overviewhv/classes/ms/lecture.ms...overview 1. history of modelling and simulation 2. modelling and...
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Overview
1. History of Modelling and Simulation
2. Modelling and Simulation Concepts
3. Levels of Abstraction
4. Experimental Frame
5. Validation
6. Studying a mass-spring system
7. The Modelling and Simulation Process
McGill, 10 September, 2000 [email protected] CS522: Modelling and Simulation 1/38
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Modelling and simulation: past
(1950–): Numerical simulations: numerical analysis, statistical analysis,
simulation languages (CSSL, discrete-event world views).
focus: performance, accuracy
(1981–): Artificial Intelligence: model = knowledge representation
Use AI techniques in modelling, AI uses simulation (“deep” knowledge)
focus: knowledge
(1988–): Object-oriented modelling and simulation
focus: object orientation, later “agents”, non-causal modelling
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Modelling and simulation: past, present, future
(1993–): Multi-formalism, Multi-paradigm (2001 –)
1. Do it right (optimally) the first time (market pressure)
2. Complex systems: multi-formalism
3. Hybrid: continuous-discrete, hardware/software
4. Exchange (between humans/tools) and re-use (validated model)
5. User focus: do not expect user to know details
(software: glueing of components), need for tools
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Real-Worldentity
BaseModel
System S
only study behaviour inexperimental context
experimentwithin context
Model M
Simulation ResultsExperiment Observed Data
within context
simulate= virtual experiment
Model Basea-priori knowledge
validation
REALITY MODEL
GOALS
Modelling and Simulation Process
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Behaviour Morphism
Real System Abstract Model
Experiment Results Simulation Results
experiment virtual experiment
modelling/abstraction
abstraction
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Verification and Validation
Cause Effect
Input Output
System
ConceptualModel
SimulationModel
BehaviouralValidation
ConceptualModel
Validation
Verification
StructuralValidation
Popper: Falsification, Confidence
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Successfulending
Type I Errorending
Unsuccessfulending
Type II ErrorEnding
Type IIError
Type IError
simulationresults
accepted ?
simulationresults
accepted ?
credibilityof simulation results
certified?
credibilityof simulation results
certified?
actual problemhas no credible solution
actual problem has a credible solution
crediblesimulation model
?
formulated problemcontains
actual problem?
Type IIIError
FormulatedProblem
NO
YES
NO
YES NO
YES
NO
YES
NO
YES
NO
YES
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System, Base Model, Lumped Model
DBaseModel � DRealSystem
DLumpedModel � E � DRealSystem � E
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Experimental Frame Structure
System(real or model)
generator transduceracceptor
Experimental Frame
Frame InputVariables
Frame OutputVariables
� Programming Language Types
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"homomorphism"
"derived from"
general
restricted
more restricted
ModelsExperimental Frames
"applies to"
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Experimental Frame and Validity
Replicative Validity ( � : accuracy):
DLumpedModel � E � DBaseModel � E
Predictive Validity:
FLumpedModel � E � FBaseModel � E
Structural Validity (morphism�� ):
LumpedModel � E�� BaseModel � E
Simulator:
DSimulator � DLumpedModel
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Modelling (and Simulation) Choices
1. System Boundaries and Constraints: Experimental Frame (EF)
2. Level of Abstraction
3. Formalism(s)
4. Level of Accuracy
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System under study: T � l controlled liquid
is_full
is_emptyheat
off
cool
is_coldis_hot
fill emptyclosed
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System Boundaries (Experimental Frame)� Inputs: liquid flow rate, heating/cooling rate
� Outputs: observed level, temperature
� Contraints: no overflow/underflow, one phase only (no boiling)
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Abstraction: detailed (continuous) view, ALG +ODE formalism
Inputs (discontinuous � hybrid model):
Emptying, filling flow rate φ
Rate of adding/removing heat W
Parameters:
Cross-section surface of vessel A
Specific heat of liquid c
Density of liquid ρ
State variables:
Temperature T
Level of liquid l
Outputs (sensors):
is low is high is cold is hot
������������ �����������
�
dTdt
� 1l� W
cρA
� φT �
dldt
� φ
is low � � l � llow �
is high � � l � lhigh �
is cold � � T � Tcold �
is hot � � T � Thot �
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Abstraction: high-level (discrete) view, FSAformalism
level
temperaturecold T_in_between hot
full
l_in_between
empty (cold,empty)
emptyfill
emptyfill
cool
heat
cool
heat (hot,full)
(hot,empty)
(cold,full)
(cold,l_ib) (T_ib,l_ib) (hot,l_ib)
(T_ib,full)
(T_ib,empty)
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Levels of abstraction: trajectories (behaviour)
level
temperaturecold T_in_between hot
onoff
offoff
ofon
is_cold sensoris_hot sensor
full
l_in_between
empty
on off
off off
off on
is_full sensor
is_empty sensor
Continuous State TrajectoryDiscrete State Trajectory
fill
fill
heat
heat
different level of accuracy
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Levels of abstraction: behaviour morphism
detailed (technical) level
abstract (decision) level
abstraction
simulation
M_dM_t
trajectory
model
traj_t traj_d
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MODEL - re-use - exchange
formal checkingformal proof
automated test generation
simulationautomated generationof system/application
(code, possibly embedded)
documentation
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A Modelling and Simulation Exercise:the Mass-Spring system
WA
LL
RestLength [m]
WA
LL
position x [m]
Mass m [kg]
Mass m [kg]
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Knowledge Sources� A Priori Knowledge: Laws of Physics
� Goals, Intentions: Predict trajectory given Initial Conditions, “optimise”
behaviour, . . .
