Root Locus and Stability

One of the most direct applications of root locus is determining the range of gain K for which a closed-loop system remains stable. Since stability requires all closed-loop poles to lie in the left half of the s-plane (LHP), the root locus immediately reveals stability by showing which portions of the locus are in the LHP versus the right half plane (RHP). The gain at which the locus crosses the imaginary axis is the critical gain, and it defines the boundary between stable and unstable operation.

The relationship between root locus and stability is one of the most GATE-relevant topics in control systems. Questions often ask for the range of K for stability, the critical gain, or the gain margin, all of which can be read directly from the root locus combined with the Routh-Hurwitz criterion for precise crossing values.

Core Concept Explanation

For a stable closed-loop system, all poles must lie strictly in the LHP (negative real parts). As the gain K increases from zero, the root locus branches move through the s-plane. As long as all branches remain in the LHP, the system is stable. At the critical gain Kc, one or more branches touch the imaginary axis, placing poles on the stability boundary. For K > Kc, those branches enter the RHP and the system becomes unstable.

The gain margin (GM) is defined as the factor by which the gain can be increased from its nominal value before the system becomes unstable. In terms of root locus, if the system is designed at gain K_op (operating gain) and becomes unstable at K_c, then the gain margin is Kc / K_op. For a system designed at K = 1 and Kc = 10, the gain margin is 10, or 20 dB.

The imaginary axis crossing can be found by two methods: substituting s = j*omega into the characteristic equation and separating real and imaginary parts, or using the Routh-Hurwitz criterion to find the gain at which the auxiliary equation vanishes. Both methods give the same critical gain and the oscillation frequency at that gain.

Mathematical Expression

To find the imaginary axis crossing, substitute s = j*omega into the closed-loop characteristic equation 1 + KG(j*omega) = 0. Separate real and imaginary parts: the imaginary part equation gives the crossing frequency omega_c, and the real part equation then gives the critical gain Kc. Alternatively, using Routh array, the row that goes to zero defines the auxiliary polynomial whose roots give the crossing frequency.

Practical Understanding

The root locus provides a visual gain margin: the designer can see directly how far the operating point is from the imaginary axis crossing. A locus that bends sharply toward the imaginary axis at modest K values indicates a low gain margin, meaning the system has little tolerance for gain uncertainty or disturbance. A locus that stays deep in the LHP across a wide range of K indicates a robust, high-gain-margin system.

The root locus also reveals phase margin information indirectly: the curvature of the locus near the operating point relates to how sensitive the damping ratio is to gain changes, which corresponds to the slope of the Nyquist plot near the -1 point. This connection between root locus and frequency domain methods is a deeper insight useful for advanced GATE questions.

Solved Numerical Example

For G(s) = K / [s(s+1)(s+2)], find the critical gain Kc at which the root locus crosses the imaginary axis. The characteristic equation is s^3 + 3s^2 + 2s + K = 0. Apply Routh-Hurwitz to find Kc and the crossing frequency.

Given:
G(s) = K / [s(s+1)(s+2)]
Characteristic equation: s^3 + 3s^2 + 2s + K = 0

Why this formula applies:
Routh-Hurwitz gives critical K when a row becomes zero

Formula:
Routh array for s^3 + 3s^2 + 2s + K

Substitution:
Row s^3: 1    2
Row s^2: 3    K
Row s^1: (3x2 - 1xK)/3 = (6-K)/3
Row s^0: K

Calculation:
For stability: all first-column elements positive
(6-K)/3 > 0  -->  K < 6
K > 0
Critical gain: Kc = 6

Crossing frequency from auxiliary equation (s^2 row):
3*s^2 + K = 0 at K=6 --> 3*s^2 + 6 = 0 --> s^2 = -2 --> s = +/- j*sqrt(2)

Final Answer:
Kc = 6; Crossing frequency omega_c = sqrt(2) rad/s

Exam Tip: In GATE, for third-order systems, critical gain from root locus equals (sum of pole products taken two at a time) / (product of all poles taken one at a time), which simplifies the Routh calculation. For G(s) = K/[s(s+a)(s+b)], Kc = a*b*(a+b) / 1... actually Kc = a*b directly from the s^1 row condition. Always derive from the Routh array to avoid formula errors.

Mechanism: Root Locus Stability Analysis

• System is stable for all K values where every root locus branch lies entirely in the LHP.
• Imaginary axis crossing occurs at critical gain Kc. Find Kc using Routh-Hurwitz or by substituting s = j*omega into the characteristic equation.
• Gain margin = Kc / K_operating. Expressed in dB as 20 log10(Kc / Kop).
• At the crossing point, the auxiliary equation from Routh gives the imaginary axis poles, which equals j*omega_c (oscillation frequency at marginal stability).
• If the locus never enters the RHP for any K greater than zero, the system is stable for all positive gains (unconditionally stable from a gain perspective).

Quick Revision

• Stability from root locus: all branches must lie in LHP. Locus in RHP means unstable closed-loop system.
• Critical gain Kc: the gain at which one or more branches cross the imaginary axis into RHP.
• Find Kc using Routh-Hurwitz: set the s^1 row element to zero and solve for K.
• Crossing frequency omega_c from auxiliary equation of Routh array: set s^2 row polynomial to zero.
• Gain margin = Kc / K_op. In dB: GM = 20 log10(Kc / Kop).
• For G(s) = K/[s(s+1)(s+2)]: Kc = 6, omega_c = sqrt(2) rad/s. A standard GATE benchmark system.
• Exam trap: Gain margin is NOT the same as critical gain. GM is the ratio of Kc to the operating gain. Always divide, do not just state Kc as the margin.