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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
� Example: Batch crystallization
� 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
� Distributional robustness analysis via polynomial chaos expansions
� Example: Batch crystallization
Overview
7
� Example: Batch crystallization
� Worst-case robustness analysis via power series expansions
� Example: 2D reaction -diffusion
� Summary and future comments
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
� Large number of simulations for correct PDF (tail effect)
� Computationally very expensiveInstead
� Maps entire pdfs of model parameters to state pdfs
� Large number of simulations for correct PDF (tail effect)
� Computationally inexpensive
Surrogate model
Algebraic expression w/time-varying coefficients
� Motivation
� Distributional robustness analysis via polynomial chaos expansions
� Example: Batch crystallization
Overview
11
� Example: Batch crystallization
� Worst-case robustness analysis via power series expansions
� Example: 2D reaction -diffusion
� Summary and future comments
� 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)
� Motivation
� Distributional robustness analysis via polynomial chaos expansions
� Example: Batch crystallization
Overview
21
� Example: Batch crystallization
� Worst-case robustness analysis via power series expansions
� Example: 2D reaction -diffusion
� Summary and future comments
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
Wθ – positive-definite weighting matrix
• 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
Wθ – positive-definite weighting matrix
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
Worst-case Robustness Analysis
• 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
∆ ≤ ≤∆ ≤ ≤∆ ≤ ≤∆ ≤ ≤
Worst-case Robustness Analysis
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
� Distributional robustness analysis via polynomial chaos expansions
� Example: Batch crystallization
Overview
30
� Example: Batch crystallization
� Worst-case robustness analysis via power series expansions
� Example: 2D reaction -diffusion
� Summary and future comments
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
Example Boundary Control Problem (cont.)
34
Example Boundary Control Problem (cont.)
35
Example Boundary Control Problem (cont.)
36
Example Boundary Control Problem (cont.)
37
Example Boundary Control Problem (cont.)
38
Example Boundary Control Problem (cont.)
39
Example Boundary Control Problem (cont.)
40
Example Boundary Control Problem (cont.)
41
Example Boundary Control Problem (cont.)
42
Example Boundary Control Problem (cont.)
43
Example Boundary Control Problem (cont.)
44
� Motivation
� Distributional robustness analysis via polynomial chaos expansions
� Example: Batch crystallization
Overview
45
� Example: Batch crystallization
� Worst-case robustness analysis via power series expansions
� Example: 2D reaction -diffusion
� Summary and future comments
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)
48