1. Analysis
2. Design
3. Control
� Measurement Data
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Experimental Frame� Room Temperature, Humidity, . . .
� Frictionless, Ideal Spring, . . .
Structure Characterisation
� n� 1-order polynomial will perfectly fit n data points
� Ideal Spring: Feature = maximum amplitude constant
� Spring with Damping: Feature = amplitute decreases
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Building the model from a-priori knowledge
Newton’s Law
F � Md2∆x
dt
Ideal Spring
F � � K∆x
�d2∆xdt2
� � KM
∆x
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CLASS Spring "Ideal Spring": DAEmodel :=
{
OBJ F_left: ForceTerminal,
OBJ F_right: ForceTerminal,
OBJ RestLength: LengthParameter,
OBJ SpringConstant: SCParameter,
OBJ x: LengthState,
OBJ v: SpeedState,
F_left - F_right = - SpringConstant * (x - RestLength),
DERIV([ x, [t,] ]) = v,
EF_assert( x - RestLenght < RestLength/100),
},
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From Model to SimulationBlock-diagrams
analog computers, Continuous System Modelling Program (CSMP)
� From (algebraic) equation to Block Diagram
� Higher order differential equations
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Time-slicing simulator pseudo-code
Main program:
FOREACH block IN system
IF block is integrator
initialise block’s output with initial condition (IC)
ELSE
initialise output with 0
READ system network (graph) structure
READ integrator configuration info
step_size
communication_interval
READ experiment info
end_time
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Time-slicing simulator pseudo-code (ctd.)
Simulation Kernel Loop:
WHILE (NOT End_of_simulation) DO
Update_blocks
Output time and state variables
Update_blocks:
FOREACH block IN system
given current block inputs
(get input from output of influencer)
Calculate_block_output for block
increment current_time with stepsize
End_of_simulation:
termination condition such as
current_time >= end_time
condition(state_values) == TRUE
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Calculate_block_output: ([...] means optional)
WeightedSum
W, block_number, P1, e1[, P2, e2[, P3, e3]] ; --->
output= SUM_i(Pi*ei)
Summer
+, block_number, P1, e1[, P2, e2[, P3, e3]] ; --->
output = SUM_i(sign(Pi)*ei)
(only the sign of Pi is used)
Integrator
I, block_number, IC, e1 ; --->
output= previous_output + step_size*e1
(simple fixed-step Euler integration)
Minus (Sign Inverter)
-, block_number, e1 ; --->
output= -e1.
Multiplier
X, block_number, e1, e2 ; --->
output= e1*e2.
Divider
/, block_number, e1, e2 ; --->
output= e1/e2.
Constant
K, block_number, P1 ; --->
output= P1.
Output
O, block_number, e1 ; --->
output= e1.
(As a side-effect, the (time, e1) tuple is put
on the output stream at every communication point).
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Time Slicing: Evaluation Order
Blocks need to be sorted !
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Time Slicing: Circle Test
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; Circle Test for
; CSMP-style Time Slicing Simulator
$endtime = 100;
$timestep = ?;
$comminterval = 1.5;
I, 1, 0, 3 ; x’ is integral of x’’, IC=0
I, 2, 1, 1 ; x is integral of x’, IC=1
-, 3, 2 ; -x
O, 4, 1 ; output x’
O, 5, 2 ; output x
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Experimentation
1. Model
2. Parameters (constant for each simulation run)
3. Initial Conditions
4. Input (file, interactive, real system)
5. Output (file, plot, real system)
6. Solver Configuration
7. Experiment type (simulation, optimization, parameter estimation �
model calibration)
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Results Analysis� Accuracy (numerical)
� Model Parameters
� Model Structure
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Model Calibration: Parameter Fit
x
x_measured
s i m p l e f r i c t i o n l e s s m a s s - s p r i n g s y s t e m
t ime [s ]
0 1 2 3
po
sit
ion
[m
]
0
0.1
0.2
0.3
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From Here On . . .� Virtual Experiments: simulation, optimisation, what-if, . . .
� Validation/Falsification
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Modelling and Simulation Process
Experimental Frame Definition
Structure Characterisation
Parameter Estimation
Simulation
Validation
class of parametric model candidates
parametric model
model with meaningful parameter values
simulated measurements
validated model
a priori
knowledge
modeller’s andexperimenter’s
goals
experiment observation(measurement)
data
Information Sources Activities
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ExperimentalFrame Definition
problem formulation
Modellingand
SimulationProcess
communication
decisionmaking
refinement
versionmanagement
comm ?N
Y
"release"
formulatedproblem
objectivesquestions
requirements
communicatedresults
communicatedproblem
refined requirements
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