Reading Rockets Norm or Criterion Measurement Tool

Motion Detection and Estimation

Janusz Konrad , in The Essential Guide to Video Processing, 2009

3.5.iv Optical Flow via Regularization

Remember the regularized estimation benchmark ( 3.33). Information technology uses a translational/linear movement model at each pixel under the constant-intensity ascertainment model and quadratic error benchmark. To observe the continuous functions v _ane and v ₂, implicitly dependent on x, the functional in (iii.33) needs to be minimized, which is a problem in the calculus of variations. The Euler-Lagrange equations yield [22]:

$\lambda {\nabla }^{\text{2}}{5}_1=\left(\frac{\partial I}{\partial \mathrm{10}}{v}_1+\frac{\partial I}{\partial y}{\mathrm{five}}_{\mathrm{ii}}+\frac{\partial I}{\partial t}\right)\frac{\partial I}{\partial \mathrm{ten}},$

$\lambda {\nabla }^{\text{2}}{v}_2=\left(\frac{\partial I}{\partial x}{v}_1+\frac{\partial I}{\partial y}{v}_2+\frac{\partial I}{\partial t}\right)\frac{\partial I}{\partial y},$

where ${\nabla }^{\mathrm{ii}}={\partial }^2/\partial x^2+{\partial }^{\mathrm{ii}}/\partial y^2$ is the Laplacian operator. This pair of elliptic partial differential equations can exist solved iteratively using finite-divergence or finite-element discretization.

An culling is to codify the trouble straight in the discrete domain. So, the integral in (3.33) is replaced by a summation, whereas the derivatives are replaced past finite differences. In [22], for example, an average of first-gild differences computed over a 2 × two × ii cube was used. By differentiating this discrete cost function, a organization of equations tin can be computed and later on solved by Jacobi or Gauss-Seidel relaxation. This detached arroyo to regularization is a special instance of the MAP interpretation presented side by side.

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Introduction

Badong Chen , ... Jose C. Principe , in Arrangement Parameter Identification, 2013

i.two Traditional Identification Criteria

Traditional identification (or estimation) criteria mainly include the least squares (LS) criterion [13], minimum mean foursquare error (MMSE) criterion [14], and the maximum likelihood (ML) criterion [xv,16] [15] [xvi] . The LS criterion, defined by minimizing the sum of squared errors (an error beingness the difference between an observed value and the fitted value provided by a model), could at least dates dorsum to Carl Friedrich Gauss (1795). Information technology corresponds to the ML benchmark if the experimental errors have a Gaussian distribution. Due to its simplicity and efficiency, the LS criterion has been widely used in problems, such as interpretation, regression, and system identification. The LS criterion is mathematically tractable, and the linear LS problem has a closed form solution. In some contexts, a regularized version of the LS solution may be preferable [17]. There are many identification algorithms developed with LS criterion. Typical examples are the recursive least squares (RLS) and its variants [4]. In statistics and signal processing, the MMSE criterion is a common measure of estimation quality. An MMSE estimator minimizes the hateful square error (MSE) of the fitted values of a dependent variable. In system identification, the MMSE benchmark is often used as a criterion for stochastic approximation methods, which are a family of iterative stochastic optimization algorithms that attempt to find the extrema of functions which cannot be computed directly, but only estimated via noisy observations. The well-known least hateful square (LMS) algorithm [18–twenty] [xviii] [19] [twenty] , invented in 1960 by Bernard Widrow and Ted Hoff, is a stochastic gradient descent algorithm nether MMSE criterion. The ML criterion is recommended, analyzed, and popularized by R.A. Fisher [15]. Given a set of information and underlying statistical model, the method of ML selects the model parameters that maximize the likelihood function (which measures the degree of "agreement" of the selected model with the observed data). The ML estimation provides a unified approach to estimation, which corresponds to many well-known estimation methods in statistics. The ML parameter estimation possesses a number of attractive limiting backdrop, such as consistency, asymptotic normality, and efficiency.

The to a higher place identification criteria (LS, MMSE, ML) perform well in near practical situations, and so far are nevertheless the workhorses of organisation identification. All the same, they have some limitations. For example, the LS and MMSE capture simply the second-order statistics in the data, and may exist a poor approximation criterion, especially in nonlinear and non-Gaussian (e.one thousand., heavy tail or finite range distributions) situations. The ML benchmark requires the cognition of the conditional distribution (likelihood part) of the data given parameters, which is unavailable in many practical problems. In some complicated issues, the ML estimators are unsuitable or do non exist. Thus, selecting a new benchmark beyond second-order statistics and likelihood function is bonny in issues of arrangement identification.

In guild to take into business relationship higher order (or lower order) statistics and to select an optimal criterion for system identification, many researchers studied the non-MSE (nonquadratic) criteria. In an early piece of work [21], Sherman first proposed the non-MSE criteria, and showed that in the case of Gaussian processes, a large family of not-MSE criteria yields the same predictor as the linear MMSE predictor of Wiener. Later, Sherman's results and several extensions were revisited by Chocolate-brown [22], Zakai [23], Hall and Wise [24], and others. In [25], Ljung and Soderstrom discussed the possibility of a general error criterion for recursive parameter identification, and found an optimal criterion by minimizing the asymptotic covariance matrix of the parameter estimates. In [26,27] [26] [27] , Walach and Widrow proposed a method to select an optimal identification criterion from the least hateful fourth (LMF) family criteria. In their arroyo, the optimal choice is determined by minimizing a cost function which depends on the moments of the interfering racket. In [28], Douglas and Meng utilized the calculus of variations method to solve the optimal criterion amidst a big family of general error criteria. In [29], Al-Naffouri and Sayed optimized the error nonlinearity (derivative of the full general fault criterion) by optimizing the steady land performance. In [30], Pei and Tseng investigated the least mean $p$ -power (LMP) criterion. The fractional lower lodge moments (FLOMs) of the error take also been used in adaptive identification in the presence of impulse alpha-stable noises [31,32] [31] [32] . Other not-MSE criteria include the Thou-estimation benchmark [33], mixed norm benchmark [34–36] [34] [35] [36] , risk-sensitive criterion [37,38] [37] [38] , high-order cumulant (HOC) criterion [39–42] [39] [40] [41] [42] , and then on.

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Hydrologic Modeling

C.T. Haan , ... J.C. Hayes , in Design Hydrology and Sedimentology for Pocket-size Catchments, 1994

Other Considerations

James and Burges (1982) present an fantabulous word of parameter interpretation and estimation criteria. If a model construction is such that only a subset of parameters touch on i particular aspect of model output, those parameters may exist estimated on the basis of an optimization criteria related to that detail aspect of model output. For example a p-parameter model may be structured so that a subset p _ane of the p parameters governs runoff volumes and a 2nd nonoverlapping subset p _two governs peak flows. The parameters in p ₁ could then exist estimated on the basis of a criterion related to volumes while the parameters in p _ii could be estimated on the basis of a benchmark related to peaks.

If in this example the parameters in p ₁ and p ₂ are all estimated on the ground of peaks, the parameters in p ₁ volition exist poorly determined and likely will take a big variance. While strict segmentation of influence between parameter sets and outputs generally does not occur, it is common for some parameters to accept relatively lilliputian influence on optimization criteria and thus be poorly adamant. James and Burges (1982) define sensitivity coefficients that tin can assistance place this possibility. If the model is so used in a state of affairs where the aspect of the model that is highly dependent on the poorly defined parameters is of import, a questionable pattern may result. From the preceding situation, designing a storage reservoir with a model whose parameters were optimized on peaks would be a situation where a potentially poor design could result. Use of Eq. (13.7) partially overcomes this problem since prediction errors on both peaks and volumes can exist incorporated into parameter estimation.

As an alternative to Eq. (13.7) to include multiple objectives, a weighted criterion office could exist used. Such a function might be

(thirteen.8) $C=w_1\Sigma e_1^2+{\mathrm{west}}_2\Sigma {\mathrm{east}}_{\mathrm{ii}}^{\mathrm{two}},$

where the subscripts refer to different objectives (i.e., peaks and volumes), westward is a weight, and due east is the error. The selection of the weights is arbitrary and may be done to reflect the relative importance of the two objectives. Equation (xiii.8) differs from Eq. (13.seven) in that interaction between the two objectives is non included. For the case where k = 2, Eq. (13.vii) becomes

(xiii.9) $C=\Sigma {\mathrm{east}}_1^2\Sigma e_2^{\mathrm{ii}}-{\left(\Sigma e_1{e}_2\right)}^2.$

Equation (13.8) can be generalized to any number of objectives. It is not necessary that all e_i used in an equation like (xiii.7) or (thirteen.8) be related to menstruum. For example, east ₂ might refer to some measure of soil water content if observed data were available and the model provided estimates of soil h2o.

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IDENTIFICATION, MODEL-BASED METHODS

S.D. Fassois , in Encyclopedia of Vibration, 2001

Introduction

Parametric identification of vibrating systems is the procedure of developing finitely parametrized models for such systems based upon measured excitation and/or response signals. Typically, the excitation is strength and the response vibration displacement, velocity, or acceleration. A typical identification experiment is depicted in Effigy 1. The structural dynamics are represented past a transfer matrix G(due south), with southward indicating the Laplace transform variable. The measurable force excitation vector is {ten(t)}, while the measurable vibration response vector (forced if x(t)≠0, complimentary if x(t)≡0) is {y(t)} and is assumed to be corrupted by stochastic zero-mean noise {n(t)}, which is uncorrelated with {ten(t)} (t indicating continuous fourth dimension).

Figure 1. Typical identification experiment.

In contrast to nonparametric identification, which leads to nonparametric representations such as frequency or impulse response functions, parametric identification (also chosen model-based) leads to finitely parametrized models such equally difference/differential equation and modal models. Such models provide important benefits due to their: (1) direct human relationship with differential equation or physically significant modal representations used in engineering analysis; (ii) improved accuracy and frequency resolution; (3) compactness/parsimony of representation, that is, their ability to provide complete system label past relatively few parameters; and (4) their suitability for analysis, prediction, fault diagnosis, and control. The cost paid for these benefits includes a generally increased identification complication and dependence of the results on the assumed model form and the estimation criterion.

The Elements of Parametric Identification

The essential elements of any parametric identification method are:

1.

the information prepare

2.

the selected model grade

3.

the estimation criterion

four.

the model validation procedure

5.

the modal parameter extraction procedure.

