7.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 7/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
The behavior of a linear system is not only determined by the denominator of the system transfer function but also by the numerator. In the nonlinear case, the concept of the transfer function numerator is replaced by the concept of zero dynamics. Having a physical interpretation of zero dynamics is helpful in designing, and redesigning a dynamical system. For this reason, a bond graph approach to system inversion has been developed. This procedure is illustrated using the simple mass-spring-damper system bond graph which, although linear, exemplifies the key ideas. In particular, system inversion is performed by reversing the causality on the source-sensor component; thus the role of input and output is reversed. In this particular case, reversing the causality of the SS component implies that the causality of the I component must also be reversed as only one component can impose flow onto the 1-junction. Thus only one component, the C component, remains in integral causality. Thus the zero dynamics are first order which, in this simple case, corresponds to the fact that the transfer function of the original system has a first order numerator. Following the causal strokes, the remaining state
q
{\displaystyle q}
can be written as in terms of the input
v
{\displaystyle v}
of the inverse system
q
˙
=
v
{\displaystyle {\dot {q}}=v}
p
{\displaystyle p}
is no longer a state, but can be written as:
p
=
m
v
{\displaystyle p=mv}
the output
F
{\displaystyle F}
of the inverse system is:
F
=
k
p
+
d
v
+
p
˙
=
k
p
+
d
v
+
m
v
˙
{\displaystyle F=kp+dv+{\dot {p}}=kp+dv+m{\dot {v}}}
This approach relies on the system input and output existing on the same SS component - the input and output are said to be colocated. If this is not so, the concept of bicausality must be used.
== Bicausality == Although (standard) causality provides a powerful tool to investigate the inverse of a dynamical system described by a bond graph, it is restricted to co-located source-sensor pairs where the input and output reside on a single SS component. To remove this restriction, the concept of bicausality was introduced and applied to various inversion problems. Bicausality has also been used in the context of fault detection, analysis of dynamical system structural properties, control system design and nonlinear analysis.
=== Source sensor (SS) components ===
The SS (source-sensor) component has two possible causal configurations corresponding to Se/Df (effort source, flow sensor) and De/Sf (effort sensor, flow source). As illustrated in the figure, the SS component has two bicausal configurations corresponding to Se/Sf (effort source, flow source) and De/Df (effort sensor, flow sensor). This is represented graphically using causal half-strokes where the half-stroke on the harpoon side of the bond corresponds to flow and the half-stroke on the other side of the bond corresponds to effort. The rules for the bicausality of junctions are the same as those for causality: only one bond imposes an effort onto a 0-junction and only one bond imposes a flow onto a 1-junction. In the context of inversion, the bonds connected to I, C and components cannot be bicausal. These ideas are illustrated using an example with non-collocated source and sensor.
=== Example: inversion of non-collocated source-sensor system ===
The example is extended by adding a zero junction and a further SS component. In the mechanical case, this corresponds to inserting a force sensor and velocity source between the spring and the ground; in the electrical case it corresponds to adding a current source and voltage sensor in parallel to the capacitor. The example is simplified by setting the flow of the added flow source to zero as indicated on the bond graph. As in the collocated example, the mechanical system is considered and system input is taken to be the force
F
{\displaystyle F}
acting on the mass M. However, the output is taken to be the force
F
s
{\displaystyle F_{s}}
of the spring which is not collocated with the applied force
F
{\displaystyle F}
.
The corresponding state-space system is the same except that the output matrix C is given by:
C
=
(
k
0
)
{\displaystyle C={\begin{pmatrix}k&0\end{pmatrix}}}
reflecting the change of the sensor. The transfer function relating input
F
{\displaystyle F}
to output
F
s
{\displaystyle F_{s}}
becomes:
k
(
k
+
d
s
+
m
s
2
)
{\displaystyle {k \over {\left(k+ds+ms^{2}\right)}}}
The denominator remains unchanged, but the numerator is different reflecting the change of sensor. The system is inverted by making the output
F
s
{\displaystyle F_{s}}
an input and the input
F
{\displaystyle F}
an output. Thus the left hand SS becomes both an effort and flow sensor and the right hand SS becomes both and effort and flow source with the corresponding causalities. As the I, C and R components must retain conventional causality, the bicausality propagates as shown. Both the I and C components are in derivative causality and thus the zero dynamics are zero order which, in this simple case, corresponds to the fact that the transfer function of the original system has a zero order numerator. As in the collocated case, the transfer function of the inverse system is the reciprocal of the transfer function of the system; again, the bond graph causality method can also be used to examine the zero dynamics of nonlinear systems.
== Deriving the bond graph of mechanical, electrical and electromechanical systems == Methods for deriving the bond graph of systems in various physical domains are explained in detail in the textbooks. A detailed derivation of a laboratory electromechanical system is given in a tutorial paper. This section has methods and worked examples for some simple systems .
=== Electromagnetic === The steps for solving an Electromagnetic problem as a bond graph are as follows: