Tutorial 2, Part 2:
Calibration of a damped oscillator
Damped oscillator
• Mass m, damping c, stiffness k
and initial kinetic energy
• Equation of motion:
• Undamped eigen-frequency:
• Lehr's damping ratio D
• Damped eigen-frequency
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Tutorial: Calibration
Damped oscillator
• Time-dependent displacement
function
• Identification of the input parameters
m, k, D and Ekin to optimally fit a
reference displacement function
• Objective function is the sum of squared errors between the
reference and the calculated displacement function values
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Tutorial: Calibration
Task description
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Parametrization of the input parameters
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Parametrization of the displacement vs. time functions as signals
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Extraction of objective from output and reference signals
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Definition and evaluation of a sensitivity analysis
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Check for dependent parameters
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Identification of the damper parameters using global and local
optimization strategies
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Identification using noisy reference function
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Estimation of model representation quality
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Tutorial: Calibration
Project manager
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Open the project manager
Define project name
Create a new project directory
Copy optiSLang Examples/Oscillator into project directory
Tutorial: Calibration
Parameterization of the problem
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Start a new parametrize workflow
Define workflow name
Create a new problem specification
Enter problem file name
Tutorial: Calibration
Parameterization of the inputs
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1. Click “open file” icon in parametrize editor
2. Browse for the SLang input file oscillator.s
3. Choose file type as INPUT
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Tutorial: Calibration
3.
Parameterization of the inputs
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1.
1. Mark value of m in the input file
2. Define m as input parameter
3. Define parameter name
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Tutorial: Calibration
3.
Parameterization of the inputs
1.
1. Open parameter in parameter tree
2. Enter lower and upper bounds
(0.1 … 5.0)
3. Repeat procedure for k, D, Ekin
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Tutorial: Calibration
2.
Parameterization of the inputs
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Tutorial: Calibration
Parameterization of the inputs
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Tutorial: Calibration
Parameterization of the output signal
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2.
1. Click “open file” icon in parametrize editor
2. Browse for the SLang output file oscillator_signal.txt
3. Choose file type as OUTPUT
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Tutorial: Calibration
3.
Parameterization of the output signal
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1. Mark output object string in editor
2. Add string to repeated block marker set
3. Select “set super marker”, set start, increment,
end values and “single steps”
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Tutorial: Calibration
Parameterization of the output signal
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Mark first value of time column
Add string to a vector
Select marker
Define name of vector
Tutorial: Calibration
Parameterization of the output signal
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1. Repeat for the displacement column
2. Block marker and vectors appear in parameter tree
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Tutorial: Calibration
Definition of the output signal object
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Tutorial: Calibration
Create a new signal object
Define signal object disp_time and activate
Choose abscissa reference and define label
Add disp as signal channel
Define channel name, label and activate
Parameterization of the reference signal
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2.
1. Click “open file” icon in parametrize editor
2. Browse for the SLang output file oscillator_reference.txt
3. Choose file type as OUTPUT
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Tutorial: Calibration
3.
Parameterization of the reference signal
2.
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3.
1. Mark output object string in editor
2. Add string to repeated block marker set
3. Select “set super marker”, set start, increment,
end values and “single steps”
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Tutorial: Calibration
Parameterization of the reference signal
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Mark first value of time column
Add string to a vector
Select marker
Define name of vector
Tutorial: Calibration
Parameterization of the reference signal
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Mark first value of displacement column
Add string to a vector
Select marker
Define name of vector
Tutorial: Calibration
Parameterization of the reference signal
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Mark first value of second displacement column (noisy reference)
Add string to a vector
Select marker
Define name of vector
Tutorial: Calibration
Parameterization of the reference signal
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1. Open the reference vectors
in the parameter tree
2. Set vectors as active and constant
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Tutorial: Calibration
Definition of the reference signal object
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1. Create a new signal object
2. Define signal object disp_time_ref and activate
and set as constant
3. Choose abscissa reference and define label
4. Add disp_ref and disp_ref_noise as channels
5. Define channel name, label and activate
Tutorial: Calibration
Definition of difference from reference
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1. Create a signal function
2. Add signal function SIG_DIFF_EUCLID as difference between
solver output and reference channel
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Tutorial: Calibration
Definition of maximum values in time slots
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1. Create new signal functions
2. Add signal functions SIG_MAX_Y and SIG_MAX_Y_SLOT to get
maximum displacement values after a certain time (0, 2, 4, 6, 8s)
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Tutorial: Calibration
Definition of difference from noisy reference
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1. Create new signal functions
2. Add signal function SIG_DIFF_EUCLID as difference between
solver output and noisy reference channel
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Tutorial: Calibration
Definition of objective functions
2a.
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2b.
