John Lataire

Robustness Issues of the Best Linear Approximation of a Nonlinear System

In many engineering applications, linear models are preferred, even if it is known that the system is disturbed by nonlinear distortions. A large class of nonlinear systems, which are excited with a ldquoGaussianrdquo random excitation, can be represented as a linear system G BLA plus a nonlinear noise source Y S . The nonlinear noise source represents that part of the output that is not captured by the linear approximation. In this paper, it is shown that the best linear approximation G BLA and the power spectrum S Y S of the nonlinear noise source Y S are invariants for a wide class of excitations with a user-specified power spectrum. This shows that the alternative ldquolinear representationrdquo of a nonlinear system is robust, making its use in the daily engineering practice very attractive. This result also opens perspectives to a new generation of dynamic system analyzers that also provide information on the nonlinear behavior of the tested system without increasing the measurement time.

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.

Estimating a Non-parametric, Colored Noise Model for Linear, Slowly Time-varying Systems

This paper proposes a methodology to easily extract some valuable nonparametric information on linear slowly time-varying systems, which have been excited by multisines. More specifically, it is first explained how time-varying systems behave when excited by multisines and how a rough nonparametric idea of the speed of variation of the instantaneous frequency response function is estimated. Second, a nonparametric model of the disturbing colored noise on the measured input and output signals is estimated. The methodology circumvents the need for repeated experiments, which are, for time-varying systems, usually difficult to perform. The estimation of the disturbing noise is mandatory when identifying dynamic systems within an errors-in-variables framework. The estimated noise model is intended for system identification performed in the frequency domain, where a nonparametric noise model is easily implemented. The noise model is estimated independently of the parameters of the time-varying system under consideration. The estimate of the speed of variation of the instantaneous frequency response function gives some insight into the measured system. It might actually provide an idea of which poles or zeros are actually moving and which are not.

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.

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.

Transfer function and transient estimation by Gaussian process regression in the frequency domain

Inspired by the recent promising developments of Bayesian learning techniques in the context of system identification, this paper proposes a Transfer Function estimator, based on Gaussian process regression. Contrary to existing kernel-based impulse response estimators, a frequency domain approach is adopted. This leads to a formulation and implementation which is seamlessly valid for both continuous- and discrete-time systems, and which conveniently enables the selection of the frequency band of interest. A pragmatic approach is proposed in an output error framework, from sampled input and output data. The transient is dealt with by estimating it simultaneously with the transfer function.Modelling the transfer function and the transient as Gaussian processes allows for the incorporation of relevant prior knowledge on the system, in the form of suitably designed kernels. The SS (Stable Spline) and DC (Diagonal Correlated) kernels from the literature are translated to the frequency domain, and are proven to impose the stability of the estimated transfer function. Specifically, the estimates are shown to be stable rational functions in the frequency variable. The hyperparameters of the kernel are tuned via marginal likelihood maximisation.The comparison between the proposed method and three existing methods from the literature-the regularised finite impulse response (RFIR) estimator, the Local Polynomial Method (LPM), and the Local Rational Method for Frequency Response Function estimation-is illustrated on simulations on multiple case studies.

Estimation of Linear Parameter-Varying affine state space models using synchronized periodic input and scheduling signals

During the past decades some very interesting results have been obtained in controller synthesis using Linear Parameter-Varying (LPV) systems. However, the LPV models are commonly required to be transformed into State Space (SS) form. We tackle the LPV SS identification problem directly in the frequency domain. To the best of our knowledge, this is a novel approach. When the input and scheduling are chosen to be periodic and synchronized, the state space equations are structured and sparse in the frequency domain. The parameters of these state space equations are estimated by minimizing a weighted non-linear least squares criterion. Starting values are generated via the Best Linear Time-Invariant (BLTI) approximation. The resulting model is also valid for non-periodic scheduling and input signals.