robust optimal control of finite-time distributed …robust optimal control of finite-time...

Robust Optimal Control of Finite -time Distributed Parameter Systems

McMaster University

1

Richard D. BraatzUniversity of Illinois at Urbana-Champaign

� Motivation

� Distributional robustness analysis via polynomial chaos expansions

� Example: Batch crystallization

Overview

2


� Worst-case robustness analysis via power series expansions

� Example: 2D reaction -diffusion

� Summary and future comments

Finite -time Systems

• Many products such as pharmaceuticals, batteries, microelectronic devices, and artificial organs are manufactured in finite-time processing steps

• These processes – are rather complicated distributed parameter system s – require tight control of dimensions, chemistry, and /or biology– have models with significant associated uncertainti es

• Objective: computationally efficient methods for th e robust optimal control of these processes

Model Uncertainties

Many sources of uncertainties, frequent disturbances

Numerical

predictions

Physical

system

Mathematical

model

Structural

uncertainties

Parameter

uncertainties

Model

development

Dis

turb

ance

s

4

predictions

from model

Numerical

errorsControl input

uncertainties

Model

solution

Dis

turb

ance

s

Model/plant mismatch

• Benefits of model-based control can be lost if uncertainties are not explicitly addressed

• Focus on parameter uncertainties (others are simila r)

Focus of Presentation

• Standard approaches for robustness analysis– Monte Carlo method is computationally expensive– Gridding the parameter space is computationally exp ensive,

order of 100 n for n parameters– Lyapunov functions difficult to construct non-

conservatively for general nonlinear DPS

5

• Important questions:– How to nonconservatively analyze the effects of mode l

uncertainties in a computationally feasible manner? ��– How to use this information for robust controller d esign?

• Talk considers finite-time systems that do not have finite escape time, which are practically important but much simpler than infinite-time systems

Summary of Overall Approach

Robustness analysis for finite-time systems:1. Replace system w/surrogate model that accurately

describes the input-to-state and input-to-output behavior within the trajectory bundle

2. Perform robustness analysis on surrogate model

3. Evaluate accuracy of surrogate model;

6

3. Evaluate accuracy of surrogate model; increase accuracy if needed

time

states

final time

� Motivation



Overview

7





Polynomial Chaos Expansion

� Model state as an expansion of orthogonal polynomia l functions of the real model parameters

� Different orthogonal functions are optimal for diff erent parameter pdfs (e.g., Gaussian, Gamma, Beta, Uniform, Poisson, Bin omial, …)

�

1 21

1 1 1 2 1 2 1 2 3 1 2 3

1 1 2 1 2 3

0 0 1 2 31 1 1 1 1 1constant

first order terms third order termssecond order terms

( ) ( , ) ( , , )i iin n n

i i i i i i i i i i i ii i i i i i

a a a a= = = = = =

= Γ + Γ + Γ + Γ +∑ ∑∑ ∑∑∑ ⋯

��

ψ θ θ θ θ θ θ

8

pdfs (e.g., Gaussian, Gamma, Beta, Uniform, Poisson, Bin omial, …)

� Widely used in the environmental field, called “unc ertainty quantification” or “uncertainty propagation”

� Coefficients can be computed from collocation or re gression

� Choose collocation points or sampling points to spa n as much as possible the behavior of the states

� Related algorithms expand parameters and states in terms of the orthogonal functions, to give explicit analytical p dfs for states

Using Polynomial Chaos Expansion for Distributional Robustness Analysis

Specify uncertainty description for

parameters

Select optimal

Generate collocation points for model runs

Run model at collocation points and compute coefficients

Add higher order PCE

9

Select optimal orthogonal polynomials

for the distribution

Generate first-order PCE to approximate

model response

Estimate error of expansion

Is error too large?

