Derivative-free Parameter Optimization of Functional Mock-up Units Andersson, Christian

12 Pages·2017·2.23 MB·English

Derivative-free Parameter Optimization of Functional Mock-up Units
Andersson, Christian; Gedda, Sofia; Åkesson, Johan; Diehl, Stefan
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2012
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Citation for published version (APA):
Andersson, C., Gedda, S., Åkesson, J., & Diehl, S. (2012). Derivative-free Parameter Optimization of Functional
Mock-up Units. In [Host publication title missing] Modelica Association.
Total number of authors:
4
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LUND UNIVERSITY
PO Box 117
221 00 Lund
[phone]
Derivative-free Parameter Optimization
of Functional Mock-up Units
SofiaGeddaa,c ChristianAnderssona,c JohanÅkessonb,c StefanDiehla
aCentreforMathematicalSciences,LundUniversity,Sweden
bDepartmentofAutomaticControl,LundUniversity,Sweden
cModelonAB,Sweden
Abstract needforestimatingparameterswhichareunknownor
uncertain is put into focus. Estimating unknown pa-
Representing a physical system with a mathematical rameters in the mathematical model using optimiza-
modelrequiresknowledgenotonlyaboutthephysical tion algorithms is a commonly used approach to in-
lawsgoverningthedynamicsbutalsoabouttheparam- creasetheaccuracyofmodels. Inthispaper,wefocus
eter values of the system. The parameters can some- onparameterestimationproblemswheretheobjective
timesbemeasuredorcalculated,butsomeofthemare istominimizetheerrorbetweenthesimulatedprofiles
often difficult or impossible to obtain directly. Never ofthemathematicalmodelandmeasurementsfromthe
the less, finding accurate parameter values is crucial corresponding physical system. The objective func-
fortheaccuracyofthemathematicalmodel. tionconsidered
Estimating the parameters using optimization algo-
M
rithms which attempt to minimize the error between f(x)= ∑(ysim(t,x)−ymeas(t))2 (1)
i i
theresponsefromthemathematicalmodelandthereal
i=0
physical system is a common approach for improving
theaccuracyofthemodel. whereysim isthemodeloutputtrajectoryandymeas are
Optimization algorithms usually require informa- themeasurements. Theparameterstobeestimatedare
tion about the derivatives which may not always be x ∈ Rn, where n is the number of parameters. M is
easily available or which may be difficult to com- thenumberofmeasurementsatthetimepointsti. The
pute due to, e.g., hybrid dynamics. In such cases, optimizationproblemisthenformulatedas
derivative-free optimization algorithms offer an alter-
min f(x). (2)
nativefordesignandparameteroptimization.
x∈Rn
In this paper, we present an implementation of
subjecttothesystemdynamics,intheFMIcasegiven
derivative-free optimization algorithms for parameter
by a hybrid Ordinary Differential Equation (ODE).
estimation in the JModelica.org platform. The imple-
Additionally,theparametersmaybesubjecttobounds,
mentation allows the underlying dynamic system to
l≤x≤u.
be represented as a Functional Mock-up Unit (FMU),
This optimization problem may be solved by tran-
andthusenablesparameteroptimizationofmodelsex-
scribing the problem into a non-linear programming
ported from modeling tools compliant with the Func-
problem using either shooting methods [6] or collo-
tionalMock-upInterface(FMI).
cation methods [6]. These methods, however, both
Keywords: Derivative-freeoptimization;Parameter
use derivative information, which may be difficult or
Estimation;JModelica.org;FMI;Assimulo
expensive to compute, e.g., in the case of hybrid sys-
tems. Theideaisthentousealgorithmswhichdonot
1 Introduction depend on derivative information, such as the Nelder-
Meadsimplexmethod[7]. Inaderivative-freemethod,
Increasingly, industry rely on mathematical modeling instead of using information from the derivatives to
for evaluating and designing new machines and de- improve the solution, the objective is evaluated at a
vices. As the models grow increasingly complex, the chosen set of points which are then used to improve
the solution. How the points are chosen and which mentation is applied to two different problems where
strategyisusedtoimprovethesolutiondependsonthe thesecondisalargeindustrialexamplewhereamodel
method. Typically, computation times are longer than ofanengineiscalibrated. Finally,Section7concludes
for derivative-based methods, but on the other hand, thepaperwithasummaryandconclusions.
derivative-freemethodsofferafeasibleandrobustop-
tionwhenotheralgorithmsfail.
