Robust Optimal Control of Finite-time Distributed Parameter Systems - - PowerPoint PPT Presentation

Robust Optimal Control of Finite-time Distributed Parameter Systems

McMaster University

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Richard D. Braatz

University of Illinois at Urbana-Champaign

Motivation Distributional robustness analysis via polynomial chaos expansions Example: Batch crystallization

Overview

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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 systems – require tight control of dimensions, chemistry, and/or biology – have models with significant associated uncertainties

• Objective: computationally efficient methods for the

robust optimal control of these processes

Model Uncertainties

Many sources of uncertainties, frequent disturbances

• 4
• Model/plant

mismatch

• Benefits of model-based control can be lost

if uncertainties are not explicitly addressed

• Focus on parameter uncertainties (others are similar)

Focus of Presentation

• Standard approaches for robustness analysis

– Monte Carlo method is computationally expensive – Gridding the parameter space is computationally expensive,

• rder of 100n for n parameters

– Lyapunov functions difficult to construct non- conservatively for general nonlinear DPS

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• Important questions:

– How to nonconservatively analyze the effects of model uncertainties in a computationally feasible manner?

• – How to use this information for robust controller design?
• 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;

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• 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

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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 polynomial functions

• f the real model parameters

Different orthogonal functions are optimal for different parameter pdfs (e.g., Gaussian, Gamma, Beta, Uniform, Poisson, Binomial, …)

• 1

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

1 2 3 1 1 1 1 1 1 constant first order terms third order terms second order terms

( ) ( , ) ( , , )

i i i n n n i i i i i i i i i i i i i i i i i i

a a a a

= = = = = =

= Γ + Γ + Γ + Γ +

∑ ∑∑ ∑ ∑ ∑

⋯

• ψ

θ θ θ θ θ θ

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pdfs (e.g., Gaussian, Gamma, Beta, Uniform, Poisson, Binomial, …) Widely used in the environmental field, called “uncertainty quantification” or “uncertainty propagation” Coefficients can be computed from collocation or regression Choose collocation points or sampling points to span as much as possible the behavior of the states Related algorithms expand parameters and states in terms of the

• rthogonal functions, to give explicit analytical pdfs 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

• rder PCE

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Select optimal

• rthogonal 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

• rder PCE

terms to polynomials

YES NO Optimal orthogonal polynomials are more efficient than power series expansions

Estimation of Time-varying pdfs of States

Monte 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 expensive

Instead

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

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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 Crystallization

f(r1, r2)

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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 = temperature

( , ) ( , ) ( ) d 3 ( , ) ( , ) d

i

f f G C T B C T L t L C G 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

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1

( , ) ( , ) ( ) d ( , )0 ( , ) d ( , )

i i i i i i

f f G C T B C T L L dL t L m B 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 process

f

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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

• ff-spec product ($1-2 million/batch)

ψ Allowed variability for quality

Optimal Control Problem

• ( )

( ) ( )

• ≤

≤

subject to: IPDE process model

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

( ) ( ) ( ) ( ) ( ) ( )

• ≤

≤ ≤ ≤ ≤

30 31 32 T (°C)

Importance of Robustness Analysis

• !

"

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40 80 120 160 28 29 Time (min)

#$

% "

Optimal profile may only nominally give 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

• MC with second-order PSE

x polynomial chaos expansion – MC with nonlinear model

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first-order PSE provides rough estimate, with some bias PCE provides very good accurate quantification of pdfs

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

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Monte Carlo with second-order PSE (80,000 points) 4 min Polynomial chaos expansion (second-order) 2 sec Full Monte Carlo – very high computational requirements First-order PSE approach – computationally very attractive Second-order PSE – increased accuracy & computation PCE – low computation, high accuracy

Distributions of Key States and Objectives at the Final Batch Time

• Coeff. var.

• Coeff. var.

• Coeff. var.

