Sum-of-Squares Data-driven Robustly Stabilizing and Contracting Controller Synthesis for Polynomial Nonlinear Systems

TL;DR

This work addresses robust control for polynomial nonlinear systems by enforcing local contraction on a compact set through data-driven, sum-of-squares (SOS) methods. It combines the data informativity framework with contraction analysis to certify robust contraction under noise and state estimation errors, and develops an SOS-based synthesis to compute stabilizing, contracting controllers from data. A fixed-gain contraction criterion is established, followed by a full SOS program that yields a robust stabilizing controller $G = B^{\dagger} \Gamma / a$ when a feasible solution exists. The approach is demonstrated on a planar UAV model with unknown wind, showing measurable contraction of both nominal and off-nominal trajectories and providing explicit contraction rates and bounds on trajectory deviation, highlighting potential for online deployment during data outages. The results offer a practical pathway for reliable, data-driven control in aerospace and other domains where sensor data quality fluctuates.

Abstract

This work presents a computationally efficient approach to data-driven robust contracting controller synthesis for polynomial control-affine systems based on a sum-of-squares program. In particular, we consider the case in which a system alternates between periods of high-quality sensor data and low-quality sensor data. In the high-quality sensor data regime, we focus on robust system identification based on the data informativity framework. In low-quality sensor data regimes we employ a robustly contracting controller that is synthesized online by solving a sum-of-squares program based on data acquired in the high-quality regime, so as to limit state deviation until high-quality data is available. This approach is motivated by real-life control applications in which systems experience periodic data blackouts or occlusion, such as autonomous vehicles undergoing loss of GPS signal or solar glare in machine vision systems. We apply our approach to a planar unmanned aerial vehicle model subject to an unknown wind field, demonstrating its uses for verifiably tight control on trajectory deviation.

Figures (3)

• Figure 1: Overview of the data-driven robustly contracting controller approach (bottom), compared to a contraction-unaware approach based on expanding integral inequalities (top).
• Figure 2: Trajectory deviation between the nominal trajectory and twenty realizations of the off-nominal system's trajectory with varying initial states. The theoretically guaranteed best contraction rate is honored and outperformed in practice using the method presented here.
• Figure 3: Trajectories of the nominal and off-nominal systems under the robustly contracting controller, where the first two states (position) are plotted.

Theorems & Definitions (21)

• Definition 1: Local One-sided Lipschitz Constant
• Remark 1
• Proposition 1: Local Data-driven Contractivity
• proof
• Remark 2
• Definition 2: Sum-of-Squares Zakeri2014
• Remark 3
• Definition 3: Sum-of-Squares Program Zakeri2014
• Theorem 1
• proof