The data set consists of suitable excitation and/or response signals. The model class is a selected family of models parametrized in terms of an (unknown) parameter vector θ – for instance, the class of rational transfer function models. Upon its optimization, the estimation criterion maps the data ready into a specific value of the parameter vector θ – a common choice is the least squares criterion. The model validation procedure aims at accepting or rejecting the estimated model; in the latter instance the estimation is repeated with proper modifications. Modal parameter extraction refers to the conclusion of the modal parameters from the estimated model and the distinction of structural from inapplicable (simulated) modes. The full general identification procedure, based upon sampled signals, is outlined in Figure 2.

Figure 2. The general identification procedure.

Classification of Identification Methods

The parametric identification methods may be classified according to 1 of the first three of the foregoing essential elements. Hence, depending upon the type of information, they may be classified every bit discrete or continuous time, transient or forced response, single-input single-output (siso) or general multiple-input multiple-output (mimo). Depending upon the model class they may be classified every bit deterministic or stochastic, distributed or lumped parameter, linear or nonlinear, with the linear being further classified as complex exponential, polynomial (transfer function), or state space. Depending upon the estimation criterion they may exist broadly classified as prediction error, least squares related, correlation, and subspace methods. Depending upon the computer realization, they may be further classified as batch (in which the complete information record is used at once) or recursive (the data record is processed sequentially in fourth dimension).

This section deals with linear, lumped parameter, and batch discrete time identification based upon sampled versions of the excitation and response signals. For simplicity the siso case is treated, although the mimo example is considered within the context of the state space model course and the Eigensystem Realization Algorithm. The identified discrete fourth dimension model is later transformed back into the continuous time domain for modal parameter extraction.

Before embarking on our give-and-take of the essential elements of parametric identification methods, it is useful to note the similarity of the identification with the betoken modeling trouble (see the section on SIGNAL PROCESSING, MODEL BASED METHODS). The two become essentially identical when the force excitation {x(t)} is zero, or random unobservable uncorrelated (white) racket, and the dissonance {due north(t)} is absent-minded (Figure 1). Hence, the model classes used in bespeak modeling are either the same as those used in identification (for instance, the complex exponential model grade) or suitable subsets (for instance the AutoRegressive (AR), or AutoRegressive Moving Average (ARMA) model classes). Many identification and indicate modeling methods are thus closely related, as the underlying interpretation theory principles are mutual. Notwithstanding, the presence of measured excitation in the identification problem introduces certain requirements, such as excitation richness, low cross-correlation amongst scalar excitations, and uncorrelatedness with the corrupting racket.

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Battery country-of-charge estimation methods

Shunli Wang , ... Zonghai Chen , in Bombardment Organisation Modeling, 2021

5.ii.5 Algorithm implementation

The Kalman filtering algorithm is used in the land-of-charge estimation and other practical scenarios. Taking the minimum average square mistake as the best estimation benchmark, the state-infinite model of signal and noise is used [29–33]. Establishing the land and observation equations of the model is to deduce the human relationship between the state and ascertainment variables [34–38]. The estimated value of the previous time bespeak and the observation value of the nowadays time point are introduced to update the state variable estimation, as shown in Fig. 5.6.

Fig. five.6. Iterative prediction-correction process for the land estimation.

It is a adept choice to use extended Kalman filtering when the estimation accuracy is non high and the calculation is not considered because information technology neglects the 2nd-order and college-society terms later the Taylor serial expansion, co-ordinate to which the estimation error increases based on information technology [39–47]. Moreover, the Jacobian matrix needs to be recalculated every time the iteration is updated, which greatly increases the adding complexity of the algorithm and reduces the system performance. The country and measurement equations are described every bit shown in Eq. (5.7).

(v.7) $\left\{\begin{array}{l}{\mathrm{10}}_{\mathrm{thousand}+1}=f\left({\mathrm{10}}_k,Z_k\right)+W_g\\ Y_k=g\left(X_{\mathrm{one\; thousand}},Z_k\right)+V_k\end{array}\right.$

When k  =   0, the initial variable and its error covariance matrix tin can be calculated by twon  +   1 points that are taken as k  =   1. The calculation loop performs the following steps, which affect the predicted issue, and its selection range is more often than not fifty–100. The equation for calculating the predicted country variable is described as shown in Eq. (5.eight).

(5.8) ${\widehat{\mathrm{10}}}_k^{-}=\sum _{i=1}^{{\mathrm{Northward}}_i}{\mathrm{Due\; west}}_{i}f\left({\xi }_i,Z_{\mathrm{1000}-\mathrm{one}}\right),{\xi }_i=\mathrm{Southward}{\gamma }_i+{\hat{X}}_{k-\mathrm{one}}^{+},P_{k-1}^{+}={\mathit{SS}}^T$

Then, the Kalman filtering algorithm is used to predict, feedback, alter, and update the information circularly. So, the information are analyzed in the adding process. The Kalman gain is updated dynamically according to the error covariance of the state variable and the observation variable. The state variable is updated iteratively co-ordinate to the estimation mistake of the ascertainment variable [48–54]. Compared with other filtering algorithms, its filtering effect is stable and reliable. The calculation error covariance is described as shown in Eq. (v.ix).