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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 analysis (via PCE) can be used to revise control formulations

20 20 40 60 80 100 120 140 160 200 300 400 500 600 700 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 Distributional robustness analysis via polynomial chaos expansions Example: Batch crystallization

Overview

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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

• n real model parameters (Hölder p-norms)

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

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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

ˆ θ θ δθ = +

{ }

ˆ : , 1

p

= = + ≤ W

θ θ

ε θ θ θ δθ δθ

Wθ – positive-definite weighting matrix

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

1

, 1

n p p i p i

x x p

=

= ≥

∑

i

x x max =

∞

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,min ,max i i i

≤ ≤ θ θ θ

Wθ – positive-definite weighting matrix

• Ex: independent bounds on each parameter

,max ,min , ,max ,min

2 ˆ , , 2

i i i ii i i

p + = = = ∞ − W

θ

θ θ θ θ θ

i i

x x max =

∞

“Worst-case” Uncertainty Description

• Uncertainty described using Holder p-norms
• Ex: hyperellipsoidal uncertainty from ID expts.

{ }

ˆ : , 1

p

= = + ≤ W

θ θ

ε θ θ θ δθ δθ

Wθ – positive-definite weighting matrix

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{ }

T 1 2

ˆ ˆ : ( ) ( ) ( )

n −

= − − ≤ V

θ θ

ε θ θ θ θ θ χ α

– positive-definite covariance matrix α – confidence level χ 2 – chi-squared distribution function

2 1/2 1/2

( ( )) , 2

n

p

− −

= = W V

θ θ

χ α

• Ex: hyperellipsoidal uncertainty from ID expts.

V

θ

Using Power Series Expansion for Worst-case Robustness Analysis

Specify uncertainty description for parameters Use sensitivity analysis

• r find sample points

for model runs Compute time-varying coefficients in PSE Add higher

• rder terms to

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Generate 1st order PSE to approximate model response Estimate error of expansion Is error too large? Apply structured singular value for uncertainty analysis

• rder terms to

PSE

YES NO

. . 1

max

p

w c

θδθ

δψ δψ

≤

=

W

Worst-case Robustness Analysis

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

. . 1

( ) max ( )

p

w c t

t

≤

=

W

θδθ

δψ δψ

δψw.c. δθw.c.

Use SSV to estimate worst-case state via power series expansion (PSE) around the nominal control trajectory:

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• θ

ψ θ

∈

  ∂   =        ∂

• Verification and/or improvement of estimates (use higher
• rder or dynamic simulation using δθ

δθ δθ δθw.c.)

1 2

T

L δψ δθ δθ δθ = + + M …

where the sensitivities along the control trajectory are

• θ

ψ θ

×

∂ = ∈ ∂

          

Worst-case Robustness Analysis

• Apply structured singular value as a matrix operator:
• Can write (Braatz et al, IEEE TAC 2004)

⋯

min max

. . (N)

M max max

T w c k

L k

θ θ θ µ θ θ θ µ θ θ θ µ θ θ θ µ

δψ δθ δθ δθ δψ δθ δθ δθ δψ δθ δθ δθ δψ δθ δθ δθ

∆ ∆ ∆ ∆

≤ ≤ ≥ ≤ ≤ ≥ ≤ ≤ ≥ ≤ ≤ ≥

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

{ { { { } } } }

1(N)

: det(I N ) 0; min µ µ µ µ −

− − − ∆ ∆ ∆ ∆

= ∆ − ∆ = ∆ ∈ Γ = ∆ − ∆ = ∆ ∈ Γ = ∆ − ∆ = ∆ ∈ Γ = ∆ − ∆ = ∆ ∈ Γ

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where

• Construct N by multidimensional realization (IJRNC 97)
• Called “skewed mixed mu” in the literature (R Smith 87)
• For p ≠ 2, called “skewed mixed generalized mu”

(Jie Chen et al, IEEE TAC, 41, 1511, 2006)

