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Robust Model Predictive Control Exploiting Monotonicity Properties

Moritz Heinlein, Sankaranarayanan Subramanian, Sergio Lucia

TL;DR

The paper tackles robust model predictive control under prediction uncertainties, which is typically hampered by conservatism and high computational cost. It harnesses monotone system properties to compute tight reachable-set envelopes efficiently and introduces a partitioning scheme that injects recourse, forming a closed-loop robust MPC with linear scaling in horizon and states. By adopting mixed-monotonicity, the framework extends to general nonlinear systems via a suitable decomposition function, bridging to tube-based MPC and enabling general feedback policies. The approach is validated on a nonlinear, high-dimensional CSTR cascade (up to 25 states and 20 uncertainties) demonstrating robust constraint satisfaction and improved performance over open-loop or nominal MPC, with scalability insights discussed for larger systems.

Abstract

Robust model predictive control algorithms are essential for addressing unavoidable errors due to the uncertainty in predicting real-world systems. However, the formulation of such algorithms typically results in a trade-off between conservatism and computational complexity. Monotone systems facilitate the efficient computation of reachable sets and thus the straightforward formulation of a robust model predictive control approach optimizing over open-loop predictions. We present an approach based on the division of reachable sets to incorporate feedback in the predictions, resulting in less conservative strategies. The concept of mixed-monotonicity enables an extension of our methodology to non-monotone systems. The potential of the proposed approaches is demonstrated through a nonlinear high-dimensional chemical tank reactor cascade case study.

Robust Model Predictive Control Exploiting Monotonicity Properties

TL;DR

The paper tackles robust model predictive control under prediction uncertainties, which is typically hampered by conservatism and high computational cost. It harnesses monotone system properties to compute tight reachable-set envelopes efficiently and introduces a partitioning scheme that injects recourse, forming a closed-loop robust MPC with linear scaling in horizon and states. By adopting mixed-monotonicity, the framework extends to general nonlinear systems via a suitable decomposition function, bridging to tube-based MPC and enabling general feedback policies. The approach is validated on a nonlinear, high-dimensional CSTR cascade (up to 25 states and 20 uncertainties) demonstrating robust constraint satisfaction and improved performance over open-loop or nominal MPC, with scalability insights discussed for larger systems.

Abstract

Robust model predictive control algorithms are essential for addressing unavoidable errors due to the uncertainty in predicting real-world systems. However, the formulation of such algorithms typically results in a trade-off between conservatism and computational complexity. Monotone systems facilitate the efficient computation of reachable sets and thus the straightforward formulation of a robust model predictive control approach optimizing over open-loop predictions. We present an approach based on the division of reachable sets to incorporate feedback in the predictions, resulting in less conservative strategies. The concept of mixed-monotonicity enables an extension of our methodology to non-monotone systems. The potential of the proposed approaches is demonstrated through a nonlinear high-dimensional chemical tank reactor cascade case study.
Paper Structure (8 sections, 6 theorems, 17 equations, 4 figures, 1 table)

This paper contains 8 sections, 6 theorems, 17 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reachable Sets of Monotone Systems
  3. Robust MPC for monotone systems
  4. Extension to general systems and feedback policies
  5. Generalization via mixed-monotonicity
  6. Relation to other robust MPC approaches - general feedback policies
  7. Nonlinear, non-monotone CSTR Cascade
  8. Conclusion

Key Result

Proposition 1

The 1-step reachable set for any fixed input ${u}\in{\mathbb{U}}$ of system eq:system satisfying eq:monotone_states and eq:monotone_uncert with ${x}\in \left[{x}^-,{x}^+\right]$ and ${p}\in \left[{p}^-,{p}^+\right]$ is bounded by the interval

Figures (4)

  • Figure 1: Schematic representation of approach \ref{['eq:Closed_Loop']} in a phase plot for multiple time steps $k$ in the prediction horizon of a 2D system with 4 subregions. Each subregion $\left[{x}_k^{s-}, {x}_k^{s+} \right]$ is propagated with the input ${u}_k^s$. The subsets in the next time step are aligned to bound all propagated sets.
  • Figure 2: Methods on dividing the reachable set
  • Figure 3: Visualisation of the assumption on the terminal set ${\mathbb{X}}_f$. The rectangle ${\mathbb{X}}_{f}=\left[x_{\text{RCIS}}^{1-},{x}_{\text{RCIS}}^{4+}\right]$ is divided and each subregion is propagated with an individual input. The propagations need to lie inside ${\mathbb{X}}_f$.
  • Figure 4: Closed-loop simulation of the five tank CSTR cascade. Nominal MPC (first row), the open-loop approach (second row) and the robust closed-loop approach (third row) are compared. The first two columns compare the concentrations of the value product $c_R$ and the side product $c_S$. The third column shows the reactor temperatures $T_{R,i}$ as solid lines and the jacket temperatures $T_{J,i}$ as dashed lines. The fourth column describes the inputs $u_{A,i}$ (solid) and $u_{B,i}$ (dashed) in each tank. The last column compares the accumulated closed-loop costs. The red dashed lines in all plots display the state and input constraints.

Theorems & Definitions (14)

  • Definition 1: Monotonicity of dynamic systems
  • Remark 1
  • Proposition 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Definition 2: Decomposition function of a dynamic system
  • Proposition 2
  • Theorem 2
  • proof
  • ...and 4 more