(5.9) $P_{\mathrm{yard}}^{-}=\sum _{i=\mathrm{one}}^{N_p}{\mathrm{Westward}}_i\left[f\left({\xi }_i,Z_{\mathrm{1000}-\mathrm{one}}\right)-{\widehat{\mathrm{10}}}_{\mathrm{one\; thousand}}^{-}\right]{\left[f\left({\xi }_i,Z_{\mathrm{one\; thousand}-1}\right)-{\hat{X}}_k^{-}\right]}^T+p_w$

Based on the classical Kalman filtering algorithm, the extended algorithm is studied and derived. These algorithms can be improved, only some defects are also introduced. The Kalman gain is calculated as shown in Eq. (5.10).

(v.10) $\left\{\begin{array}{l}{\hat{Y}}_{\mathrm{yard}}=\sum _{i=\mathrm{i}}^{N_i}{W}_{i}h\left[f\left({\tilde{\xi }}_i,Z_{\mathrm{one\; thousand}}\right)\right],{\tilde{\xi }}_i=\tilde{S}{\gamma }_i+{\widehat{\mathrm{Ten}}}_{\mathrm{thou}}^{-},P_m^{-}={\widetilde{\mathrm{South}}\tilde{S}}^T\\ P_{\mathit{yy}}=\sum _{i=\mathrm{ane}}^{{\mathrm{Northward}}_i}{W}_i\left[h\left({\tilde{\xi }}_i,Z_{k-1}\right)-{\hat{Y}}_{\mathrm{chiliad}}\right]{\left[h\left({\tilde{\xi }}_i,Z_{\mathrm{grand}-1}\right)-{\hat{Y}}_{\mathrm{grand}}\right]}^T\\ P_{\mathit{xy}}=\sum _{i=1}^{N_i}{W}_i\left[h\left({\tilde{\xi }}_i,Z_{\mathrm{1000}-\mathrm{one}}\right)-{\hat{X}}_{\mathrm{grand}}^{-}\right]{\left[h\left({\tilde{\xi }}_i,Z_{k-\mathrm{one}}\right)-{\widehat{\mathrm{10}}}_m^{-}\right]}^T\\ {\mathrm{One\; thousand}}_{\mathrm{chiliad}}=P_{\mathit{xy}}{\left(P_{\mathit{yy}}+P_v\right)}^{-1}\end{array}\right.$

Due to the complex internal construction of the lithium-ion bombardment, the estimation data often show strong nonlinearity in the working process, which makes the traditional country estimation algorithm difficult to obtain the real-time and accurate state of the battery. The extended Kalman filtering algorithm is introduced to solve the issues of big error and slow convergence speed acquired by the nonlinear battery characteristics in the state-of-charge estimation procedure. Considering the country prediction, the mistake covariance is then corrected, in which the expression is described as shown in Eq. (5.eleven).

(five.11) $\left\{\begin{array}{l}{\hat{X}}_k^{+}={\widehat{\mathrm{10}}}_k^{-}+{\mathrm{Yard}}_{\mathrm{grand}}\left(Y_g-{\hat{Y}}_{\mathrm{grand}}\right)\\ P_{\mathrm{thou}}^{+}=P_g^{-}+K_{\mathrm{yard}}{P}_{\mathit{xy}}^T\end{array}\right.$

Based on the equivalent circuit model and the extended Kalman filtering algorithm, the state interpretation is realized for the lithium-ion batteries. This experiment takes a lithium cobalt bombardment as the enquiry objective, and the results are analyzed comprehensively. Its working characteristics are obtained by carrying out the mathematical description, country-of-charge estimation, and outcome analysis. Information technology is then conducted past looping the following steps. The predicted land variable can exist obtained by calculating the error covariance, and the Kalman gain is described as shown in Eq. (5.thirteen).

(5.12) $\left\{\begin{array}{l}{\hat{X}}_k^{-}=f\left({\widehat{\mathrm{10}}}_{k-\mathrm{one}}^{+},Z_{k-1}\right)\\ P_k^{-}={\hat{A}}_{\mathrm{one\; thousand}-1}{P}_{m-1}^{+}{\hat{A}}_{k-\mathrm{i}}^T+P_w\end{array}\right.\Rightarrow {\mathrm{Thou}}_k=P_k^{-}{\hat{C}}_g^T{\left({\hat{C}}_g{P}_{\mathrm{thousand}}^{-}{\hat{C}}_g^T+P_{\mathrm{five}}\right)}^{-1}$

The equivalent circuit model is established first, then a hybrid charge-discharge experiment is carried out. The model parameters can exist identified by the experimental data analysis. The estimation accuracy is loftier in the battery state estimation process for real-fourth dimension country-of-accuse estimation. The simulation model is established for analysis. The corrected state prediction and its expression are obtained accordingly, together with the corrected error covariance as shown in Eq. (5.13).

(5.13) $\left\{\begin{array}{l}{\hat{y}}_k=g\left({\hat{x}}_k^{-},u_k\right)\\ {\hat{x}}_k^{+}={\widehat{\mathrm{10}}}_{\mathrm{thou}}^{-}+{\mathrm{Grand}}_k\left(y_{\mathrm{thou}}-{\hat{y}}_{\mathrm{thou}}\right)\end{array}\right.\Rightarrow P_k^{+}=\left(E-{\mathrm{One\; thousand}}_{\mathrm{thou}}{\hat{C}}_{\mathrm{yard}}\right)P_k^{-}$

The extended Kalman filtering algorithm is used for initial state estimation to solve the convergence trouble. It is too used for real-fourth dimension online estimation to reduce computational complexity.

