[Paper Review] Robust MPC for LTI Systems with Parametric and Additive Uncertainty: A Novel Constraint Tightening Approach.

Monimoy Bujarbaruah, Ugo Rosolia+3 more|arXiv (Cornell University)|Jul 2, 2020

Advanced Control Systems Optimization54 references4 citations

TL;DR

This paper proposes a novel robust MPC framework for linear time-invariant (LTI) systems with both parametric and additive uncertainty. By leveraging optimization-based constraint tightening using known bounds on system matrices and disturbances, and with a tailored terminal cost and constraint set, the method guarantees robust constraint satisfaction and input-to-state stability of the origin under uncertainty.

ABSTRACT

We propose a novel approach to design a robust Model Predictive Controller (MPC) for constrained uncertain linear systems. The uncertain system is modeled as linear parameter varying with additive disturbance. Set bounds for the system matrices and the additive uncertainty are assumed to be known. We formulate a novel optimization-based constraint tightening strategy around a predicted nominal trajectory which utilizes these bounds. With an appropriately designed terminal cost function and constraint set, we prove robust satisfaction of the imposed constraints by the resulting MPC in closed-loop with the uncertain system, and Input to State Stability of the origin. We highlight the efficacy of our proposed approach via a numerical example.

Motivation & Objective

• To address robust constraint satisfaction in LTI systems affected by both parametric uncertainty and additive disturbances.
• To develop a constraint tightening strategy that leverages known bounds on system matrices and disturbances to ensure closed-loop robustness.
• To guarantee input-to-state stability (ISS) of the origin under the proposed MPC scheme.
• To design a terminal cost and terminal constraint set that ensure recursive feasibility and robustness.
• To demonstrate the effectiveness of the approach through a numerical example.

Proposed method

• The method models the uncertain system as linear parameter-varying with additive disturbances, using known bounds on system matrices and disturbance terms.
• It introduces an optimization-based constraint tightening technique applied around a predicted nominal trajectory to account for uncertainty.
• The approach employs a tailored terminal cost function and terminal constraint set to ensure recursive feasibility and robust constraint satisfaction.
• Robustness is proven via Lyapunov-based arguments, establishing input-to-state stability (ISS) of the origin.
• The constraint tightening is computed offline using the known uncertainty bounds, enabling online computational efficiency.
• The controller is designed to handle both parametric uncertainty in system matrices and bounded additive disturbances simultaneously.

Experimental results

Research questions

• RQ1 How can constraint tightening be systematically designed to ensure robust satisfaction of constraints under both parametric and additive uncertainty?
• RQ2 What terminal cost and constraint set are required to guarantee recursive feasibility and stability in the presence of uncertainty?
• RQ3 Can a robust MPC framework be developed that ensures input-to-state stability for LTI systems with mixed uncertainty types?
• RQ4 How does the proposed constraint tightening strategy compare in conservatism and performance to existing robust MPC approaches?
• RQ5 What is the impact of uncertainty bounds on the robustness and feasibility of the MPC controller?

Key findings

• The proposed MPC scheme guarantees robust satisfaction of state and input constraints under both parametric and additive uncertainty.
• Input-to-state stability (ISS) of the origin is proven for the closed-loop system using the designed terminal cost and constraint set.
• The constraint tightening strategy effectively accounts for uncertainty bounds without requiring online robust optimization over the full uncertainty set.
• The method ensures recursive feasibility through the tailored terminal ingredients, enabling long-horizon control.
• A numerical example demonstrates the efficacy and practicality of the proposed approach in handling mixed uncertainty types.
• The approach achieves robustness with potentially reduced conservatism compared to traditional robust MPC methods.