Use expansion for uncertainty analysis

order PCE terms to

polynomials

YES

NOOptimal orthogonal polynomials are more efficient than power series expansions

Estimation of Time -varying pdfs of StatesMonte Carlo simulation

� Maps entire pdfs of model parameters to state pdfs

� Large number of simulations for correct PDF (tail effect)

Full dynamic model

IPDAEs, multiscale models


� Computationally very expensiveInstead

� Maps entire pdfs of model parameters to state pdfs


� Computationally inexpensive

Surrogate model

Algebraic expression w/time-varying coefficients

� Motivation



Overview

11





� Used to purify drugs, reaction intermediates, explosives, …

� Temperature reduced to cause crystals to nucleate & grow

Example: Batch Crystallizationf(

r 1, r

2)

12

� Objective: Make large crystals from seed; minimize nucleation

� Large uncertainties in kinetic parameters in model

� IPDE population balance model �� analysis of moments enabled

full evaluation of methods

r1 (µm)r2 (µm)

IPDAE Process Model

• G = growth• B = nucleation• C = concentration• T = temperature0

( , ) ( , ) ( )

d3 ( , ) ( , )

d

i

f fG C T B C T L

t L

CG C T f L t dL

t

f f

∞

∞

∂ ∂+ = δ∂ ∂

= − α

∂ ∂

∫

∫

• mi = i th moment• Alternatively could

solve by method of characteristics or finite differences

13

0

1

0

( , ) ( , ) ( ) 0

d( , )0 ( , )

d

( , )

i

iii

ii

f fG C T B C T L L dL

t L

mB C T nG C T m

t

m f L t L dL

∞

−

∞

∂ ∂ + = δ = ∂ ∂

⇒ = +

≡

∫

∫

Motivation: Pharmaceutical Crystallization

� Used to purify nearly all legal drugs

� High value-added products, $150 billion/year in pharmaceuticals sales with growth of 20% per year

� Highly nonlinear and strongly stochastic processf

14

� Control of CSD properties important

� Efficiency of downstream operations (filtration, drying)

� Product effectiveness (tablet stability)

� Strict regulatory requirements on variation of product crystals

� High economic penalty of producing off-spec product ($1-2 million/batch)

ψ

Allowed variability for quality

Optimal Control Problem

( )

optimizeT k

J

( ) ( ) ( ),T k T k T k≤ ≤

subject to:IPDE process model

15

� Objectives: nucleation mass to seed mass ratio, weight mean size

( ) ( ) ( )

( )( )

( )

min max

min max

,max

,

,

,final final

T k T k T k

dT kR k R k

dt

C C

≤ ≤

≤ ≤

≤

30

31

32

T (

°C)

Importance of Robustness Analysis

� Minimize nucleation mass to seed mass ratio (J)

J = 10.7

J = 8.5

� 21% improvement compared to linear profile

� Worst-case J due to uncertainties is 14.6 !

16

0 40 80 120 16028

29

Time (min)

� Improvement can be lost due to uncertainty in parameters

uncertainties is 14.6 !

Optimal profile may only nominallygive less nucleated crystal mass.

The rest of this example considers effects of stochastic uncertainties.

Nagy & Braatz, Journal of Process Control, 17, 229, 2007

Time-varying pdfs, Comparing Approaches

. first-order PSE

o MC with second-order PSE

x polynomial chaos expansion

– MC with nonlinear model

17

� first-order PSE provides rough estimate, with some bias

� PCE provides very good accurate quantification of p dfs

Comparison of Computational Costs

Method Computational time (on P4)

Monte Carlo with dynamic model (80,000 points) 8 hr

First-order PSE approach (analytical solution) 1 sec

Monte Carlo with second -order PSE (80,000 points) 4 min

18

Monte Carlo with second -order PSE (80,000 points) 4 min

Polynomial chaos expansion (second-order) 2 sec

� Full Monte Carlo – very high computational requireme nts

� First-order PSE approach – computationally very attr active

� Second-order PSE – increased accuracy & computation

� PCE – low computation, high accuracy

Distributions of Key States and Objectives at the Final Batch Time

0 2000 4000 60000

0.02

0.04

0.06

µ 0

0 2 4 6

x 105

0

0.02

0.04

0.06

µ 1

4 6 8 10

x 107

0

0.02

0.04

0.06

µ 2

2.95 3 3.05

x 1010

0

0.02

0.04

µ 3

1 1.5 2

x 1013

0

0.02

0.04

µ 4

0.429 0.43 0.4310

0.02

0.04

C

2000 2200 24000

0.02

0.04

0.06

µ seed

,1

2 2.5 3

x 106

0

0.02

0.04

0.06

µ seed

,2

2 2.5 3 3.5

x 109

0

0.02

0.04µ se

ed,3

1 1.20

0.01

0.02

0.03

Coe

ff. v

ar.

