Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints (1304.1998v2)

Abstract: Stability analysis and control of linear impulsive systems is addressed in a hybrid framework, through the use of continuous-time time-varying discontinuous Lyapunov functions. Necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas. Extensions to stability of aperiodic systems under minimum, maximum and ranged dwell-times are then derived. By exploiting further the particular structure of the stability conditions, the results are non-conservatively extended to quadratic stability analysis of linear uncertain impulsive systems. These stability criteria are, in turn, losslessly extended to stabilization using a particular, yet broad enough, class of state-feedback controllers, providing then a convex solution to the open problem of robust dwell-time stabilization of impulsive systems using hybrid stability criteria. Relying finally on the representability of sampled-data systems as impulsive systems, the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled-data systems are straightforwardly solved using the same ideas. Several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach.

• The paper outlines convex conditions for robust stability using continuous-time Lyapunov functions in hybrid impulsive and sampled-data systems.
• It extends stability analysis to both periodic and aperiodic impulses by applying minimum, maximum, and ranged dwell-time constraints for effective state-feedback control.
• Numerical results demonstrate improved computational efficiency and practical implications for designing advanced control systems under uncertainty.

Overview of the Paper: Convex Conditions for Robust Stability Analysis and Stabilization of Linear Aperiodic Impulsive and Sampled-Data Systems Under Dwell-Time Constraints

The paper authored by Corentin Briat provides an in-depth exploration of stability analysis and control strategies for linear impulsive systems, particularly in scenarios involving aperiodic impulses and sampled-data systems imposed with dwell-time constraints. The research is conducted within a hybrid framework utilizing continuous-time time-varying discontinuous Lyapunov functions, which are crucial for addressing the inherent complexities of systems with both continuous and discrete dynamics.

Key Contributions

1. Stable Periodic and Aperiodic Systems: Initially, the paper addresses the stability conditions for impulsive systems when impulses occur periodically. It then extends these conditions to handle more complex scenarios where impulsions are aperiodic, introducing minimum, maximum, and ranged dwell-times.
2. Non-Conservative Stability Criteria: By uncovering the particular structure of stability conditions, the research extends results to quadratic stability analysis of linear uncertain impulsive systems. These stability conditions are further losslessly applied for stabilization purposes using state-feedback controllers.
3. Robust Dwell-Time Stabilization: A significant contribution of the paper is offering a robust solution to the open problem of stabilizing impulsive systems with hybrid stability criteria under specified dwell-time constraints. This result is non-trivial given the challenges posed by aperiodic and uncertain system parameters.
4. Sampled-Data Systems Representation: The paper demonstrates that sampled-data systems can be effectively modeled as impulsive systems. Consequently, the robust stability analysis and stabilization of such systems—both periodic and aperiodic—are managed using the same foundational principles established for impulsive systems.

Numerical Results and Implications

Several examples presented in the paper substantiate the effectiveness and computational efficiency of the proposed stabilization methods. For instance, the paper reports strong numerical results that showcase the approach's capability to handle complex stability scenarios with reduced computational costs compared to existing methods such as looped-functionals and summation-based functionals.

The practical implications of this research are substantial, impacting the design and analysis of hybrid systems in control systems engineering. The results have direct applications across fields where systems exhibit both continuous and discrete behaviors, such as networked control systems, sampled systems, and systems with state resets.

Theoretical Developments and Future Directions

The theoretical advancements presented in this paper pave the way for further investigation into robust control strategies for hybrid systems. The use of quadratic Lyapunov functions and dwell-time constraints provides a robust framework that could be extended to non-linear systems, multi-agent systems, and systems with stochastic elements.

Additionally, the concepts introduced here could inspire novel computational methods for real-time control systems, leveraging advancements in optimization and computational mathematics to handle increasingly complex control problems efficiently.

In conclusion, Corentin Briat’s paper makes considerable progress in the stability analysis and control of impulsive and sampled-data systems under dwell-time constraints. The methods proposed not only address fundamental theoretical issues but also promise practical solutions in various applied domains of control engineering. The availability of robust and computationally efficient techniques as presented could significantly enhance the capability of modern control systems to operate under uncertainty and irregular conditions.