Root locus plot of coupled parameter

Question

I try to plot the root locus plot of $G(s) = \frac{(s+1)(s+\beta)}{(s-1)(s+3)^2}$ with respect to $\beta$, but could not factor out the parameter $\beta$ which is required for the sketch. Any idea how to proceed?

Answer 1

A root locus plots the closed loop poles as a function of the variable, in this case $\beta$. For a normal root locus the open loop transfer function could be written as

$$ G(s) = \frac{k\,N(s)}{D(s)}, $$

with $N(s)$ and $D(s)$ polynomials in $s$. The associated closed loop is given by

$$ G_c(s) = \frac{k\,N(s)}{k\,N(s)+D(s)}, $$

where the root locus plots the roots of the denominator for different $k$ and thus the values for $s$ such that $k\,N(s)+D(s)=0$.

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In your case the closed loop transfer function is given by

$$ G_c(s) = \frac{(s+1)(s+\beta)}{(s+1)(s+\beta)+(s-1)(s+3)^2} $$

and thus the root locus is considering

\begin{align} (s+1)(s+\beta)+(s-1)(s+3)^2 &= 0, \\ \beta\,(s+1)+(s^3+6s^2+4s-9) &= 0. \end{align}

So the same root locus would be obtained if you would use $N(s)=(s+1)$, $D(s)=(s^3+6s^2+4s-9)$ and using the normal way of drawing the root locus.