Transcript Continuous System Modeling

Advanced Simulation Technologies Conference
Smart Product Modeling:
Dealing with the Issues
of System Complexity
François E. Cellier, Ph.D.
Professor and Director of Undergraduate Studies
Department of Electrical & Computer Engineering
University of Arizona
April 18-22, 2004
Start of presentation
Advanced Simulation Technologies Conference
Smart Product Modeling
• Smart Product Modeling (SPM) is a term that
has been coined to denote modeling support
for the design and development of complex
systems.
• It is often used in the context of rapid
prototyping of alternate system designs.
• In SPM, models of alternate system
components become interchangeable in ways
that resemble physical system components.
April 18-22, 2004
Start of presentation
Advanced Simulation Technologies Conference
Manufacturing Life Cycle
• In car manufacturing, the development life cycle of
a new model is now two years. In the first year, the
features of the new model are designed, whereas in
the second year, the production of the new model is
being designed.
• Many of the components of a car are prefabricated,
and they are often outsourced. Your new “allAmerican” car may feature a German engine and a
Japanese transmission.
• Design alternatives must be chosen, before even a
real prototype of the new model has been produced.
April 18-22, 2004
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Advanced Simulation Technologies Conference
SPM and Life Cycle Reduction
• Complex systems can only be designed in a
short time and with limited resources, if
many design decisions can be made on the
basis of simulation models alone.
• Components are supposed to be delivered
together with simulation models describing
them.
• The object-oriented modeling of physical
systems provides a means to accomplish that
goal.
April 18-22, 2004
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Advanced Simulation Technologies Conference
Organization of Knowledge
Let us consider the example of an
electrical resistor :
Taxonomy
i
R
vb
va
Iconic
representation
u
Potentials: va, vb
Voltage: u : u = va
- vb
Detailed
description
Current: i : u = R · i
April 18-22, 2004
The object-oriented modeling
paradigm stores the four items
characterizing an object in a
single place. These are: the
taxonomy
(naming
and
organization),
the
iconic
representation (for graphical
modeling),
the
detailed
mathematical
description
(either in the form of equations
or in the form of a topology),
and the lexical information (a
verbal description).
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The Causality of the Model Equations
U0
i
R
Identical
Objects
U0 = f(t)
i = U0 / R
+
Different
Equations
-
I0
I0
I0 = f(t)
u = R· I0
 The causality of the equations must not be
R
-
predetermined. It can only be decided upon
after the analysis of the system topology.
April 18-22, 2004
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Basic Requirements of OO Modeling
• Physical objects should be representable by
mathematical graphical objects.
• The
graphical
objects
should
be
topologically connectable.
• The mathematical models should be
hierarchically describable. To this end, it
must be possible to represent networks of
coupled objects again as graphical objects.
April 18-22, 2004
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Example of a Topological Description

model Circuit1
SineVoltage U0(V=10, freqHz=2500);
Resistor R1(R=100);
Resistor R2(R=20);
Capacitor C(C=1E-6);
Inductor L(L=0.0015);
Ground Ground;
equation
connect(U0.p, R1.p);
connect(R1.n, C.p);
connect(R2.p, R1.n);
connect(U0.n, C.n);
connect(Ground.p, C.n);
connect(L.p, R1.p);
connect(L1.n, Ground.p);
connect(R2.n, L.n);
end Circuit1;
April 18-22, 2004
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Advanced Simulation Technologies Conference
Graphical Information (Annotation)
package CircuitLib
annotation (Coordsys( extent=[0, 0; 504, 364], grid=[2, 2], component=[20, 20]));
model Circuit1
annotation (Coordsys( extent=[-100, -100; 100, 100], grid=[2, 2], component=[20, 20]));
Modelica.Electrical.Analog.Sources.SineVoltage U0(V=10, freqHz=2500)
annotation (extent=[-80, -20; -40, 20], rotation=-90);
