Observability of a System in State Space form

Question

as we all know, each system has infinite state space representations. My question is, whether the observability/controllability (and other common properties) are a properties of the SYSTYEM or PARTICULAR STATE SPACE representation of the system.

e.g.

I test for observability of 1 particular state space representation and does it imply that every other state space representation has the same observabiliy (is/is not observable)?

Answer 1

I think what you are looking for the invariance of controllability/observability under coordinate change. Let's look at the system

$$\begin{align} \dot{x} &= Ax + Bu \\ y &= Cx \end{align}$$

Now define $z = T^{-1} x$ for some $T$. Then the system becomes

$$\begin{align} \dot{z} &= T^{-1}ATz + T^{-1}Bu \\ y &= CTz \end{align}$$

Now we need to prove that the second system is controllable/observable if and only if the first system is controllable/observable.

Actually, it is easy to see directly from the controllability/observability matrices of both systems, since $T$ is a full-rank matrix.

Answer 2

You can test neither for observability nor for controllability given only the transfer function.

Consider the system

$\frac {d}{dt}x_1=u$

$\frac {d}{dt}x_2=1$

$y=x_1$

The transfer function is $ G (s)=\frac {1}{s} $. Given this transfer function, there is no way to find out that the state $ x_2 $ ever existed. In a (minimal) state space representation you would only have one state corresponding to $ x_1 $. It is the very concept of unobservablility being the reason for this: You can "observe" experimentally the transfer function. If you could find out about an unobservable state by properties of the transfer function...well, then it would be not unobservable, wouldn't it?

There are many algorithms out there for constructing possible state space representations given only the transfer function. You might want to Google "minimal state space representation", maybe in connection to "Matlab". Virtually all of these algorithms will return observable state space representations. You can then simply add an additional state being not connected to the input nor the output (in thes same way as $ x_2 $ in my example) and you have an algorithm to construct non-observalble and non-controllable representations.