Root Locus: Get Help with State-Variable Feedback Design

[formulajoe]

I had to use state-variable feedback to design a position control system for a motor. I have to design the controller using root locus methods.
The gain that is varied is located in the feedback loop as shown in the picture.
I am completely lost as to how to generate a root locus with the gain in the feedback loop.
Can anybody give me a general form to get into so I can draw the root locus?

 

[Corneo]

Let the output of the summer going into G(s) be called E(s). Y(s) is then E(s)G(s). But E(s) = V(s)-K(s+1)Y(s). So

$$Y(s) = E(s)G(s) = [V(s) - K(s+1)Y(s)] G(s)$$
$$Y(s) = \frac {E(s)G(s)}{1 + K(s+1)G(s)}$$

Then you draw the root locus plot of G(s) (s+1).

 

[ravenprp]

Hi there,

First of all, it's great that you are using state-variable feedback to design a position control system for a motor. This is a powerful technique that can help you achieve accurate and stable control.

In terms of designing the controller using root locus methods, there are a few steps that you can follow to generate the root locus. Let's break it down into a few key points:

1. Determine the transfer function of your system: The first step is to determine the transfer function of your system, which relates the output (position) to the input (control signal). This transfer function can be derived from the state-space representation of your system.

2. Identify the poles and zeros of your system: Once you have the transfer function, you can identify the poles and zeros of your system. The poles are the values of s (the Laplace variable) that make the denominator of the transfer function equal to zero. The zeros are the values of s that make the numerator of the transfer function equal to zero.

3. Plot the poles and zeros on the s-plane: The s-plane is a coordinate system where the real axis represents the real part of the complex variable s and the imaginary axis represents the imaginary part. Plot the poles and zeros of your system on the s-plane.

4. Determine the open-loop transfer function: The open-loop transfer function is the transfer function of your system without any feedback. In your case, the gain in the feedback loop is the only component in the feedback path, so the open-loop transfer function will simply be the transfer function of your system multiplied by the gain.

5. Draw the root locus: The root locus is a plot of the locations of the closed-loop poles as the gain in the feedback loop is varied from 0 to infinity. To draw the root locus, you can use a general form for the transfer function that includes the gain in the feedback loop. For example, if your transfer function is G(s), the general form would be KG(s), where K is the gain.

6. Analyze the root locus: The root locus will give you information about the stability and performance of your system. For example, if the root locus crosses the imaginary axis, it means that your system will have oscillations. If the root locus crosses the right half of the s-plane, it means that your system will be unstable.

I hope this general guide helps you get started with drawing the root locus for your system. If you need further assistance