How to draw root locus

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Jan 20, 2024 03:23 AM
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Drawing a root locus diagram is a method used in control theory to analyze how the roots of a system (i.e., the poles of the transfer function) change as a particular parameter, typically the gain , varies from zero to infinity. The root locus provides insight into the stability and transient behavior of a control system. Here's a step-by-step guide to drawing a root locus:

1. Determine the Open-Loop Transfer Function

First, you need the open-loop transfer function of the system, typically in the form , where is the forward path transfer function and is the feedback path transfer function.

2. Find the Poles and Zeros

Identify the poles and zeros of the transfer function. Poles are the values of that make the denominator of zero, and zeros are the values of that make the numerator zero.
  • Plot the Poles and Zeros: On the complex plane (s-plane), mark the poles with an 'X' and zeros with an 'O'.

3. Determine the Real Axis Segments

Parts of the real axis will be part of the root locus. These segments can be determined by applying the angle criterion. The root locus exists on segments of the real axis where the total phase shift (angle) contributed by the poles and zeros to a point on the real axis is an odd multiple of 180 degrees.

4. Calculate Asymptotes

As approaches infinity, the loci of the roots that go off to infinity approach straight lines, known as asymptotes. The number of asymptotes equals the number of poles minus the number of zeros. The angles of the asymptotes are given by:
where is the number of poles, is the number of zeros, and is an integer (0, 1, 2, ...).
The centroid () of the asymptotes is given by:

5. Determine Breakaway and Break-in Points

Breakaway points (where loci leave the real axis) and break-in points (where they come to the real axis) are found by solving , where is the gain at a particular point on the root locus.

6. Calculate the Departure and Arrival Angles

For complex poles and zeros, calculate the angles at which the loci depart from poles or arrive at zeros. This involves summing the angles made by the line segments from all other poles and zeros to the pole or zero in question.

7. Sketch the Root Locus

Using the information from the above steps, sketch the root locus:
  • Start from the poles and move towards the zeros.
  • If the number of poles minus the number of zeros to the right of a point on the real axis is odd, then that segment of the real axis is part of the root locus.
  • Follow the real axis segments and asymptotic directions.
  • Use the breakaway and break-in points, and departure and arrival angles to guide the sketch.

7.1. Understanding the Rule of Sketching a Root Locus:

  1. Poles and Zeros: In the context of root locus, poles are the values of for which the open-loop transfer function goes to infinity, and zeros are the values of for which the open-loop transfer function becomes zero.
  1. Rule for Real Axis Segments: For any point on the real axis, count the number of poles and zeros to the right of that point. If the difference (poles minus zeros) is an odd number, then that point lies on the root locus.
  1. Why This Rule Works: The root locus diagram is constructed based on the principle that the total phase angle around the loop of a feedback system must be an odd multiple of 180 degrees (or ) for the system to be on the verge of instability. When you move along the real axis, the phase contributions of poles and zeros change. If you have an odd number of poles minus zeros to the right, it means that the total phase angle around the loop (as seen from that point) is shifting in such a way that it fulfills the angle criterion (odd multiple of 180 degrees).

8. Analyze Stability and Behavior

Finally, use the root locus to analyze the stability and transient behavior of the system. The location of the roots in the left-half plane (LHP), right-half plane (RHP), and on the imaginary axis gives insights into the system's stability and response characteristics.

Software Tools

While hand-drawing root locus plots is educational, in practice, software tools like MATLAB, Python (with control system libraries), or specialized control design software are often used for accuracy and efficiency.
Remember, drawing a root locus plot requires practice and a good understanding of the underlying principles. Each system can present unique challenges, so it's important to approach each plot methodically.
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