A looped-functional approach for robust stability analysis of linear impulsive systems (1207.1893v1)

Abstract: 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.

• The paper introduces a groundbreaking looped-functional framework that translates discrete Lyapunov conditions into continuous LMIs to assess robust stability.
• It formulates distinct LMI conditions to rigorously characterize minimal, maximal, and ranged dwell-time stability in linear impulsive systems.
• The approach extends seamlessly to uncertain systems by ensuring computationally efficient stability guarantees beyond traditional exponential constraints.

An Academic Review of "A looped-functional approach for robust stability analysis of linear impulsive systems"

The paper under review presents an innovative methodological framework for the robust stability analysis of linear impulsive systems. Such systems, characterized by state discontinuities at discrete time points, appear in various applications across fields like control systems, epidemiology, and power electronics. The authors introduce a looped-functional approach, leveraging non-monotonic Lyapunov functions to formulate Linear Matrix Inequalities (LMIs) that exclude exponential terms. This methodological advancement enables a more flexible characterization of dwell-times for certain and uncertain systems while being extendable to a broader class of impulsive systems than previous methods.

Key Contributions and Methodology

The primary contribution lies in the development of a novel framework that translates a discrete-time Lyapunov stability condition into a continuous-time setting using looped-functionals. This is pivotal as it allows for the consideration of non-monotonic Lyapunov functions, a significant departure from traditional stability analysis methods that necessitate strictly decreasing continuous-time Lyapunov functions.

1. Looped-Functional Framework: The authors define a class of looped-functionals that satisfy specific boundary conditions and possess certain differentiability properties. This framework aids in transforming the stability analysis into a set of LMIs, enabling the characterization of dwell-time stability without reliance on exponential bounded conditions.
2. Stability Analysis: They introduce separate results for periodic impulses, minimal dwell-time, maximal dwell-time, and ranged dwell-time. For each case, the conditions ensuring asymptotic stability are translated into feasible LMIs, providing sufficient conditions that are particularly useful for linear impulsive systems with uncertainties.
3. Robustness Against Uncertain Systems: The method extends naturally to systems with uncertain parameters, modeled as belonging to given polytopes. The LMI conditions derived are affine in the dwell-time, ensuring computational tractability and applicability to a wide range of uncertain systems.

Numerical Results and Implications

The paper includes exemplary cases illustrating the efficacy of their approach. In particular, comparisons with existing methods underscore the proposed scheme's enhanced capability in accounting for systems where neither the continuous-time dynamics nor the discrete-time jump matrix are independently stable. This improved capture of system structure allows for a more accurate estimation of dwell-time intervals that ensure stability.

1. Maximal Dwell-Time: The methodology provides lower bounds on dwell-times which, although slightly conservative compared to direct spectral radius conditions, offer infinitely precise LMI-based solutions applicable to uncertain systems with improved computational efficiency.
2. Minimal Dwell-Time and Arbitrary Impulses: The results highlight the method's effectiveness in situations where high-frequency impulses may destabilize a system. The method's transition to arbitrary impulse sequences also demonstrates its flexibility and broader applicability.

Theoretical and Practical Implications

The theoretical implications of this work are manifold. The authors successfully move the domain of hybrid system stability analysis beyond the traditional exponential-bounded constraints, addressing previously intractable cases involving uncertainties and providing a novel perspective on dwell-time stability.

In practical terms, the extensions to uncertain systems and the affine dependence of the derived conditions on system matrices and dwell-times encourage applications in fields requiring precise stability guarantees such as networked control systems and sampled-data systems.

Future Directions

The paper opens several avenues for future research. Potential developments could involve exploring higher-order Lyapunov functions, incorporating adaptive control laws for real-time stability adjustments, or extending this framework to nonlinear setups and systems exhibiting Zeno behavior. Applications in emerging hybrid technology fields might also benefit from this robust methodological base.

In conclusion, the looped-functional approach signifies a substantial stride in the robustness analysis for impulsive systems, meeting the need for comprehensive yet computationally viable solutions in handling system uncertainties and varying impulses.