This paper presents a new frequency domain identification technique to estimate multivariate Linear Parameter-Varying (LPV) continuous-time state space models, where a periodic variation of the parameters is assumed or imposed. The main goal is to obtain an LPV state space model suitable for control, from a single parameter-varying experiment. Although most LPV controller synthesis tools require continuous time state space models, the identification of such models is new. The proposed identification method designs a periodic input signal, taking the periodicity of the parameter variation into account. We show that when an integer number of periods is observed for both the input and the scheduling, the state space model representation has a specific, sparse structure in the frequency domain, which is exploited to speed up the estimation procedure. A weighted non-linear least squares algorithm then minimizes the output error. Two initialization methods are explored to generate starting values. The first approach uses a Linear Time-Invariant (LTI) approximation. The second estimates a Linear Time Variant (LTV) input-output differential equation, from which a corresponding state space realization is computed. (C) 2016 Elsevier Ltd. All rights reserved.