Resilient Control: Ensuring Safety in Faulty Systems

by

Author: Denis Avetisyan

A new graph-based approach guarantees the safety of control systems even when faced with communication or computational failures.

This work introduces a novel barrier function framework leveraging graph theory for safety verification and control synthesis in weakly-hard constrained, lossy hybrid systems.

Despite advances in control systems, ensuring safety remains a critical challenge when faced with inevitable communication or computational failures. This is addressed in ‘Safety for Weakly-Hard Control Systems via Graph-Based Barrier Functions’, which introduces a novel framework for verifying safety and synthesizing controllers for systems subject to bounded failures over time. The core contribution lies in a graph-based barrier function approach, offering robust safety guarantees even under unreliable conditions. Could this methodology pave the way for more resilient and dependable autonomous systems operating in real-world environments?

The Inevitable Drift: Embracing Imperfection in Control

Conventional control systems are frequently built on the assumption of flawless data transmission and precise execution of commands, a scenario rarely encountered in real-world applications. This idealized premise struggles to accommodate the inherent imperfections of physical systems, network delays, sensor noise, or computational limitations. Consider robotic surgery, autonomous vehicles, or even sophisticated manufacturing processes – each relies on a cascade of interconnected components, where even minor disruptions in communication or actuator performance can lead to significant deviations from the intended behavior. Consequently, designs predicated on perfect operation often prove brittle and unreliable when confronted with the inevitable realities of imperfect information and execution, necessitating more robust control strategies that explicitly account for these limitations.

Traditional control systems often operate under the assumption of flawless execution and communication, a condition rarely met in real-world applications. Weakly-hard constraints represent a shift in this paradigm, acknowledging that some level of failure is inevitable and even permissible. Instead of striving for absolute perfection, these constraints define acceptable limits to system deviations – bounded ‘losses’ in performance or adherence to desired states. This approach is particularly valuable in complex systems where complete reliability is cost-prohibitive or technically impossible. By explicitly allowing for controlled failures within defined boundaries, weakly-hard constraints enable the design of more robust and practical control strategies, enhancing overall system resilience without demanding unattainable levels of precision. The key lies in designing systems that can gracefully manage these anticipated losses, maintaining functionality even when faced with imperfect conditions.

Robust control design increasingly centers on acknowledging that complete operational certainty is unattainable; systems inevitably experience periods of compromised functionality, termed `LossSequence`s. These aren’t catastrophic failures, but rather bounded deviations from ideal performance – a sensor temporarily losing signal, an actuator exhibiting diminished response, or a communication channel experiencing intermittent disruption. Effectively managing these sequences requires a shift from striving for absolute precision to prioritizing graceful degradation. Control algorithms must be designed not only to achieve desired outcomes under nominal conditions, but also to maintain stability and acceptable performance during and following these predictable, yet unavoidable, losses. Analyzing the characteristics of potential `LossSequence`s – their frequency, duration, and impact on system dynamics – allows engineers to proactively implement strategies like redundancy, fault tolerance, and adaptive control, ultimately building systems that are resilient and reliable even when faced with imperfect execution.

Mapping the Permissible: A Graph-Based Safety Framework

The GraphRepresentation utilizes a directed graph to encode the system’s operational constraints and permissible failure modes. Nodes within the graph represent the system’s state variables, and directed edges signify allowed transitions between states based on control inputs and physical limitations. Constraints, such as input limits or state boundaries, are represented as edges that prevent transitions to unsafe states. Allowable failures are explicitly modeled by defining specific terminal states and associated transitions that, while representing a deviation from nominal behavior, remain within defined safety bounds. This structured representation allows for formal verification of safety properties and enables the construction of control barriers that guarantee safe operation even in the presence of disturbances and model uncertainties.

GraphBasedBarrierFunctions utilize the GraphRepresentation to formally define a safe region within the system’s state space. These functions assign non-negative values to states, with the value approaching zero as the system nears a constraint boundary, thereby quantifying the distance to failure. A key aspect is the consideration of permissible losses; the graph encodes not just absolute constraints, but also the extent to which certain state variables can deviate from ideal values before violating safety criteria. This allows for controlled degradation or temporary excursions, defining a broader, more practical safe region than would be possible with strict, hard constraints. The function’s output serves as a certificate of safety, demonstrably ensuring that the system remains within acceptable bounds given the encoded constraints and loss tolerances.

To address the computational demands of verifying safety in complex systems, simplified variants of the `GraphBasedBarrierFunction` (GBF) have been developed. The `1StepGBF` utilizes a single-step lookahead, reducing the computational burden by focusing on immediate consequences of system states. Further optimization is achieved with the `dGBF`, which leverages derivative calculations to approximate the GBF’s value, enabling efficient monitoring of safety margins without requiring full graph traversals. Crucially, these simplifications do not compromise safety guarantees; formal verification techniques ensure that the reduced computational cost is achieved while maintaining provable safety within the defined operational constraints and permissible loss parameters.

Formalizing Resilience: Synthesis and Verification of Safe Operation

The GraphBasedBarrierFunctions serve as a central component within both the ControllerSynthesis and SafetyVerification frameworks, providing a unified approach to system design and analysis. These functions, defined over the state space represented as a graph, facilitate the formulation of control objectives and safety constraints. Specifically, they are utilized during controller synthesis to shape the control law and ensure desired system behavior, while in safety verification they allow for the formal assessment of whether the system adheres to specified safety criteria. This integration streamlines the development process, enabling consistent constraint handling and efficient verification of synthesized controllers against safety specifications.

