Is there a way to find a state space representation of a non-proper transfer function? In the case of a PID controller the transfer function is:
$\frac{K_d s^2 + K_p s + K_i}{s}$
What would be the state space representation of this transfer function?
I know Fist Companion Form method handles strictly proper transfer functions:
$\frac{1}{s^n + a_1 s^{(n-1)} + … + a_n}$
Second Companion From handles:
$\frac{s^{(n-1)} + b_1 s^{(n-2)} +… + b_n}{s^n + a_1 s^{(n-1)} + … + a_n}$
And Jordan Canonical From handles:
$\frac{b_0 s^{n} + b_1 s^{(n-1)} +… + b_n}{s^n + a_1 s^{(n-1)} + … + a_n}$
Is there a method to find the state space representation for non-proper transfer functions?:
$\frac{b_0 s^m + b_1 s^{(m-1)} + … + b_m}{s^n + a_1 s^{(n-1)} + … + a_n}$with $m>n$
If there is no general method, is there a state space model for the PID transfer function?
Best Answer
With the standard representation $$\begin{align} \dot{x} &= Ax + Bu \\ y &= Cx + Du \end{align}$$ you cannot represent a non-proper transfer function. However you can add $\dot{u}$ term to do this like $$\begin{align} \dot{x} &= Ax + Bu \\ y &= Cx + Du + E \dot{u} \end{align}$$ So to represent the PID controller, you can use $$\begin{align} \dot{x} &= K_i u \\ y &= x + K_p u + K_d \dot{u} \end{align}$$