Optimal control of finite increments of model factors based on sensitivity analysis

The article addresses the topical inverse problem of target-oriented control: determining the necessary finite changes to the system's input factors to achieve a desired target state, as opposed to the classical direct problem of forecasting. To solve it, a new methodological approach is proposed. This approach is based on sensitivity analysis utilizing the Lagrange mean value theorem. This framework allows for moving beyond local linearization to precisely account for nonlinear effects and factor interactions under substantial, practically observed changes. The key scientific result is the development of a universal iterative algorithm, which, for a given mathematical model, determines the vector of finite changes for the controllable factors that ensures the required increment in the output indicator with minimal total cost of the introduced changes and within given constraints. At each iteration step, the model's gradient (sensitivity estimate) is computed at an intermediate point, whose position is sequentially refined, and an auxiliary constrained optimization problem is solved. The practical efficiency and operability of the proposed method are verified using a numerical example with the nonlinear Ishigami model. The algorithm successfully found the optimal control action, ensuring high accuracy in achieving the target.

1. Hamby D.M. A review of techniques for parameter sensitivity analysis of environmental models. Environmental Monitoring and Assessment. 1994;32(2):135–154. https://doi.org/10.1007/BF00547132

2. Campolongo F., Saltelli A., Tarantola S. Sensitivity analysis as an ingredient of modeling. Statistical Science. 2000;15(4):377–395. https://doi.org/10.1214/ss/1009213004

3. Sarrazin F., Pianosi F., Wagener Th. Global Sensitivity Analysis of environmental models: Convergence and validation. Environmental Modelling & Software. 2016;79:135–152. https://doi.org/10.1016/j.envsoft.2016.02.005

4. Saltelli A., Ratto M., Andres T., et al. Global Sensitivity Analysis: The Primer. Chichester: John Wiley & Sons; 2008. 304 p.

5. Pujol G. Simplex-based screening designs for estimating metamodels. Reliability Engineering & System Safety. 2009;94(7):1156–1160. https://doi.org/10.1016/j.ress.2008.08.002

6. Hamby D.M. A comparison of sensitivity analysis techniques. Health Physics. 1995;68(2):195–204. https://doi.org/10.1097/00004032-199502000-00005

7. Box G.E.P., Meyer R.D. An analysis for unreplicated fractional factorials. Technometrics. 1986;28(1):11–18. https://doi.org/10.1080/00401706.1986.10488093

8. Dean A., Lewis S. Screening: Methods for Experimentation in Industry, Drug Discovery, and Genetics. New York: Springer; 2006. 332 p. https://doi.org/10.1007/0-387-28014-6

9. Iooss B., Lemaître P. A review on global sensitivity analysis methods. In: Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications. New York: Springer; 2015. P. 101–122. https://doi.org/10.1007/978-1-4899-7547-8_5

10. Sobol' I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation. 2001;55(1-3):271–280. https://doi.org/10.1016/S0378-4754(00)00270-6

11. Sobol' I.M. Sensitivity estimates for nonlinear mathematical models. Mathematical Modelling and Computational Experiments. 1993;1(4):407–414.

12. Homma T., Saltelli A. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering & System Safety. 1996;52(1):1–17. https://doi.org/10.1016/0951-8320(96)00002-6

13. Hoeffding W. A class of statistics with asymptotically normal distribution. The Annals of Mathematical Statistics. 1948;19(3):293–325. https://doi.org/10.1214/aoms/1177730196

14. Efron B., Stein C. The jackknife estimate of variance. The Annals of Statistics. 1981;9(3):586–596. https://doi.org/10.1214/aos/1176345462

15. Lamboni M., Kucherenko S. Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables. Reliability Engineering & System Safety. 2021;212. https://doi.org/10.1016/J.RESS.2021.107519

16. Rana S., Ertekin T., King G.R. An efficient assisted history matching and uncertainty quantification workflow using Gaussian processes proxy models and variogram based sensitivity analysis: GP-VARS. Computers & Geosciences. 2018;114:73–83. https://doi.org/10.1016/J.CAGEO.2018.01.019

17. Ratto M., Castelletti A., Pagano A. Emulation techniques for the reduction and sensitivity analysis of complex environmental models. Environmental Modelling & Software. 2012;34:1–4. https://doi.org/10.1016/j.envsoft.2011.11.003

18. Borgonovo E., Castaings W., Tarantola S. Model emulation and moment-independent sensitivity analysis: An application to environmental modelling. Environmental Modelling & Software. 2012;34:105–115. https://doi.org/10.1016/j.envsoft.2011.06.006

19. Sysoev A., Ciurlia A., Scheglevatych R., Blyumin S. Sensitivity analysis of neural network models: Applying methods of analysis of finite fluctuations. Periodica Polytechnica Electrical Engineering and Computer Science. 2019;63(4):306–311. https://doi.org/10.3311/PPee.14654

20. Blyumin S.L., Borovkova G.S., Serova K.V., Sysoev A.S. Analysis of finite fluctuations for solving big data management problems. In: Proceedings of the 2015 9th International Conference on Application of Information and Communication Technologies (AICT), 14–16 October 2015, Rostov-on-Don, Russia. IEEE; 2015. P. 48–51. https://doi.org/10.1109/ICAICT.2015.7338514