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A multi-dimensional switched system or multi-mode multi-dimensional ($M^3D$)system extends the classic switched system by allowing different subsystemdimensions. The stability problem of the $M^3D$ system, whose state transitionsat switching instants can be discontinuous due to the dimension-varyingfeature, is studied. The discontinuous state transition is formulated by anaffine map that captures both the dimension variations and the state impulses,with no extra constraint imposed. In the presence of unstable subsystems, thegeneral criteria featuring a series of Lyapunov-like conditions for thepractical and asymptotic stability properties of the $M^3D$ system are providedunder the proposed slow/fast transition-dependent average dwell time framework.Then, by considering linear subsystems, we propose a class of parametricmultiple Lyapunov functions to verify the obtained Lyapunov-like stabilityconditions and explicitly reveal a connection between the practical/asymptoticstability property and the non-vanishing/vanishing property of the impulsiveeffects in the state transition process. Further, the obtained stabilityresults for the $M^3D$ system are applied to the consensus problem of the openmulti-agent system (MAS), whose network topology can be switching andsize-varying due to the migrations of agents. It shows that through a propertransformation, the seeking of the (practical) consensus performance of theopen MAS with disconnected digraphs boils down to that of the (practical)stability property of an $M^3D$ system with unstable subsystems.

Stability of Multi-Dimensional Switched Systems with an Application to Open Multi-Agent Systems