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A note on finite-time and fixed-time stability
Wenlian Lu, Xiwei Liu, Tianping Chen
TL;DR
Finite-time and fixed-time stability are examined through an inverse-problem formulation because finite-time convergence is practically relevant. The paper derives necessary and sufficient conditions, proposes applications to synchronization and consensus, and states a general fixed-time result, while its illustrative extensions retain specific assumptions on the governing functions.
Problem
Finite-time convergence is of practical interest, but existing analyses rely substantially on candidate Lyapunov functions and their convergence properties.
Method
The paper treats finite-time stability as an inverse problem involving t(V) and develops a general approach for synchronization and consensus.
Results
The paper gives necessary and sufficient conditions for finite-time and fixed-time stability and states fixed-time convergence by ω1 + ω2 under its general theorem.
Takeaways & Limitations
The general approach provides a framework for finite-time and fixed-time synchronization and consensus, from which some existing results can be derived.
Takeaways & Limitations
The stated extensions assume conditions such as positive µ1 and µ2 for V(t) > 0 and µ1(0) = 0, with related examples using a discontinuous monotone-nondecreasing µ(V).
Abstract
from arXiv · showhide
In this letter, by regarding finite-time stability as an inverse problem, we reveal the essence of finite-time stability and fixed-time stability. Some necessary and sufficient conditions are given. As application, we give a new approach for finite-time and fixed-time synchronization and consensus. Many existing results can be derived by the general approach.
1 Introduction
The letter reframes finite-time and fixed-time stability as an inverse-function problem, deriving conditions and applying them to synchronization and consensus. It also relates convergence behavior to the behavior of the governing function near zero and, for fixed-time convergence, at infinity.
- Motivation: Finite-time stability is motivated by practical interest in convergence over a finite interval rather than asymptotic stability.The paper describes finite-time behavior as more physically realizable than convergence over infinite time.
- Applications: The framework is applied to finite-time and fixed-time synchronization and consensus, and the conclusion states that some existing results can be derived from it.The consensus case appears as a special case of the synchronization formulation.
- Inverse-problem formulation: The proposed approach treats finite-time convergence through the inverse function t(V), rather than directly through V(t).Because V(t) is decreasing, the trajectory can be represented using time as a function of the state measure.
- Stability conditions: Finite-time stability is necessary and sufficient when the relevant integral near V = 0 satisfies the criterion stated in Proposition 1.The proposition characterizes convergence to zero after a time depending on the initial value.
- Stability conditions: Fixed-time stability additionally requires the integral behavior associated with V = ∞, yielding a convergence-time bound independent of the initial value.The paper distinguishes fixed-time convergence from finite-time convergence by its uniform upper bound.
- General theorem: The general Dini-derivative condition combines two functions so that V(t) reaches zero by the sum ω1 + ω2, establishing fixed-time stability.The proof handles both initial-value regimes, V(0) ≤ 1 and V(0) ≥ 1.
2 Applications: Finite-time and fixed-time synchronization and consensus
The paper applies its finite-time and fixed-time stability results to synchronization and consensus in nonlinear networks, including systems on strongly connected undirected graphs. The approach yields finite-time and fixed-time synchronization and consensus results, including settling-time results and existing results as consequences.
- The applications consider nonlinear node couplings on strongly connected undirected graphs for finite-time and fixed-time synchronization and consensus.
- Under linear coupling, Lyapunov analysis gives exponential convergence rather than finite-time convergence.The cited condition is V˙(t) ≤ −αV(t), corresponding to µ(V)=V.
- Finite-time synchronization: Replacing the linear coupling with nonlinear coupling provides a route to finite-time synchronization.The paper introduces a nonlinear coupled network with N nodes and positive scalar α.
- Finite-time synchronization: Theorem 2 establishes synchronization for the nonlinear network under a symmetric, irreducible coupling matrix with nonnegative off-diagonal entries.The proof differentiates a Lyapunov function and invokes the preceding stability theorem.
- Fixed-time synchronization: For fixed-time synchronization, the paper combines the Lyapunov analysis with Theorem 2 and provides the settling time.The fixed-time result is stated for the corresponding nonlinear network setting.
- Consensus: When f(·)=0 and n=1, the synchronization problem reduces to finite-time and fixed-time consensus.The paper then gives finite-time and fixed-time consensus results for nonlinear consensus models.
3 Conclusion
The paper reframes finite-time stability as an inverse problem, derives necessary and sufficient conditions for finite-time and fixed-time stability, and applies them to synchronization and consensus. It also reports new results and notes that many existing results follow from the general approach.
- Finite-time stability is treated as an inverse problem to reveal the essence of finite-time and fixed-time stability.
- The paper gives necessary and sufficient conditions for finite-time and fixed-time stability.
- A new approach is provided for finite-time and fixed-time synchronization and consensus, with additional results.
- Many existing results can be derived as direct consequences of the general approach.