基于Householder变换的贪婪正交最小二乘辨识算法

针对含有未知时滞的多输入受控自回归系统模型的时滞与参数辨识问题,基于Householder变换探讨一种贪婪正交最小二乘辨识算法.首先,由于各输入通道的时滞未知,通过设置输入数据回归长度对系统模型进行过参数化,得到一个含有稀疏参数向量的高维辨识模型;其次,为了避免最小二乘算法中对高维协方差矩阵的求逆运算,利用Householder变换对信息矩阵进行正交分解,推导基于Householder变换的正交最小二乘算法;然后,为了提高辨识效率,降低辨识成本,推导基于Householder变换的贪婪准则,进而得到基于Householder变换的贪婪正交最小二乘辨识算法,该算法能够在少量采样数据的条件下获得稀疏参数向量的估计值;最后,根据估计的稀疏参数向量的结构得到系统时滞估计.仿真结果表明了所提出算法的有效性.     

时变参数的一种快速跟踪最小二乘估计方法

给出时变参数的一种快速跟踪最小二乘估计方法，该方法对测量数据同时引入指数加权因子和矩形窗，综合了渐消记忆法和限定记忆法的优点，利用抗病态的正交变换法求解估计值，以减小权固子和窗长度，提高了估计值的跟踪速度。     

时变参数的一种快速跟踪最小二乘估计方法

给出时变参数的一种快速跟踪最小二乘估计方法．该方法对测量数据同时引入指数加权因子和矩形窗，综合了渐消记忆法和限定记忆法的优点．利用抗病态的正交变换法求解估计值，以减小权固子和窗长度，提高了估计值的跟踪速度．     

Optimal unambiguous discrimination of pure qudits

Linearly independent pure quantum states can be discriminated unambiguously, while linearly dependent states cannot. We use a physical accessible unitary transformation to map the nonorthogonal quantum states onto a set of orthogonal ones so that measuring the output states can discriminate the initial states with the deterministic and inconclusive results. The failure states that give an inconclusive result are linearly dependent ones. In finding the optimal unambiguous discrimination (UD), we show that a main constraint condition that the determinant constructed by the complex inner products of the failure states must be zero, along with two additional conditions, can provide solutions to the problem of the optimal UD for pure qudits. For any d, we give one analytical solution as all the Berry phases being zero. We also derive the lowest bound of the total failure probability of the optimal UD.     

2 optimal semistable stabilisation for linear discrete-time dynamical systems with applications to network consensus

In this article, we develop ?₂ semistability theory for linear discrete-time dynamical systems. Using this theory, we design ?₂ optimal semistable controllers for linear dynamical systems. Unlike the standard ?₂ optimal control problem, a complicating feature of the ?₂ optimal semistable stabilisation problem is that the closed-loop Lyapunov equation guaranteeing semistability can admit multiple solutions. An interesting feature of the proposed approach, however, is that a least squares solution over all possible semistabilising solutions corresponds to the ?₂ optimal solution. It is shown that this least squares solution can be characterised by a linear matrix inequality minimisation problem. Finally, the proposed framework is used to develop ?₂ optimal semistable controllers for addressing the consensus control problem in networks of dynamic agents.     

Exact analytical solutions for non-selfadjoint variable-topology shape optimization problems: perforated cantilever plates in plane stress subject to displacement constraints. Part I

Lurie (1994, 1995a, b) proved recently that variabletopology shape optimization of perforated plates in flexure for non-selfadjoint problems leads to rank-2 microstructures which are in general nonorthogonal. An extension of the same optimal microstructures to perforated plates in plane stress will be presented in Part II of this study. Using the above microstructure, the optimal solution is derived in this part for cantilever plates in plane stress, which are subject to two displacement constraints. For low volume fractions the above solutions are shown to converge to the known truss solutions of Birkeret al. (1994). The problem of homogenizing the stiffness of nonorthogonal rank-2 microstructures is also discussed.     

