Root Locus Techniques

Why it matters

Root Locus Techniques are crucial in control systems engineering as they provide a graphical method to analyze and design control systems. By understanding how the poles of a system change with varying parameters, engineers can predict system behavior and ensure stability and performance.

Key ideas

• Root Locus Definition: A root locus is a path traced by the roots of a characteristic equation in the s-plane as a system parameter (usually gain) is varied.
• Purpose: It helps in determining the stability of a system and designing controllers to meet specific performance criteria.
• Construction Rules: The root locus is constructed using specific rules, such as starting at poles and ending at zeros, and the number of branches equals the number of poles.
• Breakaway and Break-in Points: Points on the real axis where multiple branches of the root locus meet or diverge.
• Asymptotes: Lines that the root locus approaches as the gain tends to infinity, determined by the number of poles and zeros.

Formulas

• Characteristic Equation: 1 + G(s)H(s) = 0
  • G(s): Open-loop transfer function
  • H(s): Feedback transfer function
• Asymptote Angle: θ = (2q + 1)180° / (P - Z)
  • q: Integer (0, 1, 2,...)
  • P: Number of poles
  • Z: Number of zeros

Worked example

Assume negative feedback and K ≥ 0. Given: Open-loop transfer function G(s)H(s) = K / (s(s+2)(s+4))

1. Characteristic Equation: 1 + K / (s(s+2)(s+4)) = 0
2. Find Poles: s = 0, s = -2, s = -4
3. Root Locus Construction:
   • Start at poles: s = 0, -2, -4
   • No finite zeros, so locus ends at infinity.
4. Asymptotes Calculation:
   • Number of poles P = 3, Number of zeros Z = 0
   • Asymptote angles: θ = (2q + 1)180° / (3 - 0) for q = 0, 1, 2
   • Angles: 60°, 180°, 300°

Final Answer: The root locus starts at s = 0, -2, -4 and approaches asymptotes at angles 60°, 180°, 300°.

The centroid is (0−2−4)/3 = −2. For positive K, real-axis segments are (−∞,−4) and (−2,0). From K = −s(s+2)(s+4), dK/ds = 0 gives s = −2 ± 2√3/3. Only s ≈ −0.8453 lies on a positive-gain segment. The characteristic polynomial is s³+6s²+8s+K. Routh analysis gives strict stability for 0 < K < 48; at K = 48, poles include ±j√8.

Common mistakes

• Incorrectly identifying the number of poles and zeros.
• Miscalculating asymptote angles.
• Forgetting to consider the effect of zeros at infinity.

For GATE EC

Questions often involve constructing root loci for given transfer functions, determining stability, and finding gain values for specific pole locations. Practice constructing root loci and calculating breakaway points and asymptotes.

Quick check

1. What is the purpose of a root locus?
2. How do you determine the number of branches in a root locus?
3. What are asymptotes in the context of root locus?

Answers: 1. To analyze system stability and design controllers. 2. Equal to the number of poles. 3. Lines that the root locus approaches as gain tends to infinity.