On Uniform Observability of Linear Time-Varying Systems With Respect to Subspaces and Its Applications

Recently, the notion of uniform observability with respect to a subspace was proposed in [21] and was utilized to solve the synchronization problem for single-integrator multi-agent systems with time-varying matrix-weighted couplings. This paper further explores the concept of uniform observability of a linear time-varying system with respect to a subspace. We will first establish the invariance property of uniform observability with respect to some invariant subspace under time-varying output feedback control. Then, based on this invariance property, we will establish an exponential convergence result with respect to an invariant subspace for a class of linear switched systems. Leveraging this exponential convergence result, we will solve the exponential synchronization problem for general linear multi-agent systems over jointly connected switching networks. It is noted that the exponential synchronization problem was also studied recently by some more sophisticated methods. Compared with these methods, our method is much simpler. Moreover, our method leads to an explicit characterization of the exponential synchronization rate.