ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
An Introduction to Optimal Control
of Partial Differential Equations
with Real-life Applications
Hans Josef Pesch
Chair of Mathematics in Engineering Sciences University of Bayreuth, Bayreuth, Germany
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
multi-beam welding
weld seam
hot crack
main laser
beam
mushy zone
weld pool
compression
additional beams
solidification
Motivation: Optimal placement of laser beams to avoid hot cracking
Semi-infinite optimization problem
for an elliptic PDE with state constraints
[Karkhin, Ploshikin]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Motivation: Optimal placement of laser beams to avoid hot cracking
op
en
ing
dis
pla
ce
me
nt
weld pool region
hot crack criterium
limit
zoom
liquidus and solidus
isotherms surrounding
the mushy zone
[Petzet]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Motivation: Minimum fuel transcontinental flights at hypersonic speeds
Europe - USA in 2 hrs / Europe - Australia in 4.5 hrs
ODE
PDE
2 box constraints
1 control-state constraint
1 state constraint
quasilinear heat equation
non-linear boundary conditions
coupled with ODE
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Motivation: Minimum fuel transcontinental flights at hypersonic speeds
velocity [m/s] altitude [10,000 m] flight path angle [deg]
temperature [K] temperature [K] temperature [K]
1st layer 2nd layer 3rd layer
limit
temperature
1000 K
on a boundary arc
[s]
[s]
[Wächter, Chudej, LeBras]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Motivation: Optimal load changes for fuel cell systems
Molten Carbonate Fuel Cell
cell
stack
Hotmodule [CFC Solutions, IPF Berndt]
2002-2005
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
catalytic
burner
mixer
anode inlet
cathode inlet
anode
exhaust
cathode
exhaust
exhaust air inlet
recirculation
2D cross-flow design
CO32-
Motivation: Optimal load changes for fuel cell systems
28 quasi-linear partial integro-differential-algebraic equations
with non-standard non-linear boundary conditions
[Sundmacher]
[Heidebrecht]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Outline
A glimpse on the theory
A glimpse on the numerics
An application: MCFC
Conclusions
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Outline
A glimpse on the theory
A glimpse on the numerics
An application: MCFC
Conclusions
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Elliptic optimal control problem
with distributed control
An example: optimal stationary temperature distribution
subject to
tracking functional Tikhonov regularization set of admissible controls
A simple elliptic optimal control problems
Lions (since 1970s), Casas (1987-), Tröltzsch (1980-)
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
An example: optimal stationary temperature distribution
subject to
A simple elliptic optimal control problems
Elliptic optimal control problem
with boundary control
tracking functional Tikhonov regularization set of admissible controls
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
subject to
An example: Optimal stationary temperature distribution
Elliptic optimal control problem
with distributed control
Necessary conditions
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
with linear and continuous solution operator subject to
An example: Optimal stationary temperature distribution
Elliptic optimal control problem
with distributed control
Optimization problem in Hilbert space
Necessary conditions
Necessary condition: variational inequality
bilinear form linear form
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Necessary condition: variational inequality
Optimization problem in Hilbert space
Necessary conditions
adjoint operator
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Description with the adjoint solution operator
Necessary conditions
Description with the adjoint state
adjoint state pointwise
evaluation
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
The formal Lagrange technique
Defining the Lagrange function and twice formal integration by parts
Differentiation in the direction of , resp.
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Optimality system: semi-linear elliptic, distributed + boundary control
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Optimal Control
of PDE
Functional
Analysis
Partial
Differential
Equations
Optimization
in Banach spaces
Numerics of
PDE
Numerical
Methods
of Optimization
High Performance
Scientific Computing
Numerical
Methods of
Linear Algebra
Parallel
Numerical
Methods
Challenges in PDE constrained optimization
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Outline
A glimpse on the theory
A glimpse on the numerics
An application
Conclusions
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Methods for PDE constrained optimization
The general problem
The aims
concepts for real-life application
small constant effort of simulation
effort of optimization
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
capture
as much
structure
of ( P )
as possible
on discrete
level
( Ph )
First Discretize then Optimize vs. First Optimize then Discretize
First discretize then optimize (FDTO) or DIRECT
First optimize then discretize (FOTD) or INDIRECT
Questions
appropriate choice of and ansatz for ?
appropriate choice of and ansatz for ?
appropriate ansatz for adjoint variables and multipliers?
