Direction of arrival tracking for signals with known waveforms based on block least squares techniques

In this paper, we address the problem of tracking DOA of multiple moving targets with known signal source waveforms and unknown gains in the presence of Gaussian noise using a nonuniform linear array. Herein, we make use of the fact that the output of each sensor can be described as a linear regression model whose coefficients each contain a pair of DOA and gain information corresponding to one target. These coefficients are determined by solving a linear least squares (LS) problem and then updated recursively based on a block QR decomposition recursive least squares (QRD-RLS) technique or a block regularized LS technique. Since the coefficients from different sensors have the same amplitude but variable phase information for the same signal, along with simple algebraic manipulations the well-known generalized least squares (GLS) are used to obtain an asymptotically-optimal DOA estimate without requiring a search over a large region of the parameter space. Computer simulations show that the proposed DOA tracking techniques when applied to a sparse antenna array can provide a better tracking performance than some of the existing methods do.     

A new look at Akaike’s Bayesian information criterion for inverse ill-posed problems

Akaike’s Bayesian information criterion (ABIC) has been widely used in inverse ill-posed problems. Little has been done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution of measurements for ABIC under the assumption of normal distributions and show that the principle of ABIC is to statistically estimate the variances of measurements and prior data by maximizing the marginal distribution of measurements. The determination of the regularization parameter with ABIC is essentially equivalent to estimating the relative weighting between measurements and prior data. We prove that ABIC theoretically would produce a biased estimate of the variance of measurements. Since the prior mean is generally unknown but arbitrarily treated as zero in inverse ill-posed problems, ABIC is shown to fail to produce any reasonable estimate for the prior variance. Although ABIC is constructed under the Bayesian framework, it essentially plays more or less the same role as biased regularization from the frequentist’s point of view. ABIC error evaluation cannot be performed under the Bayesian framework but should be more appropriately done with the frequentist’s standpoint in terms of mean squared errors. ABIC is sensitive to prior distributions. In the case of non-informative prior distribution, ABIC leads to the conventional weighted least squares (LS) estimate of parameters and cannot be used to solve inverse ill-posed problems. It is not linked to the regularization parameter but only straightforwardly produces an unbiased estimator for the noise level of measurements, which is only applicable numerically for well-posed problems but not for inverse ill-posed problems. Numerical simulated examples are used to demonstrate the statistical performances of ABIC.     

非齐次等式约束线性回归模型回归系数的综合条件岭估计

提出非齐次等式约束线性回归模型回归系数的一个新的有偏估计,即综合条件岭估计,讨论了综合条件岭估计的性质,在一定的条件下,综合条件岭估计的样本总方差、均方误差、均方误差矩阵均分别小于约束最小二乘估计的相应误差.在综合条件岭估计下,条件岭估计和条件根方估计为其特例,从而统一了条件岭估计和条件根方估计的理论.     

The data-filtering based bias compensation recursive least squares identification for multi-input single-output systems with colored noises

For the multi-input single-output (MISO) system corrupted by colored noise, we transform the original system model into a new MISO output error model with white noise through data filtering technology. Based on the newly obtained model and the bias compensation principle, a novel data filtering-based bias compensation recursive least squares (BCRLS) identification algorithm is developed for identifying the parameters of the MISO system with colored noise disturbance. Unlike the exiting BCRLS method for the MISO system (see, in Section 3), without computing the complicated noise correlation functions, still the proposed method can achieve the unbiased parameters estimation of the MISO system in the case of colored process noises. The proposed algorithm simplifies the implementation of and further expands the application scope of the existing BCRLS method. Three numerical examples clearly illustrate the validity of and the good performances of the proposed method, including its superiority over the BCRLS method and so on.     

