State space representation for non-proper transfer function

Question

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?

Answer 1

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 $$\begin{align} \dot{x} &= K_i u \\ y &= x + K_p u + K_d \dot{u} \end{align}$$

Answer 2

I don't think you can just go about finding the state space system of the PID controller.

The PID controller is in a feedback loop, yes? The feedback loop plus PID (plus the plant) controller constitutes the system.

When you take into the account of the entire system, your transfer function will become $$G = \frac {CP} {1+ CP}$$ $P$ is your plant, $C$ is your PID controller And the system is proper.

Refer to pg 23 for details. http://www.lehigh.edu/~eus204/Teaching/ME433/lectures/lecture01_handout.pdf