Load Matching, Grid Interaction, and Advanced Control - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 321

6.4.3 Linear Models
A linear model is one satisfying the properties of superposition and
scalability (or homogeneity) (Ogata, 2002). These properties imply that
(a) the resulting effect of different inputs is the sum of their individual
effects, and (b) multiplying an input by a factor results in multiplying its
corresponding effect by the same factor. This can be expressed
mathematically with an operator H mapping inputs into outputs.
Superposition implies that
(6.14)
Scalability implies
(6.15)
These two properties significantly facilitate finding the response due to
several complex inputs acting simultaneously. When the system parameters
correspond to a mapping H that does not change over time, the system is
called linear-time invariant (LTI).
Physical systems can often be approximated with one or more ordinary
differential equations (ODEs) with constant coefficients (Ogata, 2002;
Spiegel, 1980):
(6.16)
The solutions of such an equation satisfy the conditions of linearity (i.e.,
superposition and homogeneity). This type of differential equation is found,
for example, in circuits with linear components (resistors, capacitors, and
inductors) . 6) In an RC circuit, the order of the differential equation(s)
corresponds to the number of capacitors.
6.4.3.1 Continuous-Time Transfer Functions
In an LTI system, such as the one shown in Eq. (6.16) , it is possible to
describe the relationship between an input u and an output y in a simple
manner. The transfer function between the input and the output is defined
as the Laplace transform of the output divided by the Laplace transform of
the input, when all initial conditions are equal to zero.