Feedback characteristic polynomial controller design of 3-D systems in state-space

For a general state-space model of three-dimensional (3-D) systems the characteristic polynomial (eigenvalue) control problem via state and output feedback is considered. A frequency domain approach is employed which in the scalar input case leads to a set of necessary and sufficient conditions. The multi-input problem is treated by assuming that the state or output feedback gain matrix is expressed as the dyadic product ⊙F = ⊙ ⊙f^T of a column vector ⊙β and a row vector ⊙f^T. This assumption leads to an equivalent scalar input problem β which is directly solved by using the scalar input results. Concerning the dynamic feedback compensator design problem, the important particular case of proportional plus integral plus derivative (PID) control is considered and treated by essentially the same algorithm, which leads to a linear algebraic system in the unknown parameters, along with some constraint equations upon the closed-loop characteristic polynomial sought.     

Filtering based parameter estimation for observer canonical state space systems with colored noise

This paper surveys the identification of observer canonical state space systems affected by colored noise. By means of the filtering technique, a filtering based recursive generalized extended least squares algorithm is proposed for enhancing the parameter identification accuracy. To ease the computational burden, the filtered regressive model is separated into two fictitious sub-models, and then a filtering based two-stage recursive generalized extended least squares algorithm is developed on the basis of the hierarchical identification. The stochastic martingale theory is applied to analyze the convergence of the proposed algorithms. An experimental example is provided to validate the proposed algorithms.     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

Identification of errors-in-variables ARX models using modified dynamic iterative PCA

Identification of autoregressive models with exogenous input (ARX) is a classical problem in system identification. This article considers the errors-in-variables (EIV) ARX model identification problem, where input measurements are also corrupted with noise. The recently proposed Dynamic Iterative Principal Components Analysis (DIPCA) technique solves the EIV identification problem but is only applicable to white measurement errors. We propose a novel identification algorithm based on a modified DIPCA approach for identifying the EIV-ARX model for single-input, single-output (SISO) systems where the output measurements are corrupted with coloured noise consistent with the ARX model. Most of the existing methods assume important parameters like input-output orders, delay, or noise-variances to be known. This work’s novelty lies in the joint estimation of error variances, process order, delay, and model parameters. The central idea used to obtain all these parameters in a theoretically rigorous manner is based on transforming the lagged measurements using the appropriate error covariance matrix, which is obtained using estimated error variances and model parameters. Simulation studies on two systems are presented to demonstrate the efficacy of the proposed algorithm.     

Hierarchical recursive least squares parameter estimation of non-uniformly sampled Hammerstein nonlinear systems based on Kalman filter

This paper focuses on parameter estimation problems for non-uniformly sampled Hammerstein nonlinear systems. By combining the lifting technique and state space transformation, we derive a nonlinear regression identification model with different input and output updating rates. Furthermore, the unmeasurable state vector is estimated by Kalman filter, and by using the hierarchical identification principle, we develop a hierarchical recursive least squares algorithm for estimating the unknown parameters of the identification model. Finally, illustrative examples are given to indicate that the proposed algorithm is effective.     

Some properties of a class of biased regression estimators

For the linear statistical model y = Xb + e, X of full column rank estimates of b of the form (C + X′X)⁺X′y are studied, where C commutes with X′X and Q⁺ is the Moore-Penrose inverse of Q. Such estimators may have smaller mean square error, component by component than does the least squares estimator. It is shown that this class of estimators is equivalent to two apparently different classes considered by other authors. It is also shown that there is no C such that (C + X′X)⁺X′Y = My, in which My has the smallest mean square error, component by component. Two criteria, other than tmse, are suggested for selecting C. Each leads to an estimator independent of the unknown b and σ². Subsequently, comparisons are made between estimators in which the C matrices are functions of a parameter k. Finally, it is shown for the no intercept model that standardizing, using a biased estimate for the transformed parameter vector, and retransforming to the original units yields an estimator with larger tmse than the least squares estimator.     

Hierarchical recursive generalized extended least squares estimation algorithms for a class of nonlinear stochastic systems with colored noise

This paper considers the parameter identification problem of a bilinear state space system with colored noise based on its input-output representation. An input-output representation of a bilinear state-space system is derived for the parameter identification by eliminating the state variables in the model, and a recursive generalized extended least squares algorithm is presented for estimating the parameters of the obtained model. Furthermore, a three-stage recursive generalized extended least squares algorithm is proposed for reducing the computational cost. The validity of the proposed method is evaluated through a numerical example.     