• This approach generalizes to analyze robustness in

finite-time nonlinear time-varying systems

min max

(N) k θ θ θ µ θ θ θ µ θ θ θ µ θ θ θ µ∆

∆ ∆ ∆

≤ ≤ ≥ ≤ ≤ ≥ ≤ ≤ ≥ ≤ ≤ ≥

{ { { { } } } }

diag ⋯ , , ,

r r c

δ δ δ δ ∆ = ∆ ∆ ∆ = ∆ ∆ ∆ = ∆ ∆ ∆ = ∆ ∆

• The worst-case state/output and parameter vector

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

• θδθ

δψ δθ

≤

=

• δψδθ

Computation of SSV Problems

( )

• −

 

( )

( )

• −

−

• 28
• For higher orders, compute polynomial-time upper

and lower bounds, e.g., for p = 2 using iteration/LMIs:

( )

( )

• θ

δψ

− − − =

     =      

∑

• (

)

( )

( )

( )

• θ

θ θ

δθ

− − − − =

=±            

∑

• 1

max ( ) (N) inf ( )

D D

N DND ρ µ σ ρ µ σ ρ µ σ ρ µ σ

− − − − ∆ ∆ ∆ ∆ ∆=∆ ∆=∆ ∆=∆ ∆=∆ ∆∈Γ ∆∈Γ ∆∈Γ ∆∈Γ ∆∈Γ ∆∈Γ ∆∈Γ ∆∈Γ

∆ ≤ ≤ ∆ ≤ ≤ ∆ ≤ ≤ ∆ ≤ ≤

Worst-case Robustness Analysis

Illustration of SSV approach for second-order PSE:

min max

. .

M max

T w c

L

θ θ θ θ θ θ θ θ θ θ θ θ

δψ δθ δθ δθ δψ δθ δθ δθ δψ δθ δθ δθ δψ δθ δθ δθ

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

= + = + = + = + where

ˆ j j

L

θ θ θ θ θ θ θ θ

ψ ψ ψ ψ θ θ θ θ

= = = =

∂ ∂ ∂ ∂ = = = = ∂ ∂ ∂ ∂

2 ˆ

Mij

i j

ψ ψ ψ ψ θ θ θ θ θ θ θ θ ∂ ∂ ∂ ∂ = = = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

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ˆ i j θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ

= = = =

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⇔ ⇔ ⇔ ⇔

( ( ( ( ) ) ) )

N

max

∆ ∆ ∆ ∆

≥ ≥ ≥ ≥k k µ µ µ µ

where

N M M M M

T T T

kw k k z z L w z z Lz                 = = = =         + + + + + + + +                

( ( ( ( ) ) ) )

1 max min 2

w θ θ θ θ θ θ θ θ = − = − = − = −

( ( ( ( ) ) ) )

1 max min 2

z θ θ θ θ θ θ θ θ = + = + = + = +

{ { { { } } } }

diag , , ∆ = ∆ ∆ ∆ = ∆ ∆ ∆ = ∆ ∆ ∆ = ∆ ∆

r r c

δ δ δ δ

Braatz et al, IEEE TAC, 2004

Motivation Distributional robustness analysis via polynomial chaos expansions Example: Batch crystallization

Overview

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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, 2007

• This example shows worst-case analysis of the effects
• f parametric uncertainties on boundary control

problems for finite-time DPS using series expansions

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Example Boundary Control Problem

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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Example Boundary Control Problem (cont.)

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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 worst-case robustness analysis of finite-time distributed parameter systems

Based on power series or polynomial chaos expansions Analysis results are computed at all times during the process operation Computationally very efficient compared to Monte Carlo or gridding

Provided two simple examples; see papers for applications to

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Provided two simple examples; see papers for applications to microelectronics, suspension polymerization, pharmaceutical crystallization, and lithium-ion batteries (brahms.scs.uiuc.edu) Have incorporated methods into robust model-based controller 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

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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 results 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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