Can a state space model have changing state size over time?

Question

I have worked with state space models in relation to Kalman estimation. Here I have always seen state space models with fixed state size over time, i.e. the state transition matrix is square. Let us for example define: $$x_{t+1} = A_t x_t + B_t u_t\\ y_t = C_t x_t + D_t v_t$$ Then $A_t$ is $n \times n$.

I suppose I could easily allow the state to change size at time $t_\text{change}$ by allowing $A_{t_\text{change}}$ to be of size $m \times n$ where $m\neq n$. Is this "allowed" in state space models in general? And does this entail any problems in applying a Kalman filter to estimate the state $x_t$ for $t$ "across" the point $t_\text{change}$?

Answer 1

Short answer: Yes but no.

Long answer:

If a system matrix is rectangular then it means either;

• There are more number of states than the number of their derivatives which is not meaningful. If you have a system such as a jump linear system or something that couples to the initial system and extends the state space, you can still work with interconnections of square systems that are switched on/off (say, event-based) without resorting back to this esoteric method

or

• There are more differential equations than the number of states which imply that some of those states can be considered as exogenous inputs which in turn can be reconciled by taking those terms to your B matrix and leaving A matrix square again.

So in short, with square state matrix you don't have any loss of generality.