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State interpretation strategy for continuous-fourth dimension systems with time-varying delay via a novel 50-Chiliad functional

Feisheng Yang , ... Peipei Kang , in Control Strategy for Time-Delay Systems, 2021

Abstract

This chapter focuses on studying the state estimation trouble for continuous-time systems with fourth dimension-varying delay and constructing a proper Lyapunov–Krasovskii functional (LKF), which is crucial for deriving less conservative estimation criteria. The main thought of previous LKFs are by employing more state data, and all they are equanimous of positive definite terms. In this chapter, nosotros propose a delay-product-type LKF with negative definite terms and gauge its derivative by the third-lodge Bessel–Legendre (B-50) based integral inequality together with mixed convex combination approaches. Based on the novel LKF, we obtain the desired estimator gain matrices and the $H_{\infty }$ performance index by solving a prepare of linear matrix inequalities (LMIs). Finally, nosotros requite numerical examples to demonstrate the effectiveness of the proposed method.

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Predictive variable structure filter

Lu Cao , ... Bing Xiao , in Predictive Filtering for Microsatellite Command System, 2021

7.1 Introduction

The main feature of the predictive filters is that they predict and correct the modeling mistake online and in real-time. Moreover, the modeling error or measurement noises handled past the predictive filters are not restricted to the Gaussian distribution. Although the predictive filters presented in Part Ii achieve improve estimation performance for the system states than the classical predictive filters, their performance depends on the option of the weighting matrix ${\mathit{West}}_E$ in (2.44) . When applying those filters in practical engineering, the weighting matrix is called based only on the engineer's experience. This limits their practicability. Motivated past solving this shortcoming, the Minimum Modeling Error (MME) interpretation benchmark will exist extended and three advanced predictive filtering approaches are presented in Part III.

Since the controller adult by using Variable Structure Control (VSC) theory has advantages of rapid response, robustness to parameter variations and uncertainties, and low computational costs (Shtessel et al., 2014), the VSC theory has been adopted to design nonlinear filters with high-accurateness state interpretation even in the presence of modeling mistake. Introducing the VSC to design filtering approaches was firstly seen in Habibi et al. (2002) and Habibi and Burton (2003). The VSC was applied to design the gain matrix of the Kalman filters. By doing this, the robustness of the Kalman filters is greatly improved. Post-obit this methodology, many VSC-based filtering schemes were presented (Habibi, 2006, 2007; Al-Shabi et al., 2013). Nonetheless, they were all developed in the framework of the Kalman filtering theory framework. Hence, they are unable to solve the inherent drawbacks of the Kalman filters.

Based on the MME estimation criterion and fully taking the advantages of predictive filters and the VSC theory, a predictive variable structure filter is presented in this affiliate. This filtering approach is different from the existing filtering theory. Information technology is designed with the assumption of the "Gaussian process" for the modeling errors eliminated, and information technology does not depend on the weighting matrix in the classical filtering theory. It has a great capability of estimating the modeling errors to update the system model without precise arrangement representation. Moreover, this filter does non require the nonlinear systems and their measurements to be linearized without accuracy loss of high-order items, which ensures the accurateness of the system's model.

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Addressing Accommodation and Learning in the Context of Model Predictive Control With Moving-Horizon Estimation

D.A. Copp , J.P. Hespanha , in Command of Complex Systems, 2016

2.1 Moving-Horizon Interpretation

In MHE the electric current state of the system x _t at time t is estimated past the solving of a finite-horizon online optimization problem with apply of a finite number of past measurements [nine]. If nosotros consider a finite horizon of L fourth dimension steps, then the objective of the MHE problem is to find an approximate of the current state x _t so equally to minimize a criterion of the form

(3) $\begin{array}{l}\hfill \sum _{\mathrm{due\; south}=t-L}^t{\eta }_s\left(y_s-g_{\mathrm{due\; south}}(x_{\mathrm{south}})\right)+\sum _{s=t-L}^{t-\mathrm{one}}{\rho }_{\mathrm{south}}(d_s),\end{array}$

given the system dynamics (one). The functions η _south (⋅) and ρ _s (⋅) are assumed to have nonnegative values. This is similar to the MHE criterion considered in [9, 10].

If the system dynamics besides include uncertain model parameters, as in Eq. (2), the MHE problem tin can be formulated so as to estimate both the current state x _t and the uncertain parameter θ. Then the MHE problem tin exist written as

