A Lyapunov-Based Small-Gain Theorem for Fixed-Time ISS: Theory, Optimization, and Games
Abstract
We develop a Lyapunov-based small-gain theorem for establishing fixed-time input-to-state stability (FxT-ISS) guarantees in interconnected nonlinear dynamical systems. The proposed framework considers interconnections in which each subsystem admits a FxT-ISS Lyapunov function, providing robustness with respect to external inputs. We show that, under an appropriate nonlinear small-gain condition, the overall interconnected system inherits the FxT-ISS property. In this sense, the proposed result complements existing Lyapunov-based smallgain theorems for asymptotic and finite-time stability, and enables a systematic analysis of interconnection structures exhibiting fixed-time stability. To illustrate the applicability of the theory, we study feedback-based optimization problems with time-varying cost functions, and Nash-equilibrium seeking for noncooperative games with nonlinear dynamical plants in the loop. For both problems, we present a class of non-smooth gradient or pseudogradient-based controllers that achieve fixed-time convergence without requiring time-scale separation and using real-time feedback. Numerical examples are provided to validate the theoretical findings.
Summary
This paper introduces a Lyapunov-based small-gain theorem for establishing fixed-time input-to-state stability (FxT-ISS) guarantees in interconnected nonlinear dynamical systems. The core idea is to analyze interconnected systems where each subsystem possesses a FxT-ISS Lyapunov function, thereby providing robustness against external inputs and interconnections. The paper demonstrates that under a suitable nonlinear small-gain condition, the entire interconnected system inherits the FxT-ISS property. This complements existing small-gain theorems for asymptotic and finite-time stability, offering a more comprehensive framework for analyzing systems exhibiting fixed-time stability. The authors showcase the applicability of their theoretical framework by studying two specific problems: feedback-based optimization with time-varying cost functions and Nash-equilibrium seeking for noncooperative games with nonlinear dynamical plants. For both problems, they propose a class of non-smooth gradient or pseudo-gradient-based controllers that achieve fixed-time convergence using real-time feedback, eliminating the need for time-scale separation or passivity-based assumptions. Numerical examples are provided to validate the theoretical findings and demonstrate the effectiveness of the proposed controllers. This work matters to the field because it provides a powerful tool for analyzing and designing fixed-time stable systems, particularly in complex interconnected scenarios where traditional methods are insufficient.
Key Insights
- •Novel Lyapunov-based small-gain theorem: The paper provides a new theorem for FxT-ISS, extending existing small-gain results for asymptotic and finite-time stability.
- •No time-scale separation: The proposed controllers for feedback optimization and Nash equilibrium seeking achieve fixed-time convergence without requiring time-scale separation, a common limitation in previous work.
- •Non-smooth controllers: The paper presents a class of non-smooth gradient-based controllers that can handle the complexities of interconnected systems.
- •Relaxed gain function requirements: The paper relaxes the restrictions on gain functions compared to previous work, allowing for a broader class of admissible interconnections.
- •Application to feedback optimization: The paper presents a state-of-the-art result in feedback-based optimization, achieving fixed-time convergence for time-varying cost functions with plants in the loop (Theorem 2).
- •Application to Nash equilibrium seeking: The paper achieves fixed-time stability for Nash equilibrium seeking in potential games with dynamic plants, a novel result in this area (Theorem 3).
- •Tradeoff between subsystems: The small gain condition (e.g. inequality (30)) highlights a tradeoff between the influence of state and input deviations in interconnected subsystems.
Practical Implications
- •Real-world applications: The results can be applied to a wide range of engineering domains, including robotics, power systems, and networked control systems, where fixed-time stability is crucial.
- •Feedback optimization: Practitioners can use the proposed controllers to design feedback optimization schemes for dynamical systems with time-varying objectives, achieving faster and more robust convergence.
- •Game theory: The Nash equilibrium seeking results can be used to design distributed control algorithms for multi-agent systems, enabling agents to learn and adapt to each other's actions in a fixed-time manner.
- •Controller design: Engineers can use the Lyapunov-based small-gain theorem to systematically design controllers for interconnected systems, ensuring fixed-time stability and robustness.
- •Future research directions: The paper opens up several avenues for future research, including extensions to more general classes of interconnected systems, the development of adaptive control schemes, and the application of the results to specific engineering problems.