(654b) Design of Optimal Sensor Network Based On Economic Objectives

In a typical chemical process, there are several thousands of variables that need to be monitored. However, it is feasible to measure only a small subset of the candidate measurements. The problem of measurement selection in process plants has received considerable attention from the process systems engineering community [1] and other fields [2]. Traditional sensor network design objectives have been to optimize some metric based on observability, controllability, cost, reliability, precision of estimation etc. However, these metrics are inherently incommensurate, and it is difficult to rationalize these incommensurate objectives. Of late attempts have been to select measurements based on the process economics from control [3], plant-wide or self-optimizing control [4], fault diagnosis [5], precision perspectives [6].

In this contribution, we address the problem of selecting sensors for steady state processes when the sensors are corrupted by random Gaussian errors. A subset of the process variables are directly measured and the rest are inferred from their dependencies on measured variables using a process model. Rather than minimizing average measurement error or any other traditional objective, we choose measurements based on process economics. To this end, we follow the idea that any deviation from optimal operation results in a loss [4]. Expanding the profit function about the optimal point, we obtain an approximation for the profit function that is accurate upto second order [4]. This approximation is used to compute the loss function. This is shown to be equivalent to a weighted sum of error variances (and covariances) of individual measurements, where the weights truly reflect the economic importance of each individual measurement and their interactions.

This loss function is used to design an optimal sensor network based on economic objectives. It is shown that the resulting problem can be expressed as a convex integer programming problem with linear cost function and Linear Matrix Inequality (LMI) constraints. Hence, the problem can be solved by a branch and bound method with a convex problem being solved at each stage of the branching process. This convex problem can be solved efficiently using CVX [7,8], a freely available software for solving convex optimization problems. This method is illustrated with case studies.

References

1. D. J. Chmielewski, T. Palmer and V. Manousiouthakis..On the Theory of Optimal Sensor Placement. AIChE J., 48(5), 1001-1012, 2002.

2. S. Joshi and S. Boyd. Sensor Selection via Convex Optimization. IEEE Trans. Sig. Proc., 57(2), 451-462, 2009.

3. Chmielewski D. J. and Peng J. K. Covariance Based Hardware Selection-Part I: Globally Optimal Actuator Selection. IEEE Trans. Cont. Sys. Tech., 14(2), 355- 361, 2006.

4. V. Alstad, S. Skogestad, E.S. Hori, Optimal measurement combinations as controlled variables. Journal of Process Control, 19, 138?148, 2009.

5. S. Narasimhan and R. Rengaswamy. Quantification of Performance of Sensor Networks for Fault Diagnosis. AIChE J., 53(4), 902{917, 2007.

6. Bagajewicz M., Markowski M. and Budek A. Economic Value of Precision in the Monitoring of Linear Systems. AIChE J., 51(4), 1304-1309, 2005.

7. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming. 2008. (web page and software). [Online]. Available: http://stanford.edu/ boyd/cvx

8. M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs. In Recent Advances in Learning and Control (tribute to M. Vidyasagar), V. Blondel, S. Boyd, and H. Kimura, Eds. Springer Verlag, 2008.