Modelica.Electrical.Analog.Basic.Resistor R1(R=100) annotation (extent=[ -40, 20; 0, 60], rotation=-90);
Modelica.Electrical.Analog.Basic.Capacitor C(C=1E-6) annotation (extent=[-40, -60; 0, -20], rotation=-90);
Modelica.Electrical.Analog.Basic.Resistor R2(R=20) annotation (extent=[0, -20; 40, 20]);
Modelica.Electrical.Analog.Basic.Inductor L(L=0.0015) annotation (extent=[40, 20; 80, 60], rotation=-90);
Modelica.Electrical.Analog.Basic.Ground Ground annotation (extent=[0, -100; 40, -60]);
equation
connect(U0.p, R1.p) annotation (points=[-60, 20; -60, 60; -20, 60], style(color=3));
connect(R1.n, C.p) annotation (points=[-20, 20; -20, -20], style(color=3));
connect(R2.p, R1.n) annotation (points=[0, 0; -20, 0; -20, 20], style(color=3));
connect(U0.n, C.n) annotation (points=[-60, -20; -60, -60; -20, -60], style(color=3));
connect(Ground.p, C.n) annotation (points=[20, -60; -20, -60], style(color=3));
connect(L.p, R1.p) annotation (points=[60, 60; -20, 60], style(color=3));
connect(L.n, Ground.p) annotation (points=[60, 20; 60, -60; 20, -60], style(color=3));
connect(R2.n, L.n) annotation (points=[40, 0; 60, 0; 60, 20], style(color=3));
end Circuit1;
end CircuitLib;
April 18-22, 2004
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Models in Modelica
• Models in Modelica consist of a description of
their model structure as well as a description of
their embedding in the model environment:
model Model name
Description of the model embedding;
equations
Description of the model structure;
end Model name;
April 18-22, 2004
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Model Structure in Modelica
• The model structure in Modelica consists either of a set of
equations, a description of the model topology, or a
combination of the two types of model structure
descriptions.
• A topological model description is usually done by
dragging and dropping model icons from graphical model
libraries into the modeling window. These models are then
graphically interconnected among each other.
• The stored textual version of the topological model
consists of a declaration of its sub-models (model
embedding), a declaration of its connections (model
structure), as well as a declaration of the graphical
description elements (Annotation).
April 18-22, 2004
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Model Topology in Modelica
Instance name
controller
model MotorDrive
PI
PI
controller;
n=100
Motor
motor;
Class name
Gearbox
gearbox(n=100);
Shaft
Jl(J=10);
Tachometer wl;
Modifier
equation
connect(controller.out, motor.inp);
connect(motor.flange , gearbox.a);
Connection
connect(gearbox.b
, Jl.a);
connect(Jl.b
, wl.a);
connect(wl.w
, controller.inp);
end MotorDrive;
Connector
wr
April 18-22, 2004
motor
Start of presentation
Jl=10
wl
Advanced Simulation Technologies Conference
Resistors in Modelica
vp
i
R
vn
u
connector Pin
Voltage v;
flow Current i;
end Pin;
model Resistor "Ideal resistor"
Pin p, n;
Voltage u;
parameter Resistance R;
equation
u = p.v - n.v;
p.i + n.i = 0;
R*p.i = u;
end Resistor;
type ElectricPotential = Real
(final quantity="ElectricPotential",
final unit="V");
type Voltage = ElectricPotential;
April 18-22, 2004
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Similarity Between Different Elements
vp
i
R
vn
model Resistor "Ideal resistor"
Pin p, n;
Voltage u;
parameter Resistance R;
equation
u = p.v - n.v;
p.i + n.i = 0;
R*p.i = u;
end Resistor;
vn
model Capacitor "Ideal capacitor"
Pin p, n;
Voltage u;
parameter Capacitance C;
equation
u = p.v - n.v;
p.i + n.i = 0;
C*der(u) = p.i;
end Capacitor;
u
vp
i
C
u
April 18-22, 2004
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Partial Models and Inheritance
vp
i
vn
u
vp
i
R
vn
model Resistor "Ideal resistor"
extends OnePort;
parameter Resistance R;
equation
R*p.i = u;
end Resistor;
vn
model Capacitor "Ideal capacitor"
extends OnePort;
parameter Capacitance C;
equation
C*der(u) = p.i;
end Capacitor;
u
vp
i
C
u
partial model OnePort
Pin p, n;
Voltage u;
equation
u = p.v - n.v;
p.i + n.i = 0;
end OnePort;
April 18-22, 2004
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Courtesy Toyota Tecno-Service
Decomposition and Abstraction
bearing1
C12=0.12
C6=0.12
demultiplex
bearing2
C8=0.12
C4=0.12
bearing4
shaftS=2e-3
shaftS1=2e-3
S
S
C5=0.12
C11=0.12
