# state space model) how to add a constant term in a state vector?

I am trying to construct a state-space model and ran into an issue regarding how to deal with a constant term. I have three equations as follows:

1. $$\pi_t =\tilde\rho_{t} \pi_{t-1}+\nu_t^{\pi}$$ - AR(1) process
2. $$\tilde\rho_t =\rho +q_{t}$$ -- $$\tilde\rho_t$$ is sum of a constant $$\rho$$ and time-varying $$q_{t}$$.
3. $$q_t =q_{t-1} +\nu_t^{q}$$ - random walk process

In my setting, $$\pi_t,\tilde\rho_t,q_t$$ are unobservables so they can be tracked by signal extractions(The Kalman filter). The observation equation is $$s_t =\pi_t +\epsilon_t$$. $$\begin{pmatrix} \begin{bmatrix} 1 &0 &0& 0\\ 0 & 1 & 0& 0 \\ 0 & -1 &1 & -1 \\ 0 & 0 &0 & 1 \end{bmatrix} \begin{bmatrix} \pi_{t} \\ q_{t} \\ \tilde\rho_t\\ \rho \end{bmatrix}= \begin{bmatrix} \tilde\rho_t &0 &0& 0\\ 0 & 1 & 0& 0 \\ 0 & 0 &0 & 0 \\ 0 & 0 &0 & 1 \end{bmatrix} \begin{bmatrix} \pi_{t-1} \\ q_{t-1} \\ \tilde\rho_{t-1}\\ \rho \end{bmatrix} + \begin{bmatrix} \nu_t^{\pi} \\ \nu_t^{q} \\ 0\\ 0 \end{bmatrix} \end{pmatrix}$$

Agents aim to learn the values of these parameters over time. My question centers on how to construct the state vector in this context. I have attempted to do so as described above, but I am uncertain about whether to include the true value of $$\rho$$ in the state vector. This is because agents are not able to infer the true $$\rho$$; even after processing the signal, they can only learn $$\tilde\rho_t = \rho+q_t$$