(309e) Output-Feedback Predictive Control for Stochastic Nonlinear Systems

Model predictive control (MPC) is widely used for control of complex systems since it is systematically able to handle multivariable dynamics and constraints on the system states and inputs. Robustness to modeling uncertainties and system disturbances is a desired aspect in a MPC algorithm. While the receding-horizon implementation imparts a certain degree of robustness to MPC, nominal MPC formulations cannot systematically account for the effect of stochastic uncertainties in the system [1]. Stochastic MPC (SMPC) with chance constraints provides a systematic framework to address this issue by explicitly incorporating the probabilistic descriptions of uncertainties in the optimal control problem [2]. This work presents an output-feedback stochastic MPC approach for nonlinear systems that are subject to probabilistic modeling uncertainties and system disturbances. The output-feedback approach accounts for the fact that in most engineering systems all the system states cannot be regularly measured, and there is a need for reconciliation of system states from the available noisy measurements.

In this work, the generalized polynomial chaos (gPC) framework [3] is used for computationally efficient propagation of the uncertain initial conditions and parametric uncertainties in the system model through the nonlinear system dynamics. The main challenge in using the gPC framework is to enable efficient propagation of the stochastic system disturbances since this requires a large number of basis functions in the polynomial chaos (PC) expansions for describing the independent time-varying system disturbances. To overcome this limitation, this work uses a two-step approach [4] for propagation of the parametric uncertainties and additive disturbances in the space of the PC expansion coefficients under the assumption that these coefficients can be adequately described by a Gaussian distribution. The output-feedback approach is implemented by using a gPC-based histogram filter [5], which is a Bayesian state and parameter estimation algorithm for estimating the posterior probability density functions of the states and uncertain parameters using the available measurements. The constraints on the states are reformulated as a joint chance constraint (JCC), which is satisfied in with a certain probability. A tractable implementation of the JCC is obtained by using a sample-based approach [6].

The proposed SMPC approach is implemented for control of the thermal effects of an atmospheric pressure plasma jet (APPJ) simulation case [7]. The system model consists of seven states (three of carrier gas temperature, three of carrier gas composition and one of target surface temperature) of which four are measured and three are unmeasured. The control objective is to regulate the target surface temperature. Chance constraints are imposed on the carrier gas temperature and surface temperature. The simulation results indicate that closed-loop control of the plasma device enables achieving the desired target surface temperature in the presence of system disturbances and modeling uncertainties, while maintaining the constraint satisfaction above the pre-specified probability level.

References:

1. Mayne, D. Q. Model predictive control: Recent developments and future promise. Automatica, 50:2967â??2986, 2014.
2. Mesbah, A. Stochastic model predictive control: An overview and perspectives for future research. IEEE Control Systems Magazine, accepted, 2016.
3. Xiu, D. and Karniadakis, G. E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal of Scientific Computation, 24:614â??644, 2002.
4. Konda, U., Singla, P., Singh, T. and Scott, P. D. State uncertainty propagation in the presence of parametric uncertainty and additive white noise. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, 133(5):051009-1â??051009-10, 2011.
5. Bavdekar, V. A. and Mesbah, A. A polynomial chaos-based nonlinear Bayesian approach for estimating state and parameter probability density functions. In Proceedings of the American Control Conference, accepted, Boston, 2016.
6. Alamo T., Tempo, R. and Luque, A. On the sample complexity of probabilistic analysis and design methods. In Perspectives in Mathematical System Theory, Control, and Signal Processing, pages 39â??50. Springerâ??Verlag, Berlin Heidelberg, 2010.
7. Gidon, A., Graves, D. B. and Mesbah, A. 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.