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A Framework for Quasi Time-Optimal Nonlinear Model Predictive Control with Soft Constraints

Joe Ismail, Steven Liu

TL;DR

The paper addresses time-optimal control for flexible mechatronic systems, focusing on avoiding excitation of undesired vibrations in a stacker crane. It introduces a soft-constrained nonlinear MPC framework that predicts vibration frequencies and imposes inequality constraints on undesired frequencies via a polynomial surrogate of resonance hypersurfaces, thereby enlarging the feasible set compared to hard constraints. The method uses an eight-state dynamic model derived from a two-beam representation and a frequency-prediction mechanism based on lift position $y_l$ and mass $m_l$, enabling a quasi-time-optimal objective with slack-based penalties for frequency violations. Numerical results on a stacker crane and a fixed-frequency 2-DOF system demonstrate improved vibration damping and reduced bang-bang behavior while retaining near-minimal transition times, with practical implications for robust, fast positioning of flexible payloads.

Abstract

In many mechatronic applications, controller input costs are negligible and time optimality is of great importance to maximize the productivity by executing fast positioning maneuvers. As a result, the obtained control input has mostly a bang-bang nature, which excite undesired mechanical vibrations, especially in systems with flexible structures. This paper tackles the time-optimal control problem and proposes a novel approach, which explicitly addresses the vibrational behavior in the context of the receding horizon technique. Such technique is a key feature, especially for systems with a time-varying vibrational behavior. In the context of model predictive control (MPC), vibrational behavior is predicted and coped in a soft-constrained formulation, which penalize any violation of undesired vibrations. This formulation enlarges the feasibility on a wide operating range in comparison with a hard-constrained formulation. The closed-loop performance of this approach is demonstrated on a numerical example of stacker crane with high degree of flexibility.

A Framework for Quasi Time-Optimal Nonlinear Model Predictive Control with Soft Constraints

TL;DR

The paper addresses time-optimal control for flexible mechatronic systems, focusing on avoiding excitation of undesired vibrations in a stacker crane. It introduces a soft-constrained nonlinear MPC framework that predicts vibration frequencies and imposes inequality constraints on undesired frequencies via a polynomial surrogate of resonance hypersurfaces, thereby enlarging the feasible set compared to hard constraints. The method uses an eight-state dynamic model derived from a two-beam representation and a frequency-prediction mechanism based on lift position and mass , enabling a quasi-time-optimal objective with slack-based penalties for frequency violations. Numerical results on a stacker crane and a fixed-frequency 2-DOF system demonstrate improved vibration damping and reduced bang-bang behavior while retaining near-minimal transition times, with practical implications for robust, fast positioning of flexible payloads.

Abstract

In many mechatronic applications, controller input costs are negligible and time optimality is of great importance to maximize the productivity by executing fast positioning maneuvers. As a result, the obtained control input has mostly a bang-bang nature, which excite undesired mechanical vibrations, especially in systems with flexible structures. This paper tackles the time-optimal control problem and proposes a novel approach, which explicitly addresses the vibrational behavior in the context of the receding horizon technique. Such technique is a key feature, especially for systems with a time-varying vibrational behavior. In the context of model predictive control (MPC), vibrational behavior is predicted and coped in a soft-constrained formulation, which penalize any violation of undesired vibrations. This formulation enlarges the feasibility on a wide operating range in comparison with a hard-constrained formulation. The closed-loop performance of this approach is demonstrated on a numerical example of stacker crane with high degree of flexibility.

Paper Structure

This paper contains 8 sections, 9 equations, 4 figures.

Table of Contents

  1. NUMERICAL CASE STUDY: Stacker Crane
  2. Dynamic
  3. Time-Varying Resonance Frequencies
  4. Inequality Constraints: Over- and Under approximation
  5. Frequencies Prediction
  6. Results: Stacker Crane
  7. Additional case study: Fixed Boundary Frequencies
  8. CONCLUSION

Figures (4)

  • Figure 1: flexible stacker crane (right), illustration (left)
  • Figure 2: resonance frequency (blue), approximation (black), boundaries (gray)
  • Figure 3: active vibration damping control for STC with soft NMPC due to $\mathbb {P}_N(x_0, .)$ (left) and $\mathbb {P}^Q_N(x_0, .)$ (right)
  • Figure 4: active vibration damping control with time-optimal (solid gray) and soft-constrained time-optimal (dotted) NMPC

Theorems & Definitions (1)

  • Remark 1