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457996 Stochastic Nonlinear Model Predictive Control Using Adaptive Polynomial Chaos: Application to an Atmospheric Pressure Plasma Jet

**Stochastic Nonlinear Model Predictive Control Using Adaptive Polynomial Chaos: Application to an Atmospheric Pressure Plasma Jet**

Model predictive control (MPC) is a control strategy that can effectively handle competing control objectives for multivariable systems in the presence of constraints [1,2]. Key to MPC is a dynamic system model to predict future behavior so that optimal control action can be computed. Standard MPC considers deterministic system models, whereas in reality probabilistic uncertainties in model parameters, disturbance, and measurement noise are ubiquitous. Stochastic model predictive control (SMPC) provides a framework for systematically incorporating the probabilistic description of system uncertainties into the optimal control problem and enabling constraint handling in a probabilistic sense (e.g., see [3] and the references therein). However, efficient propagation of probabilistic uncertainties through system dynamics poses a key challenge in SMPC, particularly for nonlinear systems.

Stochastic nonlinear MPC (SNMPC) approaches using generalized polynomial chaos (gPC) for uncertainty propagation have been reported in [4, 5, 6]. gPC is a framework that approximates stochastic states and parameters as a weighted sum of orthogonal polynomials [7,8]. The orthogonality property of the polynomials allows for efficient on-line propagation of the statistics of state distributions or, alternatively, efficient construction of the state distributions. One major shortcoming of the present nonlinear SMPC approaches employing gPC is, however, the inability to account for time-varying stochastic disturbances. With time-varying stochastic disturbances, the disturbance realization (and possibly its underlying distribution) may change in time. Considering each disturbance realization as a stochastic variable in gPC results in intractable formulations for real-time control since the number of terms in the gPC expansion increases factorially with the number of stochastic variables. Uncertainty propagation approaches that can handle time-varying stochastic disturbances in the context of polynomial chaos remains an open problem.

This work presents an SNMPC approach based on adaptive polynomial chaos (aPC) framework [9] for systems with time-invariant and time-varying probabilistic uncertainties. The distinguishing characteristic of aPC is that instead of selecting one set of orthogonal polynomial basis functions for all times, the basis functions are dynamically adapted at each sampling time based on the moments of the states. A primary reason that gPC cannot handle time-varying probabilistic uncertainties is due to the fact that constant orthogonal polynomial basis functions cannot accurately describe the distribution of stochastic states over long prediction horizons and when the states have been corrupted by several realizations of time-varying uncertainties. By updating the polynomial basis functions to account for the changes in the state distribution, aPC can effectively handle time-varying additive noise regardless of its distribution. This aPC-based SNMPC approach has been demonstrated for high-performance control of an atmospheric pressure plasma jet [10]. Simulation results show that the SNMPC approach is able to successfully achieve the control objectives while satisfying the state constraints in the presence of stochastic disturbances.

**References**

[1] M. Morari and J. H. Lee, “Model predictive control: Past, present and future,” *Computers & Chemical Engineering*, vol. 23, pp. 667-682, 1999.

[2] D. Q. Mayne, “Model predictive control: Recent developments and future promise,” *Automatica*, vol. 50, pp. 2967-2986, 2014.

[3] A. Mesbah, “Stochastic model predictive control: An overview and perspectives for future research," *IEEE Control Systems Magazine, In Press, *2016.

[4] A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz, “Stochastic nonlinear model predictive control with probabilistic constraints,” in *Proceedings of the American Control Conference*, pp. 2413–2419, Portland, Oregon, 2014.

[5] L. Fagiano and M. Khammash, “Nonlinear stochastic model predictive control via regularized polynomial chaos expansions,” in *Proceedings of the 51st IEEE Conference on Decision and Control*, pp. 142–147, Maui, 2012.

[6] V. A. Bavdekar and A. Mesbah, “Stochastic nonlinear model predictive control with joint chance constraints,” in *Proceedings of the 1**0th IFAC Symposium on Nonlinear Control Systems*, accepted, Monterey, 2016.

[7] N. Wiener, “The homogeneous chaos,” *American Journal of Mathematics*, vol. 60, pp. 897–936, 1938.

[8] D. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,” *SIAM Journal of Scientific Computation*, vol. 24, pp. 619–644, 2002.

[9]. S. Oladyshkin and W. Nowak, “Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion,” *Reliability Engineering & System Safety*, vol. 106, pp. 179-190, 2012.

[10] D. Gidon, D. B. Graves, and A. Mesbah, “Model predictive control of thermal effects of an atmospheric pressure plasma jet for biomedical applications," in *Proceedings of the American Control Conference*, accepted, Boston, 2016.

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