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Tube-based Robust Model Predictive Control for a Distributed Parameter System Modeled as a Polytopic LPV (extended version)

Joe Ismail, Steven Liu

TL;DR

The paper addresses active vibration damping for a stacker crane modeled as a distributed parameter system by embedding nonlinear dynamics into a polytopic LPV framework and applying tube-based MPC. It achieves online computation via a simple convex QP, while a robust disturbance-invariant tube guarantees constraint satisfaction, stability, and recursive feasibility. Key contributions include (1) a polytopic quasi-LPV representation with offline-tube design, (2) a nominal MPC with a dual-mode feedback and a soft-constraint extension to handle resonance, and (3) theoretical discussion of stability/feasibility through MCPI/MPI concepts and MRPI terminal design. The approach yields a computationally efficient, robust control scheme for STCs, validated against NMPC on a numerical case study showing good performance under uncertainty and model lumping.

Abstract

Distributed parameter systems (DPS) are formulated as partial differential equations (PDE). Especially, under time-varying boundary conditions, PDE introduce force coupling. In the case of the flexible stacker crane (STC), nonlinear coupling is introduced. Accordingly, online trajectory planning and tracking can be addressed using a nonlinear model predictive control (NMPC). However, due to the high computational demands of a NMPC, this paper discusses a possibility of embedding nonlinearities inside a linear parameter varying (LPV) system and thus make a use of a numerically low-demanding linear MPC. The resulting mismatches are treated as parametric and additive uncertainties in the context of robust tube-based MPC (TMPC). For the proposed approach, most of the computations are carried out offline. Only a simple convex quadratic program (QP) is conducted online. Additionally a soft-constrained extension was briefly proposed. Simulation results are used to illustrate the good performance, closed-loop stability and recursive feasibility of the proposed approach despite uncertainties.

Tube-based Robust Model Predictive Control for a Distributed Parameter System Modeled as a Polytopic LPV (extended version)

TL;DR

The paper addresses active vibration damping for a stacker crane modeled as a distributed parameter system by embedding nonlinear dynamics into a polytopic LPV framework and applying tube-based MPC. It achieves online computation via a simple convex QP, while a robust disturbance-invariant tube guarantees constraint satisfaction, stability, and recursive feasibility. Key contributions include (1) a polytopic quasi-LPV representation with offline-tube design, (2) a nominal MPC with a dual-mode feedback and a soft-constraint extension to handle resonance, and (3) theoretical discussion of stability/feasibility through MCPI/MPI concepts and MRPI terminal design. The approach yields a computationally efficient, robust control scheme for STCs, validated against NMPC on a numerical case study showing good performance under uncertainty and model lumping.

Abstract

Distributed parameter systems (DPS) are formulated as partial differential equations (PDE). Especially, under time-varying boundary conditions, PDE introduce force coupling. In the case of the flexible stacker crane (STC), nonlinear coupling is introduced. Accordingly, online trajectory planning and tracking can be addressed using a nonlinear model predictive control (NMPC). However, due to the high computational demands of a NMPC, this paper discusses a possibility of embedding nonlinearities inside a linear parameter varying (LPV) system and thus make a use of a numerically low-demanding linear MPC. The resulting mismatches are treated as parametric and additive uncertainties in the context of robust tube-based MPC (TMPC). For the proposed approach, most of the computations are carried out offline. Only a simple convex quadratic program (QP) is conducted online. Additionally a soft-constrained extension was briefly proposed. Simulation results are used to illustrate the good performance, closed-loop stability and recursive feasibility of the proposed approach despite uncertainties.

Paper Structure

This paper contains 18 sections, 1 theorem, 19 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Mathematical Notations
  4. Nonlinear Dynamic
  5. Polytopic LPV Representation
  6. Problem Formulation
  7. State Decomposition
  8. Disturbance Invariant Tube
  9. Nominal MPC
  10. Soft Constrainted MPC
  11. Discussion on stability and feasibility
  12. Maximal Control Invariant Sets
  13. Maximal Positive Invariant Set
  14. Asymptotically Stability
  15. Trajectory Planning and Tracking Control
  16. ...and 3 more sections

Key Result

Proposition IV.1

(RAS): The set $\mathcal{Z} \times \{0\}$ is a robust asymptotic stable for the decomposition $(z_{k+1} = A_0 z_k + B_0 v_k, e_{k+1} = A_c e_k + w_k)$ in the positive invariant set $\mathcal{Z} \times \bar{\mathcal{X}}$.

Figures (6)

  • Figure 1: flexible stacker crane
  • Figure 2: MPI sets (dotted), MCPI sets (dashed)
  • Figure 3: Projection of TMPC on $x_c$ and $\omega_t$ along the horizon
  • Figure 4: Projection of TMPC on $y_l$ and $\omega_t$ along the horizon
  • Figure 5: Projection of TMPC on $x_c$ and $\omega_t$ along the horizon
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition II.1
  • Remark 1
  • Definition III.1
  • Definition IV.1
  • Proposition IV.1
  • proof