Data-Driven Compositional Construction of Symbolic Models for Large-Scale Networks with Unknown Dynamics and Interconnection Topology

Core Concepts

This paper proposes a novel data-driven approach to construct symbolic models for large-scale interconnected networks with unknown dynamics and interconnection topologies, enabling formal control synthesis with correctness guarantees, without relying on traditional small-gain conditions.

Abstract

Bibliographic Information:

Samari, B., Nejati, A., & Lavaei, A. (2024). Data-Driven Control of Large-Scale Networks with Formal Guarantees: A Small-Gain Free Approach. arXiv preprint arXiv:2411.06743.

Research Objective:

This paper addresses the challenge of designing controllers for large-scale interconnected networks with unknown mathematical models and interconnection topologies. The authors aim to develop a data-driven approach that leverages the concept of alternating (sub-)bisimulation functions to construct symbolic models of such networks and synthesize controllers guaranteeing desired behaviors.

Methodology:

The authors propose a divide-and-conquer strategy, treating the network as an interconnection of individual subsystems. They formulate a robust optimization problem (ROP) to capture the conditions for alternating sub-bisimulation functions (ASBF) between each subsystem and its symbolic model. To handle the unknown dynamics, they collect data from subsystem trajectories and introduce a scenario optimization program (SOP) derived from the ROP. By solving the SOP, they obtain ASBFs for individual subsystems. Subsequently, they propose a novel data-driven compositional condition to construct an alternating bisimulation function (ABF) for the entire network based on the ASBFs of its subsystems.

Key Findings:

• The proposed data-driven approach eliminates the need for system identification, directly utilizing data to construct symbolic models and similarity relations.
• The data-driven compositional condition enables the construction of an ABF for the network without requiring knowledge of the interconnection topology or relying on traditional small-gain conditions.
• The approach significantly reduces the sample complexity compared to existing methods, scaling linearly with the number of agents instead of exponentially with network size.

Main Conclusions:

The paper presents a practical and scalable framework for data-driven control of large-scale networks with formal guarantees. By leveraging ASBFs and a novel compositional condition, the approach overcomes the limitations of existing methods that require precise model knowledge and suffer from scalability issues.

Significance:

This research significantly contributes to the field of symbolic control by providing a data-driven framework applicable to complex, large-scale networks with unknown dynamics and interconnection topologies. This opens up new possibilities for formally verifying and controlling a wide range of real-world systems, such as automated vehicles, biological processes, and energy infrastructures.

Limitations and Future Research:

While the paper provides a comprehensive framework, future research could explore:

• Relaxing the assumption of Lipschitz continuity for unknown dynamics.
• Investigating the impact of noise and uncertainties in the collected data on the correctness guarantees.
• Extending the approach to handle continuous-time systems and hybrid systems with both continuous and discrete dynamics.

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Data-Driven Control of Large-Scale Networks with Formal Guarantees: A Small-Gain Free Approach

by Behrad Samar... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06743.pdf

Deeper Inquiries

How can this data-driven approach be extended to handle time-varying interconnection topologies, where agents can join or leave the network dynamically?

Extending this data-driven approach to handle time-varying interconnection topologies, where agents can dynamically join or leave the network, presents a significant challenge but also an exciting research direction. Here's a breakdown of potential strategies:
1. Dynamically Updating the ABF:

Local Updates: Instead of reconstructing the entire ABF whenever the topology changes, focus on local updates. When an agent joins or leaves, update the ASBFs of its neighboring agents and re-evaluate the compositional condition (5.1) locally. This requires a mechanism to detect topology changes, possibly through communication between agents or a centralized network monitor.
Time-Varying ASBFs: Investigate the use of time-varying ASBFs, Vi(qi, xi, ˆxi, k), where 'k' represents the discrete time step. This allows the ASBF to adapt to changes in the interconnection topology over time. The challenge lies in formulating suitable update rules for these time-varying ASBFs and ensuring the stability and convergence of the overall system.
2. Robustness to Topology Changes:

Worst-Case Analysis: If the set of possible topology changes is known a priori (e.g., a limited number of agents joining or leaving), perform a worst-case analysis. Construct ASBFs and verify the compositional condition for the worst-case interconnection topology, guaranteeing stability for all possible network configurations.
Adaptive Learning: Employ online learning techniques to adapt the ASBFs and the compositional condition in real-time as the topology changes. Reinforcement learning could be a promising avenue, where agents learn to adjust their ASBF parameters based on the observed network dynamics and rewards associated with maintaining stability.
3. Switching Systems Framework:

Modeled as Switching Systems: Model the time-varying network as a switched system, where each switching mode corresponds to a different interconnection topology. Techniques from switched system theory, such as multiple Lyapunov functions or dwell-time analysis, could be applied to analyze stability and design controllers for the overall switched network.
Challenges:

Data Efficiency: Dynamically changing topologies might require frequent updates to the ASBFs, potentially demanding a large amount of data. Developing data-efficient learning algorithms is crucial.
Computational Complexity: Real-time adaptation of ASBFs and verification of the compositional condition can be computationally expensive, especially for large-scale networks. Distributed and computationally efficient algorithms are essential.

