Ebrahim Louarroudi

Brief paper: Non-parametric estimate of the system function of a time-varying system

The task of identifying an unknown dynamic system is made easier with prior knowledge on its behaviour. Using a frequency domain approach, the non-parametric maximum likelihood estimator of the system function, associated with the time-dependent impulse response of a time-varying system, is constructed. This is accomplished by the use of a simple linear least squares fitting algorithm, applied to the spectral response of the system to a multisine excitation. The noise variance on the system function is estimated simultaneously, and modelling errors can be detected, as illustrated on a simulation example.

Detecting a Time-Varying Behavior in Frequency Response Function Measurements

This paper provides data-driven tools to detect and quantify approximately the influence of the time variation of a system under test in classical frequency response function (FRF) measurements. To achieve this, the best linear time-invariant approximation of a linear time-varying system is defined and is estimated using existing FRF estimators. An analysis of the residuals of the latter estimation reveals the frequency band in which the contributions from the time variation dominates the disturbing measurement noise and, thus, is significant. All concepts are illustrated on a simulation and real measurement examples.

Nonparametric Tracking of the Time-Varying Dynamics of Weakly Nonlinear Periodically Time-Varying Systems Using Periodic Inputs

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 multivariable (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.

Detection and Quantification of the Influence of Time Variation in Frequency Response Function Measurements Using Arbitrary Excitations

This paper presents a nonparametric method for detecting and quantifying the influence of time variation in frequency response function measurements. The method is based on the estimation of the best linear time-invariant (BLTI) approximation of a linear time-variant (LTV) system from known input, noisy output data. The key idea consists in reformulating the single-input, single-output time-variant problem as a multiple-input, single-output time-invariant problem. In addition to the BLTI approximation of the LTV system, the contribution of the disturbing noise, the leakage error, and the time-varying effects at the output is also quantified. As such, the approximation error of the time-invariant framework is known.

Harmonic impedance spectra identification from time-varying bioimpedance: theory and validation

The harmonic impedance spectra (HIS) of a time-varying bioimpedance Z(ω, t) is a new tool to better understand and describe complex time-varying biological systems with a distinctive periodic character as, for example, cardiovascular and respiratory systems. In this paper, the relationship between the experimental setup and the identification framework for estimating Z(ω, t) is set up. The theory developed applies to frequency response based impedance measurements from noisy current-voltage observations. We prove theoretically and experimentally that a voltage source (VS) and a current source (CS) analogue front end-based measurement lead, respectively, to a closed-loop and an open-loop HIS identification problem. Next, we delve into the estimation of the HIS by treating Z(ω, t), on the one hand, as a linear time-invariant (LTI) system within a short time window; and, on the other hand, as a linear periodically time-varying (PTV) system within the entire measurement interval. The LTI approach is based on the short-time Fourier transform (STFT), while the PTV approach relies on the information that is present in the skirts of the voltage and/or current spectra. In addition, direct and indirect methods are developed for estimating the HIS by using simple as well as more sophisticated techniques. Ultimately, the HIS and their uncertainty bounds are estimated from real measurements conducted on a periodically varying dummy impedance.

Nonparametric time-variant frequency response function estimates using arbitrary excitations

The time-variant frequency response function (TV-FRF) uniquely characterises the dynamic behaviour of a linear time-variant (LTV) system. This paper proposes a method for estimating nonparametrically the dynamic part of the TV-FRF from known input, noisy output observations. The arbitrary time-variation of the TV-FRF is modelled by Legendre polynomials. In opposition to existing solutions, the proposed method is applicable to arbitrary inputs.

Detecting and Quantifying the Nonlinear and Time-Variant Effects in FRF Measurements Using Periodic Excitations

Frequency response function (FRF) measurements are very often used to get a quick insight into the dynamic behavior of complex systems; even if it is known that these systems are only approximately linear and time-invariant. Therefore, it is important to detect and quantify the deviation from the ideal linear time-invariant framework that is inherent to the concept of an FRF. This paper presents a method to detect and quantify the nonlinear and time-variant effects in FRF measurements using periodic excitations. The proposed method can handle noisy input, noisy output data, and nonlinear time-variant systems operating in feedback.

Detection and Quantification of the Influence of Time Variation in Closed-Loop Frequency-Response-Function Measurements

Recently, a method has been developed for detecting and quantifying the time variation in frequency response function (FRF) measurements using arbitrary excitations. The following basic assumptions have been made: 1) The input is known exactly (generalized output error stochastic framework), and 2) the time-variant system operates in an open loop. The latter excludes any interaction between the time-variant system and the generator/actuator. In this paper, we extend the results of the work by Pintelon to noisy input-output observations (errors-in-variables stochastic framework) of time-variant systems operating in a closed loop.

Estimation of nonparametric harmonic transfer functions for linear periodically time-varying systems using periodic excitations

In this paper a nonparametric estimation procedure is proposed in order to identify continuous (discrete)-time, linear periodically time-varying (LPTV) systems. Further, multisine excitations are applied onto a LPTV system such that the system transient effects will vanish completely when the system is operating in steady state. The key idea is to decompose a LPTV system into an (in)finite series of transfer functions, the so-called harmonic transfer functions (HTF). From an identification point of view, the parallel structure, which consists of a weighted sum of the HTFs, is truncated to a desirable order. A high quality estimate of the nonparametric HTFs with its uncertainty embedded in an errors-in-variables framework is then obtained from only one experiment; making use of methods, the so-called local polynomial method (LPM), that are recently developed for multivariable linear time invariant systems. The effectiveness of the LPM will be first pointed up through simulations before a real system will be identified. The methodology is then eventually demonstrated on a real-life periodically time-varying electronic system.

Non-parametric best linear time invariant approximation of a linear time-varying system

The consequences of estimating the frequency response function of a continuous-time, linear time-varying (LTV) system with tools for linear time invariant (LTI) systems are studied. To this end, the best linear time invariant approximation of an LTV system is defined, and is related to a general model for LTV systems. A recently introduced frequency response function estimation method for LTI systems is used to compute the best linear time invariant approximation, the properties of which are discussed. An analysis of the residual error specifies whether the system under consideration is time-varying or not. Also, the frequency band where the contributions from the time variation are higher than the noise floor is determined. All concepts are illustrated on simulation examples.