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 plot that shows the trajectories of the poles of a closed-loop control system as a system parameter (usually gain) is varied.
• Purpose: It helps in determining the stability of the system and designing controllers to meet specific performance criteria.
• Construction Rules: The root locus is constructed using specific rules, such as starting at open-loop poles and ending at open-loop zeros, and the number of branches equals the number of poles.
• Breakaway and Break-in Points: Points on the real axis where branches of the root locus leave or enter the real axis.
• Asymptotes: Lines that the root locus approaches as the gain tends to infinity, determined by the formula involving the number of poles and zeros.

Formulas

These standard rules assume negative feedback, real-coefficient loop transfer K L(s), K ≥ 0 and P > Z for the asymptote formulas. Branches that do not end at finite zeros go to infinity. Use k = 0 through P − Z − 1 for the distinct asymptote directions.

• Angle of Asymptotes = (2k + 1) * 180° / (P - Z)
  • k: Integer (0, 1, 2,...)
  • P: Number of poles
  • Z: Number of zeros
• Centroid = (Sum of Real Parts of Poles - Sum of Real Parts of Zeros) / (P - Z)
  • P: Number of poles
  • Z: Number of zeros

Worked example

Given a system with open-loop transfer function G(s)H(s) = K / (s(s+2)(s+4)), find the breakaway point on the real axis.

1. Identify poles and zeros: Poles at s = 0, -2, -4; no zeros.
2. Characteristic equation: 1 + K / (s(s+2)(s+4)) = 0 leads to s(s+2)(s+4) + K = 0.
3. Differentiate with respect to s: d/ds [s(s+2)(s+4)] = 3s^2 + 12s + 8.
4. Solve for s: Set 3s^2 + 12s + 8 = 0 and solve for s.
   • Using quadratic formula: s = [-12 ± sqrt(12^2 - 4*3*8)] / (2*3)
   • s = [-12 ± sqrt(144 - 96)] / 6
   • s = [-12 ± sqrt(48)] / 6
   • s = [-12 ± 6.93] / 6
   • s = -0.8453 or s = -3.1547
5. Select valid breakaway point: Only s = -0.8453 lies between s = 0 and s = -2.

Final Answer: Breakaway point is -0.8453.

Common mistakes

• Misidentifying the number of poles and zeros, leading to incorrect asymptotes and centroid calculations.
• Incorrectly applying the root locus construction rules, such as the angle of departure or arrival.
• Forgetting to check the validity of breakaway and break-in points on the real axis.

For GATE EE

• Questions often involve plotting the root locus for a given transfer function and determining stability or gain margins.
• Practice identifying poles and zeros, calculating breakaway points, and determining asymptotes.

Quick check

1. What is the purpose of a root locus plot?
2. How do you determine the number of branches in a root locus?
3. What is the formula for the centroid of asymptotes?

Answers: 1. To analyze system stability and design controllers. 2. Equal to the number of poles. 3. (Sum of Real Parts of Poles - Sum of Real Parts of Zeros) / (P - Z).