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Efficient Multidimensional Regularization for Volterra Series Estimation
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Abstract
This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models.
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Many systems exhibit a quasi linear or weakly nonlinear behavior during normal operation, and a hard saturation effect for high peaks of the input signal. The proposed benchmark is an example of this type of non-linear system. On top of this, only a short data record is available for the parameter estimation step.
In this paper a nonparametric time-domain estimation method of linear time-varying systems from measured noisy data is presented. The challenge with time-varying systems is that the time-varying two-dimensional (2-D) impulse response functions (IRFs) are not uniquely determined from a single set of input and output signals as in the case of linear time-invariant systems. Due to this nonuniqueness, the number of possible solutions is growing quadratically with the number of samples. To decrease the degrees of freedom, user-defined (adjustable) constraints will be imposed. In this particular case, it is imposed that the entire 2-D IRF is smooth. This is implemented by a 2-D kernel-based regularization. This regularization is applied over the system time (the direction of the impulse responses) and simultaneously over the global time (representing the system memory). To illustrate the efficiency of the proposed method, it is demonstrated on a measurement example as a conceptual instrument for measuring and analyzing time-varying systems.
Engineers and scientists want a reliable mathematical model of the observed phenomenon for understanding, design and control. System identification is a tool which allows the user to build models of dynamic systems from experimental noisy data. This is an interdisciplinary science which connects the world of control theory, data acquisition, signal processing, statistics, time series analysis and many other areas. In modeling and measurement techniques it is commonly assumed that the observed systems are linear time-invariant. This point of view is acceptable as long as the time variations of the systems are negligible. However, in some cases, this assumption is not satisfied and it leads to a very low accuracy of the estimates. In those cases, advanced modelling is needed taking into account the time-varying behavior of the model. In this thesis a very important class of systems, namely, the linear time varying systems are considered. The importance of these systems can be seen through some application examples. A good example from the electrical field is, for example a non-compensated transistor (in an operational amplifier) with a shifting offset-voltage: the higher the temperature, the higher the offset drift. The offset variations influence the system parameters and result in a time-varying behavior. The changing bio-impedance in the heart is also a good example from biomedical sciences. In chemistry, an interesting example can be the impedance changing due to the pitting corrosion in metals. It is already shown that LTV systems can be described by a two dimensional impulse response function. The challenge is that the time-varying two dimensional impulse response functions are not uniquely determined from a single set of input and output signals – like in the case of linear time invariant systems. Due to this non-uniqueness, the number of possible solutions is growing quadratically with the number of samples. To decrease the degrees of freedom, user-defined adjustable constraints will be imposed. This will be implemented by using two different approaches. First, a special two dimensional regularization technique is applied. The second implementation technique uses generalized two dimensional smoothing B-splines. Using the proposed methods high quality models can be built. This thesis involves the theoretical and implementational questions of the time-varying system identification.
This contribution presents a smoothing technique for the identification of systems with a smooth impulse response function. Using a generalized and modified B-spline based methodology, an impulse response function can be estimated in the time domain or a frequency response function in the frequency domain. With respect to the system dynamics, it is possible 1) to reduce the disturbing noise 2) to decrease the effect of transients 3) to reduce the number of model parameters.
A brief overview of vortex-induced vibration (VIV) of bridge decks is presented highlighting special VIV features concerning bridge decks. A popular VIV model (Van de Pol type model) for bridge decks is examined in detail. Alternatively, a truncated Volterra series based nonlinear oscillator is introduced to model the VIV system. Typical features of VIV such as the “limit cycle oscillation” (LCO), “frequency shift”, “hysteresis” and “beat phenomenon” are parsimoniously and accurately captured in the proposed nonlinear model. As a functional expansion of a nonlinear system, Volterra series is convenient for estimating the linear and nonlinear contributions to VIV. It is demonstrated that the relative contribution of nonlinear effects in VIV is around 50 percent of the total response for a range of bridge cross-sections. The efficacy of Volterra series as a reduced order model (ROM) in capturing aerodynamic nonlinearities eliminates the need for reliance on conventional phenomenological models as it promises to offer a unified framework for nonlinear wind effects on long span bridges, e.g. VIV, buffeting and flutter.
This paper presents a new approach to identify time-varying ARX models by imposing a penalty on the coefficient variation.