2 Background
In this paper, we evaluate three derivative-free op-
timization algorithms for parameter estimation avail-
2.1 TheFunctionalMock-upInterface
able in the JModelica.org platform: the Nelder-Mead
simplexmethod,thedifferentialevolutionmethodand TheFunctionalMock-upInterface[1]definesanopen
a genetic algorithm. Based on this evaluation, the standardformodelexchange. Theintentionistoallow
Nelder-Mead algorithm seems most appropriate to exchange of models between different modeling and
solve the class of parameter optimization problems simulationtools. Thestandarddescribesmodelsashy-
considered. brid ODEs with state, step and time events. A model
Themaincontributionofthepaperisanimplemen- that implements the FMI standard is called a Func-
tation of the Nelder-Mead simplex algorithm. The al- tionalMock-upUnitandisdistributedasacompressed
gorithm supports parameter bounds and parallel eval- directorycontainingasharedobjectfileorsourcecode
uationoffunctionevaluationswhereFMUmodelsare containingthemodelequations,andasetoffunctions
loadedandsimulated. fordataaccess, andanXMLfile, whichdescribesthe
We also briefly present the underlying packages model parameters and variables. The standard has re-
FMI Library (FMIL), PyFMI1 and ASSIMULO2. ceived a significant amount of attention among ven-
These packages are part of JModelica.org, but also dors since the release in 2010 and currently there are
available stand-alone, and are used for simulating the 34 environments that support or plan to support the
model response. In Figure 1, an overview of the standard.
interaction between the packages in JModelica.org
when solving a derivative-free optimization problem
2.2 Optimizationtools
isshown.
There exist many tools for optimization of complex
Functional Mock-up Unit
systems, both in the public domain and commercially
JModelica.org available. Broadly, therearethreedifferentcategories
ofoptimizationtools,althoughthescopeissometimes
Model
response overlapping. In Model integration tools the problem
DFO PyFMI ASSIMULO
ofinterfacingseveraldesigntoolsintoaasinglecom-
putation environment, where analysis, simulation and
Parameter
guess optimizationcanbeperformedisaddressed. Examples
FMIL
include ModelCenter [23], OptiY [22], modeFRON-
TIER [12], and iSIGHT [10]. Such environments are
Figure 1: Overview of the interaction between the
capable of integrating several simulation and design
packages in JModelica.org when solving a derivative-
tools into one computational chain, where the results
freeoptimizationproblem.
are optimized. The integrated tools may be hetero-
geneous in the sense that they model different phys-
The paper is outlined as follows. In Section 2, the
ical domains by means of different algorithms. Due
Functional Mock-up Interface is presented together
to this heterogeneity amongst supported tools, opti-
with an overview of optimization tools. In Section 3,
mization algorithm that does not exploit derivatives
anintroductiontotheJModelica.orgplatformisgiven
or model structure such as sparsity is commonly em-
together with the simulation package ASSIMULO as ployed. Model integration tools typically also have
wellasthePythonpackagePyFMIforinteractionwith
strongsupportformodelapproximationandvisualiza-
FMUs. Next, derivative-free optimization algorithms
tion.
areintroduced,followedbyadescriptionoftheimple-
Many modeling and simulation tools has optimiza-
mentation in JModelica.org. In Section 6, the imple-
tion add-ons, e.g., Dymola [9], gPROMS [24], Jaco-
1http://www.pyfmi.org bian [19], and OMOptim [18]. The level of support
2http://www.assimulo.org for optimization in this category differs between the
tools. Dymola,forexample,offersadd-onsforparam- tion oriented Modelica environment for op-
eter identification and design optimization [11, 20]. timization and simulation of complex dy-
gPROMS on the other hand, offers support for so- namic systems, built on well-recognized
lution of optimal control problems and has the ad- technology and supporting major plat-
ditional benefit in comparison with Modelica tools forms.”
