Load Matching, Grid Interaction, and Advanced Control - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 326

exogenous input (ARX), and the autoregressive moving-average model with
exogenous input (ARMAX). These models have the following structures
(Ljung, 2010) for multiple inputs:
(6.33)
(6.34)
In which “nu” is the number of inputs and nk i is the number of time steps of
delay associated with the i th input. The difference between both models is
how they deal with noise. The ARMAX model applies a particular treatment
to the noise e ( t ).
The main problem in system identification is the determination of the
coefficients of the polynomials so that they represent as accurately as
possible the actual system. Input and output data, obtained from
measurements or simulations, can be introduced in system identification
software tools (Ljung, 2010; National Instruments, 2004) that significantly
facilitate this task.
6.4.3.4 State-Space Representation
Yet another equivalent (but more compact) depiction of linear systems is
given by the state-space representation . This type of model, commonly
applied in MIMO systems, makes use of a set of variables containing all
the relevant information that fully determines the state of the system. It is
well known that a particular solution of a differential equation requires as
many initial conditions as the order or the system (e.g., a first-order ODE
requires one initial condition; a second-order ODE requires two). Similarly,
in a state-space model there are as many state variables as needed 8) to fully
describe the system at time t = 0. Although the possibilities for selecting
state variables are literally infinite, in reality some representations provide
more insight or information on the system. For example, in RC circuits the
temperature of the nodes with capacitances can be a convenient choice for
state variables.