Two different coefficient normalizations are compared and a method to solve the two corresponding optimization problems is
proposed.
The fault diagnosis schemes are roughly divided into two major categories: signal analysis and modeling method. Compared with the signal analysis approaches, the availability of suitable models of machines could be exploited more conveniently, which allows a more precise and reliable fault identification. In this paper, a modeling fault diagnostic approach based on Volterra series for rotating machinery is presented. After a brief review of the Volterra series theory, the process of the new diagnostic approach and the identification method of Volterra kernels are introduced in detail. The computer simulation verifies its feasibility and validity. Then, this novel method based on Volterra modeling is employed for the fault diagnosis of a rotor-bearing system. Through investigating the changes of generalized frequency response functions of the system under different states, the faults are identified successfully. Finally, the laboratory experimental results verify its effectiveness.
Discrete-time Volterra models play an important role in many application areas. The main drawback of these models is their parametric complexity due to the huge number of their pa-rameters, the kernel coefficients. Using the symmetry pro-perty of the Volterra kernels, these ones can be viewed as symmetric tensors. In this paper, we apply tensor decomposi-tions (PARAFAC and HOSVD) for reducing the kernel para-metric complexity. Using the PARAFAC decomposition, we also show that Volterra models can be viewed as Wiener mo-dels in parallel. Simulation results illustrate the effectiveness of tensor decompositions for reducing the parametric com-plexity of cubic Volterra models.
A new approach to designing sets of ternary periodic signals with different periods for multi-input multi-output system identification is described. The signals are pseudo-random signals with uniform nonzero harmonics, generated from Galois field GF( q ), where q is a prime or a power of a prime. The signals are designed to be uncorrelated, so that effects of different inputs can be easily decoupled. However, correlated harmonics can be included if necessary, for applications in the identification of ill-conditioned processes. A design table is given for q les 31. An example is presented for the design of five uncorrelated signals with a common period N = 168 . Three of these signals are applied to identify the transfer function matrix as well as the singular values of a simulated distillation column. Results obtained are compared with those achieved using two alternative methods.
Methods for designing two types of periodic ternary input signal used to identify a system in the presence of both noise and nonlinear distortions have been described here. Signals of the first type have even harmonics suppressed, to eliminate errors in odd-order estimates from even-order distortions, and vice-versa. Signals of the second type have harmonic multiples of both two and three suppressed, to further reduce errors from nonlinear distortions. For both types of signal, three design criteria are defined. The first criterion allows the signal energy-amplitude ratio to be maximised, the second allows the signal spectrum uniformity to be maximised and the third allows a compromise to be made between the first two criteria. With the methods described, ternary signals of both types with a very wide range of periods can be obtained for use in this application. Software for computer-aided design of the signals is available on the Internet. The signals and their periods are given in tables, and an example is used to show how they are applied.
Decomposing dynamical systems in terms of orthogonal expansions enables the modelling/approximation of a system with a finite length expansion. By flexibly tuning the basis functions to underlying system characteristics, the rate of convergence of these expansions can be drastically increased, leading to highly accurate models (small bias) being represented by few parameters (small variance). Additionally algorithmic and numerical aspects are favourable.
This paper describes the new developments in Mathwork's SYSTEM IDENTIFICATION TOOLBOX to be run with MATLAB. There are three main additions to the new version: (1) Frequency domain input/output data as well as frequency response data can be directly used to fit models. (2) Simple continuous-time process models of the type Static Gain, Time Constant, Dead Time can be directly estimated as a new model object class. (3) A function advice can be applied both to data sets and to estimated models, in order to provide guidance and advice through the sometime complex identification process.
Nonlinear problems have drawn great interest and extensive attention from engineers, physicists and mathematicians and many other scientists because most real systems are inherently nonlinear in nature. To model and analyze nonlinear systems, many mathematical theories and methods have been developed, including Volterra series. In this paper, the basic definition of the Volterra series is recapitulated, together with some frequency domain concepts which are derived from the Volterra series, including the general frequency response function (GFRF), the nonlinear output frequency response function (NOFRF), output frequency response function (OFRF) and associated frequency response function (AFRF). The relationship between the Volterra series and other nonlinear system models and nonlinear problem solving methods are discussed, including the Taylor series, Wiener series, NARMAX model, Hammerstein model, Wiener model, Wiener-Hammerstein model, harmonic balance method, perturbation method and Adomian decomposition. The challenging problems and their state of arts in the series convergence study and the kernel identification study are comprehensively introduced. In addition, a detailed review is then given on the applications of Volterra series in mechanical engineering, aeroelasticity problem, control engineering, electronic and electrical engineering.