(4) $\begin{array}{l}\hfill \underset{{\widehat{x}}_{t-L}\in \mathcal{X},{\widehat{d}}_{t-L:t-1}\in \mathcal{D},\widehat{\theta }\in \Theta }{min}\sum _{\mathrm{due\; south}=t-\mathrm{50}}^t{\eta }_{\mathrm{due\; south}}\left(y_s-{\mathrm{one\; thousand}}_s({\widehat{x}}_{\mathrm{southward}},\widehat{\theta })\right)+\sum _{s=t-L}^{t-1}{\rho }_s({\widehat{d}}_s),\end{array}$

where the initial state x _t−50 is constrained to belong to the set $\mathcal{Ten}$ , each chemical element of the input disturbance sequence d _t−L:t−1 is assumed to vest to the set $\mathcal{D}$ , and the uncertain parameter θ is known to belong to the set Θ. Throughout this affiliate, given two times t ₁ and t ₂ with t _ane < t _two, we apply the notation $x_{t_1:t_{\mathrm{ii}}}$ to denote the time series ${\mathrm{ten}}_{t_{\mathrm{one}}},x_{t_1+1},\dots ,x_{t_2-\mathrm{one}},x_{t_2}$ . An guess of the current country is and then determined from the dynamics (2) given the known past control inputs applied u _t−50:t−1 and estimates of the initial country ${\widehat{\mathrm{ten}}}_{t-\mathrm{50}}$ , the input disturbance sequence ${\widehat{d}}_{t-L:t-1}$ , and the uncertain parameter $\widehat{\theta }$ . The optimization (4) is solved once again at each time t in a receding-horizon way.

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Processing in the feature and model domains

Jinyu Li , ... Yifan Gong , in Robust Automatic Spoken communication Recognition, 2016

iv.2.one General Model Accommodation for GMM

As in Equation 3.35, model-domain methods only adapt the model parameters to fit the distorted speech signal. The model adaptation tin can operate in either supervised or unsupervised mode. In supervised style, the correct transcription of the adapting utterances is available. It is used to guide model adaptation to obtain the adapted model, $\widehat{\mathrm{\Lambda }}$ , used to decode the incoming utterances. In unsupervised manner, the correct transcription is not available, and usually 2-pass decoding is used. In the first pass, the initial model Λ is used to decode the utterance to generate a hypothesis. Ordinarily i hypothesis is good enough. The gain from using a lattice or N-best list to represent multiple hypotheses is limited (Padmanabhan et al., 2000). Then, $\widehat{\mathrm{\Lambda }}$ is obtained with the model adaptation process and used to re-decode the utterance to generate the concluding consequence.

Popular speaker adaptation methods such equally maximum a posteriori (MAP) (Gauvain and Lee, 1994) and its extensions such as structural MAP (SMAP) (Shinoda and Lee, 2001), MAP linear regression (MAPLR) (Siohan et al., 2001), and SMAP linear regression (SMAPLR) (Siohan et al., 2002), may not exist a practiced fit for most noise-robust speech communication recognition scenarios where only a very limited amount of adaptation data is available, for example, when only the utterance itself is used for unsupervised adaptation. Most popular methods use the maximum likelihood estimation (MLE) criterion (Gales, 1998; Leggetter and Woodland, 1995). Discriminative adaptation is also investigated in some studies (He and Chou, 2003; Wu and Huo, 2002; Yu et al., 2009). Dissimilar MLE accommodation, discriminative adaptation is very sensitive to hypothesis errors (Wang and Woodland, 2004). Every bit a consequence, most discriminative adaptation methods only work in supervised mode (He and Chou, 2003; Wu and Huo, 2002). Special processing needs to exist used for unsupervised discriminative adaptation. In Yu et al. (2009), a speaker-independent discriminative mapping transformation (DMT) is estimated during training. During testing, a speaker-specific transform is estimated with unsupervised MLE, and the speaker-independent DMT is then applied. In this way, discriminative adaptation is implicitly practical without the strict dependency on a correct transcription.

In the post-obit, popular MLE accommodation methods will be reviewed. All these general accommodation methods are widely used in robustness tasks. Maximum likelihood linear regression (MLLR) is proposed in Leggetter and Woodland (1995) to conform model mean parameters with a class-dependent linear transform

(4.thirty) $\begin{array}{l}\hfill {\mathit{\mu }}_y(\mathrm{yard})=\mathbf{A}(r_m){\mathit{\mu }}_x(m)+\mathbf{b}(r_{\mathrm{grand}}),\end{array}$

where μ _y (m) and μ _x (thousand) are the clean and distorted hateful vectors for Gaussian component m, and r _grand is the corresponding regression class.A(r _m ) and b(r _m ) are the regression-course-dependent transform and bias to exist estimated, which tin can be put together every bit Westward(r ₁₀₀₀ ) = [A(r _m )b(r _m )]. One extreme case is that if every regression grade just contains ane Gaussian, then r _yard = grand. Some other extreme example is that if in that location is only a single regression class, then all the Gaussians share the same transform as A(r _grand ) = A and b(r _m ) = b.

The expectation-maximization (EM) algorithm (Dempster et al., 1977) is used to get the maximum likelihood solution of Due west(r ₁₀₀₀ ). First, an auxiliary Q function for an utterance is defined

(4.31) $\begin{array}{l}\hfill Q(\widehat{\mathrm{\Lambda }};\mathrm{\Lambda })=\sum _{t,m}{\gamma }_t(g)log{p}_{\widehat{\mathrm{\Lambda }}}({\mathbf{y}}_t|m),\end{array}$

where $\widehat{\mathrm{\Lambda }}$ denotes the adjusted model, and γ _t (chiliad) is the posterior probability for Gaussian component m at time t.West(r _m ) tin can be obtained by setting the derivative of Q w.r.t.W(r _thousand ) to 0. A special case of MLLR is the point bias removal algorithm (Rahim and Juang, 1996), where the but single transform is simply a bias. The MLE criterion is used to estimate this bias, and it is shown that signal bias removal is meliorate than CMN (Rahim and Juang, 1996).

