Robust Optimal Control of Finite-time Distributed Parameter Systems Richard D. Braatz University of Illinois at Urbana-Champaign 1 McMaster University
Overview � Motivation � Distributional robustness analysis via polynomial chaos expansions � Example: Batch crystallization � Example: Batch crystallization � Worst-case robustness analysis via power series expansions � Example: 2D reaction-diffusion � Summary and future comments 2
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 ��������� ����������� ������ ����������� ������������� ����� ���������� ������������ ������������ ��� ������� ���������� �������� Model/plant mismatch ���������� �������������� ������ ������������� • Benefits of model-based control can be lost if uncertainties are not explicitly addressed • Focus on parameter uncertainties (others are similar) 4
Focus of Presentation • Standard approaches for robustness analysis – Monte Carlo method is computationally expensive – Gridding the parameter space is computationally expensive, order of 100 n for n parameters – Lyapunov functions difficult to construct non- conservatively for general nonlinear DPS • 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 5
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; 3. Evaluate accuracy of surrogate model; increase accuracy if needed states final time time 6
Overview � Motivation � Distributional robustness analysis via polynomial chaos expansions � Example: Batch crystallization � Example: Batch crystallization � Worst-case robustness analysis via power series expansions � Example: 2D reaction-diffusion � Summary and future comments 7
Polynomial Chaos Expansion � Model state as an expansion of orthogonal polynomial functions of the real model parameters i i i n n n 1 2 ∑ ∑∑ ∑ ∑ ∑ 1 ψ = Γ + Γ θ + Γ θ θ + Γ θ θ θ + a a a a ⋯ ( ) ( , ) ( , , ) � i i i i i i i i i i i i 1 2 3 0 0 1 1 1 2 1 2 1 2 3 1 2 3 = = = = = = i i i i i i 1 1 1 1 1 1 �� � �� � � ����� � ������ � ���� ���� � constant 1 1 2 1 2 3 first order terms third order terms second order terms � Different orthogonal functions are optimal for different parameter pdfs (e.g., Gaussian, Gamma, Beta, Uniform, Poisson, Binomial, …) 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 orthogonal functions, to give explicit analytical pdfs for states 8
Using Polynomial Chaos Expansion for Distributional Robustness Analysis Generate collocation points for model runs Specify uncertainty description for Run model at parameters collocation points and compute coefficients Add higher Select optimal Select optimal order PCE order PCE orthogonal polynomials terms to Estimate error of for the distribution polynomials expansion Generate first-order YES PCE to approximate Is error too model response large? Optimal orthogonal NO polynomials are more efficient Use expansion for than power series expansions uncertainty analysis 9
Estimation of Time-varying pdfs of States Monte Carlo simulation Full dynamic model IPDAEs, multiscale models � Maps entire pdfs of model parameters to state pdfs � Large number of simulations for correct PDF (tail effect) � Large number of simulations for correct PDF (tail effect) � Computationally very expensive Instead Surrogate model Algebraic expression w/time- varying coefficients � Maps entire pdfs of model parameters to state pdfs � Large number of simulations for correct PDF (tail effect) � Computationally inexpensive
Overview � Motivation � Distributional robustness analysis via polynomial chaos expansions � Example: Batch crystallization � Example: Batch crystallization � Worst-case robustness analysis via power series expansions � Example: 2D reaction-diffusion � Summary and future comments 11
Example: Batch Crystallization � Used to purify drugs, reaction intermediates, explosives, … f(r 1 , r 2 ) � Temperature reduced to cause crystals to nucleate & grow � Objective: Make large crystals r 2 (µm) r 1 (µm) from seed; minimize nucleation � Large uncertainties in kinetic parameters in model � IPDE population balance model � � � � analysis of moments enabled full evaluation of methods 12
IPDAE Process Model ∂ ∂ f f • G = growth + = δ G C T B C T L ( , ) ( , ) ( ) ∂ ∂ t L • B = nucleation ∞ C d ∫ • C = concentration = − α G C T f L t dL 3 ( , ) ( , ) t d • T = temperature 0 ∞ ∞ ∂ ∂ ∂ ∂ f f f f i i ∫ ∫ + = δ = G C T B C T L L dL ( , ) ( , ) ( ) 0 ∂ ∂ t L 0 m d i i • m i = i th moment ⇒ = + B C T nG C T m ( , )0 ( , ) − i 1 t d • Alternatively could ∞ solve by method of i ∫ ≡ m f L t L dL ( , ) i characteristics or 0 finite differences 13
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 � Control of CSD properties important ψ � Efficiency of downstream operations (filtration, drying) � Product effectiveness (tablet stability) Allowed � Strict regulatory requirements on variability variation of product crystals for quality � High economic penalty of producing off-spec product ($1-2 million/batch) 14
Optimal Control Problem �������� � � � � � subject to: IPDE process model ( ) ( ) ( ) ( ) ( ) � ( ) ≤ ≤ ≤ ≤ � � � � � � � � � � � � � ��� ��� ( ) �� � ( ) ( ) ≤ ≤ � � � � � ��� ��� �� ≤ � � � ����� ����� ���� � Objectives: nucleation mass to seed mass ratio, weight mean size 15
Importance of Robustness Analysis � ������������������������� 32 ���������������������� ������� � ��� ������������ 31 �������������������������� T ( ° C) �������� 30 � ����!�������������� ����������������� ���� " ����������������� ���� " 29 � #���������������$����������� �������������%�������������� 28 0 40 80 120 160 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 16
Time-varying pdfs, Comparing Approaches . first-order PSE o MC with second-order PSE x polynomial chaos expansion – MC with nonlinear model � first-order PSE provides rough estimate, with some bias � PCE provides very good accurate quantification of pdfs 17
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