Solve
large scale
NLP
Solve
coupled PDE
system
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
First Discretize then Optimize vs. First Optimize then Discretize
First discretize then optimze (FDTO): replace all quantities of the infinite dimensional optimization problem
by finite dimensional substitutes and solve an NLP
First optimze then discretize (FOTD): Derive optimality conditions of the infinite dimensional system,
discretize the optimality system and find solution of the discretized
optimality system
In general
Ideal: discrete concept for which both approaches commute Discontinuous Galerkin methods
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Outline
A glimpse on the theory
A glimpse on the numerics
An application
optimal control of a molten carbonate fuel cell
process control via model reduction techniques
Conclusions
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
catalytic
burner
mixer
anode inlet
cathode inlet
anode
exhaust
cathode
exhaust
exhaust air inlet
recirculation
Configuration and function of MCFC
2D cross-flow design
controllable
controllable
controllable
load changes
input
boundary
conditions
by ODAE
slow
state
variable
fast very fast
algebraic
[Heidebrecht]
[Sundmacher]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
0 0.1 1.1 11.1 111.1 1111.1
using control
optimal control
simulation 0.8 sec
0 0.1 1.1 11.1 111.1 1111.1 scaled time
using controls
optimal control
simulation 0.4 sec
scaled time
cell voltage 0.7 0.6 for a load change
[Sternberg]
Optimal load changes. Computation by FDTO
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
anode gas temperature cathode gas temperature
[2.8 ≈ 560 °C]
[3.2 ≈ 680 °C]
Numerical results: simulation of load change (FDTO)
reforming reactions are endothermic
oxidation reaction is exothermic reduction reaction is endothermic
flow directions
[Chudej, Sternberg]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
[2.8 ≈ 560 °C]
[3.2 ≈ 680 °C]
solid temperature
Numerical results: simulation of load change (FDTO)
flow directions
in anode
and cathode
[Chudej, Sternberg]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
[2.8 ≈ 560 °C]
[3.2 ≈ 680 °C]
solid temperature
Numerical results: simulation of load change (FDTO)
state constraint would be desirable
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
with
Pareto performance index:
Numerical results: optimal control of fast load change (FDTO)
while temperature gradients stay small
fast
slow
on
on
0.7 0.6
instead of
state constraint
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
How to apply adjoint-based methods
on real-life problems?
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
O2
N2
U
e-
CO32-
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
Recirculation
Exhaust
1D counter-flow design
Air inlet
Reforming reaction CH4 + 2 H2O CO2 + 4 H2
Configuration of MCFC for 1D counter-flow design
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
O2
N2
U
e-
CO32-
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
Cathode
Elektrolyte
CH4
H2O
Anode
Anode gas channel
Cathode gas channel
Configuration and function of MCFC
Recirculation
Exhaust
Mixer
Catalytic
burner
Oxidation reaction
1D counter-flow design
Air inlet
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
U
e-
CO32-
O2
N2
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
Recirculation
Exhaust
Reduction reaction
1D counter-flow design
Air inlet
Configuration of MCFC for 1D counter-flow design
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
U
e-
O2
N2
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
Recirculation
Exhaust
1D counter-flow design
Air inlet
CO32-
Configuration of MCFC for 1D counter-flow design
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
U
e-
O2
N2
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
Recirculation
Exhaust
Reactant
1D counter-flow design
Air inlet
Fuel gas
CO32-
Configuration of MCFC for 1D counter-flow design
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
CO32-
O2
N2
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
U
e-
Recirculation
Exhaust
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
only ions can move
through electrolyte
German Federal
Pollution
Control
Act: Air
1D counter-flow design
Air inlet
Configuration of MCFC for 1D counter-flow design
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
1D counter-flow design
CO32-
O2
N2
½O2 + CO2 + 2e- CO32-
CH4 + H2O CO + 3H2
CO + H2O CO2 + H2
H2 + CO32- H2O + CO2 + 2e-
CO + CO32- 2CO2 + 2e-
U
e-
Recirculation
Exhaust
CH4
H2O
Cathode
Anode
Elektrolyte
Mixer
Catalytic
burner
Anode gas channel
Cathode gas channel
Air inlet
Configuration of MCFC for 1D counter-flow design
controls
4
2
1
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
The equations: gas channels and solid
molar fractions
gas temperature
solid temperature
molar flow densities
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
The equations: burner and mixer
The catalytic burner
is fed by the anode
and cathode outlet
The mixer is described
by a system of ODAE
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
The equations: potential fields
currents
current densities
cell voltage
potentials
input data
for load changes
plus appropriate initial and boundary conditions for all equations
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Numerical methodology for forward solver (fact sheet)
• States: smooth in space direction, but high gradients in time:
semi-discretization in space (N fixed grid points)
upwind formulas to preserve the conservation laws
adaptive time steps
large scale index 1 DAE system of dimension 14N – 6
fully implicit multistep variable order method ode15s (MATLAB)
with simplified Newton method for the non-linear systems
and Jacobian by numerical or automatic differentiation
• Choice of consistent initial data by computing stationary initial values
by a multi-level discretization (from coarse to fine grids)
State solver
Numerical solution via “first optimize then discretize” (1st part)
[Rund]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Numerical simulation of a load change
zoom
[Rund]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Optimal control problem (fact sheet)
• Choice of controls:
Initial values enter the mixing chamber
obey box constraints
• Choice of objective functional:
Minimize L2 – distance to target temperature (reduce overheating)
Minimize L2 – value of temperature gradient (reduce thermal stress)
Minimize L2 – distance to target cell current (fast load change)
Minimize control costs (regularization)
Pareto optimal control problem
Assumption on existence of optimal solution
(because of non-linearity)
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Necessary conditions (fact sheet)
• Assumption on existence of multipliers of sufficient regularity
formal Lagrange technique (67 multipliers)
• Derivation of directional derivatives
partial integration, differentiation, separate contribution of objective,
symbolic or automatic differentiation of source terms
• Variational argument
structure of adjoint system (type of PDE/ODE/DAE preserved)
with reverse time
partial derivatives of states in source terms due to quasilinearity
ODEs with spatial integrals in their r.h.s.
