11 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 6/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
Choose any source (SS, Se or Sf) and assign the required causality. Immediately extend the causal implications though the bond graph as far as possible using the constraints on the junction elements (0 and 1) and the TF and GY elements. Repeat step 1 for all of the source components. Choose any storage element (C or I) and assign the integral causality. Immediately extend the causal implications though the bond graph as far as possible using the constraints on the junction elements (0 and 1) and the TF and GY elements. Repeat step 3 until all C and I elements have been assigned causality. The figure shows the result of this procedure on the simple example. In particular, the I component imposes flow causality onto the 1-junction thus the one junction imposes flow causality on to the other three components: SS, C and R. This means that the C, R and SS components impose effort causality onto the one junction; thus the C component is in integral causality and the R component corresponds to the assignment
e
R
:=
R
f
{\displaystyle e_{R}:=Rf}
. Although this procedure can be accomplished manually for small systems, a computer-based approach is more generally useful. There are three possible results of this procedure.
It completes with all C and I components in integral causality and all bond causalities assigned; the resultant system is a causal bond graph and can be converted into a state-space system. All bonds have causality assigned but one or more C or I component has derivative causality. All C and I components have integral causality, but some bond do not have causality assigned. Whereas case 1 leads to ordinary differential equations (ode) in state-space form, case 2 gives rise to differential-algebraic equations (dae). Depending on the context, it may be better to reconsider the underlying physical system to avoid daes, or it may be possible to reduce the dae to an ode or the daes can be solved using an appropriate dae solver.
==== Alternative causality. ==== The sequential causal assignment procedure (SCAP) is designed to derive state-space equations, suitable for computation or control systems analysis. However, there are other forms of causal analysis designed to address other issues, including:
bicausality derivative causality alternative equation formulations, including those of Hamilton and Lagrange
=== State-space equations === A bond graph system representation contains the constitutive equations of each component, embedded in the structure of bond and junctions. The question of how to manipulate a set of equations into a form suitable for analogue computation was posed, and partially answered by Lord Kelvin. This approach underlies the conversion of a bond graph model to a state-space representation suitable for digital computation. The bond graph uses the notion of (bond graph) causality to provide a systematic and constructive way to investigate whether a state-space representation exists, and, if so, what is that representation; this causality approach is well suited to computational implementation and has an intuitive representation on the bond graph itself using the causal stroke notation. A causal bond graph can be put into state-space form if:
every bond has a causal stroke and every component has allowed causality all C and I components are in integral causality. The sequential causal assignment procedure (SCAP) provides a graphical approach to determining whether a system represented by a bond graph has a state-space representation. The bond graph is analogous to a number of different systems; for clarity, the mechanical analogue is used as an example, but exactly the same method applied to electrical and other physical domains. It is convenient to choose the system states as the integrated variables corresponding to the components in integral causality; in the case of the simple example, there are two states:
p
{\displaystyle p}
corresponding to the mass M and
q
{\displaystyle q}
corresponding to the spring. The system inputs and outputs correspond to the Se, Sf and SS components. In this case, there is one input and one output corresponding to the SS component: the effort (applied force
F
{\displaystyle F}
) and the measured flow (velocity
v
{\displaystyle v}
). Guided by the causal strokes, the state derivatives (effort applied to I components, flow applied to C components) can be written in terms of the states. Thus:
q
˙
=
f
=
p
m
p
˙
=
F
−
e
c
−
e
r
=
F
−
k
q
−
d
f
=
F
−
k
q
−
d
p
m
{\displaystyle {\begin{aligned}{\dot {q}}&=f={\frac {p}{m}}\\{\dot {p}}&=F-e_{c}-e_{r}=F-kq-df=F-kq-d{\frac {p}{m}}\end{aligned}}}
This can be written in standard linear state space for as:
x
˙
=
A
x
+
B
u
y
=
C
x
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx\end{aligned}}}
where the state
x
{\displaystyle x}
input
u
{\displaystyle u}
and output
y
{\displaystyle y}
are given by:
x
=
(
q
p
)
,
u
=
F
and
y
=
v
{\displaystyle x={\begin{pmatrix}q\\p\end{pmatrix}},\;u=F{\text{ and }}y=v}
and the matrices
A
{\displaystyle A}
,
B
{\displaystyle B}
and
C
{\displaystyle C}
are given by:
A
=
(
0
1
m
−
k
−
d
m
)
,
B
=
(
0
1
)
,
C
=
(
0
1
m
)
{\displaystyle A={\begin{pmatrix}0&{\frac {1}{m}}\\-k&-{\frac {d}{m}}\end{pmatrix}},\;B={\begin{pmatrix}0\\1\end{pmatrix}},\;C={\begin{pmatrix}0&{\frac {1}{m}}\end{pmatrix}}}
The state equation has three parameters: the mass
m
{\displaystyle m}
, the spring constant
k
{\displaystyle k}
and the damping constant
d
{\displaystyle d}
. Using the usual Laplace transform notation of control systems theory, the system transfer function relating the input
u
{\displaystyle u}
to output
y
{\displaystyle y}
is:
s
(
k
+
d
s
+
m
s
2
)
{\displaystyle {s \over {\left(k+ds+ms^{2}\right)}}}
Although it may be instructive to derive these formulae by hand, it is often convenient to make use of computer algebra to perform the same function.
=== System inversion ===