Switched systems have recently become the focus of research attributed to the multimodality of systems and variation in the external environment. They are a class of hybrid dynamical systems common in practical applications involving switching between several systems and subsystems. Like most dynamical systems, the stability problem is the most critical aspect of switched systems. Accordingly, several methods for investigating the stability of switched systems under different switching conditions have been proposed. They include multiple Lyapunov function (MLF) methods applied for constrained switching and quadratic Lyapunov function and lie algebra mostly used for arbitrarily switching. However, it is more challenging to perform stability analysis with constrained signals than with arbitrary signals. Therefore, overcoming this challenge is of great research interest to improve switched systems’ stability and operational efficiency.
Lately, a combination of quadratic Lyapunov function, lie algebra and MLF have resulted in more promising strategies such as dwell time and convex combinations for time-driven and event-driven strategies, respectively. In particular, the dwell time is considered natural, intuitive, and therefore suitable for describing slow switching. However, due to its too restrictive nature, average dwell time (ADT) has been proposed. ADT has been under continuous improvement considering the differences of subsystems, resulting in mode-dependent ADT (MDADT). Despite the progress, ADT and MDADT schemes only consider certain aspects of switched systems while ignoring others. The introduction of Ф-dependent average dwell time (Ф DADT) method has emerged as a promising solution to this challenge. By combining the ADT and MDADT frameworks, it is capable of covering unnoticed situations that were previously ignored. It also aids the establishment of a limit form of ФDADT to address the shortcoming associated with both limiting ADT and ФDADT by expanding the feasible region for stability analysis.
Considering the nonlinearity of switched systems, research shows that Lyapunov theory remains the most appropriate for studying the stability of switched systems. The problem, however, is the enhancement of the MLF method to analyze stability based on the limit form of ФDADT. Motivated by the previous findings, Associate Professor Qiang Yu from Shanxi Normal University and Professor Guisheng Zhai from Shibaura Institute of Technology generalized various exiting methods to develop a novel limit inferior ФDADT method for analyzing the stability of switched systems. The work involved determining the sufficient stability conditions under LФDADT scheme and establishing the relationship between different dwell time schemes. The authors also provided an expression of the switched systems’ controller design and stability conditions for further analysis. Finally, three numerical examples were provided to validate the feasibility of the proposed LΦDADT scheme. Their work is currently published in the journal, International Journal of Robust and Nonlinear Control.
The authors obtained new sufficient stability conditions based on the MLF and LФDADT scheme. The stability and stabilization criteria for switched systems were specifically denoted as linear matrix inequalities that are not only compatible with the existing schemes but also easy to manipulate, providing a unified processing sketch and a less conservative switching. Additionally, the obtained stability results exhibited lower bounds and larger regions of switching signals than the existing ones.
In summary, a new LФDADT scheme based on the concepts of existing methods: ADT, limiting ADT, MDADT, and ФDADT was reported to overcome the shortcomings of the existing methods and enhance the stability analysis of switched systems. Furthermore, the superiority of the developed technique was confirmed through the three numerical examples. In a statement to Advances in Engineering, Associate Professor Yu mentioned their study provided new opportunities for developing more efficient methods for analyzing the stability of switched systems.
Yu, Q., & Zhai, G. (2020). A limit inferior ‐dependent average dwell time approach for stability analysis of switched systems. International Journal of Robust and Nonlinear Control, 31(2), 565-581.