Stability of Stochastic Switched Systems

• Working with Dr. Benjamin Webb, Camille Carter, and David Reber

• April 2020 to April 2021

• Paper "A Simple Stability Criterion for Dynamical Systems with Stochastic Switching and-or Stochastic Time-Delays" published in  Nonlinearity

Summary of Research 

• Studied stochastic switched systems, and found conditions for stability

• Extended linear stability results to the nonlinear regime

• Investigated the effect of time delays on a system

Talks 

• February 2021, BYU Student Research Conference (won "Best Undergraduate Presentation" in session)

Paper Abstract

In many natural and technological systems the rule that governs the system's dynamics changes over time. In such switched systems the system's switching can be a significant source of instability. Here we give a simple sufficient criterium to determine if an i.i.d. stochastically switched system is stable in expectation. This method extends recent results for linear switched systems to nonlinear switched systems. It also extends results known for general switched systems giving improved results for systems with i.i.d. stochastic switching. The paper also considers the effects of time-delays on the stability of switched systems. Such time delays, which are intrinsic to any real-world system, can also have a destabilizing effect on the system's dynamics. Previously, it has been shown that if a dynamical system is intrinsically stable, which is a stronger form of global stability, then it maintains its stability even when time-delays are introduced into the system. Here we extend this notion to stochastically switched systems. We refer to this type of stability as patient stability and give a simple sufficient criterium under which such systems are patiently stable, i.e. cannot be destabilized by time delays. Both criteria introduced in this paper side step the need to use Lyapunov, Linear Matrix Inequalities, and Semi-Definite Programming-type methods. Our examples in this paper demonstrate the simplicity of these criteria.