Identification of Continuous-Time Deterministic System utilizing Orthonormal Basis Functions and Sample Data

In hands-free telephone or video conference application, there exists an acoustic feedback coupling between the loudspeaker and microphone in an enclosed environment, which creates the acoustic echo. FIR filters are commonly used in acoustic echo cancellers because of their simple structure. However, in this paper, the Kautz and Laguerre filter structures are shown to be more efficient echo cancellers than the FIR filters, because they can describe accurately the acoustic echo system with fewer parameters. These filters are built from their respective orthonormal Kautz and Laguerre basis functions. The proposal is motivated by some theoretical and numerical results that the time-varying acoustic echo path is basically due to its time-varying zeros and not its time-invariant acoustical poles. Therefore, the poles of the Kautz and the Laguerre filters are estimated, and can be kept fixed or updated occasionally if required. The poles are estimated by a batch Gauss-Newton algorithm. Then, the coefficients of the Kautz and Laguerre filters can be estimated by most recursive algorithms that are suitable for linear regression models, e.g., the normalized LMS algorithm. Generally, it is shown that the proposed Kautz and Laguerre filters, as the filter structures in an acoustic echo canceller, have better convergence and tracking properties than the FIR and IIR filters.

The Ingenuity time-of-flight (TF) PET/MR is a recently developed hybrid scanner combining the molecular imaging capabilities of PET with the excellent soft tissue contrast of MRI. It is becoming common practice to characterize the system's point spread function (PSF) and understand its variation under spatial transformations to guide clinical studies and potentially use it within resolution recovery image reconstruction algorithms. Furthermore, due to the system's utilization of overlapping and spherical symmetric Kaiser-Bessel basis functions during image reconstruction, its image space PSF and reconstructed spatial resolution could be affected by the selection of the basis function parameters. Hence, a detailed investigation into the multidimensional basis function parameter space is needed to evaluate the impact of these parameters on spatial resolution. Using an array of 12 × 7 printed point sources, along with a custom made phantom, and with the MR magnet on, the system's spatially variant image-based PSF was characterized in detail. Moreover, basis function parameters were systematically varied during reconstruction (list-mode TF OSEM) to evaluate their impact on the reconstructed resolution and the image space PSF. Following the spatial resolution optimization, phantom, and clinical studies were subsequently reconstructed using representative basis function parameters. Based on the analysis and under standard basis function parameters, the axial and tangential components of the PSF were found to be almost invariant under spatial transformations (~4 mm) while the radial component varied modestly from 4 to 6.7 mm. Using a systematic investigation into the basis function parameter space, the spatial resolution was found to degrade for basis functions with a large radius and small shape parameter. However, it was found that optimizing the spatial resolution in the reconstructed PET images, while having a good basis function superposition and keeping the image representation error to a minimum, is feasible, with the parameter combination range depending upon the scanner's intrinsic resolution characteristics. Using the printed point source array as a MR compatible methodology for experimentally measuring the scanner's PSF, the system's spatially variant resolution properties were successfully evaluated in image space. Overall the PET subsystem exhibits excellent resolution characteristics mainly due to the fact that the raw data are not under-sampled/rebinned, enabling the spatial resolution to be dictated by the scanner's intrinsic resolution and the image reconstruction parameters. Due to the impact of these parameters on the resolution properties of the reconstructed images, the image space PSF varies both under spatial transformations and due to basis function parameter selection. Nonetheless, for a range of basis function parameters, the image space PSF remains unaffected, with the range depending on the scanner's intrinsic resolution properties.

On numerically reliable frequency-domain system identification: new connections and a comparison of methods

Numerical experiments involving galerkin and collocation methods for linear integral equations of the first kind

A fuzzy clustering approach is developed to select pole locations for Orthonormal Basis Functions (OBFs), used for identification of Linear Parameter Varying (LPV) systems. The identification approach is based on interpolation of locally identified Linear Time Invariant (LTI) models, using globally fixed OBFs. Selection of the optimal OBF structure, that guarantees the least worst-case local modelling error in an asymptotic sense, is accomplished through the fusion of the Kolmogorov n-width (KnW) theory and Fuzzy c-Means (FcM) clustering of observed sample system poles.