A Layer-by-Layer Least Squares based Recurrent Networks Training Algorithm: Stalling and Escape

The limitations of the least squares based training algorithm is dominated by stalling problem and evaluation error by transformation matrix to obtain an unacceptable solution. This paper presents a new approach for the recurrent networks training algorithm based upon the Layer-by-Layer Least Squares based algorithm to overcome the aforementioned problems. In accordance with our proposed algorithm, all the weights are evaluated by the least squares method without the evaluation of transformation matrix to speed up the rate of convergence. A probabilistic mechanism, based upon the modified weights updated equations, is introduced to eliminate the stalling problem experienced by the pure least squares type computation. As a result, the merits of the proposed algorithm are capable of providing an ability of escaping from local minima to obtain a good optimal solution and still maintaining the characteristic of fast convergence.     

Orthogonal vs. uncorrelated least squares discriminant analysis for feature extraction

In this paper, a new discriminant analysis for feature extraction is derived from the perspective of least squares regression. To obtain great discriminative power between classes, all the data points in each class are expected to be regressed to a single vector, and the basic task is to find a transformation matrix such that the squared regression error is minimized. To this end, two least squares discriminant analysis methods are developed under the orthogonal or the uncorrelated constraint. We show that the orthogonal least squares discriminant analysis is an extension to the null space linear discriminant analysis, and the uncorrelated least squares discriminant analysis is exactly equivalent to the traditional linear discriminant analysis. Comparative experiments show that the orthogonal one is more preferable for real world applications.     

A Jacobi-type method for computing balanced realizations

A new numerical scheme for computing balancing coordinate transformations in linear systems theory is presented. The method is closely related to the Jacobi method for diagonalizing symmetric matrices. Here the minimization of the sum of traces of the Gramians by orthogonal and nonorthogonal Jacobi-type rotations is considered. The algorithm is shown to be globally convergent to a balancing transformation that arranges the Hankel singular values in a prescribed ordering. Local quadratic convergence of the algorithm is shown.     

On the efficiency of the orthogonal least squares training methodfor radial basis function networks

The efficiency of the orthogonal least squares (OLS) method for training approximation networks is examined using the criterion of energy compaction. We show that the selection of basis vectors produced by the procedure is not the most compact when the approximation is performed using a nonorthogonal basis. Hence, the algorithm does not produce the smallest possible networks for a given approximation error. Specific examples are given using the Gaussian radial basis functions type of approximation networks.     

Genetic evolution of radial basis function coverage usingorthogonal niches

A well-performing set of radial basis functions (RBFs) can emerge from genetic competition among individual RBFs. Genetic selection of the individual RBFs is based on credit sharing which localizes competition within orthogonal niches. These orthogonal niches are derived using singular value decomposition and are used to apportion credit for the overall performance of the RBF network among individual nonorthogonal RBFs. Niche-based credit apportionment facilitates competition to fill each niche and hence to cover the training data. The resulting genetic algorithm yields RBF networks with better prediction performance on the Mackey-Glass chaotic time series than RBF networks produced by the orthogonal least squares method and by k-means clustering.     

Data filtering based least squares algorithms for multivariable CARAR-like systems

This paper focuses on the identification problem of multivariable controlled autoregressive autoregressive (CARAR-like) systems. The corresponding identification model contains a parameter vector and a parameter matrix, and thus the conventional least squares methods cannot be applied to directly estimate the parameters of the systems. By using the hierarchical identification principle, this paper presents a hierarchical generalized least squares algorithm and a filtering based hierarchical least squares algorithm for the multivariable CARAR-like systems. The simulation results show that the two hierarchical least squares algorithms are effective.     

Improvement of grey models by least squares

In this paper, we present an approach to the least squares solution to grey Verhulst model, and verify its feasibility by numerical examples. We also present the least squares solutions of grey models GM (1, 1) and GM (2, 1). For the convenience of applications in expert systems, the parameters computing formulas of grey models are also presented here. We carry out some numerical examples to examine the modeling precision of grey models in conventional way and in least squares. The numerical results reveal that the modeling precision of grey models in least squares is always better than that in conventional way.     

