
48 Cyber-Physical Systems: From Theory to Practice
Mathematically, it is equivalent to the following representation:
E
1
F
2
e
1
f
2
+
F
1
E
2
f
1
e
2
= 0. (2.21)
The difference lies in the fact that the positions of E
2
and F
2
are interchanged. This elementary
column operation interchanges a column in E(x) with a corresponding column in F(x). From bond
graph theory, we know that source elements and nonlinear elements have fixed causality; sources
of effort have effort-out-flow-in causality, sources of flow have flow-out-effort-in causality, and
nonlinear elements have a causality that is dependent on their constituent equations. As a result,
the Dirac structure ...