We present an extension of the method of eigenmode decomposition of Gramians proposed earlier for the small-signal stability analysis of dynamical systems. In this paper we derive the spectral decomposition for finite Gramians on any time interval and with arbitrary initial conditions. These expansions allow both a stability analysis of non-stationary systems and a monitoring of instability development in unstable systems. Eigen components in the expansion of Gramians on a finite interval of time we calledfinite sub-Gramians. Because each sub-Gramian is associated with a particular eigenvector, the sources of instability can be easily localized and tracked in real time. We perform a simulation experiment in which finite sub-Gramians are used to analyze the development of instability arising in an actual power grid on Russky Island when it is disconnected from the mainland network.