In the last years, the success of kernel-based regularisation techniques in solving impulse response modelling tasks has revived the interest on linear system identification. In this work, an alternative perspective on the same problem is introduced. Instead of relying on a Bayesian framework to include assumptions about the system in the definition of the covariance matrix of the parameters, here the prior knowledge is injected at the cost function level. The key idea is to define the regularisation matrix as a filtering operation on the parameters, which allows for a more intuitive formulation of the problem from an engineering point of view. Moreover, this results in a unified framework to model low-pass, band-pass and high-pass systems, and systems with one or more resonances. The proposed filter-based approach outperforms the existing regularisation method based on the TC and DC kernels, as illustrated by means of Monte Carlo simulations on several linear modelling examples.
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.
Recently, a new kernel-based approach for identification of time-invariant linear systems has been proposed. Working under a Bayesian framework, the impulse response is modeled as a zero-mean Gaussian vector, with covariance given by the so called stable spline kernel. Such a prior model encodes smoothness and exponential stability information, and depends just on two unknown parameters that can be determined from data via marginal likelihood optimization. It has been shown that this new regularized estimator may outperform classical system identification approaches, such as prediction error methods. This paper extends the stable spline estimator to identification of time-varying linear systems. For this purpose, we include an additional hyperparameter in the model noise, showing that it plays the role of a forgetting factor and can be estimated via marginal likelihood optimization. Numerical experiments show that the new proposed algorithm is able to well track time-varying systems, in particular effectively detecting abrupt changes in the process dynamics.
In recent works, a new regularized approach for linear system identification has been proposed. The estimator solves a regularized least squares problem and admits also a Bayesian interpretation with the impulse response modeled as a zero-mean Gaussian vector. A possible choice for the covariance is the so called stable spline kernel. It encodes information on smoothness and exponential stability, and contains just two unknown parameters that can be determined from data via marginal likelihood (ML) optimization. Experimental evidence has shown that this new approach may outperform traditional system identification approaches, such as PEM and subspace techniques. This paper provides some new insights on the effectiveness of the stable spline estimator equipped with ML for hyperparameter estimation. We discuss the mean squared error properties of ML without assuming the correctness of the Bayesian prior on the impulse response. Our findings reveal that many criticisms on ML robustness are not well founded. ML is instead valuable for tuning model complexity in linear system identification also when impulse response description is affected by undermodeling.
Structural damages result in nonlinear dynamical signatures that can significantly enhance their detection. An original nonlinear damage detection approach is proposed that is based on a cascade of Hammerstein models representation of the structure. This model is estimated by means of the Exponential Sine Sweep Method from only one measurement. On the basis of this estimated model, the linear and nonlinear parts of the outputs are estimated, and two damage indexes (DIs) are proposed. The first DI is built as the ratio of the energy contained in the nonlinear part of an output versus the energy contained in its linear part. The second DI is the angle between the subspaces obtained from the nonlinear parts of two sets of outputs after a principal component analysis. The sensitivity of the proposed DIs to the presence of damages as well as their robustness to noise is assessed numerically on spring–mass–damper structures and experimentally on composite plates with surface-mounted PZT-elements. Results demonstrate the effectiveness of the proposed method to detect a damage in nonlinear structures and in the presence of noise.
Most of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.
There has been recently a trend to study linear system identification with high order finite impulse response (FIR) models using the regularized least-squares approach. One key of this approach is to solve the hyper-parameter estimation problem that is usually nonconvex. Our goal here is to investigate implementation of algorithms for solving the hyper-parameter estimation problem that can deal with both large data sets and possibly ill-conditioned computations. In particular, a QR factorization based matrix-inversion-free algorithm is proposed to evaluate the cost function in an efficient and accurate way. It is also shown that the gradient and Hessian of the cost function can be computed based on the same QR factorization. Finally, the proposed algorithm and ideas are verified by Monte-Carlo simulations on a large data-bank of test systems and data sets.
