(684e) Economic MPC with Local Optimality

Model predictive control (MPC), or receding horizon control, refers to a control methodology that approximates the solution of a constrained infinite-horizon optimal control problem by solving finite-horizon optimal control problems in a receding horizon fashion. Generally, close-loop stability and performance of MPC cannot be guaranteed because of the finite horizon. A common approach to overcome the shortsightedness of MPC is to employ a terminal cost (often with terminal constraint). An ideal terminal cost to ensure stability and performance of MPC is the one characterizing the infinite-time cost-to-go of the system. However, for generic nonlinear systems, such a terminal cost can be very difficult to design, if it exists at all. An alternative to still ensure the stability of nonlinear MPC is to employ a terminal cost which is essentially an upper bound on the infinite-time cost-to-go (see e.g., [1-3]). Specifically, the terminal cost is constructed as a control Lyapunov function (CLF) which is compatible with the stage cost. Similar approach has been extended to the design of economic model predictive control (EMPC) which optimizes general economic cost functions to achieve better economic performance [4].

While the existing framework is superior for its stability property, with enlarged stability region compared to MPC/EMPC without terminal condition, it may not be favorable in terms of performance. In the effort to construct the CLF type terminal cost in which nonlinearity of the system is explicitly handled, local optimality of MPC is often lost. On the other hand, researches on the intrinsic properties of EMPC reveal that EMPC with a finite horizon is inherently stabilizing under certain conditions [5-6]. Thus it is conceivable that if a sufficiently large horizon is used or if the system state is close to the steady state, stability of MPC/EMPC is less of a concern. In these cases, the existing framework could be overly conservative.

In the present work, we provide an alternative framework for EMPC which is in favor of the performance. We design a terminal cost which preserves the local optimality for EMPC. First we show under certain conditions that for systems optimally operated at steady state, the optimal control is locally approximate to an LQR controller. The proposed terminal cost is constructed as the value function of the LQR controller plus an extra linear term. For conventional NMPC with quadratic cost functions, the linear term can be dropped. We show that EMPC with the proposed terminal cost and a sufficiently large horizon is stabilizing and locally behaves like the LQR controller. Estimation of the region of attraction is also provided.

References

[1] H. Chen and F. Allgöwer. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34:1205â??1217, 1998.

[2] A. Jadbabaie, J. Yu, and J. Hauser. Stabilizing receding horizon control of nonlinear systems: a control Lyapunov function approach. In Proceedings of the American Control Conference, San Diego, pages 1535â??1539, 1999.

[3] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36:789â??814, 2000.

[4] R. Amrit, J. B. Rawlings, and D. Angeli. Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control, 35:178â??186, 2011.

[5] L. Grüne. Economic receding horizon control without terminal constraints. Automatica, 49:725â??734, 2013

[6] D. Limón, T. Alamo, F. Salas, and E. F. Camacho. On the stability of constrained MPC without terminal constraint. IEEE Transactions on Automatic Control, 51:832â??836, 2006.