to provide support for partial differential equations
JModelica.orgprovidescompilersfortheModelica
(PDEs). Toolsinthiscategoryusuallysupportasetof
language and the extension Optimica [25]. For sim-
derivative-basedandderivative-freeoptimizationalgo-
rithms. Optimization problems are typically formu- ulations, the Python package ASSIMULO is used for
bothsimulatingODEsandDAEs. Dynamicoptimiza-
latedbymeansofgraphicaluserinterfaces.
tion is available using direct local collocation algo-
In the third category there are numerical packages
rithms based on the DAE formulation of the model.
for dynamic optimization, often developed as part of
The user interaction with JModelica.org is based on
researchprograms. ExamplesareACADO[21],Mus-
theprogramminglanguagePython.
codII[28],andDynoPC[17]. Suchpackagesaretyp-
Included in JModelica.org are packages that can
ically focused on efficient implementation of an op-
alsobeusedstand-alone. Inthefollowingsubsections,
timization algorithm for a particular class of dynamic
systems. Also,detailedinformationaboutthemodelto thepackagesFMILibrary,PyFMIandASSIMULOare
presented.
optimize is generally required in order for such algo-
rithms to work, including accurate derivatives and in
some cases also sparsity patterns. While these pack- 3.1 FMILibrary
ages offer state of the art algorithms, they typically
FMI Library (FMIL) is a C package designed for
come with simple or no user interface. Their usage
working with FMUs and serving as support for ap-
is therefore limited due to the effort required to code
plications interfacing the FMI. The package con-
the model and optimization descriptions. A notable
tainsconvenientmethodsfordecompressingofFMUs,
example is CasADi [4], which provides an efficient
parsingXMLinformationandconnectingthebinary5.
AD kernel, interfaces to numerical optimization algo-
ThelibrarysupportsFMI1.0formodelexchangeand
rithmsandacomprehensiblePythoninterfaceforcus-
forco-simulationsandisintendedforcustomintegra-
tomdevelopmentofdynamicoptimizationalgorithms.
tion of FMI technology in applications. FMIL is also
CasADi also support import of Modelica models in
usedasabasisofthePythonpackagePyFMI.
XMLformat,see[5].
The approach presented in this paper falls into
3.2 PyFMI
the category of additions to modeling and simulation
tools. Specifically,modelsexportedfromFMIcompli- PyFMI[2]isapackageforinteractingwithFMUsus-
ant tools can be optimized. The presented algorithm ing Python, based on the FMI Library. It provides
uses Python scripting as a means to formulate opti- convenienthigh-levelfunctionsforinteractingwithan
mization problems, and in this respect it differs from, FMU, retrieving values and accessing variable infor-
e.g.,theapproachtakeninDymola. mation from the XML information. Additionally, a
low-levelmappingofthefunctionsspecifiedinthein-
terface can also be accessed. A model can be loaded
3 JModelica.org
and made available from Python using the following
JModelica.org3 [26] is a platform for modeling, sim- Pythoncode:
ulation and optimization of complex physical sys- #Import the model class
tems primarily based on the Modelica4 modeling lan- from pyfmi import FMUModel
guage. JModelica.org is a community-based open-
#Load the model into Python
sourceprojectstartedatLundUniversitywiththefol-
model = FMUModel("bouncingBall.fmu")
lowingaim:
PyFMIalsoprovidesaconnectiontothesimulation
“To offer a community-based, free, package ASSIMULO and thus enables access to state-
open-source, accessible, user and applica- of-the-art solvers such as CVode and IDA from the
Sundials suite, capable of simulating hybrid systems.
3http://www.jmodelica.org
4http://www.modelica.org 5http://www.jmodelica.org/FMILibrary
A simulation is performed by using the simulate δ Operationtype
method.
−1 insidecontraction
2
#Simulate the model using Assimulo 1 outsidecontraction
2
res = model.simulate(final_time=10)
1 reflection
2 expansion
3.3 ASSIMULO Table1: Differentδ-valueswithcorrespondingopera-
tiontypes.
