What is the range of values for stability using the Routh-Hurwitz criterion?

[Dustinsfl]

Homework Statement

Find the stability of $s^3 + 3s^2 + 2(1 + K_p)s + 2K_i = 0$.

Homework Equations

Routh-Hurwitz criterion

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 $13.5$?

 

[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.

 

[Dustinsfl]

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.

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

 

[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.

 

[rezeew]

I would like to point out that the Routh-Hurwitz criterion is a mathematical tool used to determine the stability of a system based on its characteristic equation. It is important to note that this criterion only provides a necessary condition for stability, not a sufficient one. Therefore, further analysis and testing may be needed to fully determine the stability of a system.

Regarding the specific problem at hand, the conclusion that $\frac{K_i}{K_p} < 13.5$ can be obtained by setting the expression $\frac{3}{K_p} + 3$ to a maximum value of 13.5, which occurs when $K_p = \frac{1}{2}$. Therefore, the range of values for $K_i$ and $K_p$ that satisfy the Routh-Hurwitz criterion and ensure stability is $0 < K_i < 13.5K_p$ and $K_p > \frac{1}{2}$. However, as mentioned earlier, further analysis and testing may be needed to fully determine the stability of the system.