Frequency domain, parametric estimation of the evolution of the time-varying dynamics of periodically time-varying systems from noisy input–output observations

• Frozen and instantaneous transfer function determine the speed of time-variations.
• The equivalence of different LTP models is performed at the level of the HTFs.
• Consistent estimators are derived for LTP systems from noisy input–output data.
• Successful demonstration of the identification scheme on real measurements.

This paper presents a frequency domain, parametric identification method for continuous- and discrete-time, slow linear time-periodic (LTP) systems from input–output measurements. In this framework, the output as well as the input is allowed to be corrupted by stationary noise (i.e. an errors-in-variables approach is adopted). It is assumed that the system under consideration can be excited by a broad-band periodic signal with a user-defined amplitude spectrum (i.e. a multisine), and that the periodicity of the excitation signal TexcTexc can be synchronized with the periodicity of the time-variation TsysTsys (i.e. Texc/Tsys∈QTexc/Tsys∈Q), such that the system reaches a steady state (a periodic solution). TsysTsys is also known as the pumping period. Once the parametric estimation of the time-evolution of the system parameters has been performed, the system model is evaluated at the level of the instantaneous transfer function (also known as system function, or parametric transfer function), which rigorously characterizes LTP systems. If the dynamics of the LTP system are slowly varying or the system is linear parameter varying (LPV), a frozen transfer function approach is provided to easily visualize and assess the quality of the estimated model. To give the estimated quantities a quality label, uncertainty bounds on the model-related quantities (such as the time-periodic (TP) system parameters, the frozen transfer function, the frozen resonance frequency, etc.) are derived in this paper as well. Besides, a clear distinction between the instantaneous and the frozen transfer function concept is made, and both can be estimated with the proposed identification scheme. The user decides which transfer function definition suits best its purpose in practice. Finally, the identification algorithm is applied to a simulation example and to real measurements on an extendible robot arm.