The variance of the noisy speech signal besides changes with the introduction of noise every bit shown in Department 3.2. Hence, in addition to transforming Gaussian hateful parameters with Equation 4.30, it is better to too transform Gaussian covariance parameters (Gales, 1998; Gales and Woodland, 1996) as

(4.32) $\begin{array}{l}\hfill {\mathbf{\Sigma }}_y(\mathrm{grand})=\mathbf{H}(r_m){\mathbf{\Sigma }}_{\mathrm{ten}}(m){\mathbf{H}}^T(r_m).\end{array}$

A 2-stage optimization is normally used. First, the hateful transform W(r _{one thousand} ) is obtained, given the current variance. Then, the variance transform H(r _m ) is computed, given the current mean. The whole process can be done iteratively. The EM method is used to obtain the solution, which is washed in a row-by-row iterative format.

Constrained MLLR (CMLLR) (Gales, 1998) is a very popular model adaptation method in which the transforms of the mean and covariance, A(r _grand ) and H(r _{one thousand} ), are constrained to be the same:

(4.33) $\begin{array}{ll}\hfill {\mathit{\mu }}_y(m)& =\mathbf{H}(r_g)({\mathit{\mu }}_{\mathrm{10}}(m)-\mathbf{g}(r_m)),\hfill \end{array}$

(four.34) $\begin{array}{ll}\hfill {\mathbf{\Sigma }}_y(\mathrm{thou})& =\mathbf{H}(r_{\mathrm{yard}}){\mathbf{\Sigma }}_x(m){\mathbf{H}}^T(r_m).\hfill \end{array}$

Rather than adapting all model parameters, CMLLR tin can be efficiently implemented in the feature space with the following relation

(4.35) $\begin{array}{l}\hfill \mathbf{y}=\mathbf{H}(r_{\mathrm{thousand}})(\mathbf{x}-\mathbf{g}(r_m)),\end{array}$

or

(iv.36) $\begin{array}{l}\hfill \mathbf{x}=\mathbf{A}(r_m)\mathbf{y}+\mathbf{b}(r_{\mathrm{thousand}}),\end{array}$

with A(r _m ) = H(r _m )⁻¹ and b(r _m ) = g(r _m ). The likelihood of the distorted speech y can now be expressed equally

(iv.37) $\begin{array}{l}\hfill p(\mathbf{y}|\mathrm{chiliad})=|\mathbf{A}(r_m)|\mathcal{Due\; north}(\mathbf{A}(r_{\mathrm{one\; thousand}})\mathbf{y}+\mathbf{b}(r_m);{\mathit{\mu }}_x(m),{\mathbf{\Sigma }}_{\mathrm{10}}(m))\end{array}$

As a issue, CMLLR is also referred to equally characteristic infinite MLLR (fMLLR) in the literature. Annotation that betoken bias removal is a special class of CMLLR with a unit of measurement scaling matrix.

In Saon et al. (2001a), fMLLR and its project variant (fMLLR-P) (Saon et al., 2001b) are used to adapt the audio-visual features in noisy environments. Adaptation needs to accumulate sufficient statistics for the examination information of each speaker, which requires a relatively large number of adaptation utterances.

As reported in Cui and Alwan (2005) and Saon et al. (2001a), general adaptation methods such as MLLR and fMLLR in noisy environments yield moderate improvement, but with a large gap to the performance of noise-specific methods (Li et al., 2007a, 2009) on the same task. Noise-specific compensation methods ordinarily change model parameters by explicitly addressing the nature of the distortions caused by the presence of dissonance. Therefore, they can accost the racket-robustness issue improve. The representative methods in this sub-category are parallel model combination (PMC) (Gales, 1995) and model-domain vector Taylor series (Moreno, 1996). Some representative dissonance-specific bounty methods will be discussed in detail in Chapter 6.

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Heating-Related Oscillations and Noise

In Flow-induced Vibrations (Second Edition), 2014

6.1.2.2 Evaluation methods

It is difficult to accurately predict the occurrence of combustion oscillations. However, some evaluation methods have been proposed and are used in practice.

6.1.2.2.ane Rayleigh'due south benchmark [ane]

Rayleigh considered the piece of work washed on the system during one cycle of oscillation, and wrote the post-obit equation for the energy alter, assuming that the volume alter is proportional to the estrus release rate:

(six.1) $\mathrm{\Delta }\mathrm{Due\; east}=\oint pq\mathrm{d}t\propto \oint p\stackrel{\cdot }{Q}\mathrm{d}t\propto {\stackrel{\cdot }{Q}}_0{p}_0\cos \theta$

Here, $\stackrel{\cdot }{Q}$ is the oestrus release charge per unit, p the pressure, θ the phase angle, and the subscript 0 indicates amplitude. When ΔE>0, oscillations occur. For example, suppose the deportation x in Fig. vi.2 is given past ten=A sin ωt, then the velocity is $u\equiv \stackrel{\cdot }{x}=A\omega \cos \omega t$ and the pressure p∝−A sin ωt. If Q∝u, that is $\stackrel{\cdot }{Q}\propto \stackrel{\cdot }{u}=-A{\omega }^{\mathrm{two}}\sin \omega t$ , then:

$\mathrm{\Delta }\mathrm{Due\; east}=\oint p\stackrel{\cdot }{Q}\mathrm{d}t\propto {\omega }^2{A}^2\oint {\sin }^2\omega t\mathrm{d}t\mathbf{>}0$

This means the system is unstable and oscillations occur. In Rayleigh's criterion, estimation of the phase angle θ is difficult. Furthermore, hateful flow effects are neglected.