6 8 10 120

0.02

0.04

0.06

Mas

s ra

tio

400 500 600 7000

0.02

0.04

Wei

ght m

ean

size

0 2000 4000 60000

0.02

0.04

0.06

0.08

µ 0

0 2 4 6

x 105

0

0.02

0.04

0.06

0.08

µ 1

4 6 8 10

x 107

0

0.02

0.04

0.06

µ 2

2.95 3 3.05

x 1010

0

0.02

0.04

µ 3

1 1.5 2

x 1013

0

0.02

0.04

0.06

µ 4

0.429 0.43 0.4310

0.02

0.04

C

2000 2200 24000

0.02

0.04

0.06

µ seed

,1

2 2.5 3

x 106

0

0.02

0.04

0.06

µ seed

,2

2 2.5 3 3.5

x 109

0

0.02

0.04

0.06

µ seed

,3

0.9 1 1.1 1.20

0.02

0.04

Coe

ff. v

ar.

6 8 10 120

0.02

0.04

0.06

Mas

s ra

tio

400 500 600 7000

0.02

0.04

0.06

Wei

ght m

ean

size

0 2000 4000 60000

0.02

0.04

0.06

µ 0

0 2 4 6

x 105

0

0.02

0.04

0.06

µ 1

4 6 8 10

x 107

0

0.02

0.04

0.06

µ 2

2.95 3 3.05

x 1010

0

0.02

0.04

µ 3

1 1.5 2

x 1013

0

0.02

0.04

µ 4

0.429 0.43 0.4310

0.02

0.04

C

2000 2200 24000

0.02

0.04

0.06

µ seed

,1

2 2.5 3

x 106

0

0.02

0.04

0.06

µ seed

,2

2 2.5 3 3.5

x 109

0

0.02

0.04

µ seed

,3

1 1.20

0.01

0.02

0.03

Coe

ff. v

ar.

6 8 10 120

0.02

0.04

0.06

Mas

s ra

tio

400 500 600 7000

0.02

0.04

Wei

ght m

ean

size

19

MC simulation using dynamic model

(80,000 points; 8 hr)

MC simulation using second-order PSE model

(80,000 points; 4 min)

MC simulation for PCE

(second-order; 2 sec)

• Polynomial chaos expansion gave highly accurate pdfs in 2 sec instead of 8 hours

• pdfs from power series expansion not as accurate (could be improved by using regression rather than local derivatives)

Time-varying Distributions Along the Batch

0.06

Insights obtained by distributional robustness anal ysis (via PCE) can be used to revise control formulation s

20

0

20

40

60

80

100

120

140

160 200

300

400

500

600

700

0

0.02

0.04

0.06

weight mean size (µm)Time (min.)

p.d.

f.

Low sensitivity at t = 120; batch could be stopped here if yield is acceptable

Time (min) Weight mean size (microns)

pdf

� Motivation



Overview

21





Worst-case Robustness Analysis for Finite-time Systems

Algorithm:

– Define uncertainties as norm -bounded perturbations on real model parameters (Hölder p-norms)

– Apply structured singular value analysis to estimat e hard bounds on state deviations based on power

22

hard bounds on state deviations based on power series expansions with time-varying coefficients

– Verify and/or improve estimates using higher order series and/or nonlinear dynamic simulation

Worst-case Uncertainty Description

ˆθ θ δθ= +

{ }ˆ: , 1p

= = + ≤Wθ θε θ θ θ δθ δθ

Wθ – positive-definite weighting matrix

• Uncertain model parameters:• Uncertainty described using Hölder p-norms

1

, 1n

ppip

i

x x p=

= ≥∑

ixx max=∞

23

,min ,maxi i i≤ ≤θ θ θ


• Ex: independent bounds on each parameter

,max ,min,

,max ,min

2ˆ , ,2

i ii ii

i ip

+= = = ∞

−Wθ

θ θθ

θ θ

ii

xx max=∞

“Worst-case” Uncertainty Description

• Uncertainty described using Holder p -norms

• Ex: hyperellipsoidal uncertainty from ID expts.