1. Create new objective function
2. Define objective as difference between solver output and
reference channel (second objective using noisy reference)
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Tutorial: Calibration
Parameterization of the problem
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1. Close parametrization editor
2. Check overview for inputs
3. Check overview for outputs
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Tutorial: Calibration
Parameterization of the problem
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1. Check overview for signals
2. Check overview for objectives
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Tutorial: Calibration
Design Of Experiments (DOE)
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1. Start a new DOE workflow
2. Define workflow name and workflow identifier
3. Enter problem file name
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Tutorial: Calibration
Design Of Experiments (DOE)
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1. Enter solver call (slang –b oscillator.s)
2. Start DOE evaluation with 100 LHS samples
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Tutorial: Calibration
Design Of Experiments (DOE)
100 samples
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Coefficient of Determination of quadratic approximation of
rmse_all_diff is very low (38% CoD)
Single values as maximum in time slot can be approximated
much better (96% - 99% CoD)
Tutorial: Calibration
Meta-Model of Optimal Prognosis (MOP)
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Start a new MOP workflow
Define workflow name
Define workflow identifier
Choose DOE result file
Tutorial: Calibration
Meta-Model of Optimal Prognosis (MOP)
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1. CoP settings (sample splitting or cross validation)
2. Investigated approximation models
3. Set DCoP (accepted reduction in prediction quality to simplify
model) to 0.01
4. Filter settings
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Tutorial: Calibration
Meta-Model of Optimal Prognosis (MOP)
100 samples
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Coefficient of Prognosis of rmse_all_diff is very low (45% CoP) and
only m and k are found to be significant
CoP of maximum values in time slot are much better (95% - 99%
CoD) and all inputs are indicated to be significant
Tutorial: Calibration
Meta-Model of Optimal Prognosis (MOP)
100 samples
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500 samples
2000 samples
Coefficient of Prognosis of rmse_all_diff increases if number of
samples is increased (from 45% to 84%) and additionally Ekin
becomes significant
 Sensitivity study of objective function itself may require many
samples due to a certain complexity
 Analysis of single values may be more efficient
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Tutorial: Calibration
Meta-Model of Optimal Prognosis (MOP)
100 samples
Full
m
k
19% 44%
500 samples
D
Ekin
Full
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72%
m
k
Ekin
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21%
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56%
8%
43%
RMSE
45%
Max0
99%
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42%
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57%
99%
Max2
97%
7%
41%
9%
47%
99%
10% 45%
Max4
97%
15% 42% 18% 29%
98%
15% 44% 16% 30%
Max6
98%
23% 36% 23% 23%
99%
20% 41% 22% 24%
Max8
95%
23% 28% 35% 16%
96%
19% 36% 26% 20%
• All inputs are significant for
at least some of the output
values
 Identification of all input
parameters is generally
possible
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Tutorial: Calibration
27% 53%
D
-
46%
Evolutionary algorithm (global search)
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Start a new NOA workflow
Define workflow name and workflow identifier
Enter problem file name
Choose optimization algorithm (EA with global search as default)
Enter solver call (slang –b oscillator.s) and start workflow
Tutorial: Calibration
Evolutionary algorithm (global search)
1.
1. Choose start population size
2. Keep defaults for Selection,
Crossover and Mutation
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Tutorial: Calibration
Evolutionary algorithm (global search)
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Global optimization converges to small
difference between output and reference
Tutorial: Calibration
Dependent parameters
Run 1: RMSE=0.183
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Run 2: RMSE=0.434
Different optimization runs lead
to different parameter sets with
similar differences
Tutorial: Calibration
Dependent parameters
Reason for non-unique solution:
• The parameters Ekin and m as well as k and m appear only
pair-wisely in the displacement function
 Only the ratio between Ekin and m as well as k and m can be
identified
 We keep the value of m as constant
General procedure:
• Check designs from DOE with almost equal objective values
• Or perform multiple global optimization runs
• Sensitivity indices quantify the global influence of each input,
But: the dependency between input parameters with respect to
the minimum objective values can not be identified
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Tutorial: Calibration
Modify the parametrization
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Start a new parametrize workflow
Define workflow name
Create a copy and modify it
Enter original problem file name
Enter new problem file name
Tutorial: Calibration
Modify the parametrization
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Open the parameter m in the parameter treee
Modify reference value
Set parameter as constant
Close parametrization editor and check inputs
Tutorial: Calibration
Evolutionary algorithm (global search)
Run 1: RMSE=1.587
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Run 2: RMSE=0.287
Different optimization runs
lead to similar parameter sets
with similar differences
 No parameter dependencies
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Tutorial: Calibration
Run 3: RMSE=0.769
Gradient-based optimization
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Start a new Gradient-based workflow
Define workflow name and workflow identifier
Enter problem file name
Choose optimization method
Enter solver call (slang –b oscillator.s)
Start gradient workflow
Tutorial: Calibration
Gradient-based optimization
1.
1. Decrease size of differentiation
interval
2. Choose best design from
EA optimization as start value
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Tutorial: Calibration
2.
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Gradient-based optimization
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Local gradient-based optimization
gives exact reference values for inputs
Fitting is perfect (almost zero rmse)
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Tutorial: Calibration
Identification using noisy reference
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Measurements are more or less precise
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Reference displacement function is disturbed by Gaussian noise
with zero mean and standard deviation of 0.1 m
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Second objective is used for parameter identification
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Again global + local optimization with reduced input parameter
set k, D and Ekin
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Tutorial: Calibration
Identification using noisy reference
Evolutionary Algorithm
(global search)
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Tutorial: Calibration
Identification using noisy reference
Gradient based
(local search)
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Measurements errors may reduce the
identification quality
The accuracy of the identified parameters
depends on the number of measurements
and the sensitivity of the parameter
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Tutorial: Calibration