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Constraint-adaptive MPC for large-scale systems: Satisfying state constraints without imposing them

S. A. N. Nouwens, B. de Jager, M. M. Paulides, W. P. M. H. Heemels

TL;DR

These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem, which can significantly reduce the computational complexity with often only a slight impact on the closed-loop performance.

Abstract

Model Predictive Control (MPC) is a successful control methodology, which is applied to increasingly complex systems. However, real-time feasibility of MPC can be challenging for complex systems, certainly when an (extremely) large number of constraints have to be adhered to. For such scenarios with a large number of state constraints, this paper proposes two novel MPC schemes for general nonlinear systems, which we call constraint-adaptive MPC. These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem. Carefully selecting the included constraints can significantly reduce, as we will demonstrate, the computational complexity with often only a slight impact on the closed-loop performance. Although not all (state) constraints are imposed in the on-line optimization, the schemes still guarantee recursive feasibility and constraint satisfaction. A numerical case study illustrates the proposed MPC schemes and demonstrates the achieved computation time improvements exceeding two orders of magnitude without loss of performance.

Constraint-adaptive MPC for large-scale systems: Satisfying state constraints without imposing them

TL;DR

These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem, which can significantly reduce the computational complexity with often only a slight impact on the closed-loop performance.

Abstract

Model Predictive Control (MPC) is a successful control methodology, which is applied to increasingly complex systems. However, real-time feasibility of MPC can be challenging for complex systems, certainly when an (extremely) large number of constraints have to be adhered to. For such scenarios with a large number of state constraints, this paper proposes two novel MPC schemes for general nonlinear systems, which we call constraint-adaptive MPC. These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem. Carefully selecting the included constraints can significantly reduce, as we will demonstrate, the computational complexity with often only a slight impact on the closed-loop performance. Although not all (state) constraints are imposed in the on-line optimization, the schemes still guarantee recursive feasibility and constraint satisfaction. A numerical case study illustrates the proposed MPC schemes and demonstrates the achieved computation time improvements exceeding two orders of magnitude without loss of performance.

Paper Structure

This paper contains 14 sections, 2 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. State-of-the-art
  4. Contributions
  5. MPC setup and problem formulation
  6. System description
  7. MPC setup
  8. Problem statement
  9. ca-MPC under controlled invariance of original constraint set
  10. CA-MPC with terminal sets
  11. Double integrator case study
  12. System and constraint definition
  13. Results
  14. Conclusion

Key Result

Theorem 2

Consider the ca-MPC scheme eq:reduced_mpc_scheme with $\mathbb{X}$ controlled invariant for eq:system_description with the input set $\mathbb{U}$. If for all states $\bm{x}\in\mathbb{X}$ and corresponding reduced constraint set $\mathbb{X}_r(\bm{x})$, the delta set $\Delta(\bm{x})$ satisfies the pro then the feasible set is equal to $\mathbb{X}$, i.e., $\mathbb{X}_F^r = \mathbb{X}$ and eq:reduced_

Figures (4)

  • Figure 1: Illustrating ca-MPC when $\mathbb{X}$ is controlled invariant. Clearly, $\bm{x}_{1|k}$cannot be outside of $\mathbb{X}$ while $\bm{x}_{2|k},\dots,\bm{x}_{N|k}$ are not necessarily constrained to $\mathbb{X}$ as illustrated by $\bm{x}_{3|k}$.
  • Figure 2: Illustrating ca-MPC with a terminal set and parameterized by $\mathcal{X}^\star_{k-1}$. Note that $\mathbb{X}_r(\mathcal{X}^\star_{k-1})$ is constructed as detailed in Remark \ref{['rem:invariant_reduced_constraint_set']} for illustration purposes. Critically, the reduced constraint set and delta sets ensure $\bm{x}_{i|k}\in\mathbb{X}$ for $i=1,\dots,N$.
  • Figure 3: The closed-loop state trajectory using the regular MPC solution and the ca-MPC solution.
  • Figure 4: The percentage of constraints and the computation time with respect to the full MPC solution over time. When the state approaches the boundary of $\mathbb{X}$ the number of constraints and computation time increase.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 2
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7