Nonlinear adaptive estimation with application to sinusoidal identification

Parameter estimation of a sinusoidal signal in real-time is encountered in applications in numerous areas of engineering. Parameters of interest are usually amplitude, frequency and phase wherein frequency tracking is the fundamental task in sinusoidal estimation. This thesis deals with the problem of identifying a signal that comprises n (n ≥ 1) harmonics from a measurement possibly affected by structured and unstructured disturbances. The structured perturbations are modeled as a time-polynomial so as to represent, for example, bias and drift phenomena typically present in applications, whereas the unstructured disturbances are characterized as bounded perturbation. Several approaches upon different theoretical tools are presented in this thesis, and classified into two main categories: asymptotic and non-asymptotic methodologies, depending on the qualitative characteristics of the convergence behavior over time. The first part of the thesis is devoted to the asymptotic estimators, which typically consist in a pre-filtering module for generating a number of auxiliary signals, independent of the structured perturbations. These auxiliary signals can be used either directly or indirectly to estimate—in an adaptive way—the frequency, the amplitude and the phase of the sinusoidal signals. More specifically, the direct approach is based on a simple gradient method, which ensures Input-to-State Stability of the estimation error with respect to the bounded-unstructured disturbances. The indirect method exploits a specific adaptive observer scheme equipped with a switching criterion allowing to properly address in a stable way the poor excitation scenarios. It is shown that the adaptive observer method can be applied for estimating multi-frequencies through an augmented but unified framework, which is a crucial advantage with respect to direct approaches. The estimators’ stability properties are also analyzed by Input-to-State-Stability (ISS) arguments. In the second part we present a non-asymptotic estimation methodology characterized by a distinctive feature that permits finite-time convergence of the estimates. Resorting to the Volterra integral operators with suitably designed kernels, the measured signal is processed, yielding a set of auxiliary signals, in which the influence of the unknown initial conditions is annihilated. A sliding mode-based adaptation law, fed by the aforementioned auxiliary signals, is proposed for deadbeat estimation of the frequency and amplitude, which are dealt with in a step-by-step manner. The worst case behavior of the proposed algorithm in the presence of bounded perturbation is studied by ISS tools. The practical characteristics of all estimation techniques are evaluated and compared with other existing techniques by extensive simulations and experimental trials.Open Acces

Frequency domain descriptions of linear systems

This thesis begins by applying Lagrange interpolation to linear systems theory. More specifically, a stable, discrete time linear system, with transfer function G(z), is interpolated with an FIR transfer function at n equally spaced points around the unit circle. The L∞ error between the original system and the interpolation is bounded, the bound going to zero exponentially fast as n -> ∞. A similar result applies to unstable systems except that the interpolating function is a non-causal FIR transfer function . The thesis then considers Hilbert transforms from interpolation data. Given the real part of a stable transfer function evaluated at n equally spaced points around the unit circle, the Hilbert transform from interpolation data reconstructs the complete frequency response, real and imaginary parts, at all frequencies, to within a bounded L∞ error. The error bound goes to zero exponentially fast as n -> ∞. Also considered is the gain-phase problem from interpolation data. This is the same as the Hilbert transform from interpolation data, except that magnitude interpolation data instead of real part interpolation data is given. Two constructions for the gain phase problem from interpolation data are given , and L∞ error bounds derived . In both cases, the error bounds go to zero exponentially fast as n -> ∞. Application of Kalman filters to short-time Fourier analysis then follows. This contains a new method in Kalman filtering called covariance setting. The filters derived from covariance setting generalize the discrete Fourier transform. They offer a design trade-off between noise smoothing and transient response time, are recursive, and are of similar computational complexity to the discrete Fourier transform. Combining the Kalman filters for short-time Fourier analysis and Lagrange interpolation gives a new method of adaptive frequency response identification. A feature of this method is the L∞ error bound between the original system and the identified model. Using recent analysis on the inherent frequency weighting in identification algorithms shows the superiority of this new method over previous adaptive frequency response identification schemes. Finally, model reduction for unstable systems is considered. Given an unstable rational function of high McMillan degree, an approximation of lower McMillan degree, but with the same number of unstable poles, is constructed. An L∞ error bound between the original transfer function and approximation is derived. Such an approximation has application to control systems

Frequency Estimation of Periodic Signals: an Adaptive Observer approach

This paper deals with an adaptive observer methodology for estimating the frequency of an unknown periodic signal with non-zero fundamental component corrupted by unstructured and bounded uncertainties. The proposed estimator is characterized by Input-to-State Stability (ISS) with respect to the bounds on the higher order harmonics and on the unstructured disturbance. The influence of each tuning parameter is analyzed through a specified ISS asymptotic bound. The effectiveness of the proposed technique is shown through numerical simulations