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A model predictive control framework with robust stability guarantees under unbounded disturbances

Johannes Köhler, Melanie N. Zeilinger

TL;DR

This work proposes an alternative strategy: relaxing the initial state constraint with a penalty for open-loop asymptotically stable nonlinear systems and shows how to relax this condition to Lyapunov stable systems; stabilizable systems; and utilizing an implicit characterization of the Lyapunov function.

Abstract

To address feasibility issues in model predictive control (MPC), most implementations relax state constraints by using slack variables and adding a penalty to the cost. We propose an alternative strategy: relaxing the initial state constraint with a penalty. Compared to state-of-the-art soft constrained MPC formulations, the proposed formulation has two key features: (i) input-to-state stability and bounds on the cumulative constraint violation for unbounded disturbances; (ii) close-to-optimal performance under nominal operating conditions. The idea is initially presented for open-loop asymptotically stable nonlinear systems by designing the penalty as a Lyapunov function, but we also show how to relax this condition to: i) Lyapunov stable systems; ii) stabilizable systems; and iii) utilizing an implicit characterization of the Lyapunov function. In the special case of linear systems, the proposed MPC formulation reduces to a quadratic program, and the offline design and online computational complexity are only marginally increased compared to a nominal design. Numerical examples demonstrate benefits compared to state-of-the-art soft-constrained MPC formulations.

A model predictive control framework with robust stability guarantees under unbounded disturbances

TL;DR

This work proposes an alternative strategy: relaxing the initial state constraint with a penalty for open-loop asymptotically stable nonlinear systems and shows how to relax this condition to Lyapunov stable systems; stabilizable systems; and utilizing an implicit characterization of the Lyapunov function.

Abstract

To address feasibility issues in model predictive control (MPC), most implementations relax state constraints by using slack variables and adding a penalty to the cost. We propose an alternative strategy: relaxing the initial state constraint with a penalty. Compared to state-of-the-art soft constrained MPC formulations, the proposed formulation has two key features: (i) input-to-state stability and bounds on the cumulative constraint violation for unbounded disturbances; (ii) close-to-optimal performance under nominal operating conditions. The idea is initially presented for open-loop asymptotically stable nonlinear systems by designing the penalty as a Lyapunov function, but we also show how to relax this condition to: i) Lyapunov stable systems; ii) stabilizable systems; and iii) utilizing an implicit characterization of the Lyapunov function. In the special case of linear systems, the proposed MPC formulation reduces to a quadratic program, and the offline design and online computational complexity are only marginally increased compared to a nominal design. Numerical examples demonstrate benefits compared to state-of-the-art soft-constrained MPC formulations.
Paper Structure (26 sections, 81 equations, 2 figures, 3 tables)

This paper contains 26 sections, 81 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Proposed formulation
  4. Closed-loop theoretical analysis: ISS, constraint satisfaction, and performance
  5. Weighted projection
  6. Input-to-state stability
  7. Constraint satisfaction
  8. Performance analysis
  9. Constrained linear quadratic regulator
  10. Comparison to soft-constrained MPC
  11. Numerical examples
  12. Comparison to linear soft-constrained MPC
  13. Nonlinear system subject to large disturbances
  14. Conclusion
  15. Relax Assumption \ref{['ass:increm_stab']}
  16. ...and 11 more sections

Figures (2)

  • Figure 1: Linear system: Closed-loop trajectories initialized at $x_0=c\cdot [0.832,1]^\top$ with $c=1$ (left), $c=1.520$ (middle), $c=4$ (right). The state constraint $\mathbb{X}$ is shaded. Only feasible MPC solutions are plotted.
  • Figure 2: Nonlinear system subject to disturbances: Closed-loop water level $h_1$ and $h_3$ for different MPC formulations. State constraints and time points $k_i$, $i\in\mathbb{I}_{[1,3]}$, are shown as black dotted lines. The MPC schemes are stopped in case of infeasibility, which is highlighted with a circle with a cross $\times$.