Building on previous efforts dedicated to educing the flow motions associated with low-angle sound emission from turbulent jets ,2 , the current paper proposes a strategy for the reduced-order dynamical modelling of these motions. The technique comprises a dynamic Ansatz obtained by Galerkin projection of the Navier-Stokes equations onto a set of orthonormal spatial basis functions; however, rather than directly computing the coefficients of the resulting evolution equation, which is only possible when full flow information is available, the coefficients are here identified. The described approach, though validated here using partial flow information from a numerical simulation, is thus applicable in an experimental context. A key objective of such modelling is to obtain a dynamical model with as few degrees of freedom as possible, this being motivated by the need to perform real-time sensing and control. The number of degrees of freedom is dictated by the number of basis functions deemed necessary for accurate and robust reproduction of the dynamics, and this raises the question regarding what kind of basis functions are best suited to a given problem. The empirical eigenfunctions obtained by Proper Orthogonal Decomposition (hereafter POD) are frequently used, largely on account of the optimality property of this kind of decomposition. POD is used in this work; however, rather than applying it to the full flow solution, it is applied to the reduced complexity, sound-producing flow skeleton identified in the previous studies . These studies showed that the number of modes necessary to capture a given percentage of fluctuation energy of the acousticallyimportant coherent structures is an order of magnitude less than the number necessary to capture the same percentage of fluctuation energy of the turbulence. The POD modes used here can thus be considered as acoustically-optimised. A reduced-order model (twenty degrees of freedom) is constructed that reproduces the dynamics of the essential sound-producing flow motions. The approach combines the flow calibration technique of Perret et al. and Tikhonov regularisation such as proposed by Cordier et al. . The model successfully computes the evolution of the flow motions over a short time horizon (about six convective time units), and a further intrinsic stabilisation technique is then implemented in order to enable long-time prediction. Sound radiation is computed by means of the jittering wavepacket source Ansatz of Cavalieri et al., as per Kerherve et al., and good agreement is obtained with the sound field of the original simulation, validating the modelling approach with respect to sound source dynamics.

This paper concerns the identification of MISO (multiple inputs single output) Wiener systems. For each input-output path, the linear dynamics are modeled by a set of orthonormal basis functions (OBFs). The static nonlinearity is modeled through a multivariate polynomial. The parameters of the model are the coefficients of this polynomial. In this paper, an identification procedure for SISO (single input single output) Wiener systems is extended towards MISO Wiener systems. The poles of the OBFs are estimated using an extension of the best linear approximation (BLA) towards MIMO (multiple input multiple output) systems. As the number of parameters can increase significantly compared to the SISO case, a parameter reduction step, first developed for the SISO case, is extended in this paper to the MISO case. In each set of OBFs, one OBF is replaced by the BLA of the input-output path corresponding to that set. It is shown that in this way the number of relevantly contributing terms in the multivariate polynomial is significantly reduced. Simulation results show a major reduction of the number of parameters, with only a minor increase in the rms error on the simulated output.

Presents a framework for fuzzy modeling of dynamic systems using orthonormal basis functions in the representation of the model input signals. The main objective of using orthonormal bases is to overcome the task of estimating the order and time delay of the process. The result is a nonlinear moving average fuzzy model which, consequently, has no feedback of prediction errors. Although any technique of fuzzy modeling can be used in the proposed framework, a relational approach is considered. The performance of fuzzy models with orthonormal basis functions is illustrated by examples and the results are compared with those provided by conventional fuzzy models and Volterra models.