planetary1=110/50
planetary2=110/50
planetary3=120/44
April 18-22, 2004
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Heterogeneous Modeling Formalisms
k2
i
qddRef
qdRef
qRef
1
1
S
S
k1
cut joint
r3Control
r3Motor
i
axis6
tn
r3Drive1
1
qd
qdRef
Kd
q: angle
qd: angular velocity
qdd: angular acceleration
S
rel
0.03
Jmotor=J
w Sum
+1
+1
-
rate2
rate3
b(s)
340.8
a(s)
S
joint=0
spring=c
S
iRef
axis4
gear=i
fric=Rv0
0.3
sum
axis3
rate1
tacho2
b(s)
b(s)
a(s)
a(s)
tacho1
PT1
g5
q
qd
axis2
C=0.004*D/w m
Rd1=100
Rp1=200
Rd2=100
Ri=10
-
-
+
cut in
-
dif f
+
+
pow er
OpI
Ra=250 La=(250/(2*D*w m))
Rp2=50
-
Kv
axis1
Vs
Rd4=100
emf
iRef
Rd3=100
qRef
pSum
axis5
y
x
inertial
g3
g1
hall2
hall1
g2
qd
g4
w
r
q
April 18-22, 2004
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Mechanical Connectors
connector Frame
Position r0[3]
"Distance of the frame from the inertial system";
Real S[3, 3]
"Transformation matrix of the frame to the inertial system";
Velocity v[3]
"Absolute velocity of the frame";
AngularVelocity w[3]
"Absolute angular velocity of the frame";
Acceleration a[3]
"Absolute acceleration of the frame";
AngularAcceleration z[3] "Absolute angular acceleration of the frame ";
flow Force f[3]
"Force acting on the frame";
flow Torque t[3]"Torque acting on the frame";
end Frame;
Because of the sign conventions, an empty output frame (
must always be connected to a full input frame ( ).
April 18-22, 2004
)
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Mechanical Bodies I
• Mechanical bodies define the D’Alembert
Principle for the sum of acting forces and torques.
rCM
Frame
Center of
Gravity
model BodyBase "Inertia and mass properties of a rigid body";
extends Frame a;
Mass m;
Position rCM[3] "Distance from frame to center of gravity";
Inertia I[3, 3];
equation
f = m*(a + cross(z, rCM) + cross(w, cross(w, rCM)));
t = I*z + cross(w, I*w) + cross(rCM, f);
end BodyBase;
The coordinates of the frames are first converted to the center of gravity.
The D’Alembert Principle is then formulated for the center of gravity.
The resulting force f and torque t are finally transformed back to the frame by means of their relative
movement under introduction of the accompanying centripetal and Coriolis forces.
April 18-22, 2004
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Mechanical Bodies II
model Body
"Rigid body with one cut";
extends Frame_a;
parameter Position rCM[3]={0,0,0}
"Vector from frame_a to center of mass, resolved in frame_a;
parameter Mass m=0
"Mass of body [kg]";
parameter Inertia I11=0
"(1,1) element of inertia tensor";
parameter Inertia I22=0
"(2,2) element of inertia tensor";
parameter Inertia I33=0
"(3,3) element of inertia tensor";
parameter Inertia I21=0
"(2,1) element of inertia tensor";
parameter Inertia I31=0
"(3,1) element of inertia tensor";
parameter Inertia I32=0
"(3,2) element of inertia tensor";
BodyBase body;
equation
connect (frame_a, body.frame_a);
body.m = m;
body.rCM = rCM;
body.I = [I11, I21, I31; I21, I22, I32; I31, I32, I33];
end Body
April 18-22, 2004
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Mechanical Bodies III
model Body
"Rigid body with one cut";
extends Frame_a;
parameter Position rCM[3]={0,0,0}
"Vector from frame_a to center of mass, resolved in frame_a;
parameter Mass m=0
"Mass of body [kg]";
parameter Inertia I11=0
"(1,1) element of inertia tensor";
parameter Inertia I22=0
"(2,2) element of inertia tensor";
parameter Inertia I33=0
"(3,3) element of inertia tensor";
parameter Inertia I21=0
"(2,1) element of inertia tensor";
parameter Inertia I31=0
"(3,1) element of inertia tensor";
parameter Inertia I32=0
"(3,2) element of inertia tensor";
BodyBase body;
equation
connect (frame_a, body.frame_a);
body.m = m;
body.rCM = rCM;
body.I = [I11, I21, I31; I21, I22, I32; I31, I32, I33];
end Body
Information
extracted from
type declaration
April 18-22, 2004
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Advanced Simulation Technologies Conference
Mechanical Bodies IV

Coordinate
transformation
frame a  frame b
Body calculated
relative to frame a
Bodies with more than two joints have to be constructed by the modeler using
additional frame translations. Such elements are not available in the MBS library as
pre-designed modules.
