The complementary root locus applies when K varies from 0 to negative infinity in the open-loop transfer function KG(s)H(s).
• The angle condition shifts to even multiples of 180 degrees (0, 360, ...), in contrast to odd multiples for the standard locus.
• Real axis rule: count of poles plus zeros strictly to the right must be even for the point to lie on the complementary locus.
• Asymptote angles are 2q x 180/(n-m) degrees; the centroid formula is identical to that of the standard root locus.
• Together, the standard and complementary root loci form the complete root locus for all real K, covering the entire s-plane picture.

Quick Revision

• Complementary root locus: K varies from 0 to negative infinity.
• Angle condition: angle of G(s)H(s) = 2q x 180 degrees (even multiples).
• Real axis rule: even number of poles plus zeros to the right of test point.
• Asymptote angles: 2q x 180 / (n-m); centroid formula unchanged from standard locus.
• Magnitude condition is identical for both standard and complementary loci.
• Exam trap: Do not apply odd-count rule to complementary locus — that belongs only to standard root locus.
• Standard + Complementary loci together describe all possible closed-loop pole positions for all real values of K.