Abstract: The governor system of hydropower unit includes dead zone and amplitude limiter, which is a typical non-smooth dynamic system and also a linear switched system. If the control parameters are not set reasonably, the switched system would cause switching oscillation phenomenon. So far, the governor system of hydropower unit has encountered multiple times ultra-low frequency oscillation problems related to governor dead-zone and amplitude limiter. To this end, the describing function method is mainly and currently used to analyze the stability of the governor system. The describing function method is a frequency domain engineering approximation method, that requires certain prerequisites to be met in practical applications. To solve the problem of ultra-low frequency oscillation, this paper proposes a new and time-domain state-space based stability analysis method with theoretical proof. The proposed method can be used for stability analysis of general smooth dynamic systems as well as discontinuous differential systems. Applying this method to the stability analysis and optimization of control parameters for hydropower governor system can completely avoid the ultra-low frequency oscillation problem. The example result verifies the effectiveness of the proposed new method.