(706e) Stochastic Predictive Control with Closed-Loop Model Adaptation: Application to a Cold Atmospheric Plasma Jet

Authors: 
Ehlinger, V., University of California at Berkeley
Bavdekar, V., University of California - Berkeley
Gidon, D., University of California - Berkeley
Mesbah, A., University of California - Berkeley
Predictive controllers offer many advantages, including handling of multivariable dynamics, system constraints and competing control objectives [1]. However, the model quality often degrades over time due to system uncertainties or a limited validity range of the model as the system evolves (e.g., see [2] and the references therein). The key objective of this work is to address the problem of model uncertainty in the context of predictive control of stochastic systems. This work presents a stochastic MPC approach with integrated experiment design (iX-SMPC) for linear systems subject to parametric uncertainties and additive disturbances. The proposed approach allows for simultaneously controlling the system and probing the system dynamics to generate informative data for estimation of the uncertain model parameters. Hence, the designed control inputs have some form of dual control feature [3].

A control-oriented input design cost function is incorporated into a stochastic optimal control problem to allow for tuning the quality of model adaptation toward meeting a prespecified control performance level [4]. The closed-loop data generated by iX-SMPC can then be used for identifying the (posterior) probability distribution of model parameters and, consequently, facilitate online model maintenance during predictive control. The generalized polynomial chaos framework [5] is used to obtain a deterministic surrogate for the stochastic optimal control problem with integrated input design. For the case of expectation-type state constraints, a quadratic programming (QP) program is derived for the iX-SMPC approach. A gPC-based histogram filter [6], which relies on Bayesian estimation, is designed to re-estimate the probability density functions of the uncertain parameters using the closed-loop data obtained from the system.

The proposed iX-SMPC algorithm is implemented on a cold atmospheric plasma (CAP) jet. CAP jets have found extensive applications in materials processing and biomedical applications [7], [8]. This work considers the argon-CAP jet presented in [9]. The control objective is to maintain the temperature of a target surface (e.g., tissue), in contact with the plasma jet, at a desired setpoint while fulfilling various system constraints in the presence of parametric uncertainties and system disturbances. The system constraints pertain to safe and reproducible operation of the device. The control inputs have a dual feature of controlling the system dynamics and generating informative data for parameter estimation. Monte Carlo simulations were carried out to evaluate the closed-loop performance of the control approach under different uncertainty realizations. The results indicate that the designed optimal inputs can generate informative closed-loop data such that the estimated probability distribution of the unknown model parameters converges to its true value. The simulation results suggest that the control approach can effectively realize the control objectives in the presence of the intrinsic system stochasticity.

References

[1] M. Morari and J. H. Lee, "Model predictive control: past, present and future," Computers and Chemical Engineering, vol. 23, pp. 667-682, 1999.
[2] A. Mesbah, X. Bombois, M. Forgione, H. Hjalmarsson and P. M. J. Van den Hof, "Least costly closed-loop performance diagnosis and plant re-identification," International Journal of Control, vol. 88, no. 11, pp. 2264-2276, 2015.
[3] A. A. Feldbaum, "Dual-control thoery I," Automatic Remote Control, vol. 21, pp. 874-880, 1961.
[4] C. A. Larsson, M. Annergren, H. Hjalmarsson, C. R. Rojas, X. Bombois, A. Mesbah and P. E. Moden, "Model predictive control with integrated experiment design for output error systems," in Proceedings of the European Control Conference, pp. 3790-3795, Zürich, 2013.
[5] D. Xiu and G. E. Karniadakis, "The Wiener-Asky polynomial chaos for stochastic differential equations," SIAM Journal on Computing, vol. 24, pp. 619-644, 2002.
[6] V. Bavdekar and A. Mesbah, "A polynomial chaos-based nonlinear Bayesian approach for estimating state and parameter probability distribution functions," In Proceedings of the American Control Conference, Accepted, Boston, 2016.
[7] A. Schutze, J. Y. Jeong, S. E. Babyan, J. Park, G. S. Selwyn and R. F. Hicks, "The Atmospheric-Preseure Plasma Jet: A Review and Comparison to Other Plasma Sources," IEEE Transactions on Plasma Science, vol. 26, no. 6, pp. 1685-1694, 1998.
[8] H. Lee, G. Park, Y. Seo, Y. Im, S. Shim and H. Lee, "Modelling of atmospheric pressure plasmas for biomedical applications," Jounal of Physics D: Applied Physics, vol. 44, no. 5, 2011.
[9] D. Gidon, D. B. Graves and A. Mesbah, "Model predictive control of thermal effects of an atmospheric plasma jet for biomedical applications," In Proceedings of the American Control Conference, Accepted, Boston, 2016.