Control systems can be daunting, especially when tackling complex university-level assignments. One challenging topic within this field is the concept of Stability Analysis using the Nyquist Criterion. This blog will guide you through a sample question on this topic, explaining the concepts in detail and providing a step-by-step method for solving the problem. By the end, you'll be well-equipped to approach similar questions with confidence.

Question: Given a unity feedback control system with an open-loop transfer function G(s)=K/[s(s+1)]​, determine how the system's stability changes as the gain $K$ varies. Use the Root Locus method to analyze and describe the behavior of the system.

Understanding the Root Locus Method

The Root Locus method is a graphical technique used to analyze how the roots of a control system's characteristic equation change with varying values of the system gain $K$. This method helps determine the stability of the system as the gain changes.

Key Concepts:

1. Root Locus Plot: This is a plot showing the paths of the system poles in the s-plane as the gain $K$ varies from 0 to infinity. The plot provides insight into how the system's stability changes with different gains.

2. Stability: A system is stable if all its poles (roots of the characteristic equation) are located in the left half of the s-plane. If any pole is in the right half of the s-plane, the system is unstable.

Step-by-Step Guide to Solving the Sample Question

1. Define the Open-Loop Transfer Function: For the given system, the open-loop transfer function is G(s)=K/s(s+1)

2. Determine the Characteristic Equation: The characteristic equation of the closed-loop system with unity feedback is given by:

   1+G(s)=0

   Substituting $G(s)$ into the equation gives:

   1+K/[s(s+1)]=0

   Simplify to find:

   s(s+1)+K=0

3. Sketch the Root Locus:

   • Identify the Poles and Zeros: For $G(s)$, the poles are at s=0 and s=−1. There are no zeros.
   • Plot the Poles and Zeros: Mark these points on the s-plane.
   • Draw the Root Locus: As $K$ varies from 0 to infinity, draw the paths that the poles will follow. This involves sketching the loci from the poles to infinity. The paths typically start at the poles and move outward.

4. Analyze Stability:

   • Initial and Final Locations: At K=0, the poles are at s=0 and s=−1. As $K$ increases, observe how the poles move. For the system to remain stable, all poles must be in the left half of the s-plane.
   • Determine Stability Range: Identify the value of $K$ where any poles cross into the right half of the s-plane. This transition indicates instability.

5. Describe the Behavior:

   • For small values of $K$, the system is stable as all poles are on the left side of the s-plane.
   • As $K$ increases, monitor the root locus plot to see if any poles move to the right half of the s-plane, indicating the onset of instability.

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Conclusion

Analyzing stability using the Root Locus method can provide valuable insights into how a control system behaves as gain varies. By following the steps outlined in this blog, you can effectively determine the stability of a system and understand the impact of changing gain. For personalized assistance with your control system assignments, consider reaching out to our expert team for tailored support and guidance.