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Economic Model Predictive Control for Periodic Operation: A Quadratic Programming Approach

Jose A. Borja-Conde, Juan M. Nadales, Filiberto Fele, Daniel Limon

TL;DR

This paper tackles economic optimization for periodically operating constrained systems by developing a single-layer economic model predictive control (E-MPC) that remains computationally light for real-time implementation. It recasts the original nonquadratic E-MPC into a quadratic program via a first-order Taylor expansion around a feasible trajectory, enabling a gradient-based, Newton-like online update that preserves recursive feasibility and convergence to the economically optimal periodic trajectory. Theoretical results establish controllability-based recursive feasibility, stability, and convergence, while a ball-and-plate numerical example demonstrates the method’s ability to follow a periodic reference and reduce economic cost relative to a tracking MPC. The approach offers a practical, solver-friendly solution for embedded platforms, synthesizing DRTO-like economic optimization with real-time MPC in a single layer for periodic operation.

Abstract

Periodic dynamical systems, distinguished by their repetitive behavior over time, are prevalent across various engineering disciplines. In numerous applications, particularly within industrial contexts, the implementation of model predictive control (MPC) schemes tailored to optimize specific economic criteria was shown to offer substantial advantages. However, the real-time implementation of these schemes is often infeasible due to limited computational resources. To tackle this problem, we propose a resource-efficient economic model predictive control scheme for periodic systems, leveraging existing single-layer MPC techniques. Our method relies on a single quadratic optimization problem, which ensures high computational efficiency for real-time control in dynamic settings. We prove feasibility, stability and convergence to optimum of the proposed approach, and validate the effectiveness through numerical experiments.

Economic Model Predictive Control for Periodic Operation: A Quadratic Programming Approach

TL;DR

This paper tackles economic optimization for periodically operating constrained systems by developing a single-layer economic model predictive control (E-MPC) that remains computationally light for real-time implementation. It recasts the original nonquadratic E-MPC into a quadratic program via a first-order Taylor expansion around a feasible trajectory, enabling a gradient-based, Newton-like online update that preserves recursive feasibility and convergence to the economically optimal periodic trajectory. Theoretical results establish controllability-based recursive feasibility, stability, and convergence, while a ball-and-plate numerical example demonstrates the method’s ability to follow a periodic reference and reduce economic cost relative to a tracking MPC. The approach offers a practical, solver-friendly solution for embedded platforms, synthesizing DRTO-like economic optimization with real-time MPC in a single layer for periodic operation.

Abstract

Periodic dynamical systems, distinguished by their repetitive behavior over time, are prevalent across various engineering disciplines. In numerous applications, particularly within industrial contexts, the implementation of model predictive control (MPC) schemes tailored to optimize specific economic criteria was shown to offer substantial advantages. However, the real-time implementation of these schemes is often infeasible due to limited computational resources. To tackle this problem, we propose a resource-efficient economic model predictive control scheme for periodic systems, leveraging existing single-layer MPC techniques. Our method relies on a single quadratic optimization problem, which ensures high computational efficiency for real-time control in dynamic settings. We prove feasibility, stability and convergence to optimum of the proposed approach, and validate the effectiveness through numerical experiments.
Paper Structure (14 sections, 33 equations, 3 figures, 1 algorithm)

This paper contains 14 sections, 33 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Problem Formulation
  4. Economic MPC for Changing Periodic Operation
  5. Gradient-Based Approach
  6. Proposed algorithm
  7. Stability and Convergence Analysis
  8. Recursive feasibility:
  9. Stability and convergence:
  10. Numerical example
  11. Scenario 1 -- Pure path-following problem
  12. Scenario 2 -- Economic optimization problem
  13. Comparative Analysis
  14. Conclusions

Figures (3)

  • Figure 1: Pure path-following problem. Ball and plate system controlled through Algorithm \ref{['alg:approximatedMPC']}. A star-shaped periodic trajectory, which is unreachable in some places, is set as a reference. The ball converges to the optimal periodic trajectory obtained by solving the DRTO problem \ref{['eq:D']}.
  • Figure 2: Economic MPC problem. Ball and plate system controlled by Algorithm \ref{['alg:approximatedMPC']}. A star-shaped periodic trajectory---partly unfeasible---is set as a reference (same as in Fig. \ref{['fig:experiment1_position']}). In this case, the economic objective is to minimize the energy demand of the actuators (hence the acceleration of the plate). The ball converges to the same optimal periodic trajectory obtained by solving the DRTO problem \ref{['eq:D']}.
  • Figure 3: Comparison between the proposed single-layer periodic E-MPC, and a standard MPC for periodic reference tracking (see periodictracking_limon), in following the optimal DRTO periodic trajectory $({x}^{\circ},\mathbf{u}^{\circ})$ determined by setting \ref{['eq:Da']} as per \ref{['eq:cost_sc2']}. Both schemes converge to the same periodic cost evolution (as expected); however, the proposed single-layer E-MPC achieves a more economic solution in the sense of \ref{['eq:cost_sc2']} (achieving an average cost of $3.46$ against the $3.71$ for the tracking MPC).