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Crystallization Control: feedback policy from data

code published for "Data-Driven Modeling and Dynamic Programming Applied to Batch Cooling Crystallization"

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Overview

This code has been posted in association with the manuscript titled "Data-Driven Modeling and Dynamic Programming Applied to Batch Cooling Crystallization" by D. J. Griffin, M. A. Grover, Y. Kawajiri, and R. W. Rousseau. The main function is ObtainPolicy.m. This is a MATLAB function that takes experimental data on crystallization dynamics and outputs a feedback control policy for reaching a target crystal mass and chord count. For more details, please consult the associated manuscript.

Running ObtainPolicy

To run this function, the m-files in the subfunctions folder must be in the same path. In addition, CVX (Software for Disciplined Convex Programming [1],[2]) is required. This must be installed and in the appropriate path: http://cvxr.com/cvx/download/. Note: for commercial use with non-free solvers, such as MATLAB, please obtain the appropriate license (http://cvxr.com/cvx/licensing/).

Viewing and Using the Output Policy

The output policy can be viewed as a time-varying colormap using viewPolicy.m. The output policy can be interpreted using inputFromPolicy.m.

Example Data

Example data has been provided in ExampleData.mat. Once the required functions have been added to the appropriate directory in MATLAB, the policy given for Test Run 2 can be obtained by opening the data file (so that each input variable is specified in the Workspace) and then entering the Command >> [Pol]=ObtainPolicy(Xtr,Utr,dXtr,dt,Dt,xTarget,N,rho,gamma,Grid).

Inputs to ObtainPolicy

There are a number of inputs to ObtainPolicy. The inputs specify: the training data, the length of the time steps, the target position in mass-count space, the batch run time, adjustable parameters in the optimization formulation, and the space-input discretization to use.

REQUIRED

Xtr - Contains training data 'positions'.

Utr - Contains training data inputs.

dXtr - Contains training data 'movements'.

dt - The time interval between measurements [min.].

Dt - The time interval over which the input is held constant [min.].

xTarget - The target position [count, mass].

N - The number of time intervals in the control run.

OPTIONAL

rho - Parameter that weights the input-effort cost (default: 25).

gamma - Parameter that weights the running distance-to-target (default: 5).

Grid - Structure containing the grid spacing for: count (Grid.c), mass (Grid.m), supersaturation (Grid.s), and the count "normalizing scale" (Grid.scale).

Note on Computation Time

Depending on input data and discretization, the function may require substantial computation time (on the order of 30 minutes for the example data with the given Grid). The modeling step takes the longest. Prompts and visuals have been added as progress checks. These should be commented-out to run the function unattended.

Bibliography

[1] Michael Grant and Stephen Boyd. CVX: Matlab software for disciplined convex programming, version 2.0 beta. http://cvxr.com/cvx, September 2013.

[2] Michael Grant and Stephen Boyd. Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control (a tribute to M. Vidyasagar), V. Blondel, S. Boyd, and H. Kimura, editors, pages 95-110, Lecture Notes in Control and Information Sciences, Springer, 2008. http://stanford.edu/~boyd/graph_dcp.html.