Post on 11-Mar-2021
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Flanders Centre / Laboratory of Postharvest Technology www.vcbt.be
Dynamic process simulation for design, operation and control
Pieter Verboven
Aim
Learn about implementing dynamic modelsHands on:
Program dynamic models in SimulinkUse predefined examples
Demonstrations of other models
it don’t mean a thing if it ain’t got that swingDuke Ellington
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
Dynamic equation systems
Simple systemsAlgebraic equations
Continuous systemsDifferential equations
Discrete systemsDifference equations
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Simple systems
systems without a state
output at a time instance only depends on the input at that time instancesystem response is always the same for the same input, independent of timeuses:
algebraic equations: y=f(u)logic relations
u(t) (input) y(t) (output)
example of a simple system
510
52
<==>=
tifuy
tifuy
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Continuous systems
systems with a state
state depends on the inputs and previous values of the state continuouslyuses differential equations
e.g. 1st order explicit ordinary differential equation (ODE)
x(t)statesu(t) (input) y(t) (output)
00 )()),(),((
)),(),((
xtx
ttutxgy
ttutxfdtdx
==
=
example of a continous system
0)0(
25
===
+−==′
txxy
xdtdx
x
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Discrete systems
systems with a state
state depends on the inputs and previous valuesof the state at discrete time stepsuses difference equations
x(t)statesu(t) (input) y(t) (output)
tnnt
xx
nnunxgny
nnunxfnx
∆===
−−−=
*)(
)0()),(),(()(
)1),1(),1(()(
0
example of a discrete system
0.100.2)1(5.0)0(
)()()(5.0)1(*5.1)2(
=∆===
−+=+
t
x
x
nxny
nxnxnx
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Example in the agro-food chain
Large-scale cooling facility
Condenser
Compressor
Expansion valve
Evaporator
Condenser
Compressor
Expansion valve
Evaporator
Refrigeration unit Gas handling unit
Cells
-+
Buffer tank
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Room temperature
Relative humidity
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O2 concentration
Product temperature
Firmness loss
Respiration heat generation
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Application:loading strategybatch wisestep wise
OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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Numerical solution methods
Continous systems: solution of ODEs
Analytical solutions only for simple ODEsNumerical solution
make assumption about behaviour of variable in between time points, e.g., linear variationInsert the approximation into the equationsolve the resulting system of algebraic equations in a discrete number of time points
00 )(
),('
yty
ytfy
==
Solver algorithmsexplicit solvers
forward Eulervery inexpensivelow accuracy
Adams-Bashfortuse old derivative functionsinexpensivemoderate accuracy
Runge-Kutta methodsderivative function at new intermediate pointsexpensive (higher order)very accurate
not suited for stiff problems
)(
)(
1
1'
iii
iii
i
ytfyy
yft
yyy
∆+=
=∆−=
+
+
j
n
jii
jjij
i
kayy
tkyfk
tyfk
�+=
∆+=∆=
+
−
11
1
1
)(
)(
α
)(0
1 ji
n
jjii yfatyy −
=+ �∆+=
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stiff problem
explicit solver implicit solver
2.01
15
0
=>==
−=′
ctdx
yxy
implicit solvers for stiff systemsAdams-Moulton
backward Euler & Crank-Nicolson (Trapezium)conditionally stableuse new and old derivative functionshigher order very expensive
( ) 2)()(
)(
11
111
++
+++
+∆=−∆=−=∇
iiii
iiii
yfyftyy
ytfyyy
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Backward Differentiation Formulas (BDFs)use more old variable valueshigher order very stableexpensive
Numerical Differentiation Formulas (NDFs)improved BDFs (add estimate of truncation error)more efficient than BDFs
use iteration methods to solve non-linear equationuse with variable time step: superior performance
1112
11
112
1
2
)(21
:2
−++
++
+++
+−=∇
−=∇
∆=∇+∇
=
iiii
iii
iii
yyyy
yyy
ytfyy
k
11
111
)(1
++
++=
∇+∆=∇� ik
kiim
k
m
yytfym
κγ
)(1
111
++=
∆=∇� iim
k
m
ytfym
Solvers available in SimulinkNon-stiff
fixed stepode1: forward Eulerode4: 4th order Runge-Kuttaode5: Runge-Kutta
variable stepode45: Runge-Kutta basedode23: Runge-Kutta based
Stiffvariable step
ode15s: NDFs or BDFsode23s: linear implicit methodode23t: trapezoidal rule based
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
Simulink
software package for modelling, simulation and analyzing dynamic systems
linear and non-linearcombined continuous and discrete systemsgraphical interfaceblock diagramsblock libraries (pre-defined and user-developed)modular – hierarchicalcoupling to MATLAB
example_simple.mdl
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Simulink in a nut shell
from within MATLABclick on iconcommand line simulink
list of thematic block libraries openscreate a new modeldrag and drop blocks from library to modellink the blocks with line arrowsdefine parameters for all blocks (double click)set time spanstart simulationanalyse results
ExampleProgram the lumped heat transfer equation
ρ: density [kg m-3]V: volume [m3]cp: specific heat [J kg-1 °C-1]h: heat transfer coefficient [W m-2 °C-1]A: surface area [m2]Q: heat generation [W kg -1]T: temperature [°C]t: time [s]
( ) VQTThAtT
Vc ap ρρ +−−=∂∂
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Use integrator block
Integration is done numerically by means of thespecified solver
� +=t
t
ydttyfty0
0),()(
example_lumped_heat_transfer.mdl
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example_lumped_heat_transfer.mdl
example_lumped_heat_transfer.mdl
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Notesmodel parameters are defined as constants: linearmodel
specific heat/ heat generation can be a function of T: use, e.g., look-up table, function block
can include discontinuities
example_lumped_heat_transfer_nonlinear.mdl
