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466990 Adaptive Multiple-Model Stochastic Predictive Control

*stochastic*(describing uncertainty with probability masses or densities) [1] or

*robust*(describing uncertainty in terms of bounded, deterministic sets) [2]. Some of these approaches are adaptive, meaning the controller adjusts the control law as new information is revealed (e.g., [3,4]). A problem that has received less attention is that of an uncertain model structure; this situation may arise in a variety of problems, including when there is uncertainty regarding which physical phenomena are taking place, and when it is unclear what model order is required to adequately describe the process dynamics.

In this talk we present an adaptive multiple-model predictive control strategy with active learning of model structure for stochastic nonlinear systems. The controller considers a set of model hypotheses, each with different structural form. The uncertain time-invariant parameters in the model hypotheses are described with probability density functions (PDFs), and the temporal evolution of the state probability distributions is predicted using the generalized polynomial chaos framework [5]. The uncertain parameters in each hypothesis are estimated online [6] and their distributions are updated as new measurements become available. The measurements are also used to estimate the validity probability of each model hypothesis using Bayesian statistics, providing the controller with a metric for the degree to which each hypothesis agrees with the data. In order to effectively determine which hypothesis best describes the system, we include a measure of model-hypothesis similarity in the control formulation and specify that the controller attempts to decrease the predicted output distribution overlap, thereby facilitating the task of reconciling the next data point with one of the models. In the talk, we review some dissimilarity metrics and argue that the Kolmogorov distance [7] is best suited for our purposes. This results in actively adaptive control with respect to structural uncertainty, or suboptimal dual control in the sense of Feldbaum [8]. That is, the MPC algorithm simultaneously controls the process and performs active model discrimination.

We demonstrate the performance of the proposed adaptive multiple-model stochastic predictive control strategy through implementation on an atmospheric pressure plasma jet (based on [9]). The problem includes two model hypotheses, one for laminar flow and another for turbulent flow in the jet, and three uncertain parameters in each model hypothesis. The controller inputs will not only regulate the system dynamics, but also excite the system to effectively identify the flow regime (i.e., model hypothesis) in the presence of model parameter uncertainty.

[1] Mesbah, A. (2016). Stochastic model predictive control: An overview and perspectives for future research. *IEEE Control Systems Magazine.* Accepted.

[2] Bemporad, A., & Morari, M. (1999). Robust model predictive control: A survey. In *Robustness in identification and control*, pp. 207–226. Springer.

[3] Tanaskovic, M., Fagiano, L., Smith, R., & Morari, M. (2014). Adaptive receding horizon control for constrained MIMO systems. *Automatica*, 50(12), 3019–3029.

[4] Heirung, T.A.N., Ydstie, B.E., & Foss, B. (2015). Dual MPC for FIR systems: Information anticipation. In *Advanced Control of Chemical Processes*, pp. 1034–1039. Whistler, Canada.

[5] Xiu, D., & Karniadakis, G.E. (2002). The Wiener-Askey polynomial chaos for stochastic differential equations. *SIAM Journal on Scientific Computing*, 24(2), 619–644.

[6] Bavdekar, V.A., & Mesbah, A. (2016). A polynomial chaos-based nonlinear Bayesian approach for estimating state and parameter probability density functions. In *American Control Conference*. Boston, MA. To appear.

[7] Gibbs, A.L., & Su, F.E. (2002). On choosing and bounding probability metrics. *International Statistical Review*, 70(3), 419–435.

[8] Feldbaum, A.A. (1961). Dual-control theory. I. *Automation and Remote Control*, 21(9), 874–880.

[9] Gidon, D., Graves, D.B., & Mesbah, A. (2016). Model predictive control of thermal effects of an atmospheric pressure plasma jet for biomedical applications. In *American Control Conference*. Boston, MA. To appear.

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