April 18-22, 2004
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Mechanical Bodies V

Geometry for
the animation.
Geometry for the computation of mass and inertia matrix (not
represented graphically, since modeled by means of equations).
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An Example I
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An Example II
April 18-22, 2004
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An Example III
Cut joint
Kinematic loop
April 18-22, 2004
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An Example IV
Equations after expansion of the
matrix expressions
Elimination of trivial equations of
the type: a = b
Remaining equations after the symbolic
transformation.
April 18-22, 2004
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An Example V
April 18-22, 2004
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Multi-body Systems
• A multi-body system (MBS) consists of a combination of
mechanical parts, connected to each other to support
motion in three-dimensional space.
Hmm! Maybe this is not yet the most luxurious
model… but abstraction is everything after all.
April 18-22, 2004
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Properties of OO Modeling Environment
• The object-oriented modeling environment supports
both flexible topological and hierarchical
interconnection mechanisms.
• The environment lends itself to information hiding.
The details of the models are hidden behind iconic
model representations.
• The environment contains no domain-specific
knowledge. The entire domain-specific knowledge
is encoded in the domain libraries, such as the MBS
library.
April 18-22, 2004
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Passive Solar Space Heating
• An experimental building
with passive solar heating is
shown here from three sides.
• Solar radiation through the
walls, the windows, and the
ceiling is to be modeled.
• Losses are also being
modeled, including the losses
through the slab.
April 18-22, 2004
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Top-level Model
Iconic representation
Topological representation
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Living Room Model
Iconic representation
Topological representation
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The Interior Wall
April 18-22, 2004
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The Window
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The Solar Position
April 18-22, 2004
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The Solar-House Package
April 18-22, 2004
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Simulation Results
April 18-22, 2004
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Knowledge Abstraction
• The object-oriented modeling metaphor lends
itself to knowledge abstraction.
• Thereby, it enables the modeler to formulate
domain-specific knowledge in the language
of the domain expert.
• Object-oriented models are thus well suited
for communication of knowledge between
modelers and domain experts.
April 18-22, 2004
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The Hemodynamic System
The heart chambers and blood vessels
are containers of blood.
Each
container is a storage of mass, thus
contains a C-element.
The C-elements are partly non-linear,
and in the case of the heart chambers
even time-dependent.
The mSE-element on the left side
represents the residual volume of the
vessel.
The mSE-element on the right side
represents the thoracic pressure,
which is influenced by the breathing.
April 18-22, 2004
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The Hemodynamic System II
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The Hemodynamic System III
April 18-22, 2004
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The Heart
The heart contains the four
chambers, as well as the four major
heart valves, the pulmonary and
aorta valves at the exits of the
ventricula, and the mitral and
triscuspid valves between the atria
and the corresponding ventricula.
The sinus rhythm block programs
the contraction and relaxation of
the heart muscle.
The heart muscle flow symbolizes
the coronary blood vessels that are
responsible for supplying the heart
muscle with oxygen.
April 18-22, 2004
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The Thorax
The thorax contains the
heart and the major blood
vessels.
The table lookup function
at the bottom computes the
thoracic pressure as a
function of the breathing.
The arterial blood is
drawn in red, whereas the
venous blood is drawn in
blue.
Shown on the left are the
central nervous control
signals.
April 18-22, 2004
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Summary
• The Object-oriented modeling paradigm enables us
to organize and encapsulate knowledge about a
system.
• The methodology allows us to manage and
communicate this knowledge in natural ways, using
terminology that is familiar to the domain expert.
• Each individual component model, be it a leaf model
or a topological description of subsystems, can be
kept small and manageable.
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Summary
• In the process of compilation, the models are
merged to form a large and monolithic set of
equations.
• To obtain efficient simulation models, it may
even be necessary to merge the model
equations and the simulator equations
(integration algorithms).
• These equations are then further processed by
pruning out knowledge that is not needed for
the task at hand.
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Summary
• The resulting simulation codes are as efficient
as if not more efficient than the best spaghetti
code special-purpose simulators of the past.
• The technology is already widely used in the
automotive industries, but is also making
headway in other application domains.
• At the University of Arizona, the technology is
taught in a senior/graduate level class to
roughly 25 students per year.
April 18-22, 2004
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