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457017 Development of an Integrated Framework for Stochastic Model Predictive Control with Moving Horizon Estimation

Previous contributions on output feedback SMPC considered observers with fixed gain [5], time-varying gain [6] and based on Kalman filter [7]. The integration between MHE and robust MPC was performed by deriving an error bound when an unconstrained observer is applied to linear systems with bounded disturbances, then proposing a tube controller formulation [8]. However, these previous studies did not consider the case of MHE coupled with SMPC, which is the main novelty of this work. Specifically, here we extend the work in [9] to the case of output feedback through the use of an MHE scheme [10]. The main contribution in the proposed SMPC formulation corresponds to a procedure to handle probabilistic constraints despite of the estimation error. The proposed approach considers that the exact value of the current state is unknown, but lies inside a set centered at the estimated state and shaped according to an error set calculated similarly as in [8]. The size of this set is used to tighten the probabilistic constraints, as in the tube-based control strategies. The implementation of such constraints is translated in a deterministic form based on the cumulative density function of the disturbances.

In this presentation, we show the theoretical results on recursive feasibility and closed-loop stability guarantees in the integrated setup of SMPC with MHE. In particular, the stability analysis follows the general idea of standard MPC that uses a feasible shifted solution to prove the convergence of the control cost. Therefore, the estimated and the real state are driven towards an invariant set around the set point. The resulting controller is less conservative than the purely robust counterpart due to the relaxation of constraints. Simulation examples of the output-feedback controller show that the derived error bounds are accurate and illustrate the convergence to neighborhoods of the set point.

**References**

[1] L. Magni, D. Pala, and R. Scattolini, “Stochastic model predictive control of constrained linear systems with additive uncertainty,” in *Proceedings of the European Control Conference 2009*, Budapest, Hungary, pp. 2235–2240, 2009.

[2] P. D. Couchman, M. Cannon, and B. Kouvaritakis, “Stochastic MPC with inequality stability constraints,” *Automatica*, vol. 42, no. 12, pp. 2169–2174, 2006.

[3] D. Van Hessem and O. Bosgra, “Stochastic closed-loop model predictive control of continuous nonlinear chemical processes,” *Journal of Process Control*, vol. 16, no. 3, pp. 225–241, 2006.

[4] J. Yan and R. R. Bitmead, “Incorporating state estimation into model predictive control and its application to network traffic control,” *Automatica*, vol. 41, no. 4, pp. 595–604, 2005.

[5] M. Cannon, Q. Cheng, B. Kouvaritakis, and S. V. Raković, “Stochastic tube MPC with state estimation,” *Automatica*, vol. 48, no. 3, pp. 536–541, 2012.

[6] M. Farina, L. Giulioni, L. Magni, and R. Scattolini, “An approach to output-feedback MPC of stochastic linear discrete-time systems,” *Automatica*, vol. 55, pp. 140–149, 2015.

[7] P. Hokayem, E. Cinquemani, D. Chatterjee, F. Ramponi, and J. Lygeros, “Stochastic receding horizon control with output feedback and bounded controls,” *Automatica*, vol. 48, no. 1, pp. 77–88, 2012.

[8] D. Sui, L. Feng, and M. Hovd, “Robust Output Feedback Model Predictive Control for Linear Systems via Moving Horizon Estimation,” in *2008 American Control Conference*, Seattle, Washington, pp. 453–458, 2008.

[9] B. F. Santoro and D. Odloak, “Stochastic Model Predictive Control for Output Tracking with Bounded Control Inputs,” in *2014 AIChE Annual Meeting*, Atlanta, USA, 2014.

[10] F. V Lima and J. B. Rawlings, “Nonlinear stochastic modeling to improve state estimation in process monitoring and control,” *AIChE Journal*, vol. 57, no. 4, pp. 996–1007, 2011.

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