An Introduction to Optimal Control of Partial Differential ...An Introduction to Optimal Control of...

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

[email protected]

mailto:[email protected]







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]



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]



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



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]



Motivation: Optimal load changes for fuel cell systems

Molten Carbonate Fuel Cell

cell

stack

Hotmodule [CFC Solutions, IPF Berndt]

2002-2005



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]



Outline

A glimpse on the theory

A glimpse on the numerics

An application: MCFC

Conclusions



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-)



An example: optimal stationary temperature distribution

subject to

A simple elliptic optimal control problems


with boundary control

tracking functional Tikhonov regularization set of admissible controls



subject to

An example: Optimal stationary temperature distribution



Necessary conditions



with linear and continuous solution operator subject to

An example: Optimal stationary temperature distribution



Optimization problem in Hilbert space


Necessary condition: variational inequality

bilinear form linear form



Necessary condition: variational inequality

Optimization problem in Hilbert space


adjoint operator



Description with the adjoint solution operator


Description with the adjoint state

adjoint state pointwise

evaluation



The formal Lagrange technique

Defining the Lagrange function and twice formal integration by parts

Differentiation in the direction of , resp.



Optimality system: semi-linear elliptic, distributed + boundary control



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



Outline



An application

Conclusions



Methods for PDE constrained optimization

The general problem

The aims

concepts for real-life application

small constant effort of simulation

effort of optimization



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



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



Outline



An application

optimal control of a molten carbonate fuel cell

process control via model reduction techniques

Conclusions



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]



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



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]



[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

solid temperature


flow directions

in anode

and cathode

[Chudej, Sternberg]



[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

solid temperature


state constraint would be desirable



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



How to apply adjoint-based methods

on real-life problems?



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



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


Air inlet



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


Air inlet




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


Air inlet

CO32-




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


Air inlet

Fuel gas

CO32-




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


Air inlet





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


controls

4

2

1



The equations: gas channels and solid

molar fractions

gas temperature

solid temperature

molar flow densities



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



The equations: potential fields

currents

current densities

cell voltage

potentials

input data

for load changes

plus appropriate initial and boundary conditions for all equations



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]



Numerical simulation of a load change

zoom

[Rund]



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)



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]



Lagrangian



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



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]



Numerical results “first optimize then discretize”

[Rund]

load change after 0.1. sec

regularization:

41 lines in space

767 time steps



Aim for process control

How to apply optimal solutions

in practise?



• 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



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:



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



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



Scheme for state estimator for discrete measurements

process

input

MCFC sensors

y

Simulator

state

sensor models

y

measurement

?

observer

observer correction

+

-

MCFC model



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



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



Conclusions

Concerning theory: already well developed

Concerning numerics: still improving

Concerning applications: has to be intensified

one always abuts against limits


July 3-7, 2012, Biblioteca Francescana, Ravello, Italy Thank you for your attention

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