To ensure system stability and safety under weakly-hard (WH) constraints, the controller synthesis and safety verification frameworks utilize optimization techniques including Sum-of-Squares Programming (SOS) and Linear Matrix Inequality (LMI) methods. SOS programming relaxes polynomial inequalities into a sum of squares of polynomials, allowing for verification of non-linear constraints, while LMI formulations convert stability and performance requirements into a set of linear matrix inequalities that can be efficiently solved. These techniques are particularly relevant for handling WH constraints, which allow for temporary violations under specific conditions, requiring robust analysis to guarantee overall system safety. The feasibility of these optimization approaches was demonstrated through numerical case studies employing WH constraint parameters of (r,s) = (2,4), (3,7), and (3,5).

The verification process is augmented through the application of an SMTsolver, which rigorously checks adherence to pre-defined safety criteria during both controller synthesis and safety verification. This constraint solving methodology was successfully implemented and validated via numerical case studies employing weakly-hard (WH) constraints. Specifically, the system demonstrated successful operation under parameter sets defined by (r,s) values of (2,4), (3,7), and (3,5), indicating robustness across a defined operational range and confirming the effectiveness of the SMTsolver in ensuring safety compliance.

Embracing the Inevitable: Robust Actuator Strategies for Fault Tolerance

The control framework incorporates distinct actuator strategies designed to mitigate the impact of communication or execution failures. When an actuator disconnects, the system can immediately transition to a ZeroStrategy, effectively halting all output from that specific component-a useful approach when continued operation could be destabilizing. Alternatively, a HoldStrategy allows the actuator to maintain its last known value, providing a degree of continuity that can be crucial for maintaining overall system stability, particularly in scenarios where a brief interruption is anticipated. These strategies aren’t simply reactive measures; they are integrated into the control design, allowing the system to proactively anticipate and manage potential actuator losses, thereby enhancing the robustness and reliability of the entire system. The selection between these, or even more complex strategies, depends on the specific application and the nature of the potential failure modes.

The system’s ability to withstand actuator failures hinges on the synergistic interaction between adaptable control strategies and a graph-based safety framework. This framework doesn’t simply react to failures, but anticipates potential disruptions and integrates them into the control design. When an actuator is compromised – losing communication or failing to execute – pre-defined strategies like ZeroStrategy or HoldStrategy are automatically deployed, minimizing deviation from the desired operational parameters. Crucially, the graph-based approach allows for a holistic assessment of system connectivity; it identifies how a single failure might propagate, and proactively adjusts control inputs to maintain stability and performance across the entire system. This combination ensures not just immediate recovery, but sustained resilience, allowing the system to continue functioning safely and effectively even with compromised components.

The development of truly dependable control systems hinges on anticipating and mitigating potential component failures; this research delivers a proactive strategy for achieving that goal. By designing control frameworks that inherently account for actuator loss – be it through communication disruption or functional breakdown – systems can maintain stability and performance even when faced with adversity. Validation of this approach encompassed both linear and nonlinear – specifically polynomial – system dynamics, demonstrating broad applicability. The method’s implementation relies on a powerful suite of computational tools, leveraging Linear Matrix Inequalities (LMIs), Sum-of-Squares (SOS) programming, and Satisfiability Modulo Theories (SMT) solvers to guarantee robustness and reliability in a computationally verifiable manner.

The pursuit of safety in control systems, as detailed in this work concerning weakly-hard constraints, echoes a fundamental truth about all engineered systems: their inevitable confrontation with imperfection. This research, leveraging graph-based barrier functions, doesn’t aim to eliminate failure-an impossible task-but to manage its impact, allowing for controlled degradation rather than catastrophic collapse. As Jean-Paul Sartre observed, “Man is condemned to be free”; similarly, these systems are ‘condemned’ to operate under uncertainty, and the innovation lies in structuring that freedom – or loss of control – to maintain overall stability. The graph-based approach acknowledges the inherent fragility of interconnected systems, treating incidents not as deviations from a perfect state, but as steps towards a more robust maturity.

What Lies Ahead?

The presented work addresses safety within weakly-hard systems, framing the challenge through a graph-theoretic lens. Yet, the illusion of control persists. Barrier functions, while effective at caching stability, cannot fundamentally alter the entropic trajectory. The system will, inevitably, degrade. The pertinent question becomes not if failure occurs, but the acceptable latency of response when it does. Further exploration must acknowledge that these graphs are themselves transient representations of a dynamic reality, subject to alteration and eventual collapse.

Current methods largely treat communication and computational failures as discrete events. A more nuanced understanding requires modeling these degradations as continuous processes-a slow erosion of system fidelity. Investigations into adaptive barrier functions, capable of recalibrating in real-time to diminishing resources, seem crucial. The synthesis of controllers that anticipate loss, rather than merely reacting to it, remains a significant hurdle.

Ultimately, the pursuit of ‘safe’ systems is a temporary stay against the tide. Research should not solely focus on extending uptime, but on designing graceful degradation pathways. The goal isn’t to build systems that never fail, but systems that fail predictably and with minimized consequence-systems that acknowledge the inherent impermanence of all things.

Original article: https://arxiv.org/pdf/2601.00494.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

See also:

2026-01-05 23:56