Solved Examples of Root Locus Design for Control Systems

📚 Quick Study Guide

🔍 Root Locus: A graphical representation of the closed-loop poles as a system parameter (usually gain, $K$) is varied. 📐 Angle Criterion: A point 's' is on the root locus if the sum of angles of open-loop poles and zeros, evaluated at 's', is an odd multiple of 180 degrees: $\angle G(s)H(s) = (2k+1)180^\circ$, where $k$ is an integer. 📏 Magnitude Criterion: $|G(s)H(s)| = 1$ for 's' to be on the root locus. ➕ Breakaway/Break-in Points: Occur where multiple roots exist; found by solving $\frac{dK}{ds} = 0$. ↔️ Asymptotes: Root locus approaches asymptotes as $|s| \rightarrow \infty$. The angles are given by $\frac{(2k+1)180^\circ}{n-m}$, where $n$ is the number of poles and $m$ is the number of zeros. 🎯 Centroid: The intersection of the asymptotes on the real axis, calculated as $\frac{\sum poles - \sum zeros}{n-m}$. 🧭 Stability: The root locus indicates stability; if roots are in the left-half plane, the system is stable.

🧪 Practice Quiz

1. Question 1: Which of the following is the primary purpose of the root locus plot?
   1. A. To determine the stability of an open-loop system.
   2. B. To analyze the transient response of a system for a fixed gain.
   3. C. To show how the closed-loop poles move as a system parameter varies.
   4. D. To design a lead compensator.
2. Question 2: The angle criterion for a point to be on the root locus is given by:
   1. A. $\angle G(s)H(s) = 2k \cdot 180^\circ$
   2. B. $\angle G(s)H(s) = (2k+1) \cdot 180^\circ$
   3. C. $|G(s)H(s)| = 1$
   4. D. $|G(s)H(s)| = 0$
3. Question 3: Breakaway points on the root locus occur:
   1. A. Where the root locus crosses the imaginary axis.
   2. B. Where multiple roots of the characteristic equation exist.
   3. C. At the open-loop poles.
   4. D. At the open-loop zeros.
4. Question 4: The asymptotes of the root locus are used to determine the behavior of the root locus:
   1. A. Near the origin.
   2. B. As the system gain approaches zero.
   3. C. As $|s|$ approaches infinity.
   4. D. Near the open-loop poles.
5. Question 5: The centroid of the asymptotes is calculated as:
   1. A. The sum of the poles plus the sum of the zeros divided by (n-m).
   2. B. The sum of the zeros minus the sum of the poles divided by (n+m).
   3. C. The sum of the poles minus the sum of the zeros divided by (n-m).
   4. D. The sum of the poles minus the sum of the zeros divided by (n+m).
6. Question 6: What does the root locus indicate about the stability of a system?
   1. A. If all roots are in the right-half plane, the system is stable.
   2. B. If roots are on the jω axis, the system is always unstable.
   3. C. If roots are in the left-half plane, the system is stable.
   4. D. The root locus does not provide information about stability.
7. Question 7: The magnitude criterion for a point 's' to lie on the root locus is:
   1. A. $|G(s)H(s)| = 0$
   2. B. $|G(s)H(s)| = \infty$
   3. C. $|G(s)H(s)| = 1$
   4. D. $\angle G(s)H(s)| = (2k+1)180^\circ$