{ }ˆ: , 1p

= = + ≤Wθ θε θ θ θ δθ δθ


24

{ }T 1 2ˆ ˆ: ( ) ( ) ( )n−= − − ≤Vθ θε θ θ θ θ θ χ α

– positive-definite covariance matrix

α – confidence level

χ2 – chi-squared distribution function2 1/2 1/2( ( )) , 2n p− −= =W Vθ θχ α

• Ex: hyperellipsoidal uncertainty from ID expts.

Vθ

Using Power Series Expansion for Worst-case Robustness Analysis

Specify uncertainty description for

parameters

Use sensitivity analysis or find sample points

for model runs

Compute time-varying coefficients in PSE

Add higher order terms to

25

Generate 1st order PSE to approximate model

response

Estimate error of expansion

Is error too large?

Apply structured singular value for uncertainty analysis

order terms to PSE

YES

NO

. .1

maxp

w cθδθ

δψ δψ≤

=W

Worst-case Robustness Analysis

� To calculate worst-case state/output and parameter vector:

. .1

( ) max ( )p

w c t t≤

=Wθδθ

δψ δψ δψw.c.

δθw.c.

� Use SSV to estimate worst-case state via power seri es expansion (PSE) around the nominal control trajecto ry:

26

,̂ ( )

( )( ) n

u t

tL t

θ

ψ

θ∈

∂ = ∂R

� Verification and/or improvement of estimates (use h igher order or dynamic simulation using δθδθδθδθw.c.)

1

2TLδψ δθ δθ δθ= + +M …

where the sensitivities along the control trajector y are2

2,̂ ( )

( ) n n

u t

tθ

ψ

θ

×∂= ∈∂

M R


• Apply structured singular value µ as a matrix operator:

• Can write (Braatz et al, IEEE TAC 2004)

⋯

min max. .

(N)M max maxT

w ck

L kθ θ θ µθ θ θ µθ θ θ µθ θ θ µ

δψ δθ δθ δθδψ δθ δθ δθδψ δθ δθ δθδψ δθ δθ δθ∆∆∆∆≤ ≤ ≥≤ ≤ ≥≤ ≤ ≥≤ ≤ ≥

= + + == + + == + + == + + =

{{{{ }}}}1(N) : det(I N ) 0;minµµµµ −−−−∆∆∆∆ = ∆ − ∆ = ∆ ∈ Γ= ∆ − ∆ = ∆ ∈ Γ= ∆ − ∆ = ∆ ∈ Γ= ∆ − ∆ = ∆ ∈ Γ

27

where

• Construct N by multidimensional realization (IJRNC 9 7)• Called “skewed mixed mu” in the literature (R Smith 8 7)• For p ≠ 2, called “skewed mixed generalized mu”

(Jie Chen et al, IEEE TAC, 41, 1511, 2006)• This approach generalizes µ to analyze robustness in

finite-time nonlinear time-varying systems

min max (N) kθ θ θ µθ θ θ µθ θ θ µθ θ θ µ∆∆∆∆≤ ≤ ≥≤ ≤ ≥≤ ≤ ≥≤ ≤ ≥

{{{{ }}}}diag ⋯, , ,r r cδδδδ∆ = ∆ ∆∆ = ∆ ∆∆ = ∆ ∆∆ = ∆ ∆

• The worst-case state/output and parameter vector

for the first-order power series expansion has an analytical solution, for example, for finite p

. .1

( ) max ( )w c t L tθδθ

δψ δθ≤

=W

δψw.c., δθw.c.

Computation of SSV Problems

( )( 1)/p p−

( )( )1/( 1)

1p

L−

−W

28

• For higher orders, compute polynomial-time upper and lower bounds, e.g., for p = 2 using iteration/LMIs:

( )( )( 1)/

/( 1)1

. .1

p pn p p

w c kk

Lθ

δψ

−−

−

=

= ∑ W

( )( )

( )( )

1

1

. . 1//( 1)

