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450662 On the Robustness of Economic Nonlinear Model Predictive Control

Input-to-state stability (ISS) is used to extend stability analysis to systems with uncertainty. The property was originally described for continuous time (CT) systems in [12] and was extended to discrete time (DT) systems in [7]. Furthermore, ISS has been proposed as a framework for NMPC [8], and it provides a very convenient and natural way of thinking about robust stability. Previous work [3] provides a method for calculating ISS bounds. Input-to-state practical stability allows for a slight modification to ISS in order to be more widely applicable.

Economic NMPC allows for the use of more general objective functions in the controller. The difficulty is that standard stability results of tracking MPC cannot be used. Originally, asymptotic stability was shown by using strict dissipativity of the rotated cost function [2], but this property is not easy to show in general. However, it is possible to show a stronger property, strong convexity of the rotated cost function, by including regularization terms in the objective function. It has been shown that a sufficiently large regularization term (e.g., a tracking term) may be added to the objective in order to force the rotated cost function to be strongly convex, which leads to asymptotic stability [6]. The drawbacks of this method are that regularization weights may be difficult to calculate and very conservative, limiting economic gains. In [14] it is proposed to replace the regularization term in the objective with a stabilizing inequality constraint derived from the inherent robustness margin of an auxiliary, asymptotically stable MPC controller. It is demonstrated that this approach (that we call eMPC-sc) provides flexibility to optimize economic performance. Moreover, it is shown that this approach is related to regularization-based Lyapunov-based MPC approaches, but the cumbersome calculations required to find regularization terms are not required. The eMPC approach differs from the Lyapunov-based approach [5] in that feasibility of the stabilizing constraint can be guaranteed directly while in the Lyapunov approach the feasible set for the states needs to be adjusted. Also, a formulation for the tracking case with uncertainty with a similar stabilizing constraint is shown in [1]. Here, the stabilizing constraint is required to be feasible for a list of possible plant realizations, while we only require feasibility of the terminal constraint for the nominal plant.

In this work, we extend eMPC with a stabilizing constraint to the nonlinear case with uncertainty and state constraints. We show that a system to which this controller is applied is ISpS, which requires a slight modification to existing theory on ISpS. We show simulation results on a CSTR, as well as on a system of two distillation columns in series.

References

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[2] M. Diehl, R. Amrit, and J. Rawlings. A Lyapunov function for economic optimizing model predictive control. *IEEE Transactions on Automatic Control*, 56(3):703-707, 2011.

[3] D. Grith and L. Biegler. Trajectory bounds of input-to-state stability for nonlinear model predictive control. In *Preprints of the 9th International Symposium on Advanced Control of Chemical Processes*, pages 1028-1033, 2015.

[4] L. Grune and J. Pannek. *Nonlinear Model Predictive Control: Theory and Aglorithms*. Springer, London, 2011.

[5] M. Heidarinejad, J. Liu, and P. Christodes. Economic model predictive control of nonlinear process systems using Lyapunov techniques. *AICHE Journal*, 58(3):855-870, 2012.

[6] J. Jaschke, X. Yang, and L. Biegler. Fast economic model predictive control based on NLP-sensitivities. *Journal of Process Control*, 24:1260-1272, 2014.

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[9] G. Pannocchia, J. Rawlings, and S. Wright. Conditions under which suboptimal nonlinear MPC is inherently robust. *Systems & Control Letters*, 60:747-755, 2011.

[10] S. Qin and T. Bagdwell. A survey of industrial model predictive control technology. *Control Engineering Practice*, 1:733-764, 2003.

[11] J. Rawlings and D. Mayne. *Model Predictive Control: Theory and Design*. Nob Hill Publishing, Madison, WI, 2009.

[12] E. Sontag. Smooth stabilization implies coprime factorization. *IEEE Transactions on Automatic Control*, 34(4):435-443, Apr 1989.

[13] X. Yang and L. Biegler. Advanced-multi-step nonlinear model predictive control. *Journal of Process Contro*l, 23(8):1116-1128, 2013.

[14] V. M. Zavala. A multiobjective optimization perspective on the stability of economic MPC. *9th International Symposium on Advanced Control of Chemical Processes*, pages 975-981, 2015.

[15] V. M. Zavala and M. Anitescu. Real-time nonlinear optimization as a generalized equation. *SIAM Journal on Control and Optimization*, 48(8):5444-5467, 2010.

[16] V. M. Zavala and L. Biegler. The advanced-step NMPC controller: Optimality, stability, and robustness. *Automatica*, 45:86-93, 2009.

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