I have a transfer function:
$$G(s) =\frac{num(s)}{den(s)}$$
And if I want to simulate this transfer function with a P-controller. I need to do a feedback transfer function:
$$G_f(s) =\frac{KG(s)}{1 + KG(s)} = \frac{\frac{Knum(s)}{den(s)}}{\frac{Knum(s)+den(s)}{den(s)}} = \frac{Knum(s)}{Knum(s)+den(s)}$$
I want to find the $K \in \Re$ who gives the process unstable dynamics. To do that, I need to find the poles from: $$ 0 = Knum(s)+den(s)$$
Assume that $num(s)$ and $den(s)$ are arbitrary polynomial row vectors.
And now the question:
Is there a analytical solution to this or only numerical?
Can I sett the poles to $s_p > 0$ e.g $s_p = 0.01$ and then solve $K$? Just assume that the poles becomes positive, all of them, then solve the $K$?
Found the solution now! Only for first order system!!!
Assume that we have a discrete and time continuous feedback system and we look at the poles:
$$0=Knum(s) + den(s)$$
$$0=Knum(z) + den(z)$$
For discrete system, we say $z_1 = 1, z_2 = -1$ and then we solve $K$.
For time continuous system. We say $s = 0$ and then we solve $K$.
Example:
Assume that we have a discrete system:
$$H(z) = \frac{0.09}{z-0.81}$$
And we want to find what gain $K$ gives a unstable system.
We pick up the formula:
$$0=Knum(z) + den(z) = K0.09 + z - 0.81$$
Then we insert $z_1 = 1$
$$\frac{-1+0.81}{0.09} = K = -2.11$$
And then $z_1 = -1$
$$\frac{1+0.81}{0.09} = K = 20.11$$
So my conclusion to this is that we cannot have a negative gain in our controller. The gain need to be positive. And our maximum gain is $20.11$.
Repeat this with a time continuous system transfer function.