Load Matching, Grid Interaction, and Advanced Control - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 329

(6.43)
The output variables are selected depending on the needs of each case. For
instance, in this case, the output variable can be the indoor temperature.
Therefore
(6.44)
Then, the four matrices for this state-space representation are
(6.45)
With a state-space representation, the state of the system and the outputs of
interest can be readily found. As presented in Åström and Murray (2009),
the state at a time step k is given by
(6.46)
and the system output is
(6.47)
To recap, having the four matrices, the system output can be easily found.
Likewise, after some algebraic manipulation, it is also possible to find the
required input (e.g., heating or cooling power) to obtain a certain output.
Figure 6.12 shows some of the embodiments that linear systems can take.
In spite of technical subtleties, these representations are largely equivalent.
None of them is intrinsically better than the others, although some provide
an advantage for a given purpose in a given context. An RC circuit may be
more appropriate to provide physical insight; a thermal capacitance is more
readily associated with the properties of building materials. A state-space