Orthonormal basis functions for modelling continuous-time systems

Maximum likelihood and prediction error methods

The creep deformation process is constructed by complex interactions of multiple factors, and the measurement of creep deformation requires enormous economic costs and a long experimental time, so there is a small amount of measurement data. In such a situation, multiple models are often proposed to explain the same experimental data. The coexistence of multiple models based on different physical assumptions makes it difficult to understand the creep deformation process. The purpose of this study is to construct a framework to compare and evaluate coexistence models based on measurement data using the Bayesian model selection framework. Basically, in the creep deformation model, basis functions, usually the exponential function eat or power function tb, corresponding to the primary creep stage and the tertiary creep stage are set, and the time series of the whole creep deformation process is regressed as the linear sum of these bases. The parameters a and b of the basis functions are parameters that must also be optimized to regress the time series data. Izuno et al. proposed the Bayesian model selection method for the creep deformation process, and it was confirmed that the model with the linear term corresponding to the secondary creep stage, called the modified theta method, is a good model compared to the model without adding the linear term. To obtain the criterion of Bayesian model selection analytically, which can reduce the computational cost greatly, Izuno et al. treated only the regression coefficients as the probabilistic parameter of the creep model. In this method, parameters a and b of the basis functions are optimized by a grid search. Because the grid search requires significant computational costs if multiple basis functions are set for each creep stage, as in the model proposed by Kimura et al., it is difficult to apply the algorithm to models with multiple bases. In addition, if parameters a and b of the basis functions correlate with the regression coefficients, the method of Izuno et al. would offer a biased evaluation of the models. To improve such an existing method, we proposed a Bayesian model selection method that treats the parameters of the basis functions as probabilistic parameters. The large computational cost to treat the parameters of the basis functions as probabilistic parameters was reduced using the exchange Monte Carlo method, which is an efficient sampling method. By applying our proposed method to the evaluation of the creep deformation model, information was obtained to understand the creep deformation process deeply. For example, it was confirmed that there exists a model in which there is a correlation between the parameter of basis function and the regression coefficients.

Formulation of a Practical Algorithm for Parameter Estimation with Process and Measurement Noise

Data fusion for ballistic targets tracking using least squares

The goal of this paper is the approximation of a nonlinear, dynamic process. More specifically it is studied whether the nonlinear, dynamic process can be simplified by separating it in its linear dynamic and nonlinear static components. Focus of this work is the approximation of the dynamic proportion with orthonormal basis functions (OBFs). The approximation of the nonlinear static properties is not fixed to a specific technique, but we focus on Takagi-Sugeno fuzzy models. This approach contrasts with other publications like [10], [12], [15], where OBFs are used to model both the dynamic and nonlinear part of the process. By modeling NOBF models, the main task is the optimization of the OBF pole, because it is the hyper-parameter, which affects the model quality most. This contribution focuses on the loss functions used for the optimization of the OBF pole.The loss functions must fulfill several requirements: It has to have a unique minimum, must be computable without prior knowledge of the nonlinear component of the process, be smooth around minima and sufficiently fast to compute. We compare three approaches to approximate the loss function. The first approach is the true loss function, which is the squared error of the nonlinear model and acts as reference. The second approach incorporates the structure of local model networks [3]. Finally, we propose a new approach, which is model free and based on the gamma test. We further analyze for which type of systems the technique is suitable and compare the advantages and disadvantages of the respective approach.

An alternative method was proposed for identification of linear continuous time system with noisy output. Using Gaussian modulating filters, by numerical integration, a discrete equivalent identification model that was parametrized directly in continuous time model parameters was obtained, and the LS estimation algorithm was developed. The LS estimator is biased even with white measurement output noise, and the bias is determined by variance of output noise. On the basis of obtained output measurement noise variance, the bias was compensated and the consistent estimates of system parameters were given. Simulation which compares the standard LS with BCLS algorithm indicates that the proposed algorithm is an efficient method for noisy output continuous time system identification.