ASSIMULO [3] is a Python package for solving first
or second order explicit ordinary differential equa-
tions (ODEs) or implicit ordinary differential equa-
We shall now introduce three different derivative-
tions(DAEs).
free optimization algorithms which have been imple-
ASSIMULO combines a variety of different solvers mented or interfaced in the JModelica.org platform
written in FORTRAN, C and Python via a com- [13]: the Nelder-Mead simplex method, the differen-
mon high-level interface. The state-of-the-art solvers tialevolutionmethodandageneticalgorithm.
CVode and IDA from the SUNDIALS suite [15] as
well as RADAU5 [14] are amongst the available
4.1 TheNelder-Meadsimplexmethod
solvers.
ASSIMULO is divided into two parts, namely prob- The Nelder-Mead simplex method has obtained its
lemdefinitionsandsolvers. Aproblemdefinitionmay
name from the fact that each iteration is based on
in addition to the right-hand side of the differential a simplex. A simplex in Rn is a set of n+1 ver-
equationalsocontainforinstancetheJacobianaswell ticesx ,...,x ∈Rnsuchthatthevectorsx −x ,i=
1 n+1 i 1
as event functions in order to support simulation of
2,...,n+1arelinearlyindependent, i.e. itisagener-
hybridsystems. Theideaistoseparateinformationre-
alizationofatriangletoarbitrarydimension.
latedtoaproblemfromthesolver. Forinstance,which
IneachiterationoftheNelder-Meadalgorithm, the
statesarealgebraicisinformationthatisrelatedtothe
objective is to replace the vertex with the highest cost
problem and not the solver. In Figure 2, an overview
in the n-dimensional simplex with a better point. The
isgivenshowingtheavailableproblemdefinitionsand
verticesareorderedbyincreasingvalueof f suchthat
solvers in ASSIMULO. Also shown is the connection
f(x )≤...≤ f(x ). The new point is searched for
1 n+1
betweenthedifferentproblemformulations.
alongthelinethroughthevertexwiththehighestcost,
x ,andthecentroid,
n+1
Implicit ODE Explicit
Overdetermined ODE
1 n
Problems Implicit Explicit ODE xc= ∑xi, (3)
O ODE (2nd order) n
L i=1
U
M
SI of the remaining vertices x ,...,x . This line has the
S 1 n
A Solvers ODASSL IDA CVODE HHT-alpha equation
methods
RADAU5 GLIMDA RODAS LSODAR Newmark
RADAU5
DOPRI5 x=x +δ(x −x ), δ ∈R. (4)
c c n+1
Figure 2: Connection between the different problem
The parameter δ defines the type of the operation
formulationsandthedifferentsolversavailableinAS- performedonthesimplex. Therearefourdifferentop-
SIMULO. eration types that are performed by the algorithm: re-
flection, expansion, inside contraction or outside con-
traction,resultinginthereflectionpoint,x ,theexpan-
r
4 Derivative-free Optimization sion point, x , the inside contraction point, x , or the
e ic
outside contraction point, x respectively. Table 4.1
oc
In applications where derivatives are difficult or com- displays the δ-values corresponding to these four op-
putationally expensive to obtain, there is a need for erations. Ifnoneoftheseoperationsresultsinabetter
derivative-free optimization methods. Examples in- point than x , the simplex is shrunk toward the ver-
n+1
clude very large models which also contains hybrid tex with the lowest cost, x . That is, the n points with
1
elements. the highest costs are replaced by new points obtained
from within which the individuals live. The value f(x¯) de-
termines how fit the individual x¯ is to survive in the
1
x=x + (x −x ), i=2,...,n+1. (5) environment;alowervaluemeansabetterfit. Ateach
1 i 1
2
iteration, or generation, a new population of possi-
Thisprocedureisrepeateduntilsometerminationcri- ble solutions is produced through mutation, crossover
terionisfulfilled. Thereareusuallythreedifferentter- and selection. Mutation is a mechanism for maintain-
mination criteria, one of which has to be fulfilled in inggeneticdiversitybymodifyinganexistingsolution
orderforthealgorithmtoterminate: while crossover means combining two existing solu-
tionsintoanewone.
