Additional material to the paper "On necessity and robustness of dissipativity in economic model predictive control"

This technical report contains additional material to the paper “On necessity and robustness of dissipativity in economic model predictive control” by M. A. Müller, D. Angeli, and F. Allgöwer, IEEE Transactions on Automatic Control, 2015, 60, 1671-1676, DOI: 10.1109/TAC.2014.2361193, in particular some extensions and proofs. References and labels in this technical report (in particular Equation labels (1)–(26), references [1]–[23], and all theorem numbers etc.) refer to those in that paper

Data-driven Economic NMPC using Reinforcement Learning

Reinforcement Learning (RL) is a powerful tool to perform data-driven optimal control without relying on a model of the system. However, RL struggles to provide hard guarantees on the behavior of the resulting control scheme. In contrast, Nonlinear Model Predictive Control (NMPC) and Economic NMPC (ENMPC) are standard tools for the closed-loop optimal control of complex systems with constraints and limitations, and benefit from a rich theory to assess their closed-loop behavior. Unfortunately, the performance of (E)NMPC hinges on the quality of the model underlying the control scheme. In this paper, we show that an (E)NMPC scheme can be tuned to deliver the optimal policy of the real system even when using a wrong model. This result also holds for real systems having stochastic dynamics. This entails that ENMPC can be used as a new type of function approximator within RL. Furthermore, we investigate our results in the context of ENMPC and formally connect them to the concept of dissipativity, which is central for the ENMPC stability. Finally, we detail how these results can be used to deploy classic RL tools for tuning (E)NMPC schemes. We apply these tools on both a classical linear MPC setting and a standard nonlinear example from the ENMPC literature

A Gauss-Newton-Like Hessian Approximation for Economic NMPC

Economic Model Predictive Control (EMPC) has recently become popular because of its ability to control constrained nonlinear systems while explicitly optimizing a prescribed performance criterion. Large performance gains have been reported for many applications and closed-loop stability has been recently investigated. However, computational performance still remains an open issue and only few contributions have proposed real-time algorithms tailored to EMPC. We perform a step towards computationally cheap algorithms for EMPC by proposing a new positive-definite Hessian approximation which does not hinder fast convergence and is suitable for being used within the real-time iteration (RTI) scheme. We provide two simulation examples to demonstrate the effectiveness of RTI-based EMPC relying on the proposed Hessian approximation

Implications of dissipativity on stability of economic model predictive control—The indefinite linear quadratic case

© 2016 Elsevier B.V.In contrast to the conventional model predictive control (MPC) approach to control of a given system where a positive–definite objective function is employed, economic MPC employs a generic cost which is related to the ‘economics’ of the process as the objective function in the regulation layer. Often, stability proofs of the closed-loop system are based on strict-dissipativity of the system with respect to this objective function. In this paper, we focus on linear systems with indefinite quadratic costs. We show that while strict–dissipativity guarantees stability of the closed–loop system, it is not required. Hence we formulate a necessary and sufficient condition that guarantees asymptotic stability of the closed loop system. This condition comes down to the existence of two distinct storage functions for which the system is dissipative.This research is funded by the Federal Government of Nigeria through the Presidential Special Scholarship Scheme for Innovation and Development

A set-theoretic generalization of dissipativity with applications in Tube MPC

This paper introduces a framework for analyzing a general class of uncertain nonlinear discrete-time systems with given state-, control-, and disturbance constraints. In particular, we propose a set-theoretic generalization of the concept of dissipativity of systems that are affected by external disturbances. The corresponding theoretical developments build upon set based analysis methods and lay a general theoretical foundation for a rigorous stability analysis of economic tube model predictive controllers. Besides, we discuss practical prodecures for verifying set-dissipativity of constrained linear control systems with convex stage costs.Comment: 14 pages, 2 figure