Could the reliance on Lipschitz continuity be relaxed by employing alternative function approximation techniques or incorporating uncertainty representations into the ASBF construction?

Yes, the reliance on Lipschitz continuity can be relaxed by exploring alternative function approximation techniques or incorporating uncertainty representations into the ASBF construction. Here are some potential avenues:
1. Alternative Function Approximation:

Neural Networks: Employ neural networks to approximate the ASBFs, Vi(qi, xi, ˆxi). Neural networks offer flexibility in approximating complex nonlinear functions without requiring explicit knowledge of Lipschitz constants. Techniques like Lipschitz regularization during training can help enforce certain smoothness properties on the learned ASBFs.
Gaussian Processes: Utilize Gaussian processes (GPs) to model the ASBFs. GPs provide a probabilistic framework for function approximation, capturing uncertainty in the learned function. This uncertainty information can be incorporated into the compositional condition, leading to probabilistic guarantees.
2. Uncertainty Representation:

Set-Membership Approaches: Instead of assuming Lipschitz continuity, represent the uncertainty in the unknown dynamics fi using set-membership approaches, such as ellipsoidal or polytopic sets. This allows for a more flexible representation of uncertainty, potentially capturing non-smooth or discontinuous behavior.
Interval Analysis: Employ interval analysis techniques to propagate uncertainty in the dynamics through the ASBF conditions. This provides guaranteed bounds on the ASBF values, even in the presence of uncertainty in the system dynamics.
3. Robust Optimization:

Robust ASBF Conditions: Reformulate the ASBF conditions (3.1a)-(3.1b) as robust optimization problems, explicitly considering the uncertainty in the dynamics fi. This leads to ASBFs that are robust to variations in the system dynamics within a specified uncertainty set.
Benefits of Relaxing Lipschitz Continuity:

Handling More General Dynamics: Allows the framework to handle a broader class of systems, including those with non-smooth or discontinuous dynamics, which are common in many practical applications.
Improved Data Efficiency: By explicitly accounting for uncertainty, the reliance on large amounts of data to accurately estimate Lipschitz constants can be reduced.
Challenges:

Theoretical Guarantees: Establishing formal guarantees, such as stability or convergence, becomes more challenging when relaxing Lipschitz continuity. New theoretical tools and analysis techniques are required.
Computational Complexity: Incorporating uncertainty representations or using more complex function approximators can increase the computational complexity of ASBF construction and verification of the compositional condition.

What are the potential applications of this framework in other domains beyond control systems, such as distributed optimization or game theory, where similar challenges of scalability and unknown interactions arise?

This data-driven framework, designed to address scalability and unknown interactions in large-scale networks, holds significant promise for applications beyond control systems. Let's explore its potential in distributed optimization and game theory:
1. Distributed Optimization:

Decentralized Resource Allocation: Consider a network of agents, each with its own local objective function and constraints, aiming to collaboratively optimize a global objective. This framework can be adapted to design distributed optimization algorithms where agents exchange limited information with their neighbors while ensuring convergence to an optimal solution. The ASBFs could capture the local behavior of each agent's optimization process, and the compositional condition could guarantee convergence of the overall distributed algorithm.
Learning-Based Optimization: In scenarios where the objective functions or constraints are unknown a priori, this framework can be combined with online learning techniques. Agents can learn their local ASBFs and update them based on data collected during the optimization process, leading to adaptive and data-driven distributed optimization algorithms.
2. Game Theory:

Learning in Multi-Agent Systems: In multi-agent systems where agents have unknown payoff functions and interact strategically, this framework can be employed to design learning algorithms. Agents can learn their optimal strategies by observing the actions and payoffs of their neighbors, using ASBFs to approximate the unknown payoff landscape. The compositional condition can ensure stability and convergence of the learning dynamics in the multi-agent system.
Robust Game Design: This framework can aid in designing robust mechanisms for strategic interactions. By incorporating uncertainty representations into the ASBFs, the mechanism can be made robust to uncertainties in agents' preferences or behaviors, leading to more stable and predictable outcomes.
Key Advantages for Distributed Optimization and Game Theory:

Scalability: The compositional nature of the framework allows it to scale to large networks, handling a large number of agents or players.
Handling Unknown Interactions: The data-driven approach eliminates the need for precise knowledge of the interaction dynamics between agents, making it suitable for complex systems with unknown or partially known interactions.
Formal Guarantees: The use of ASBFs and the compositional condition provides formal guarantees on stability, convergence, or robustness, which are crucial for reliable operation of distributed optimization algorithms or game-theoretic mechanisms.
Challenges and Future Directions:

Adapting to Specific Domains: Tailoring the framework to the specific requirements of distributed optimization or game theory, such as incorporating constraints, handling different types of games, or addressing equilibrium concepts, requires further research.
Communication Efficiency: In distributed settings, communication between agents can be a bottleneck. Developing communication-efficient algorithms that leverage the compositional structure of the framework is essential.
Dynamic Environments: Extending the framework to handle dynamic environments, where the objective functions, constraints, or game rules change over time, is an important direction for future research.