• Convergence criterion for x – the simplex is suf-
ficientlysmallaccordingtoauser-providedtoler-
4.2.1 Thedifferentialevolutionmethod
ance.
The differential evolution method [27] works accord-
• Convergencecriterionfor f –thefunctionvalues
ingtothefollowingsteps.
at the simplex vertices are sufficiently close ac-
Initialization: An initial population of N individu-
cordingtoauser-providedtolerance.
als,orvectors,isgeneratedrandomlyinsidethefeasi-
bleregion.
• Termination criterion without convergence – the
Mutation: At each iteration, the population con-
maximum number of iterations or function eval-
sistsofN vectors, x ∈Rn,i=1,...,N. Foreachvec-
uationshasbeenreached. i
tor x, the target vector, a mutant vector, v ∈ Rn, is
i i
In Figure 3, two iterations of the algorithm are produced by adding the weighted difference between
shown,illustratinghowthesimplexchangesformand twovectorsinthecurrentpopulationtoathirdoneac-
position. cordingtothefollowingformula:
v =x +F(x −x ),
i r r r
1 2 3
Simplex search
6
wherer ,r ,r ∈{1,2,...,N}arerandomindices,dis-
1 2 3
4 tinct from each other and from i, and F ∈ [0,2] is a
constant.
2
Crossover: The mutant vector, v, is then recom-
i
bined with its corresponding target vector, x, through
0 i
a mixing of their elements, generating a trial vector,
2 ui ∈Rn. Thetrialvectorreceiveselementsofthemu-
tant vector with probability P∈[0,1] and elements of
4
thetargetvectorwithprobability1−P.
Selection: Thetrialvector,u,iscomparedwiththe
6 i
6 4 2 0 2 4 6
targetvector,x,andtheonegivingthelowestvalueof
i
thefitnessfunction, f, isselectedforthenextgenera-
Figure 3: Two simplex iterations where the solid tri- tion.
angle is the initial simplex which transforms into the The phases mutation, crossover and selection con-
dashedtriangle(δ =2)andthenthedash-dottriangle tinueuntilaterminationcriterionisfulfilled.
(δ =−1).
2
4.2.2 Geneticalgorithms
Ingeneticalgorithms[16]theindividualsareencoded
4.2 Evolutionaryalgorithms
as bit strings. There are various genetic algorithms
The differential evolution method and genetic algo- which differ from one another but the following is a
rithms belong to the class of evolutionary algorithms, generaldescription.
which consists of stochastic optimization algorithms Initialization: An initial population of size N is
inspired by the principles of biological evolution the- generatedrandomlyinsidethefeasibleregion.
ory. In such algorithms, each candidate solution, x¯∈ Selection: Ineachgeneration,aselectionprobabil-
Rn,representsanindividualandtheobjectivefunction, ity, p(x), is definedfor each individual, x ∈Rn. The
i i
f(x), or fitness function, represents the environment selection probability depends on the fitness function
value for the individual, f(x), a smaller value gives basedbasedonexecutiontimeandconvergencetothe
i
alargerprobability. Twoindividualsarethenselected optimalsolution.
randomlyaccordingtotheirselectionprobabilities.
Crossover: Crossover is performed on the two 6.1 Furutapendulum
selected individuals with a certain probability, the
TheFurutapendulumisasystemconsistingofahori-
crossover rate. A common choice for this probability
zontal arm driven by a motor which is connected to a
isaround0.7. Therearedifferentcrossovertechniques
vertical pendulum, see Figure 4. The system has two
butacommonapproachistorandomlychooseaposi-
degrees of freedom, namely the angle of the arm, φ,
tioninthebitstringsandswapallbitsbetweenthetwo
and the angle of the pendulum, θ. Additionally, there
stringsafterthatposition.
isfrictioninboththearmjointandthependulumjoint.
