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Artificial-reference tracking MPC with probabilistically validated performance on industrial embedded systems

Victor Gracia, Pablo Krupa, Filiberto Fele, Teodoro Alamo

TL;DR

This work addresses deploying advanced MPC for tracking on resource-constrained industrial embedded systems by proposing MPCT with an artificial reference, offset-free disturbance rejection, constraint tightening via back-off, and soft constraints to guarantee feasibility. A structure-exploiting ADMM-based solver enables efficient implementation on PLCs, while a probabilistic validation framework selects robust controller parameters and provides long-term performance guarantees. The approach is demonstrated in a hardware-in-the-loop setup controlling a nonlinear CSTR, achieving feasible constraint satisfaction and bounded computation time with modest memory usage. Overall, the paper advances practical MPC deployment in industry by delivering probabilistic performance assurances and an efficient embedded solver.

Abstract

Industrial embedded systems are typically used to execute simple control algorithms due to their low computational resources. Despite these limitations, the implementation of advanced control techniques such as Model Predictive Control (MPC) has been explored by the control community in recent years, typically considering simple linear formulations or explicit ones to facilitate the online computation of the control input. These simplifications often lack features and properties that are desirable in real-world environments. In this article, we present an efficient implementation for embedded systems of MPC for tracking with artificial reference, solved via a recently developed structure-exploiting first-order method. This formulation is tailored to a wide range of applications by incorporating essential practical features at a small computational cost, including integration with an offset-free scheme, back-off parameters that enable constraint tightening, and soft constraints that preserve feasibility under disturbances or plant-model mismatch. We accompany this with a framework for probabilistic performance validation of the closed-loop system over long-term operation. We illustrate the applicability of the approach on a Programmable Logic Controller (PLC), incorporated in a hardware-in-the-loop setup to control a nonlinear continuous stirred-tank reactor. The behavior of the closed-loop system is probabilistically validated with respect to constraint violations and the number of iterations required at each time step by the MPC optimization algorithm.

Artificial-reference tracking MPC with probabilistically validated performance on industrial embedded systems

TL;DR

This work addresses deploying advanced MPC for tracking on resource-constrained industrial embedded systems by proposing MPCT with an artificial reference, offset-free disturbance rejection, constraint tightening via back-off, and soft constraints to guarantee feasibility. A structure-exploiting ADMM-based solver enables efficient implementation on PLCs, while a probabilistic validation framework selects robust controller parameters and provides long-term performance guarantees. The approach is demonstrated in a hardware-in-the-loop setup controlling a nonlinear CSTR, achieving feasible constraint satisfaction and bounded computation time with modest memory usage. Overall, the paper advances practical MPC deployment in industry by delivering probabilistic performance assurances and an efficient embedded solver.

Abstract

Industrial embedded systems are typically used to execute simple control algorithms due to their low computational resources. Despite these limitations, the implementation of advanced control techniques such as Model Predictive Control (MPC) has been explored by the control community in recent years, typically considering simple linear formulations or explicit ones to facilitate the online computation of the control input. These simplifications often lack features and properties that are desirable in real-world environments. In this article, we present an efficient implementation for embedded systems of MPC for tracking with artificial reference, solved via a recently developed structure-exploiting first-order method. This formulation is tailored to a wide range of applications by incorporating essential practical features at a small computational cost, including integration with an offset-free scheme, back-off parameters that enable constraint tightening, and soft constraints that preserve feasibility under disturbances or plant-model mismatch. We accompany this with a framework for probabilistic performance validation of the closed-loop system over long-term operation. We illustrate the applicability of the approach on a Programmable Logic Controller (PLC), incorporated in a hardware-in-the-loop setup to control a nonlinear continuous stirred-tank reactor. The behavior of the closed-loop system is probabilistically validated with respect to constraint violations and the number of iterations required at each time step by the MPC optimization algorithm.

Paper Structure

This paper contains 16 sections, 1 theorem, 40 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement and MPC formulation
  3. Practical MPCT formulation
  4. Disturbance rejection and offset-free MPC
  5. Tightening of constraints
  6. Soft constraints
  7. Practical MPCT formulation
  8. Probabilistic validation of the controller
  9. Implementation of the MPC solver
  10. Hardware-in-the-loop results
  11. Testing system: continuous stirred-tank reactor
  12. Controller and estimator parameters
  13. Validation experiments design
  14. Performance indicators and hyperparameters
  15. Validation experiments
  16. ...and 1 more sections

Key Result

Proposition 1

Given the controllers $\kappa_i$, $i\in \mathbb{I}_1^M$, and the performance indicators $\phi^{\ell}(\cdot)$, $\ell\in\mathbb{I}_1^K$, suppose that $N_s$ i.i.d. scenarios $w_j \sim \mathcal{W}$, $j\in \mathbb{I}_1^{N_s}$, are generated. Then, for any choice of the integer $1 \leq r \leq N_s$, it hol provided that

Figures (5)

  • Figure 1: Closed-loop simulation of the CSTR controlled with $C_1$. This experiment corresponds to the $5$-th worst case of $C_1$, where $\phi^{1}(\cdot) = 0.0045$ (see Table \ref{['tab:validation_results']}).
  • Figure 2: Comparison of closed-loop simulations of the CSTR system using the controllers $C_1$ and $C_0$. A case with nonadmissible reference pairs $(x_r(k),u_r(k))$ due to back-off.
  • Figure 3: Comparison of distances from outputs to reference $y_r$ using controllers $C_1$ and $C_0$. In the legend, "PW" stands for "point-wise".
  • Figure 4: Comparison of distances from the outputs to the admissible reference $y^\circ$ using controllers $C_1$ and $C_0$.
  • Figure 5: Comparison of number of iterations and time required to obtain the input using controllers $C_1$ and $C_0$.

Theorems & Definitions (6)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4