Stability Analysis of Switched Dynamics via Commutators
Switched dynamical systems are used to model processes that switch between different modes of operation. They are typically described by a collection of continuous-time systems and a rule that orchestrates the switching between them. Some examples of switched systems include a thermostat turning the heat on and off, biological cells growing and dividing, a server switching between buffers in a queuing network, and a car’s wheels locking and unlocking on ice.
During his Center appointment Professor Liberzon will focus on a basic system property known as stability, where the system returns to a desired equilibrium configuration after being subject to small perturbations. Stability is generally not preserved under switching, and Professor Liberzon is interested in characterizing those systems—or those switching rules—for which stability is guaranteed.
Earlier research in this area formulated stability criteria for switched systems in terms of commutation relations among the systems being switched. The simplest commutation relation is when the flows of different systems commute, which means they can be activated in any order with the same end result. Stability criteria based on such commutation relations are typically fragile, however, in the sense that they are destroyed by arbitrarily small perturbations of the system data. This shortcoming renders existing results in this area largely useless
Professor Liberzon’s project will pursue novel ideas and mathematical tools to obtain new stability criteria that both capture commutation relations and are robust to perturbations. The results will open the door to new applications in engineering and several other areas.