In this paper, an original approach based on the Volterra series theory is applied in order to analyse a nonlinear single track model, which is considered to describe the vehicle dynamics behaviour in the nonlinear domain. This model is based on a polynomial approximation up to the third order of the Pacejka formula that describes the full tyre behaviour. The analysis of the model is carried out using a truncated form of the Volterra series; this allows the extraction of an analytical formulation of the nonlinear response characteristics. The analysis is focused on the extraction of the first order frequency response function expression and the understeer angle curve vs lateral acceleration, which characterises the vehicle typology and stability. The resulting equations and illustrations in both the cases are presented.
There has been a rapid increase recently in the number of software packages available to generate different types of perturbation signals for the purpose of system identification in the frequency domain. The types of signal include both computer-optimised signals, which are designed to match a specified power spectrum as closely as possible, and pseudo-random signals, which have fixed spectra. Several of these signals are now readily available directly from the World Wide Web. The objective of this paper is to review what packages are available, and from where, and to outline some of the application areas in which they might be used.
Intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques, we revisit the old problem of transfer function estimation from input–output measurements. We formulate a classical regularization approach, focused on finite impulse response (FIR) models, and find that regularization is necessary to cope with the high variance problem. This basic, regularized least squares approach is then a focal point for interpreting other techniques, like Bayesian inference and Gaussian process regression. The main issue is how to determine a suitable regularization matrix (Bayesian prior or kernel). Several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets. Our findings based on the data bank are as follows. The classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach–the prediction error method/maximum likelihood (PEM/ML) approach. If the goal is to estimate a model of given order as well as possible, a low order model is often better estimated by the PEM/ML approach, and a higher order model is often better estimated by model reduction on a high order regularized FIR model estimated with careful regularization. Moreover, an optimal regularization matrix that minimizes the mean square error matrix is derived and studied. The importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach.
A new nonparametric paradigm to model identification has been recently introduced in the literature. Instead of adopting finite-dimensional models of the system transfer function, the system impulse response is searched for within an infinite dimensional space using regularization theory. The method exploits the so called stable spline kernels which are associated with hypothesis spaces embedding information on both regularity and stability of the impulse response. In this paper, the potentiality of this approach is studied with respect to the reconstruction of sums of exponentials. In particular, first, we characterize the learning rates of our estimator in reconstructing such class of functions also exploiting recent advances in statistical learning theory. Then, we use Monte Carlo studies to illustrate the definite advantages of this new nonparametric approach over classical parametric prediction error methods in terms of accuracy in impulse responses reconstruction.
The accurate modelling of friction dynamics remains an obstacle to the identification of a number of processes. Typically models are derived based on a detailed physical understanding of the underlying process. However, this can often be time consuming and, in any case, the models must be tailored to the particular process being studied. Alternatively, we can model friction dynamics using only observations of the process. In this paper a comparison is presented of the relative value of incorporating differing levels of prior knowledge into the modelling of friction dynamics. A new approach to data-driven modelling is presented, the generalised Fock space nonlinear autoregressive with exogeneous inputs (GFSNARX) model. This model allows Volterra and polynomial NARX models of arbitrary degree to be estimated using finite data. The results of modelling friction dynamics using the GFSNARX model are compared to the DNLR Maxwell slip and AVDNN models. It is shown that, whilst the incorporation of a priori physical knowledge results in a definite improvement in model performance, appropriate data driven models can achieve comparable performance without the effort of physical modelling. However, these results depend on the friction regime being modelled. Those models incorporating prior knowledge achieved good results for all regimes whilst the GFSNARX model was best suited to the pre-sliding regime.
This work tackles the problem of modeling nonlinear systems using Volterra models based on Kautz functions. The drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions, such as the Kautz basis. The resulting model, so-called Wiener/Volterra model, can be truncated into a few terms if the Kautz functions are properly designed. The underlying problem is how to select the free-design complex poles that fully parameterize these functions. A solution to this problem has been provided in the literature for linear systems, which can be represented by first-order Volterra models. A generalization of such strategy focusing on Volterra models of any order is presented in this paper. This problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions. The aim is to minimize the number of two-parameter Kautz functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. The main result is the derivation of an analytical solution for a sub-optimal expansion of the Volterra kernels using a set of Kautz functions.
