The general formulation for a multizone state-space solution

For every node n, the heat transfer equation is

$C_n\frac{dT_n(t)}{dt} = Q_n(t) + \sum_{m=1}^M{ U_{nm} \left[ T_m(t)-T_n(t) \right] }$

For a state space representation form a continuous, linear, time-invariant system is

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

And from [1], the solution is:

$y_t= \sum_{j=0}^n(S_j u_{t-j\delta})-\sum_{j=1}^n(e_j y_{t-j\delta})$

where

\begin{align} S_0 & = CR_0 \Gamma_2+D \\ S_j & = C \left[ R_{j-1} ( \Gamma_1 -\Gamma_2 ) + R_j \Gamma_2 \right] + e_j D, for 1 \le j \le n-1 \\ S_n & = CR_{n-1} (\Gamma_1 -\Gamma_2 ) + e_n D \end{align}

## Caveat

Prior to Hassayampa (Version 3.0) Multizone offices are created by connecting several single-zone objects with a multizone object linking them together with a user-defined UA. Unfortunately, this UA is not integrated into the thermal solution but used to add or remove heat from the zones based on the temperature difference. This method works fine as long as the temperature difference between the two zones remains relatively constant. The multizone object limits the time step to avoid problem with excessive changes in the heat transfer rate. Nevertheless, this method is only an approximation and can introduce some error when large temperature fluctuations occur.

At this time, the multizone offices only support unitary HVAC. There is no support for central/VAV HVAC systems.

1. Seem 1989