Mutation: Mutation is performed by flipping bits
Due to the discontinuities introduced by the friction,
(from 0 to 1 or vice versa) at random positions in the
thesystemisnotwellsuitedforderivative-basedopti-
bit strings. The probability of flipping a bit, the mu-
mizationalgorithms.
tation rate, should be much lower than the crossover
rate.
Selection, crossover and mutation is repeated until
aterminationcriterionisreached.
5 Implementation
ThealgorithmsevaluatedinSection4,havebeenmade
availableinJModelica.org. TheNelder-Meadsimplex
algorithm has been implemented and is now provided
as part of JModelica.org, while the differential evolu-
tion algorithm and a genetic algorithm has been in-
terfaced through the OpenOpt package 6. The algo-
rithmsareavailablethroughthePythonfunctionfmin
inJModelica.org.
The method fmin requires as input the objective
function together with the initial conditions as well Figure4: TheFurutapendulum.
asoptionsforspecifyingtheintendedoptimizational-
The Furuta pendulum is modeled by a Modelica
gorithm and tolerances. In Section 6.1, it is shown
model, see Figure 5. The problem at hand is to cal-
how the objective function can be defined when the
ibratetheunknownfrictioncoefficientsofthearmand
dynamicmodeliscontainedinanFMU.
pendulum, respectively, against the given measure-
In the Nelder-Mead algorithm, support for parallel
mentsusingtheNelder-Meadsimplexalgorithm. The
evaluation of the objective function, f(x), has been
objectiveisthus
implemented. In each iteration of the algorithm, the
evaluations of the n+1 vertices are distributed over
a user-supplied number of processes, as well as the M
f(x)= ∑(φsim(t,x)−φmeas(t))2+
evaluations of the reflection, expansion and contrac- i i
i=1
tionpoints. (6)
M
Forfurtherimplementationdetails,see[13]. ∑(θsim(t,x)−θmeas(t))2
i i
i=1
wherexisavectorcontainingthefrictioncoefficients
6 Examples
forthearmandthependulumrespectively.
The measurements were generated by simulation
In [13], the different derivative-free algorithms was
of the Modelica model for the Furuta pendulum and
tested and the result indicated that the Nelder-Mead
white measurement noise was added to the outputs.
algorithm is the preferred algorithm for the tested pa-
The measurements were given for a period of 40 sec-
rameterestimationproblems. Theevaluationwasdone
onds and were contained in a data file. The data was
6http://openopt.org/ loadedintoPythonbythefollowingcode:
theta_sim = res[’pendulumJoint.phi’]
t_sim = res[’time’]
# Evaluate the objective function
y_sim = N.vstack((phi_sim,theta_sim))
obj = dfo.quad_err(t_meas,y_meas,
t_sim,y_sim)
return obj
Finally, the objective is provided to the optimization
function fmin together with the initial guess and the
parameter bounds. The initial guess, i.e., the nomi-
nalvalues,wereobtainedthroughmanualtesting. The
objectreturnedbyfmincontainstheoptimizedparam-
eterstogetherwithstatistics,suchasthenumberofit-
erationsperformed:
Figure5: AModelicamodelfortheFurutapendulum.
# Specify initial conditions (scaled)
x0 = N.array([0.012,0.002])*1e3
from scipy.io import loadmat # Lower and Upper bounds
import numpy as N lb = N.zeros(2)
ub = x0 + 10
# Load measurement data from file
data = loadmat(’FurutaData’) # Solve using the Nelder-Mead algorithm
res = dfo.fmin(furuta_dfo_cost,
# Extract data series xstart=x0,lb=lb,ub=ub,
t_meas = data[’time’][:,0] x_tol=1e-3,f_tol=1e-2)
phi_meas = data[’phi’][:,0]
theta_meas = data[’theta’][:,0] # Optimal point rescaled
[armFriction_opt,pendFriction_opt] =
y_meas = N.vstack((phi_meas,theta_meas)) res[0]/1e3
TheobjectivefunctionisdefinedasaPythonfunction The optimized parameter values were found to be
wheretheFMU,generatedbyDymola, fortheFuruta 0.010 for the arm friction coefficient and 0.0010 for
pendulumisloadedandsimulatedforgivenparameter thependulumfrictioncoefficient. Theresultisvisual-
values. ized in Figure 6, where it can be seen that the model
responseissignificantlymoreaccurateusingtheopti-
from pyfmi import FMUModel
from pyjmi.optimization import dfo mized parameters as compared to the response given
from the nominal parameters. In Figure 7, the error
# Define the objective function isshownbetweenthemeasurementsandthesimulated
def furuta_dfo_cost(x):
response using both the nominal parameters and the
#Scale down
optimizedparameters.
