[Paper Review] Ultimate Boundedness for Switched Systems with Multiple Equilibria Under Disturbances

Sushant Veer, Ioannis Poulakakis|arXiv (Cornell University)|Sep 8, 2018

Stability and Control of Uncertain Systems32 references4 citations

TL;DR

This paper establishes ultimate boundedness for switched discrete and continuous systems with multiple equilibria under external disturbances, showing that if each subsystem is Input-to-State Stable (ISS) and switching follows an average dwell-time constraint, solutions remain ultimately bounded within a compact set whose size scales monotonically with disturbance magnitude. The results generalize classical ISS for switched systems with a common equilibrium.

ABSTRACT

In this paper, we investigate the robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. It is shown that if each subsystem of the switched system is Input-to-State Stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. Furthermore, the size of this set varies monotonically with the supremum norm of the disturbance signal. It is observed that when the subsystems share a common equilibrium, ISS is recovered for solutions of the corresponding switched system; hence, the results in this paper are a natural generalization of classical results in switched systems that exhibit a common equilibrium. Additionally, we provide a method to analytically compute the average dwell time if each subsystem possesses a quadratic ISS-Lyapunov function. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive motions, each corresponding to an equilibrium point of a dynamical system, must be composed to realize a task. However, the results are relevant to a much broader class of applications, in which composition of different modes of behavior is required.

Motivation & Objective

• To analyze the robustness of switched systems with multiple equilibria under external disturbances.
• To extend Input-to-State Stability (ISS) theory to switched systems where subsystems do not share a common equilibrium.
• To establish conditions under which solutions remain ultimately bounded despite switching and disturbances.
• To provide an analytically computable average dwell-time bound when quadratic ISS-Lyapunov functions exist.
• To support motion planning in robotics by enabling robust composition of distinct dynamical behaviors (e.g., gait primitives).

Proposed method

• Utilizes Input-to-State Stability (ISS) for individual subsystems as a foundation for analyzing switched system behavior under disturbances.
• Applies average dwell-time switching signals to ensure overall system stability despite switching between subsystems with different equilibria.
• Employs multiple Lyapunov functions that may increase during switching but decrease overall under the average dwell-time constraint.
• Derives an upper bound on the state deviation from the respective subsystem equilibrium, showing dependence on initial conditions and disturbance magnitude.
• Introduces a method to analytically compute the average dwell time when each subsystem admits a quadratic ISS-Lyapunov function.
• Uses class-𝐾∞ and class-𝐾𝐿 functions to characterize the ultimate bound and transient decay behavior of the system.

Experimental results

Research questions

• RQ1 Under what switching conditions do switched systems with multiple equilibria remain ultimately bounded under external disturbances?
• RQ2 Can Input-to-State Stability (ISS) be preserved in switched systems when subsystems do not share a common equilibrium?
• RQ3 How does the size of the ultimate bound depend on the disturbance magnitude in such systems?
• RQ4 Can the average dwell-time constraint be analytically computed for systems with quadratic ISS-Lyapunov functions?
• RQ5 What is the relationship between the switching signal and the robustness of the overall system behavior?

Key findings

• If each subsystem is ISS and the switching signal satisfies an average dwell-time bound, the solutions of the switched system are ultimately bounded within a compact set.
• The size of the ultimate bound increases monotonically with the supremum norm of the disturbance signal.
• When all subsystems share a common equilibrium, the system recovers full ISS, showing that the results generalize classical ISS for switched systems.
• An analytical expression for the average dwell time can be computed if each subsystem has a quadratic ISS-Lyapunov function.
• The ultimate bound depends on the initial deviation from the respective subsystem equilibrium and scales with the disturbance magnitude via a class-𝐾∞ function.
• The transient decay of the state deviation is characterized by a class-𝐾𝐿 function, ensuring asymptotic convergence to the ultimate bound.