(705f) Fast Advanced-Step Economic MPC Based on Turnpike Properties

Recently there has been a widespread interest in economic model predictive control (MPC), where the objective is to optimize the transient performance and not just the steady-state operation. Economic MPC uses a dynamic model of the process to solve numerical optimization problems online in order to minimize a desired objective function. Realizing this task in real time requires fast optimization algorithms. As many industrial applications call for increasingly complex, detailed and large-scale process models, a major concern is the computational resources needed to solve the resulting large-scale optimization.

While advances in numerical optimization strategies have enabled us to solve increasingly larger optimal control problems (OCPs), real-time implementation is still challenging, even with today's computing power [1]. The non-negligible amount of time taken to solve the numerical optimization problem online leads to computational delays, that are known to degrade the control performance and can also destabilize the system [2] [3].

To address this issue, advanced-step NMPC approach was recently proposed, which essentially moves the computationally intensive tasks off-line [4]. This is achieved by solving the computationally intensive NLP problem offline using the future predicted states in order to solve the future OCP one time step in advance. The solution computed offline is then updated online for the actual state realization by exploiting the parametric property of the OCP using NLP sensitivities.

As pointed out in a recent survey article [5], turnpike properties are essential features when studying economic NMPC problems. In OCPs, turnpike describes the common features of the optimal trajectories under varying initial conditions and prediction horizon lengths. In other words, turnpikes are parametric in initial condition and the prediction horizon length. The use of turnpikes in the context of economic MPC is relatively new with only a handful of papers that deal explicitly with turnpikes in the context of economic NMPC. While turnpike properties have been used to show recursive feasibility, finite time convergence and asymptotic stability of economic NMPC problems [6], we use specific turnpike properties to reduce the computation time of economic MPC using the advanced step framework.

Since the turnpike property states that for irrespective of the initial conditions the turnpike occurs at the steady-state optimum and that the end pieces of the solution trajectory are identical for all initial conditions [6], we use this property to show that the sensitivity of the initial condition on the optimal trajectory tends to zero as the turnpike is reached. This essentially allows us to truncate the prediction horizon in the online update step without affecting the optimal solution trajectory, leading to a significantly smaller problem size in the online step.

We demonstrate the effectiveness of the proposed approach using two case examples. On comparison with the ideal economic MPC and the conventional advanced-step economic MPC with full prediction horizon, we show that the proposed approach is able to significantly reduce the computational delay without trading-off on the controller performance.

References

1. L. T. Biegler, Efficient solution of dynamic optimization and NMPC problems, in: Nonlinear model predictive control, Springer, 2000, pp. 219-243.
2. R. Findeisen, F. Allgöwer, Computational delay in nonlinear model predictive control, IFAC Proceedings Volumes 37 (1) (2004) 427-432.
3. O. Santos, P. A. Afonso, J. A. Castro, N. M. Oliveira, L. T. Biegler, On-line implementation of nonlinear MPC: an experimental case study, Control Engineering Practice 9 (8) (2001) 847-857.
4. M. Zavala, L. T. Biegler, The advanced-step NMPC controller: Optimality, stability and robustness, Automatica 45 (1) (2009) 86-93.
5. M. Ellis, H. Durand, P. D. Christofides, A tutorial review of economic model predictive control methods, Journal of Process Control 24 (8) (2014) 1156-1178.
6. T. Faulwasser, D. Bonvin, On the design of economic NMPC based on an exact turnpike property, IFAC-PapersOnLine 48 (8) (2015) 525-530.