Model Predictive Control with Preview: Recursive Feasibility and Stability

Abstract: This paper proposes a stabilising model predictive control (MPC) scheme with preview information of disturbance for nonlinear systems. The proposed MPC algorithm is able to not only reject disturbance by making use of disturbance preview information as necessary, but also take advantage of the disturbance if it is good for a control task. This is realised by taking into account both the task (e.g. reference trajectory) and disturbance preview in the prediction horizon when performing online optimisation. Conditions are established to ensure recursive feasibility and stability under the disturbance. First the disturbance within the horizon is augmented with the state to form a new composite system and then the stage cost function is modified accordingly. With the help of input-to-state stability theory, a terminal cost and a terminal constraint are constructed and added to the MPC algorithm with preview to guarantee its recursive feasibility and stability under a pre-bounded disturbance. Numerical simulation results demonstrate the effectiveness of the proposed MPC algorithm.

• The paper introduces an MPC method that integrates disturbance preview to robustly manage disturbance-driven nonlinear dynamics.
• It employs input-to-state stability theory and a tailored Lyapunov function to guarantee recursive feasibility and convergence.
• Simulations demonstrate enhanced performance with lower cost metrics and improved settling time compared to conventional MPC.

Model Predictive Control with Preview: Recursive Feasibility and Stability

Introduction

The paper proposes an advanced approach to model predictive control (MPC) by integrating preview information of disturbances for nonlinear systems. This method extends conventional MPC frameworks by not only rejecting disturbances but also utilizing them to potentially enhance control objectives. The authors construct conditions ensuring recursive feasibility and stability, using input-to-state stability (ISS) theory. This approach is validated with numerical simulations showcasing its effectiveness over existing MPC methods.

Model Predictive Control Framework

Problem Formulation

The central focus of this research is handling a discrete-time nonlinear system subject to bounded disturbances. The state dynamics are: $x(k+1) = f(x(k), u(k), w(k))$ with state $x(k)$, control input $u(k)$, and disturbance $w(k)$ subject to constraints. The objective of the proposed MPC is to effectively manage these disturbances while driving the system states toward a target equilibrium.

MPC Design and Augmented System

The innovation in this MPC approach arises from incorporating disturbance preview into the control strategy. This involves forming a composite system by integrating disturbance dynamics with the system state. The optimization problem is constructed over a prediction horizon, balancing task objectives with disturbance predictions.

The cost function accounts for state, input, and disturbance penalties. Terminal costs and constraints are introduced to secure the system’s stability and feasibility, specifically leveraging $\mathcal{K}_{\infty}$ functions for robustness against disturbances.

Stability and Feasibility

Key theoretical contributions include establishing recursive feasibility and input-to-state stability for the proposed framework. By designing a terminal constraint set and cost function, the authors ensure that the system retains stability under bounded disturbances, a significant improvement over traditional MPC approaches.

The paper’s stability proof rests on defining a Lyapunov function suitable for the augmented system, demonstrating decay properties that guarantee convergence and system robustness to disturbances.

Implementation Considerations

Off-line Preparation

1. Weights Selection: Determine stage cost weights for state, input, and disturbance components.
2. Assumption Verification: Ensure the system complies with the posed assumptions regarding constraints and dynamic function continuity.
3. Terminal Conditions: Calculate terminal cost weights and set constraints for stability assurance.

Online Execution

1. State Measurement: Real-time assessment of system state and disturbance preview.
2. Optimization: Solve the formulated MPC problem to obtain optimal control actions.
3. Control Application: Implement the control inputs iteratively, adjusting the system in light of measured disturbances.

Simulation and Results

Numerical simulations reflect the substantial advantages of this approach, displaying lower costs and tighter control margins in comparison with existing methods. The use of disturbance preview information in optimization allows the controller to exploit potential advantageous disturbances, enhancing performance metrics like settling time and control effort.

Conclusion

This research introduces a refined MPC framework adept at leveraging disturbance previews for enhanced system control under nonlinear conditions. By ensuring recursive feasibility and stability through properly designed terminal components, this method shows significant potential in applying MPC to real-world systems where disturbance prediction is viable. Future work may explore handling noise and estimation errors in disturbance predictions, broadening the practical applicability of this advanced MPC scheme.