Safe Navigation with Zonotopic Tubes: An Elastic Tube-based MPC Framework
Abstract
This paper presents an elastic tube-based model predictive control (MPC) framework for unknown discrete-time linear systems subject to disturbances. Unlike most existing elastic tube-based MPC methods, we do not assume perfect knowledge of the system model or disturbance realizations bounds. Instead, a conservative zonotopic disturbance set is initialized and iteratively refined using data and prior knowledge: data are used to identify matrix zonotope model sets for the system dynamics, while prior physical knowledge is employed to discard models and disturbances inconsistent with known constraints. This process yields constrained matrix zonotopes representing disturbance realizations and dynamics that enable a principled fusion of offline information with limited online data, improving MPC feasibility and performance. The proposed design leverages closed-loop system characterization to learn and refine control gains that maintain a small tube size. By separating open-loop model mismatch from closed-loop effects in the error dynamics, the method avoids dependence on the size of the state and input operating regions, thereby reducing conservatism. An adaptive co-design of the tube and ancillary feedback ensures $λ$-contractive zonotopic tubes, guaranteeing robust positive invariance, improved feasibility margins, and enhanced disturbance tolerance. We establish recursive feasibility conditions and introduce a polyhedral Lyapunov candidate for the error tube, proving exponential stability of the closed-loop error dynamics under the adaptive tube-gain updates. Simulations demonstrate improved robustness, enlarged feasibility regions, and safe closed-loop performance using only a small amount of online data.
Summary
This paper addresses the challenge of safe navigation for systems with unknown dynamics and disturbances. The authors propose an elastic tube-based Model Predictive Control (MPC) framework that iteratively refines a zonotopic disturbance set using both online data and prior knowledge. Unlike traditional tube-based MPC methods that assume perfect model knowledge, this approach identifies matrix zonotope model sets for the system dynamics and discards models inconsistent with known constraints. This fusion of offline information and limited online data improves MPC feasibility and performance. The core innovation lies in leveraging closed-loop system characterization to learn and refine control gains, maintaining a small tube size. By separating open-loop model mismatch from closed-loop effects in the error dynamics, the method reduces conservatism, avoiding dependence on the size of the state and input operating regions. The adaptive co-design of the tube and ancillary feedback ensures λ-contractive zonotopic tubes, guaranteeing robust positive invariance, improved feasibility margins, and enhanced disturbance tolerance. The paper establishes recursive feasibility conditions, proposes a polyhedral Lyapunov candidate for the error tube, and proves exponential stability of the closed-loop error dynamics under the adaptive tube-gain updates. Simulations demonstrate improved robustness, enlarged feasibility regions, and safe closed-loop performance with minimal online data.
Key Insights
- •Data-Driven Model Refinement: The paper introduces a novel approach to refine matrix zonotopes representing system dynamics by excluding models inconsistent with prior physical knowledge. This data-driven refinement reduces conservatism compared to methods relying solely on offline information.
- •Closed-Loop Error Dynamics: The error recursion separates open-loop model mismatch from closed-loop effects, enabling tighter tube propagation. This is a significant improvement over methods that lump all uncertainty into a single set, which can lead to conservative tube sizes.
- •Adaptive Tube and Feedback Gain: The method alternates between solving TMPC for a nominal trajectory and updating both the tube and the ancillary feedback. The gain is not fixed a priori but updated at each iteration, reducing worst-case conservatism relative to fixed-gain designs.
- •λ-Contractive Zonotopic Tubes: The design of a λ-contractive zonotopic tube guarantees robust positive invariance while enforcing shrinkage of the error set over time. This increases feasibility margins and enhances disturbance tolerance.
- •Recursive Feasibility and Stability: The paper derives verifiable conditions that guarantee recursive feasibility and proposes a polyhedral Lyapunov candidate for the error tube, certifying exponential stability of the closed-loop error dynamics under adaptive tube–gain updates.
- •Performance Improvement: The simulations demonstrate improved robustness, enlarged feasibility regions, and safe closed-loop performance using only a small amount of online data, suggesting a practical advantage over purely model-based or purely data-driven approaches.
- •Constraint Enforcement: The method ensures safety by tightening the state and input constraints using the tube cross-sections, so feasibility of the nominal MPC guarantees feasibility of the true closed loop for all models in the learned sets.
Practical Implications
- •Robotics and Autonomous Systems: The research has direct applications in robotics and autonomous systems, where safe navigation in uncertain environments is crucial. The proposed framework can be used to design controllers for robots operating in cluttered or dynamic environments, ensuring constraint satisfaction and robustness to disturbances.
- •Aerospace and Automotive Control: The framework is applicable to safety-critical systems in aerospace and automotive control, where hard state and input constraints must be satisfied. For example, it can be used to design controllers for aircraft or self-driving cars, ensuring safe operation in the presence of model uncertainties and disturbances.
- •Real-Time Implementation: The algorithm is designed to be computationally efficient, potentially enabling real-time implementation in embedded systems. The use of zonotopes and polyhedral computations allows for tractable online computations, making it suitable for applications with limited computational resources.
- •Data-Driven Control Design: The research opens up new avenues for data-driven control design, where controllers are learned directly from data without relying on accurate system models. The proposed framework provides a principled way to combine offline information with limited online data, improving MPC feasibility and performance.
- •Future Research Directions: Future research could focus on extending the framework to nonlinear systems, exploring different types of uncertainty sets, and developing more efficient algorithms for solving the optimization problems involved. Furthermore, investigating the trade-off between conservatism and computational complexity would be valuable.