Coupled staggered system of variational inequalities
to determine optimal control laws
no projection formulae, but
gradient of objective function (for gradient or Newton method)
[Rund]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Lagrangian
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Necessary conditions (summary)
Adjoint state solver
control
control
BC
OUT
OUT control
control
BC
BC
BC
OUT
AE
PDE
PDE
PDE
PDE
PDE
PDE OUT
AE
DAE
DAE
anode
anode
cathode
cathode
burner
burner
mixer
mixer by
variational
inequalities
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Numerical solution via “first optimize then discretize” (2nd part)
Numerical methodology for optimization (fact sheet)
• Backward sweep method: staggered solution of optimality system
efficient for many time steps (different time scales)
good initial guesses for non-linear solver
drawback: inferior convergence properties
• Choice of iterative method:
Quasi-Newton (use gradient)
superlinear convergence
no second derivatives
• SQP methods are hardly applicable (2nd order information required)
[Rund]
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Numerical results “first optimize then discretize”
[Rund]
load change after 0.1. sec
regularization:
41 lines in space
767 time steps
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Aim for process control
How to apply optimal solutions
in practise?
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
• measurable: cell voltage, gas temperatures and concentrations at anode and cathode outlet
• diserable for process control: information on spatial temperatur and concentration profiles
• solution ansatz: observer / state estimator
• Problem: complexity of model
Aim for process control
?
?? Remedy: model reduction technique
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Model reduction by POD: idea
Complete model:
Ansatz (separation of variables):
Reduced model:
orthogonal
snapshots
low order model: ODAE of index 1
Method of weighted residuals:
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
test signal
1. temperature basis function
Model reduction by POD: computation of snapshots
by the complete model
2. temperature basis function
orthogonalization
by singular value
decomposition
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Model reduction by POD: comparison of reduced vs. complete model
random variation
of cell voltage
perfect coincidence
with reference model
appropriate for
process control
[Mangold, Sheng]
#eqs. 4759 vs. 131
3200 sec vs. 82 sec
2 < N < 10
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Scheme for state estimator for discrete measurements
process
input
MCFC sensors
y
Simulator
state
sensor models
y
measurement
?
observer
observer correction
+
-
MCFC model
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
References
Focus on Theory:
Tröltzsch, F.:
Optimal Control of Partial Differential Equations:
Theory, Methods, and Applications
AMS, Graduate Studies in Mathematics, Vol. 112, 2010.
Focus on Methods:
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.:
Optimization with PDE Constraints
Mathematical Modelling: Theorie and Applications,
Vol. 23, 2008.
Borzi, A., Schulz, V.:
Computational Optimization of Systems
Governed by Partial Differential Equations
SIAM, Philadelphia, 2011.
Focus on Applications:
See my homepage: google: Hans Josef Pesch
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
The Fuel Cell Team
Prof. Kai Sundmacher Dr.-Ing.h.c. Joachim Berndt Dr.-Ing. Peter Heidebrecht
Prof. Kurt Chudej Dr. Kati Sternberg
Prof. Michael Mangold †
Dr. Armin Rund
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy
Conclusions
Concerning theory: already well developed
Concerning numerics: still improving
Concerning applications: has to be intensified
one always abuts against limits
ITN SADCO Summer School "New Trends in Optimal Control"
July 3-7, 2012, Biblioteca Francescana, Ravello, Italy Thank you for your attention
Top Related