# How to preserve or recover states meanings in canonical state-space realizations

Let assume that one would like to translate an input-output model, into a state-space one. It is well-known that the process of realization addresses this aspect. Let assume to have the following input-output ARX model

$$y(k) = \sum_{i=1}^{N_a}a_iy(k-i) + \sum_{i=1}^{N_b}b_iu(k-i)$$

in which, at the discrete time step $k$, $y$ represents the output of the model, and $u$ represents the input, while $a_i$ and $b_i$ are the model's parameters. $N_a$ and $N_b$ can be chosen independently.

This model can be translated into a suitable quartet of matrices $A,B,C,D$ such to obtain the state-space representation $$\boldsymbol{x}(k+1) = A \boldsymbol{x}(k) + B\boldsymbol{u}(k) \\ \boldsymbol{y}(k) = C \boldsymbol{x}(k) + D \boldsymbol{u}(k)$$ that will have the same transfer-function (and the same dynamics) of the aforementione ARX model.

As it is known, the realization process in not unique, i.e. there are infinite possible realizations in state-space form of the same input-output model. Among the different possibilities, canonical forms are quite adopted (controllability and observability) for different reasons.

However, one of the main drawback with these canonical forms, is that one will loose the relationship between the original input and output quantities (i.e., $y(k-i)$ and $u(k-i)$) and the states resulting from the canonical realization process of the ARX model. This usually leads to a states array $\boldsymbol{x}$ that does not necessarily reflect physical quantities.

One could always implement his/her own version of a realization scheme in order to ensure the maintaining of physical meaning of the resulting states.

My question here is: Is there a way to recover the states meaning and map them onto the original physical quantities of inputs and outputs of the ARX model when canonical forms are used for state-space realizations?

No this is not possible. And actually, it is a good practice to not to rely on this type of inferences. Well before anything they tend to get in the way and certain operations are not structure preserving such as balancing and model reduction. Unit wise they are not that useful either. For example, a noise on the measurement signal shouldn't have any units since it operates on the signal itself but not necessarily the original physical phenomenon. Hence an input from disturbance $w_i$ to output $y_j$ which happen to have Kelvins as a unit doesn't really make sense.