# Routh-Hurwitz criterion

1. Apr 8, 2014

### Dustinsfl

1. The problem statement, all variables and given/known data
Find the stability of $s^3 + 3s^2 + 2(1 + K_p)s + 2K_i = 0$.

2. Relevant equations
Routh-Hurwitz criterion

3. The attempt at a solution
By the Routh-Hurwitz stability criterion, we have
$$\begin{array}{ccc} s^3 & 1 & 2(1 + K_p)\\ s^2 & 3 & 2K_i\\ s^1 & \frac{6(1 + K_p) - 2K_i}{3} & 0\\ s^0 & 2K_i & 0 \end{array}$$
From the $s^3$ line, we have that $2(1 + K_p) > 0$; therefore, $K_p > -1$. From the $s^2$ and $s^0$ line, we have that $2K_i > 0$; thus, $K_i > 0$. From the $s^1$ line, we have that $6K_p - 2K_i + 6 > 0$; therefore,
$$\frac{K_i}{K_p} < \frac{3}{K_p} + 3.$$
I am supposed to conclude
$$0 < \frac{K_i}{K_p} < 13.5.$$
From line $s^2$, we get greater than zero, but what do I do to go from $\frac{3}{K_p} + 3$ to $13.5$?

Last edited: Apr 8, 2014
2. Apr 8, 2014

### donpacino

To be honest I do not know. I went through the routh-Hurwitz criterion and got the same results you did

If you plug in 10 for ki and 1 for kp the system is unstable (i ran it in matlab)
therefore the conclusion that 0<ki/kp<13.5 is incorrect

I recommend confirming that all the information you provided us is correct, and all the information provided to you is correct.

3. Apr 8, 2014

### Dustinsfl

Is the information from the book:
I had to use the link since the image appears too big on the site.
http://i.imgur.com/M5Ks5up.jpg?1

Last edited: Apr 8, 2014
4. Apr 9, 2014

### milesyoung

Must be an error in your book. Your Routh table is correct, so to avoid any sign changes in the first column for $K_p > 0, K_i > 0$, we must have:
\begin{align} 6(1 + K_p) - 2K_i \geq 0 \Leftrightarrow K_i \leq 3(1 + K_p) \quad (1) \end{align}
Although (1) being true is necessary and sufficient for stability, the weaker condition:
$$K_i \leq 3K_p \Leftrightarrow \frac{K_i}{K_p} \leq 3$$
is sufficient.