1384 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 5, MAY 2012 Nonparametric Tracking of the Time-Varying Dynamics of Weakly Nonlinear Periodically Time-Varying Systems Using Periodic Inputs Ebrahim Louarroudi, Student Member, IEEE, Rik Pintelon, Fellow, IEEE, and John Lataire, Member, IEEE Abstract—In this paper, a nonparametric estimation procedure is presented in order to track the evolution of the dynamics of continuous (discrete)-time (non)-linear periodically time-varying (PTV) systems. Multisine excitations are applied to a PTV system since this kind of excitation signals allows us to discriminate between the noise and the nonlinear distortion from a single experiment. The key idea is that a linear PTV system can be decomposed into an (in)finite series of transfer functions, the so- called harmonic transfer functions (HTFs). Moreover, a systematic methodology to determine the number of significant branches is provided in this paper as well. Making use of the local polynomial approximation, a method that was recently developed for multi- variable (non)-linear time invariant systems, the HTFs, together with their uncertainties embedded in an output-error framework, are then obtained from only one single experiment. From these nonparametric estimates, the evolution of the dynamics, described by the instantaneous transfer function (ITF), can then be achieved in a simple way. The effectiveness of the identification scheme will be first illustrated through simulations before a real system will be identified. Eventually, the methodology is applied to a weakly nonlinear PTV electronic circuit. Index Terms—Instantaneous transfer function (ITF), multisine excitations, nonlinear distortions, nonparametric modeling, out- put error, periodically time-varying (PTV) systems. I. I NTRODUCTION A LTHOUGH the identification framework of linear time- invariant (LTI) systems covers a vast number of applica- tions [1], [2], there exist circumstances where the linear and time-invariant conditions are not fulfilled [3], [4]. This is, for example, the case when the system properties vary arbitrarily with time [5], [6] or, in particular, periodically [7]–[10]. In this paper, we will put the emphasis on the nonparametric estimation of (non)-linear periodically time-varying [(N)LPTV] systems excited by periodic signals, and this is within an output-error framework (input is known exactly, and output is disturbed by stochastic errors). The output-error framework is Manuscript received June 20, 2011; revised September 14, 2011; accepted October 12, 2011. Date of publication December 13, 2011; date of current ver- sion April 6, 2012. This work was supported by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem METH1), and by the Belgian Federal Government (IUAP VI/4). E. Louarroudi is on a Ph.D. fellowship from the Methusalem project. The work of J. Lataire was supported by the Research Foundation-Flanders (FWO) under a Ph.D. fellowship. The authors are with the Department of Fundamental Electricity and Instru- mentation (ELEC), Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail: elouarro@vub.ac.be). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2011.2175830 useful in applications where the modeling is done from the reference signal (the signal that enters the actuator) to the output of the system [2]. The output-error problem is often used in control applications where the actuator is modeled together with the unknown system. The periodic time variations appear in practice because either the system is inherently periodically time-varying (PTV) [7]– [9] or the time variation is established by external parameters, the so-called scheduling parameters [5], [6], [11]. PTV systems can be found in a lot of engineering applications such as sig- nal processing, control, sampled data systems, multirate filter banks, channel modeling, mechanical systems, and so on. Many mechanical systems that sustain a periodic motion (e.g., gear boxes, electrical motors, fans, shafts, helicopter blades, etc.) create vibrations and radiate noise which show a PTV behavior. For instance, a twist-actuated helicopter rotor blade where the periodic time variations show up due to the aerodynamic milieu that is varying once per revolution of the rotor. In order to achieve a proper design of the controller, a good model [de- scribed by harmonic transfer functions (HTFs) or the instanta- neous transfer function (ITF)] is required between the loads and the blade activation (see [7]). A nice overview of cyclostation- ary processes, supported by real-life examples, can be found in [12]. The PTV behavior could also arise when a nonlinear system is linearized around a periodic trajectory of the set point, for instance, in power distribution networks [10]. In [10], the HTFs are used to analyze an inverter locomotive since it is problematic to tune the controller for converter switching as the design relies mostly on LTI models and a lot of experi- ence. The inverter locomotive consists of nonlinear components (transformer, line converter, dc link, and motor) that are driven by a sinusoidal external voltage source (50 Hz in Europe), which makes the inverter locomotive a nonlinear PTV system. In order to introduce robustness in the controller of the inverter locomotive, more advanced models are necessary such that the effects of the converters (creation of harmonics) can be well captured. Those different classes of PTV systems might be viewed as belonging to the same class from an identification point of view, as long as the base frequency of the time variation f sys =1/T sys is either known or can be estimated from data. The advantage of using a black box approach is that the model is also able to take into account the possible dynamics that could show up between the scheduling and the system parameters. 0018-9456/$26.00 © 2011 IEEE