How do I find a closed loop PID controller from a transfer function?

Question

I have the transfer function below:

$$T_c(s) = \frac{s+k_i}{m_0s^3+s+k_i}$$

It is assumed that $k_p=1$ and $k_d=0$. I am given that $x_l(t)=[x_l(t),y_l(t)]^T$ is the trajectory and the tracking error is $e(t)=x(t)-(x_l(t)-10)$. I found the roots in terms of $k_i$, but I can't figure out what to do next or what other information I need.

Answer 1

The PID transfer function is $H_{pid} = p+I/s + sd = 1 + I/s$. Assuming the loop is closed under unity gain and the loop TF is $L = H_{pid}T_c$ then the (SISO) closed loop is always $L/(1+L)$. Thus

$$ \frac{L}{1+L} = \frac{(s+I)(s+k_i)}{m_0s^4+s^2+k_is+(s+I)(s+k_i)} = \frac{s^2+Is+k_iI}{m_0s^4+2s^2+(I+k_i)s+Ik_i}. $$

I am not sure how the other information is relevant.