Ensuring Input/Output-to-State Stability of Switched Nonlinear Systems Under Arbitrary but Restricted Switching Signals

Core Concepts

The core message of this article is that under certain conditions on the subsystem dynamics, admissible switches between subsystems, and admissible dwell times, the switched nonlinear system is input/output-to-state stable (IOSS) under all switching signals that obey the given restrictions.

Abstract

The article studies the input/output-to-state stability (IOSS) of continuous-time switched nonlinear systems whose switching signals obey pre-specified restrictions on admissible switches between the subsystems and admissible dwell times on the subsystems. The key highlights and insights are: The authors associate a weighted directed graph with the family of subsystems and the admissible switches between them. Properties of the subsystems and the switches are captured by vertex and edge weights of this graph. The authors show that if the underlying weighted directed graph of the switched system admits a class of finite walks that satisfy a "contractivity" property, then the switched system is IOSS under all switching signals obeying the given restrictions. The authors provide sufficient conditions under which the contractivity of the relevant walks on the graph can be verified in a numerically tractable manner. The results do not cater to switching signals that dwell only on unstable subsystems, which is identified as a direction for future work.

Stats

The total duration of activation of subsystem p on the interval (s, t] is denoted as Tp(s, t). The total number of switches from subsystem p to subsystem q on the interval (s, t] is denoted as Npq(s, t).

Quotes

"IOSS property of a switched system implies that irrespective of the initial state, if the inputs and the observed outputs are small, then the state of the system will become small eventually." "Multiple IOSS-Lyapunov-like functions and graph theoretic arguments are employed in our analysis."

Key Insights Distilled From

Stabilizing switched nonlinear systems under restricted but arbitrary switching signals

by Atreyee Kund... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02596.pdf

Deeper Inquiries

What are the implications of the proposed stability conditions on the design of switching controllers for switched nonlinear systems

The proposed stability conditions have significant implications for the design of switching controllers for switched nonlinear systems. By ensuring that all switching signals obey pre-specified restrictions on admissible switches between subsystems and dwell times, the system can be guaranteed to be input/output-to-state stable (IOSS). This means that regardless of the initial state, if the inputs and observed outputs are small, the state of the system will eventually become small. In the context of designing switching controllers, these stability conditions provide a framework for developing control strategies that can maintain stability and ensure robust performance of the switched system. By incorporating the conditions into the controller design, it becomes possible to create switching policies that adhere to the restrictions on switches and dwell times, thereby preserving the IOSS property of the system. This can lead to the development of more reliable and efficient control algorithms for switched nonlinear systems.

How can the stability analysis be extended to switched systems where the subsystems are not necessarily IOSS, but satisfy other stability notions like input-to-state stability (ISS)

The stability analysis can be extended to switched systems where the subsystems are not necessarily input/output-to-state stable (IOSS), but satisfy other stability notions like input-to-state stability (ISS). In such cases, the analysis would involve adapting the stability conditions and criteria to accommodate the specific stability properties of the subsystems. For subsystems that are ISS but not necessarily IOSS, the stability analysis would focus on ensuring that the overall switched system maintains a certain level of stability and performance. By incorporating ISS properties into the analysis, the stability conditions can be adjusted to account for the different stability characteristics of the subsystems. This would involve modifying the stability criteria and requirements to align with the ISS properties of the individual subsystems, while still ensuring overall stability of the switched system.

Can the graph-theoretic approach be leveraged to study other performance measures of switched systems beyond stability, such as optimal control or energy efficiency

The graph-theoretic approach used in the stability analysis of switched systems can indeed be leveraged to study other performance measures beyond stability, such as optimal control or energy efficiency. By representing the system as a weighted directed graph and analyzing the properties of walks and cycles on the graph, it becomes possible to gain insights into various aspects of system behavior and performance. For optimal control, the graph structure can be utilized to identify optimal paths or sequences of switches that lead to desired system behavior or performance objectives. By optimizing the paths on the graph, it is possible to design control strategies that minimize cost, maximize efficiency, or achieve other optimization goals. Similarly, for energy efficiency, the graph representation can be used to analyze the energy consumption of the system under different switching scenarios. By studying the energy profiles of walks and cycles on the graph, it becomes possible to optimize the system's energy usage and design energy-efficient control strategies. This approach can help in improving the overall energy efficiency of switched systems and reducing operational costs.