This paper describes a new kernel-based approach for linear system identification of stable systems. We model the impulse response as the realization of a Gaussian process whose statistics, differently from previously adopted priors, include information not only on smoothness but also on BIBO-stability. The associated autocovariance defines what we call a stable spline kernel. The corresponding minimum variance estimate belongs to a reproducing kernel Hilbert space which is spectrally characterized. Compared to parametric identification techniques, the impulse response of the system is searched for within an infinite-dimensional space, dense in the space of continuous functions. Overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization. The proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two-step procedure that projects the nonparametric estimate onto the space of nominal models. The continuous-time derivation immediately extends to the discrete-time case. On several continuous- and discrete-time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques.
We present a novel nonparametric approach for identification of nonlinear systems. Exploiting the framework of Gaussian regression, the unknown nonlinear system is seen as a realization from a Gaussian random field. Its covariance encodes the idea of “fading” memory in the predictor and consists of a mixture of Gaussian kernels parametrized by few hyperparameters describing the interactions among past inputs and outputs. The kernel structure and the unknown hyperparameters are estimated maximizing their marginal likelihood so that the user is not required to define any part of the algorithmic architecture, e.g., the regressors and the model order. Once the kernel is estimated, the nonlinear model is obtained solving a Tikhonov-type variational problem. The Hilbert space the estimator belongs to is characterized. Benchmarks problems taken from the literature show the effectiveness of the new approach, also comparing its performance with a recently proposed algorithm based on direct weight optimization and with parametric approaches with model order estimated by AIC or BIC.
The miniaturization of GSM handsets creates nonlinear acoustical echoes between microphones and loudspeakers when the signal level is high (hands-free communication). Several methods including nonlinear cascade filters and a bilinear filter are proposed to compensate these echoes. A bilinear filter is a restricted NARMAX (nonlinear autoregressive moving average with exogenous inputs) filter. We present an evaluation based on the standard ERLE (echo return loss enhancement) measure, between a simple linear adaptive FIR filter and various nonlinear filters. These experiments are carried out first on a simulated communication system, then on experimental signals.
Acoustic echo cancellers in today's speakerphones or video conferencing systems rely on the assumption of a linear echo path. Low-cost audio equipment or constraints of portable communication systems cause nonlinear distortions, which limit the echo return loss enhancement achievable by linear adaptation schemes. These distortions are a super-position of different effects, which can be modelled either as memoryless nonlinearities or as nonlinear systems with memory. Proper adaptation schemes for both cases of nonlinearities are discussed. An echo canceller for nonlinear systems with memory based on an adaptive second order Volterra filter is presented. Its performance is demonstrated by measurements with small loudspeakers. The results show an improvement in the echo return loss enhancement of 7 dB over a conventional linear adaptive filter. The additional computational requirement for the presented Volterra filter is comparable to that of existing acoustic echo cancellers
Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.
Some recent results on the design and implementation of second-order Volterra filters are presented. The (second-order) Volterra filter is a nonlinear filter with the filter structure of (second-order) Volterra series. A simple minimum mean-square error solution for the Volterra filter is derived, based on the assumption that the filter input is Gaussian. Also, we propose an iterative factorization technique to design a subclass of the Volterra filters, which can alleviate the complexity of the filtering operations considerably. Furthermore, an adaptive algorithm for the Volterra filter is investigated along with its mean convergence and asymptotic excess mean-square error. Finally, the utility of the Volterra filter is demonstrated by utilizing it in studies of nonlinear drift oscillations of moored vessels subject to random sea waves.
Classical model validation procedures are placed at the focus of our attention. We discuss the principles by which we reach confidence in a model through such validation techniques, and also how the distance to a "true" description can be estimated this way. In particular we stress that model errors must be separated from disturbances in this process, and that consequently correlation tests between the model residuals and past inputs play a crucial role in this process.
Regularized Nonparametric Volterra Kernel Estimation
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Nonparametric Volterra series estimate of the cascaded tank
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Identification in Nuclear and Thermal Energy, Moderator Temperature Coefficient Estimation via Noise
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Cascaded Tanks Benchmark: Parametric and Nonparametric
- G Holmes
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