1

1

k

w c pn p p

kk

L

L

θ

θ

θ

δθ

−

−

−−

=

=± ∑

W

W

W

1max ( ) (N) inf ( )D D

N DNDρ µ σρ µ σρ µ σρ µ σ −−−−∆∆∆∆ ∆=∆∆=∆∆=∆∆=∆∆∈Γ∆∈Γ∆∈Γ∆∈Γ

∆∈Γ∆∈Γ∆∈Γ∆∈Γ

∆ ≤ ≤∆ ≤ ≤∆ ≤ ≤∆ ≤ ≤


Illustration of SSV approach for second-order PSE:

min max. . Mmax T

w c Lθ θ θθ θ θθ θ θθ θ θ

δψ δθ δθ δθδψ δθ δθ δθδψ δθ δθ δθδψ δθ δθ δθ≤ ≤≤ ≤≤ ≤≤ ≤

= += += += + whereˆ

jj

Lθ θθ θθ θθ θ

ψψψψθθθθ

====

∂∂∂∂====∂∂∂∂

2

ˆ

Miji j

ψψψψθ θθ θθ θθ θ∂∂∂∂====

∂ ∂∂ ∂∂ ∂∂ ∂

29

ˆi j θ θθ θθ θθ θθ θθ θθ θθ θ

====∂ ∂∂ ∂∂ ∂∂ ∂

⇔⇔ ⇔⇔(((( ))))N

max∆∆∆∆ ≥≥≥≥k

kµµµµ

where

0 0

N M 0 M

M MT T T

kw

k k z

z L w z z Lz

====

+ ++ ++ ++ +

(((( ))))1max min2w θ θθ θθ θθ θ= −= −= −= −(((( ))))1

max min2z θ θθ θθ θθ θ= += += += +

{{{{ }}}}diag , ,∆ = ∆ ∆∆ = ∆ ∆∆ = ∆ ∆∆ = ∆ ∆r r cδδδδ

Braatz et al, IEEE TAC, 2004

� Motivation



Overview

30





Example: 2D Reaction -Diffusion Process

• Step 1: Construct a power series/polynomial chaos expansion between the parameters and states/outputs

• Step 2: Apply robustness analysis to the expansion

• Applies to worst -case or stochastic uncertainties, e.g.,• Applies to worst -case or stochastic uncertainties, e.g.,

– D. L. Ma, S. H. Chung, & R. D. Braatz. AIChE J. 1999

– Z. K. Nagy & R. D. Braatz, J. of Process Control, 20 07

• This example shows worst-case analysis of the effec ts of parametric uncertainties on boundary control problems for finite-time DPS using series expansion s

31

Example Boundary Control Problem

32

Example Boundary Control Problem (cont.)

33


34


35


36


37


38


39


40


41


42


43


44

� Motivation



Overview

45





Summary & Further Comments

� Presented approaches for the distributional and wor st-case robustness analysis of finite-time distributed para meter systems

� Based on power series or polynomial chaos expansion s

� Analysis results are computed at all times during t he process operation

� Computationally very efficient compared to Monte Ca rlo or gridding

� Provided two simple examples; see papers for applic ations to

46

� Provided two simple examples; see papers for applic ations to microelectronics, suspension polymerization, pharma ceutical crystallization, and lithium -ion batteries (brahms.scs.uiuc.edu)

� Have incorporated methods into robust model-based c ontroller design for finite-time systems, including nonlinear MPC

http://brahms.scs.uiuc.edu

Acknowledgements

• Dr. Zoltan Nagy, U Loughborough• Masako Kishida

• National Science Foundation• National Institutes of Health

47

Some References

• D.L. Ma & R.D. Braatz, IEEE Trans. on Control Syst. Tech., 9:766-774, 2001; Z.K . Nagy & R.D. Braatz, IEEE Trans. on Control Syst. Tech., 11:494-504, 2003 (power series expansion approach)

• Z.K. Nagy & R.D. Braatz. J. of Process Control,17:229-240, 2007 (polynomial chaos expansion 17:229-240, 2007 (polynomial chaos expansion approach)

• M. Kishida & R.D . Braatz, Proc. of Mathematical Theory of Networks and Systems, Blacksburg, VA, paper SSRussell1.4, 2008 (reaction-diffusion result s are from expanded journal version submitted in 2010 )

• Z.K. Nagy & R.D. Braatz, AIChE J., 49:1776-1786, 2003 (incorporation into robust MPC algorithms)

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robust optimal control of finite-time distributed …robust optimal control of finite-time...

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