Improving the weighted least squares estimation of parameters in errors-in-variables models

Although the weighted least squares (LS) method is straightforward to deal with errors-in-variables (EIV) models, it results in the biased estimates of parameters and the variance of unit weight. The total least squares (TLS) method is statistically rigorous and optimal but is computationally much more demanding. This paper aims at constructing an improved weighted LS estimate of parameters from the perspective of multiplicative error models, which is expected to mainly maintain the advantages of computational simplicity of the weighted LS method and the optimality of the weighted TLS method to some extent but almost remove the bias of the weighted LS estimate and free the computational burden of the TLS method. The statistical aspects of the weighted LS estimate developed from the perspective of multiplicative error models have been analyzed and an almost unbiased estimate of the variance of unit weight proposed. Finally, an N-calibrated weighted LS estimate has been constructed from the perspective of multiplicative error models.     

Auxiliary model based recursive generalized least squares identification algorithm for multivariate output-error autoregressive systems using the decomposition technique

This paper focuses on the parameter estimation problem of multivariate output-error autoregressive systems. Based on the decomposition technique and the auxiliary model identification idea, we derive a decomposition based auxiliary model recursive generalized least squares algorithm. The key is to divide the system into two fictitious subsystems, the one including a parameter vector and the other including a parameter matrix, and to estimate the two subsystems using the recursive least squares method, respectively. Compared with the auxiliary model based recursive generalized least squares algorithm, the proposed algorithm has less computational burden. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithms.     

A parametric approach for eigenvalue assignment by static output feedback

A well-known method of eigenvalue assignment by static output feedback is improved. The main result is a parametric expression for the output feedback controller gain matrix explicitly characterized by the set of non-linear system of equations obtained for the state feedback design and the set of linear equations resulting from static output feedback consideration. In practice, it is shown that all the possible controllers can be generated for exactly assigning the prescribed eigenvalues of the nominal plant by appropriate software for solving the set of non-linear system of equations thus obtained. This in turn makes it possible to select the output feedback matrix with minimum norm or other constraints. Some numerical examples are presented to illustrate the design technique.     

相似性挖掘在时间序列数据中的应用研究

针对时间序列的数据挖掘首先需要将时间序列（Time Series）数据转换为离散的符号序列（Symbol Sequence）。在前人的基础上,将界标模型和分段线性化进行了结合,以关键点作为分段依据,以最大似然函数和最小二乘法来拟合各分段线性拟合函数;此方法的优点在于符合人体生理实验结果,考虑了时间序列中的噪声。     

Variational Bayesian adaptation of process noise covariance matrix in Kalman filtering

Adaptive Kalman filtering with unknown constant or varying process noise covariance matrix is studied. A resolution is proposed to directly estimate or tune the process noise covariance matrix in Kalman filtering using variational Bayesian technique. By state augmentation, conjugacy of the process noise covariance matrix's inverse-Wishart distribution is realized in the estimation at each time instant. The methodological development is given. Illustration examples are presented to demonstrate the improved state filtering performance and the process noise covariance tracking performance of the new method.     

A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation

Mathematical models are basic for designing controller and system identification is the theory and methods for establishing the mathematical models of practical systems. This paper considers the parameter identification for Hammerstein controlled autoregressive systems. Using the key term separation technique to express the system output as a linear combination of the system parameters, the system is decomposed into several subsystems with fewer variables, and then a hierarchical least squares (HLS) algorithm is developed for estimating all parameters involving in the subsystems. The HLS algorithm requires less computation than the recursive least squares algorithm. The computational efficiency comparison and simulation results both confirm the effectiveness of the proposed algorithms.     

Parameter estimation on linear time-varying systems

This paper studies parameter estimation for a class of linear, continuous, time-varying dynamic systems whose state-space model's matrices are affine combinations of static matrix coefficients and the aforementioned time-varying scalar parameters. It is assumed that the coefficient matrices are all known, that the state is mensurable, and that the parameters are bounded piecewise continuous functions of time. Estimation methods are developed from basic equations for a single parameter first, and later extended to multiple parameters.