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?


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


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.

enter image description here

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.