Help with Root Locus: Get into General Form for Drawing

[formulajoe]

I had to use state-variable feedback to design a control system for a motor. I first designed a deadbeat controller, which was pretty easy.
But now I have to design another 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?

 

[formulajoe]

The problem I am having is that when H(s) and G(s) are multiplied together, a zero is introduced because of the feedback loop. The location of this zero changes as the gain changes. So the gain cannot be varied without moving the zero. Thus every value of gain produces a different root locus.

 

[piercas]

I can understand your confusion about using root locus methods to design a controller for your motor. Let me provide some guidance on how to approach this problem.

Firstly, it is important to understand that the root locus method is used to analyze the closed-loop poles of a control system as the gain is varied. This means that the poles of the system will move along a specific path on the complex plane, known as the root locus, as the gain is changed. The goal is to design a controller that results in the desired closed-loop pole locations.

To generate the root locus, you will need to follow these steps:

1. Write the transfer function of your system in terms of the forward path gain, K, and the feedback path gain, H. This will give you the characteristic equation of the system, which can be written in the form of a polynomial equation.

2. Use the characteristic equation to determine the open-loop poles of the system. These are the poles of the system without any feedback.

3. Now, add the feedback loop to your system and determine the closed-loop transfer function. This can be done by using the formula: Gc(s) = G(s)/(1+G(s)H(s)), where G(s) is the open-loop transfer function and H(s) is the feedback transfer function.

4. Once you have the closed-loop transfer function, you can use it to determine the closed-loop poles of the system. These poles will depend on the value of K, the gain in the feedback loop.

5. Now, plot the open-loop poles and the closed-loop poles on the complex plane. As you vary the gain K, the closed-loop poles will move along the root locus.

6. Finally, use the general form of the root locus equation, which is given by: z = (1+GH)/(1+G), to plot the root locus on the complex plane. This equation will help you determine the path of the closed-loop poles as the gain K is varied.

I hope this general form and the steps provided will help you draw the root locus for your system. It is important to note that the specific form of the root locus equation may vary depending on the type of control system you are designing. I would suggest referring to a textbook or consulting with a control systems expert for further guidance and clarification. Good luck with your design!