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A looped-functional approach for robust stability analysis of linear impulsive systems
Corentin Briat, Alexandre Seuret
TL;DR
The paper addresses drawbacks in impulsive-system stability analysis, including exponential terms that hinder robust dwell-time characterization. It proposes looped-functionals to express discrete-time stability through a continuous-time approach, accommodating non-monotonic Lyapunov functions and yielding exponential-free conditions for certain and uncertain systems.
Problem
Existing stability conditions contain exponential terms that complicate robust dwell-time characterization and are difficult to check over dwell-time intervals.
Method
The paper uses looped-functionals to obtain a continuous-time formulation of discrete-time stability, allowing non-monotonic Lyapunov functions between impulse-times.
Results
The approach provides dwell-time stability results for known and uncertain linear impulsive systems and applies to systems with discrete-events more broadly than existing approaches.
Takeaways & Limitations
The resulting exponential-free stability conditions facilitate robustness results and provide tractable tools for analyzing uncertain linear impulsive systems.
Takeaways & Limitations
The approach may introduce conservatism through the choice of functional, and some results require stability properties such as J being Schur.
Abstract
from arXiv · showhide
A new functional-based approach is developed for the stability analysis of linear impulsive systems. The new method, which introduces looped-functionals, considers non-monotonic Lyapunov functions and leads to LMIs conditions devoid of exponential terms. This allows one to easily formulate dwell-times results, for both certain and uncertain systems. It is also shown that this approach may be applied to a wider class of impulsive systems than existing methods. Some examples, notably on sampled-data systems, illustrate the efficiency of the approach.
1. Introduction
The paper develops looped-functionals to address exponential-term difficulties in stability and robust dwell-time analysis for linear impulsive systems. The approach permits non-monotonic Lyapunov functions and extends to broader impulsive-system classes.
- Impulsive systems combine continuous flows with discrete state jumps at impulse instants and arise in fields including sampled-data and networked control.
- Periodic-impulse analysis reduces to checking the Schurness of Je^AT, but this does not extend reliably to aperiodic impulses.
- Exponential terms make robust LMIs numerically complex and hinder robust analysis, especially for block matrix uncertainties at the exponential.
- Looped-functionals translate discrete-time stability into continuous-time conditions, allowing pointwise Lyapunov decrease and non-monotonic behavior between impulses.
- The framework targets systems with discrete events or time-marks, including sampled-data and impulsive systems, and can extend to other systems with such marks.
- The resulting LMIs are affine in dwell-time and exponential-free, supporting periodic and aperiodic stability analysis for certain and uncertain systems.
2. Preliminary definitions and results
The preliminary framework lifts impulsive trajectories into variable-support function spaces and relates discrete-time Lyapunov decrease at impulse instants to continuous-time looped-functionals. This supports stability under ranged dwell-times while allowing non-monotonic Lyapunov behavior.
- The lifted state space uses continuous-function trajectories with varying support, which is necessary to represent aperiodic dwell-times.
- A looped-functional satisfies endpoint equality f(0,z,T)=f(T,z,T) and a derivative condition over each inter-impulse interval.
- Theorem 2.4 establishes equivalence between decreasing discrete-time Lyapunov values at impulse instants and existence of a suitable looped-functional.
- For impulse intervals Tk∈[Tmin,Tmax], the resulting criterion guarantees asymptotic stability for every admissible impulse sequence without accumulation points.
- Discrete-time decrease can imply boundedness and convergence of the continuous-time Lyapunov function and state norm, with Tmax unbounded when A is Hurwitz.
- The method accommodates non-monotonic Lyapunov functions, so expansive jumps or unstable flows can be offset by sufficient decrease at impulse instants.
3. Nominal stability analysis of linear impulsive systems
The looped-functional approach replaces exponential-term LMIs with tractable dwell-time conditions and extends stability analysis to broader impulsive-system classes. Nominal results cover periodic, minimal, maximal, arbitrary, and ranged dwell-times, with examples demonstrating accurate stability intervals.
- Motivation: Exponential terms make existing LMIs difficult to check over dwell-time intervals and obstruct robust stability extensions.The difficulty is tied to handling uncertainties at matrix exponentials.
- Periodic impulses: The functional formulation yields sufficient periodic-impulse stability conditions, but selecting a specific functional sacrifices necessity.Theorem 2.4 provides equivalent statements, whereas the chosen functional produces sufficient conditions only.
- Dwell-time characterizations: The approach provides minimal- and maximal-dwell-time conditions and guarantees stability for impulse sequences satisfying corresponding lower or upper dwell-time bounds.For example, stability follows for all inter-impulse distances at least T in the minimal-dwell-time result and within ε < T_k ≤ T in the maximal-dwell-time result.
- Ranged dwell-time: Ranged-dwell-time conditions guarantee asymptotic stability for every impulse sequence with T_k ∈ [T_min, T_max], while avoiding a semi-infinite-dimensional LMI check.The resulting finite-dimensional formulation applies over the entire interval.
- Scope and examples: The method applies even when neither A nor J is stable, unlike existing approaches requiring stability or anti-stability of at least one matrix.This broader applicability is demonstrated on a system whose continuous-time and jump dynamics each contain stable and unstable modes.
- Numerical examples: Example intervals include [0.1824, 0.5776] for periodic impulses and [0.1907, 0.5063] for aperiodic impulses, with the latter contained in the former.Other examples report maximal-period estimates of 0.4620 and 0.4471, and a minimal-period value of 1.2323.
4. Quadratic stability analysis of aperiodic uncertain linear impulsive systems
The section extends looped-functional stability conditions to uncertain impulsive systems, yielding tractable robust periodic, minimal, maximal, and ranged dwell-time results. The conditions preserve affine dependence and avoid exponential terms.
- The affine structure makes extending looped-functional conditions to polytopic uncertainties immediate while preserving much of the approach’s efficiency.
- Periodic impulses: Theorem 4.1 gives a periodic-impulse robust stability condition for uncertain linear impulsive systems through finitely many matrix inequalities.
- Dwell-time results: A robust minimal dwell-time condition guarantees asymptotic stability for every impulse sequence satisfying tk+1 − tk ≥ T.
- Dwell-time results: A robust maximal dwell-time condition guarantees asymptotic stability when impulse intervals satisfy ε < tk+1 − tk ≤ T.
- Dwell-time results: A robust ranged dwell-time condition guarantees asymptotic stability for impulse intervals in [Tmin, Tmax].
- Examples: For the uncertain example, the periodic maximal-period bound is 0.1148 versus 0.1155 from fine-grid analysis, while the aperiodic bound is also 0.1148.
5. Conclusion
The conclusion presents looped-functionals as a framework for uncertain linear impulsive systems that combines discrete-time stability criteria with continuous-time analysis. It highlights broader applicability and identifies extensions to nonlinear and higher-order settings as future work.
- The approach uses a continuous-time formulation to express a discrete-time stability criterion, allowing non-monotonic continuous-time Lyapunov functions.
- The framework can analyze a wider class of systems than existing approaches and has been illustrated by several examples.
- Applications to nonlinear systems, higher-order Lyapunov functions, and T-dependent Lyapunov functions are identified as future work beyond the paper’s scope.