[Solved] How do you use rlocus with multiple feedback loops?

So I have this control loop and I want to plot the root locus with the gain variable Kt but I how do you do this as rlocus expects a single feedback loop, not two where one is affected by the gain and the other is not?

Answer 1

[Paul]

Start with the correct characteristic equation:

1 + D(s)*G(s)*I(s) + D(s)*G(s)*Kt = 0 (not sure why the picture shows the last term divided by I(s) ).

Let L(s) be the "loop transfer function" that should be used to plot the root locus with respect to Kt. What is the CE written in terms of L(s) and Kt? Can the actual CE be manipulated into that form?

[Joseph Patrick]

The division by I is needed as the feedback loop pickoff point is before the I(s) block right?

I can't seem to get it in the form 1+KL(s) because the second term doesn't depend on K but apparently it can be done

edit: I see now why the I is not needed in that last term

[Paul]

Right. You want an equation in the form of 1 + Kt*L(s) = 0. Let's look at a similar problem. Suppose we have an equation N(s) + Kt * M(s) = 0 that we want to put in the form 1 + Kt*L(s) = 0 using algebraic manipulation. Because the RHS must stay equal to zero, the only options are to multiply or divide the LHS of the equation by something. So, what should N(s) + Kt*M(s) be multiplied or divided by to put it in the form 1 + Kt *L(s) ?

[Joseph Patrick]

In that situation dividing by N(s) would give 1+Kt*M(s)/N(s) where L(s) = M(s)/N(s)

But in this situation I have three terms not two, If say I divide by DGI then I have (1/DGI) + 1 + (Kt/I) which still isn't the form 1 + KtL(s)

Unless I'm missing something obvious here?

[Joseph Patrick]

Unless I divide by (1+DGI), giving 1 + K(DGKt/1+DGI) ?

L(s) = DG / 1+DGI

[Paul]

Correct.

So now you have to decide ...

if the nominal value of Kt is given and you want to know how the closed loop poles migrate based on scaling that nominal value, in which case:

L(s) = Kt*D(s)*G(s)/(1 + D(s)*G(s)*I(s))

as shown in the first line in the preceding comment, or if you want to know how the closed loop poles migrate as a function of Kt itself, in which case:

L(s) = D(s)*G(s) /(1 + D(s)*G(s)*I(s))

as in the second line in the preceding comment.

[Joseph Patrick]

Yeah, managed to get my plot

Thanks for the help :)