[Paper Review] Input-Output Stability of Barrier-Based Model Predictive Control

Panagiotis Petsagkourakis, William P. Heath+2 more|arXiv (Cornell University)|Mar 7, 2019

Advanced Control Systems Optimization27 references4 citations

TL;DR

This paper establishes input-output stability conditions for barrier-based model predictive control (MPC) of linear systems with linear and convex nonlinear constraints using integral quadratic constraints (IQCs). By constructing static and dynamic multipliers via convex optimization, the method ensures global robust stability under unstructured model uncertainty, demonstrating significant improvements in robustness over conventional MPC in numerical examples.

ABSTRACT

Conditions for input-output stability of barrier-based model predictive control of linear systems with linear and convex nonlinear (hard or soft) constraints are established through the construction of integral quadratic constraints (IQCs). The IQCs can be used to establish sufficient conditions for global closed-loop stability. In particular conditions for robust stability can be obtained in the presence of unstructured model uncertainty. IQCs with both static and dynamic multipliers are developed and appropriate convex searches for the multipliers are presented. The effectiveness of the robust stability analysis is demonstrated with an illustrative numerical example.

Motivation & Objective

• To develop sufficient conditions for global input-to-output stability in barrier-based MPC under unstructured model uncertainty.
• To extend robust stability analysis to systems with hard and soft convex constraints, including time-varying and nonlinear constraints.
• To reduce conservatism in robustness analysis by leveraging dynamic multipliers and convex optimization for multiplier computation.
• To demonstrate the effectiveness of barrier-MPC in improving robustness compared to conventional MPC through numerical validation.
• To provide a framework applicable to both time-invariant and time-varying constraints using Zames-Falb and convex-multiplier techniques.

Proposed method

• Formulates barrier-based MPC using recentered and relaxed barrier functions for hard and soft constraints, respectively.
• Applies integral quadratic constraints (IQCs) to model the controller and unstructured uncertainty, enabling robustness analysis.
• Constructs static and dynamic multipliers for IQC-based analysis, with a focus on Zames-Falb (ZF) multipliers for time-invariant constraints.
• Develops a convex optimization framework to compute multipliers efficiently, enabling tractable stability verification.
• Uses the LMI-based multiplier search methodology from Jonsson and Rantzer to unify the analysis of nonlinearities and uncertainty.
• Employs a receding horizon optimization framework with gradient recentering to maintain stability and feasibility under constraints.

Experimental results

Research questions

• RQ1 What conditions ensure global input-to-output stability for barrier-based MPC under unstructured model uncertainty?
• RQ2 How can IQCs with static and dynamic multipliers be constructed to analyze robust stability in barrier-MPC?
• RQ3 To what extent does barrier-MPC reduce conservatism in robustness analysis compared to conventional MPC?
• RQ4 Can convex optimization be used to efficiently compute multipliers that guarantee stability for time-invariant and time-varying constraints?
• RQ5 How does the performance of barrier-MPC compare to nominal MPC in terms of maximum stable gain and robustness margins?

Key findings

• Barrier-based MPC achieves a maximum stable gain of 2.913 under input uncertainty, significantly outperforming nominal MPC, which is stable only up to a gain of 1.130.
• For a fixed design parameter $ r = 0.001 $, the system remains stable across all tested initial conditions, confirming robustness in simulations.
• The C-ZF multiplier with $ N_{ZF} = 10 $ yields a minimum $ r $ of 0.0001 for $ b = 0.25 $, indicating minimal conservatism in stability prediction.
• The analysis predicts stable operation for $ b = 0.5112 $ when $ r = 0.1 $, while nominal MPC fails to predict stability for $ b > 0.0986 $, highlighting superior robustness.
• Simulations confirm that barrier-MPC remains stable at $ ho = 3.4 $, very close to the computed stability limit of 2.913, validating the theoretical bounds.
• The use of dynamic multipliers and convex optimization reduces conservatism, especially in tighter constraint cases, enabling more accurate and less conservative stability margins.