armFriction = x[0]/1e3
pendFriction = x[1]/1e3
6.2 DieselEngine
# Load the FMU Model
model = FMUModel(’Furuta.fmu’)
In this example, parameters in a model of an exhaust
# Set new parameter values gas pipe in a diesel engine is calibrated against mea-
model.set(’armFriction’, surements. ThemodelwasdevelopedinDymolausing
armFriction)
the Engine Dynamics Library and models a 13 liters
model.set(’pendulumFriction’,
Volvotruckengine[8]. Theenergyoftheexhaustgas
pendFriction)
afterthecombustionisconvertedtotorque,beforere-
# Simulate the model response leasing the gas to the purification process. In Figure
res = model.simulate(final_time=40)
8, an overview of the model is shown. The energy is
converted into torque by two turbines, shown as two
# Load simulation result
phi_sim = res[’armJoint.phi’] trapezoids, where the first drives a compressor at the
theta [rad] theta
6 101
5 MSiemausluarteiomne nnotsminal parameters 100 MMeeaassuurreemmeennttss -- NOopmtiminaall
4 Simulation optimal parameters d]10-1
32 error [ra111000---432
1 10-5
00 5 10 15 20 25 30 35 40 10-60 5 10 15 p2h0i 25 30 35 40
0.1 phi [rad] 100
Measurements - Nominal
0.0 MSiemausluarteiomne nnotsminal parameters 10-1 Measurements - Optimal
0.1
0.2 Simulation optimal parameters ad]10-2
00..43 error [r10-3
0.5
10-4
0.6
0.7 10-5
0.8 0 5 10 15 20 25 30 35 40
0 5 10 15 20 25 30 35 40 t [s]
t [s]
Figure 7: Error between the measurements and
Figure 6: Simulation profiles corresponding to the
the simulated profiles using the nominal parame-
optimized parameters (dashed-dotted), profiles result-
ters (dashed) and the optimized parameters (dashed-
ing from simulation with nominal parameter values
dotted).
)dashed)andmeasurements(solid).
air intake of the engine and the second is connected
tothedriveshaft. Additionally, therearetwogasvol-
umes which are connected to two thermal conductors
thattransportheattothesurroundingair. Theendpoint
circlesrepresenttheboundaryconditionsforthegas.
Theuncertainparametersarethethermalcapacities
inthewallsofthegasvolumestogetherwiththether-
malconductancefromgastowallinthevolumes.
Theinputsofthemodelarethegastemperatureand
pressure entering the system, angular velocity of the
Figure8: Overviewofthemodelofthedieselengine.
turbines and the gas pressure exiting the system. The
outputisthegastemperatureexitingthesystem.
Measurements are provided for the inputs and the andthethermalconductanceintheleftvolume, using
output sampled every second over a thirty minute pe- the results from the first problem. This procedure is
riod. InFigure9,theresultisshownwhensimulating used since the parameters of the first and second vol-
themodelusingnominalparametervalues. ume are correlated. Optimizing all parameters simul-
The problem is to minimize the error between the taneouslythenresultsinover-parameterization.
simulatedgastemperaturethatexitsthesystemandthe
Foreachoptimizationproblem,thefirstthirdofthe
measuredtemperature,
measurement data sequences are used for calibration
andtheremainingpartisusedforvalidation.