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example_lumped_heat_transfer_phase_change.mdl
OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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Exercise: accuracy of solversexample_lumped_heat_transfer.mdl
compare solvers at different fixed time steps.25tc, .50tc,tc,2tc,4tcrecord value at certain time
compare to using automatic time steps
ode45 at 4tc ode15s at 4tc ode15s automatic step
hAVc
tcρ=
OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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Multi-component systems
real lifeinteraction of different dynamic systems
coupled ODEsstiffness
modular-hierarchical modellinguse of (user-defined) librariesexample_cool_room_continuous.mdl
cool room
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model
r: refrigerant (coolant)c: refrigeration in vessela:aire:exteriorp:product
pppappap
pppp
eawwappaarcca
paaa
arccrcprrr
prrr
QVTTAht
TcV
TTAhTTAhTTAht
TcV
TTAhTTcmt
TcV
ρρ
ρ
ρ
+−−=∂
∂
−−−+−=∂
∂
−−−=∂
∂
)(
)()()(
)()(�
example_cool_room_continuous.mdl
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
Distributed dynamic systemsdepend also on spatial coordinatese.g., the Fourier equation for heat conduction
ρ: density [kg m-3]cp: specific heat [J kg-1 °C-1]k: thermal conductivity [W m-1 °C-1]h: heat transfer coefficient [W m-2 °C-1]Q: heat generation [W kg -1]T: temperature [°C]t: time [s]
0)0(
)(
TtT
TThTn
k
QTktT
c
a
p
==
−=∂∂−
+∇∇=∂∂ ρρ
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also have the format
but f(y,t) contains spatial partial derivativesuse numerical techniques to solve equation
finite difference methodfinite element methodfinite volume method…
generally lead to the set of first order ODEs
use ODE solver to solve set of equations
00 )(
),('
yty
ytfy
==
C u Ku fddt
+ =
example: heat conduction in a spheresolution in FEMLAB
3D
ha, Ta
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1D spherical
ha, Tacenter surface
temperature profile
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3D 1D spherical coordinates
FEMLAB in Simulink
possibility to combine ODEs and PDEsexport model from FEMLAB to MATLAB structure
general dynamiclinearised dynamicspecify inputs and outputsinitial conditions cannot be exported!
introduce structure in FEMLAB blockFEMLAB model uses ODE solver of Simulink
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example_sphere_linear.mdl
comparison of results FEMLAB - Simulink
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example_sphere_linear_plus_air_lumped.mdlstiff problem!
OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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Discrete-continuous simulationDiscrete events require that time step is accuratelycontrolled at the discontinuity
imposes new initial value problemmissing the discontinuity means wrong initiation of nextcontinuous period
fixed time solvers cannot deal with discontinuitiesMethod in simulink
zero-crossing detectioneach relevant block carries a ‘zero-crossing’ variablechanges sign at discontinuity of stateinterpolation between last and current step to localise crossingstep up to and then over discontinuity
example_lumped_heat_transfer_phase_change.mdlturn of zero-crossing in switch and integrator blocks
without detectionwith zero-crossing detection
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cool room with door openings
model
r: refrigerant (coolant)c: refrigeration in vessela:aire:exteriorp:product
pppappap
pppp
deaddeaww
appaarcca
paaa
arccrcprrr
prrr
QVTTAht
TcV
sTTAhTTAh
TTAhTTAht
TcV
TTAhTTcmt
TcV
ρρ
ρ
ρ
+−−=∂
∂×−−−−
−+−=∂
∂
−−−=∂
∂
)(
)()(
)()(
)()(�
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example_cool_room_continuous_discrete.mdl
OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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Controllers
Feedback controller
PID controller
Kp: gainTi: integral timeTd: derivation time
controller proces-
+
y
R
e u
���
����
�++= � dt
tdeTdtte
TteKu d
ip
)()(
1)(
dynamic modelling of controllersparameter tuningreal process simulation
smallchange
decreasedecreasesmallchange
Td
eliminateincreaseincreasedecreaseTi
decreasesmallchange
increasedecreaseKp
Steadyerror
Settlingtime
OvershootRise time
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example_controller.mdl
example_controller.mdl
Kp=1000 (P)
Kp=1000,Ti=0.1 (PI)
Kp=1000,Ti=0.1,Td=1000 (PID)
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cool room with PID control of air temperature
PID
example_cool_room_continuous_discrete_control.mdl
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
Parametric studies
explore the effect of changing parametervalues in the modelexample_cool_room_parametric.mdlautomatic parameter testing
run the model from the MATLAB command lineparametric_cool_room.m
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
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multi-step modelling
chain modellingproduct changes conditions/process from one step to the nextimportant for
agro-food chain quality management: chain designsolving reliability issuestraining of operatorsaids management decissions
example_chain_simulation.mdlsimulink is limited for multiple steps (max.3)!
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OutlineDynamic equation systemsNumerical solution methodsGetting started with SimulinkSimulation of a dynamic systemMulti-component systemsDistributed dynamic systemsDiscrete-continuous simulationController simulationParametric studiesMulti-step modellingCustomized visualisation
Customized visualisation
Make your model user-friendlyuse object-oriented programminguse hierarchical and modular programming
example: GUI for temperature and quality in chicory production chain
completely built in MATLAB environmentsolve ODEs for temperature and temperature-dependent quality kinetics
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it don’t mean a thing if it ain’t got that swingDuke Ellington