6.1.two.2.two Madarame's criterion

Madarame [22] developed a adding method for the energy input during 1 cycle in the Rijke tube, shown in Fig. 6.8, assuming that the heating rate Q can exist expressed as:

(6.two) $Q(\tau )=\mathrm{\Theta }{\sigma }^{m+1}{\tau }^m{\mathit{e}}^{\mathrm{ii}\sigma \tau }/m$

and representing the pressure p, velocity u, and temperature T every bit:

(6.3) $u=u_0+u_{\mathrm{i}}{\mathit{due\; east}}^{i\omega t}p=p_0+p_1{\mathit{e}}^{i\omega t}T=T_0+T_1{\mathit{due\; east}}^{i\omega t}$

The steady term T ₀ is calculated first, followed past the fluctuation term T ₁ Then, the destabilizing free energy per cycle can be obtained as follows;

(6.4) $\mathrm{\Delta }\mathrm{East}\mathbf{=}A\beta \iint {\mathrm{R}}_{\mathrm{e}}(-i{p}_1{\mathrm{east}}^{\mathit{i}\omega \mathit{t}})\frac{\partial }{\partial t}{\mathrm{R}}_{\mathrm{e}}(T_1{\mathrm{east}}^{\mathit{i}\omega \mathit{t}})dx\mathrm{d}t\mathbf{=}\frac{\pi }{\omega }{p}_1{u}_{\mathrm{ane}}A\beta \mathrm{\Theta }\mathrm{Re}(\mathrm{Thousand})$

Here, τ is the time lag from the start of firing, Θ the adiabatic combustion temperature, while σ and m are combustion parameters. A is the cross-exclusive area of the tube, β the thermal expansion coefficient, and ω the angular frequency. Re(G) indicates the real role of 1000.

The Japan Burner Inquiry Committee developed an evaluation method for applied combustion furnaces based on Madarame'due south theory. First, the natural frequencies and mode shapes are calculated using the finite element method where the chamber system is modeled using i-dimensional wave propagation theory. The energy coefficient Re(G) is calculated numerically. One time the energy coefficient is determined, Eq. (6.4) is used to decide the destabilizing energy ΔE. For practical evaluation it is recommended to employ the growth ratio (negative damping ratio) ζ _h, defined equally:

(6.5) ${\zeta }_h=\mathrm{\Delta }\mathrm{East}/(4\pi E)$

where E is the total energy of the mode considered. If ζ _h>ζ (the damping ratio of the system), the oscillation aamplitude grows. For example, Table 6.one shows results of an evaluation carried out for the mid-sized practical burner system shown in Fig. vi.14. The natural frequency and the growth ratio for each fashion are presented in Table vi.1. A large and positive ζ _h value indicates that oscillations in the corresponding mode are easily generated. The symbol X indicates cases where large oscillations occurred, while Δ indicates depression-aamplitude oscillations. No oscillations were observed in the experiments for the case indicated by O. This makes physical sense since ζ _h is negative, indicating stability.

Table 6.1. Growth ratio for each mode

Mode	Frequency (Hz)	Growth ratio (ζ h)	Stability evaluation
1	xix.eight	−5.02	O
2	45.8	−0.23	O
3	65.viii	1.22	X
4	80.2	0.04	Δ
5	167.2	−2.38	O

Figure six.14. Mid-sized furnace used in experiments.

6.1.2.2.iii Evaluation by feedback theory

Feedback theory [17] can be employed if the transfer functions of the audio-visual system Z and the combustion system A are known. If the loop gain of AZ is larger than unity and the phase angle greater than 180 degrees, that is:

(6.6) $\left|AZ\right|>\mathrm{i}\mathrm{and}\text{arg}(AZ)>180\mathrm{degrees}$

then oscillations will occur. It is important to have the appropriate transfer functions of the heating organisation for this method.

half dozen.i.2.2.4 Eisinger'due south criterion

Eisinger [24] proposed the following criterion for the Sondhauss tube and Rijke tubes shown in Fig. 6.fifteen, based on many field cases:

Figure half-dozen.15. Sondhauss and Rijke tubes.

(6.7) ${(\log \xi )}^2=\mathrm{i.52}(\log \alpha -\log {\alpha }_{\min })$

where ξ=(L−l)/fifty, and α=T _h/T _c. L is the length from the open end to the closed end, and l the altitude from the open up stop to the heat source. T _h and T _c are the temperatures at the heat source and the open end, respectively. The constant α _min=2.14 is the value at the virtually unstable position of the source, which is ξ=ane(l=L/ii). Fig. 6.16 is a plot of Eq. (6.vii). The region in a higher place the curve is unstable. In this criterion, the effect of the mean menses is not considered. The benchmark also does not agree with Madarame's theory. Even so, information technology is practical and easy to use.

Figure vi.xvi. Stability criterion nautical chart.

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Reading Rockets Norm or Criterion Measurement Tool

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