The stability of a unity-feedback control system whose open-loop transfer function is $G(s)=K/[s(s+1)(s+5)]$

Question

Question and solution from book. Regarding the solution: How do you obtain the characteristic equation? Why is it K/s(s+1)(s+5) + 1=0? Where did the 1 come from?? And then how do you go from that to s^3+6s^2+5s+K = 0?

Thanks.

Answer 1

The $1$ comes from the closed-loop transfer function.

See these WikiBook Routh-Hurwitz-Criterion notes for details.

From $G(s) = \dfrac{K}{s(s+1)(s+5)}$, we form:

$$1 + \dfrac{K}{s(s+1)(s+5)} = 0 \implies 1 + \dfrac{K}{s^3+6 s^2+5 s} = 0$$

If we find a common denominator and multiply through, we arrive at:

$$\tag 1 s^3+6 s^2+5 s + K = 0$$

The Routh table (see linked site above) is:

$$\begin{array}{c|c|c} \hline s^3 & 1 & 5 \\ s^2 & 6 & K \\ s^1 & \dfrac{30-K}{6} & 0 \\ s^0 & K & -\\ \end{array} $$

Routh's criterion states that the number of sign changes in the first column is equal to the number of roots in the right-half plane. For stability, the first column must have all entries positive.

Clearly $1$ and $6$ are positive, but the next two values are dependent on the gain parameter $K$.

We have:

$$\dfrac{30-K}{6} \gt 0, K \gt 0 \implies 0 \lt K \lt 30$$

You might want to review some more notes and examples on this method, see:

• http://www.engr.iupui.edu/~skoskie/ECE680/Routh.pdf
• http://www.me.ust.hk/~mech261/index/Lecture/Chapter_7.pdf