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Real-Time Non-Smooth MPC for Switching Systems: Application to a Three-Tank Process

Hendrik Alsmeier, Felix Häusser, Andreas Knödler, Armin Nurkanović, Anton Pozharskiy, Moritz Diehl, Rolf Findeisen

TL;DR

The paper tackles real-time MPC for piecewise-smooth switching systems by adopting a non-smooth framework (NOSNOC) that leverages Filippov differential inclusions, finite-element discretization with switch detection (FESD), and a homotopy-based MPCC solver. It formulates a discrete-time MPC problem without mixed-integer variables and demonstrates real-time feasibility on a physical three-tank process, including mode-consistent switching and constraint satisfaction under references and model mismatch. Key contributions include a dynamic complementarity system representation of switching dynamics, exact switch detection within time discretization, and real-time hardware validation using a non-smooth optimization pipeline. The results suggest practical deployment potential for non-smooth MPC on switching processes, with robustness to model mismatch and clear avenues for acceleration and extension to more complex non-smooth systems.

Abstract

Real-time model predictive control of non-smooth switching systems remains challenging due to discontinuities and the presence of discrete modes, which complicate numerical integration and optimization. This paper presents a real-time feasible non-smooth model predictive control scheme for a physical three-tank process, implemented without mixed-integer formulations. The approach combines Filippov system modeling with finite elements and switch detection for time discretization, leading to a finite-dimensional optimal control problem formulated as a mathematical program with complementarity constraints. The mathematical program is solved via a homotopy of smooth nonlinear programs. We introduce modeling adjustments that make the three-tank dynamics numerically tractable, including additional modes to avoid non-Lipschitz points and undefined function values. Hardware experiments demonstrate efficient handling of switching events, mode-consistent tracking across reference changes, correct boundary handling, and constraint satisfaction. Furthermore, we investigate the impact of model mismatch and show that the tracking performance and computation times remain within real-time limits for the chosen sampling time. The complete controller is implemented using the non-smooth optimal control framework NOSNOC

Real-Time Non-Smooth MPC for Switching Systems: Application to a Three-Tank Process

TL;DR

The paper tackles real-time MPC for piecewise-smooth switching systems by adopting a non-smooth framework (NOSNOC) that leverages Filippov differential inclusions, finite-element discretization with switch detection (FESD), and a homotopy-based MPCC solver. It formulates a discrete-time MPC problem without mixed-integer variables and demonstrates real-time feasibility on a physical three-tank process, including mode-consistent switching and constraint satisfaction under references and model mismatch. Key contributions include a dynamic complementarity system representation of switching dynamics, exact switch detection within time discretization, and real-time hardware validation using a non-smooth optimization pipeline. The results suggest practical deployment potential for non-smooth MPC on switching processes, with robustness to model mismatch and clear avenues for acceleration and extension to more complex non-smooth systems.

Abstract

Real-time model predictive control of non-smooth switching systems remains challenging due to discontinuities and the presence of discrete modes, which complicate numerical integration and optimization. This paper presents a real-time feasible non-smooth model predictive control scheme for a physical three-tank process, implemented without mixed-integer formulations. The approach combines Filippov system modeling with finite elements and switch detection for time discretization, leading to a finite-dimensional optimal control problem formulated as a mathematical program with complementarity constraints. The mathematical program is solved via a homotopy of smooth nonlinear programs. We introduce modeling adjustments that make the three-tank dynamics numerically tractable, including additional modes to avoid non-Lipschitz points and undefined function values. Hardware experiments demonstrate efficient handling of switching events, mode-consistent tracking across reference changes, correct boundary handling, and constraint satisfaction. Furthermore, we investigate the impact of model mismatch and show that the tracking performance and computation times remain within real-time limits for the chosen sampling time. The complete controller is implemented using the non-smooth optimal control framework NOSNOC

Paper Structure

This paper contains 15 sections, 16 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Non-Smooth MPC via NOSNOC
  4. MPC formulation
  5. Dynamic Complementarity System
  6. Time Discretization and Switch Detection
  7. Discrete-time MPC subproblem as an MPCC
  8. Experimental Evaluation
  9. Scenario Definition
  10. Scenario 1
  11. Scenario 2
  12. Scenario 1: Reference cascade
  13. Scenario 2: Model-mismatch
  14. Discussion
  15. Conclusion and Outlook

Figures (3)

  • Figure 1: Image of the three-tank-system used in the experiments
  • Figure 2: Scenario 1: Tank heights are shown on the top and inflows on the bottom. References show as dashed horizontal lines and region switches as dashed vertical lines. Detailed computation times are shown on the right (including CPU times for each NLP solve in the MPCC homotopy).
  • Figure 3: Scenario 2: Tank heights are shown on the top and inflows on the bottom. References show as dashed horizontal lines and region switches as dashed vertical lines. Detailed computation times are shown on the right (including CPU times for each NLP solve in the MPCC homotopy).