# State space equation from differential equation

I have very general system. I don't know whether it is electrical or mechanical or whatever. This system can be modeled by the following differential equation

$$\dot q = \frac{Tf_1-f_2}{T+1}$$

where:

• $\dot q$ would be equivalent to current in an electrical system

• $f_1$ and $f_2$ would be equivalent to current sources

• $T$ is a given constant

From what I see, if $f_1$ and $f_2$ are constant, there is no way for $\dot q$ to change. How is it possible that this system has a state space representation? And how do I get this state space representation?

• That depends on how you define your state. The easiest would just be to say your state is $\dot{q}$. Are you interested in $q$? If so, you may want to choose that as a state variable.
– Peter K.
Nov 21, 2015 at 15:27
• You have to tell us what you consider the system's input, possibly its output, and what should be considered its state(s). The way I understand your system, it doesn't have any memory, so it becomes pointless to talk about states. Nov 22, 2015 at 11:10
• @MattL. Exactly. That's what I'm saying. I don't see any states. For the same reasons. Yet our teacher claims it is possible to get State Space representation from that equation above. Only input I can think of are $f_1$ and $f_2$. I probably lack of some understanding about this. It would help to see an example of the State Space reprepresentation of this system. Nov 23, 2015 at 12:33

Assuming that the inputs are $f_1$ and $f_2$,
$$\dot q = 0 \cdot q + \begin{bmatrix} \quad\left(\frac{T}{T+1}\right)\\ - \left(\frac{1}{T+1}\right)\end{bmatrix}^\top \begin{bmatrix} f_1\\ f_2\end{bmatrix}$$