Transcript Inline Integration

Analysis and Simulation of
Variable Structure Systems
Using Bond Graphs
and Inline Integration
François E. Cellier
ETH Zürich
and
Matthias Krebs
DaimlerChrysler AG
Outline
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Motivation
Problem
Mathematical Switch Model
Diodes With Free Causality
Inline Integration
Diodes With Fixed Causality
Numerical Accuracy
Conclusions
Motivation
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Many engineering models contain discontinuities that
can be modeled by switching events.
• Although switches can be represented easily in a bond
graph, models containing ideal switches are
notoriously difficult to simulate.
• Whereas non-ideal switches resolve the conceptual
difficulties of dealing with switching events, the
resulting simulation results are sometimes highly
inaccurate.
Summary
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A new simulation algorithm was developed, based on
inline integration, that is capable of simulating
switching events reliably and accurately.
• The algorithm was implemented in Dymola/Modelica,
but could also be implemented easily in other bond
graph modeling environments.
• The paper describes the conceptual problem of
dealing with ideal switching events; it explains, how
the new approach circumvents these difficulties; and it
demonstrates the solution by means of some simple
examples.
A First Example
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Let us start with the following very simple electronic circuit.
It can be modeled and simulated easily in Dymola using the
standard Modelica library.
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The simulation does not cause any particular
difficulties.
A Second Example
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Let us now model and simulate a second, slightly modified,
electronic circuit using the standard Modelica library. The diode
is modeled as an ideal element.
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This time around, the
simulation dies with a
division by zero.
A non-ideal diode (with
a leakage resistance in
its closed position and a
leakage conductance in
its open position) solves
the problem apparently.
The default diode model in the standard
Modelica library is a leaky diode.
Bond Graph Representation
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Let us represent the two circuits by corresponding
bond graphs.
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On the former circuit, the causality on the diode is
free, whereas it is fixed on the latter.
Mathematical Switch Model
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The switch can be modeled mathematically by the
following equation:
sw =
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switch is open
{ 0,1, switch
is closed
The two possible solutions are:
and
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The left solution produces a division by zero in the
open position, whereas the right solution produces a
division by zero in the closed position.
Mathematical Switch Model II
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We conclude that a fixed computational causality of
the diode leads invariably to a division by zero in one
of the two switch positions.
• Hence any model that makes use of ideal switches
must prevent the causality of a switching element to
ever be fixed.
• We still need to analyze what happens when the
causality of the diode is allowed to change.
Diodes With Free Causality
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Let us analyze the first circuit a bit further. The
circuit can be represented by seven equations:
}
Algebraic loop in three
equations and three
unknowns (uR1, iD, uD).
Diodes With Free Causality II
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We use the tearing approach for converting the
algebraic loop to its causal form. To this end, we
solve the switch equation for iD, while substituting
uD and uR1 from the other two equations:

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This set of equations works in both switch positions.
There is no division by zero.
Free Diodes Avoid Exception
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Krebs showed in his MS thesis that:
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As long as every independent switch can change its
computational causality freely and independently of all
other independent switches, there is no division by zero
in either switch position.
 The causality of a switch can be constrained by either a
capacitor or an inductor, in which case we are
confronted with a variable index problem.
 The causality of a switch can alternatively be
constrained by a source, in which case we are
confronted with a non-physical condition, such as a
short-circuit or a floating circuit.
Inline Integration
“Inlining” an integrator means appending the solver
equation for that integrator to the set of model
equations, thereby converting the corresponding
differential equation to a difference equation.
• Individual integrators can be inlined separately.
• Different inlined integrators can use different
integration algorithms.
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Inlining The Inductor
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Let us inline the inductor that constrains the causality of the
diode in the second circuit by appending a backward Euler
formula.
uL = L · diL /dt
iL(t) = iL(t-h) + h · diL(t) /dt

iL(t) = iL(t-h) + (h/L) · uL(t)
The Causality of Inline Integration
iL(t) = iL(t-h) + (h/L) · uL(t)
Known, since computed
in the past.
This constitutes an algebraic relation between i and u.
This now looks like a resistor. Hence the causality is
now free.
When using the inline integration algorithm, the causalities of
the so integrated storage elements are being freed up.
Consequently, the division by zero problem disappears.
Diodes With Fixed Causality
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Let us analyze the second circuit a bit further. The
circuit can be represented by eight equations:
Algebraic loop in four
equations and four
unknowns (uL, iD, diD,
uD).
Diodes With Fixed Causality II
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Let us solve the algebraic loop by tearing:
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
As in the previous example,
this set of equations works in
both switch positions. There
is no division by zero.
Implementation in Dymola
Numerical Accuracy
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Let us compare the inlined inductor approach with
the leaky diode approach:
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The leaky diode uses a leakage resistance of 10-5Ω,
and a leakage conductance of 10-5 mho.
Numerical Accuracy II
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Inlined inductor
Although the leakage resistance
and leakage conductance values
are very small, the results of the
two simulations are quite
different.
• The
discrepancy disappears
leaky diode
when the leakage resistance is
reduced to 10-10Ω, and when the
leakage conductance is reduced
to 10-10 mho.
• Using fudge values in models is often dangerous, as the user
is not aware of their influence, and often not even of their
existence.
Conclusions
• It has been shown that inline integration can be used to resolve
the division-by-zero problem occurring in the simulation of
causality-constrained switches.
• The use of inline integration avoids the introduction of fudge
parameters that may falsify the simulation results.
• Inline integration can be applied to each integrator
individually, i.e., there is no need to inline all integrators, only
those that constrain switch equations.
• The bond graph technique enables the modeler to analyze the
causality constraints easily and quickly, thereby providing him
or her with guidance as to which integrators need to be inlined.
Conclusions II
• The new version of BondLib, Dymola’s bond graph library,
contains inlined inductor and capacitor models.
• Inlining has to be done manually at the current time, although
it would principally be possible to implement an algorithm that
finds automatically the integrators that need to be inlined.
• The current implementation of inlined integrators works with a
fixed step size, i.e., doesn’t tie into Dymola’s step-size control
algorithm.
• This is a pity, as another fudge parameter is being introduced
in this fashion, namely the fixed step size, h.
• Yet, this is not a conceptual problem, only an issue with the
current implementation.
The End
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