M The model was exported from Dymola as an FMU
min ∑(Tsim(t,x)−Tmeas(t))2 (7)
x∈Rn i i and thereby made available to the DFO algorithms
i=1
in JModelica.org. The two problems are then solved
subjectto x≥0 (8)
using the Nelder-Mead simplex algorithm. Figure 9
where M is the number of measurement points and n shows the resulting simulation response for the opti-
thenumberofparameters. mizedparameters. InFigure10,thecorrespondinger-
Insteadofoptimizingthefouruncertainparameters rorprofilesareshownforthecalibrationandvalidation
simultaneously, the problem is divided into two prob- data sets respectively. As can be seen, the optimized
lems. The first problem is to determine the thermal parameters significantly increase the accuracy of the
capacityandthethermalconductanceintherightvol- model. The (scaled) RMS error was decreased from
ume. Thesecondistodeterminethethermalcapacity 1.0to0.18forthecalibrationdatasetandfrom1.0to
0.36forthevalidationdataset. e.g., simulation of FMUs and comparison with mea-
surement data or algorithmic evaluation of complex
Exhaust gas temperature (scaled)
discontinuouscosts.
1.00
Measurements
Simulation with nominal parameters
Simulation with estimated parameters
0.95 8 Acknowledgments
The authors gratefully acknowledges financial sup-
0.90
portfromVinnovaunderthecontract(Projectnumber
P35278-5) and from the Lund Center for Control of
0.85
ComplexSystems,LCCC,fundedbytheSwedishRe-
searchCouncil.
0.80
References
0.75
0.0 0.2 0.4 0.6 0.8 1.0
time
[1] Functional Mock-up Interface for Model Ex-
Figure9: Simulationresultwiththeoptimizedparam-
change. Interface specification, MODELISAR,
eterstogetherwithresultusingthenominalparameter
January2010.
valuesandmeasurements.
[2] C. Andersson, J. Åkesson, C. Führer, and
M. Gäfvert. Import and export of Functional
Mock-up Units in JModelica.org. In In 8th In-
Exhaust gas temperature error (scaled)
10-1
ternational Modelica Conference 2011. Model-
10-2
data10-3 icaAssociation,2011.
Calibration 111000---654 MMeeaassuurreemmeennttss -- NOopmtiminaall [3] JC..ÅAkensdseornss.oAn,wJo.rkAbnendcrehafsosornm,uClti.boFdüyhrseyrs,teamnds
10-70.00 0.05 0.10 0.15 0.20 0.25 0.30
ODE and DAE solvers. In The Second Joint In-
10-1
10-2 ternationalConferenceonMultibodySystemDy-
data10-3 namics,2012.
Validation 1100--54 Measurements - Nominal [4] J. Andersson, J. Åkesson, and M. Diehl.
10-6 Measurements - Optimal
10-7 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CasADi—Asymbolicpackageforautomaticdif-
time ferentiation and optimal control. In S. Forth,
P. Hovland, E. Phipps, J. Utke, and A. Walther,
Figure 10: Error profiles for the calibration data set
editors, Proc. 6th International Conference
(top)andthevalidationdataset(bottom).
on Automatic Differentiation, Lecture Notes
in Computational Science and Engineering.
Springer,2012.
7 Summary
[5] Joel Andersson, Johan Åkesson, Francesco
Casella, and Moritz Diehl. Integration of casadi
An implementation of derivative-free optimization al- andjmodelica.org.In8thInternationalModelica
gorithms in JModelica.org has been presented. The Conference,March2011.
implementation has been successfully applied to two
dynamicmodelswherethedynamicsarecontainedin [6] T. Binder, L. Blank, H.G. Bock, R. Bulirsch,
a Functional Mock-up Unit. In one of the examples, W. Dahmen, M. Diehl, T. Kronseder, W Mar-
a Volvo truck engine was calibrated against measure- quardt,J.P.Schlöder,andO.v.Stryk.OnlineOp-
ment data, demonstrating the industrial applicability timizationofLargeScaleSystems,chapterIntro-
oftheapproach. ductiontomodelbasedoptimizationofchemical
The Python-based user interface enables flexible processes on moving horizons, pages 295–339.
implementation of complex cost functions involving, Springer-Verlag,BerlinHeidelberg,2001.