Supervised fuzzy clustering for rule extraction

This paper deals with the application of orthogonal transforms and fuzzy clustering to extract fuzzy rules from data. It is proposed to use the orthogonal least squares method to supervise the progress of the fuzzy clustering algorithm and remove clusters of less importance with respect to describing the data. Clustering takes place in the product space of systems inputs and outputs and each cluster corresponds to a fuzzy IF-THEN rule. By initializing the clustering with an overestimated number of clusters and subsequently remove less important ones as the clustering progresses, it is sought to obtain a suitable partition of the data in an automated manner. The approach is generally applicable to the fuzzy c-means and related algorithms. The adaptive distance norm fuzzy clustering is studied and applied to the identification of Takagi-Sugeno type rules. Both a synthetic example as well as a real-world modeling problem are considered to illustrate the working and the applicability of the algorithm     

Identification of multi-class linear and nonlinear systems

An algorithm for the identification of multi-class systems which can be described by a class of models over different operating regions is presented. The algorithm involves partitioning the raw data set using discriminant functions followed by parameter estimation. An orthogonal least squares algorithm coupled with a backward elimination procedure is employed for the parameter estimation and data partitioning processes. Provided the data elements are linearly separable, the proposed algorithm will correctly partition the data into the respective classes; parameter estimation algorithms can then be applied to estimate the models associated with each different class. Simulation studies are included to illustrate the algorithm     

A Fast Nonlinear Model Identification Method

The identification of nonlinear dynamic systems using linear-in-the-parameters models is studied. A fast recursive algorithm (FRA) is proposed to select both the model structure and to estimate the model parameters. Unlike orthogonal least squares (OLS) method, FRA solves the least-squares problem recursively over the model order without requiring matrix decomposition. The computational complexity of both algorithms is analyzed, along with their numerical stability. The new method is shown to require much less computational effort and is also numerically more stable than OLS.     

State estimation for stochastic time‐varying multisensor systems with multiplicative noises: Centralized and decentralized data fusion

In this paper, the state estimation problems, including filtering and one‐step prediction, are solved for uncertain stochastic time‐varying multisensor systems by using centralized and decentralized data fusion methods. Uncertainties are considered in all parts of the state space model as multiplicative noises. For the first time, both centralized and decentralized estimators are designed based on the regularized least‐squares method. To design the proposed centralized fusion estimator, observation equations are first rewritten as a stacked observation. Then, an optimal estimator is obtained from a regularized least‐squares problem. In addition, for decentralized data fusion, first, optimal local estimators are designed, and then fusion rule is achieved by solving a least‐squares problem. Two recursive equations are also obtained to compute the unknown covariance matrices of the filtering and prediction errors. Finally, a three‐sensor target‐tracking system is employed to demonstrate the effectiveness and performance of the proposed estimation approaches.     

基于辅助模型和数据滤波的伪线性回归系统参数估计方法

针对伪线性输出误差回归系统的辨识模型新息信息向量存在不可测变量的问题,首先通过构造一个辅助模型,用辅助模型的输出代替未知中间变量,推导得到的基于辅助模型的递推最小二乘参数估计算法计算量较大,但算法的辨识效果不佳。进一步采用估计的噪声模型对系统观测数据进行滤波,使用滤波后的数据进行参数估计,从而推导提出了基于数据滤波的递推最小二乘参数估计算法。仿真结果表明,所提算法能够有效估计伪线性回归线性输出误差系统的参数。     

Identification and reconstruction of chaotic systems using multiresolution wavelet decompositions

A new modelling framework for identifying and reconstructing chaotic systems is developed based on multiresolution wavelet decompositions. Qualitative model validation is used to compare the multiresolution wavelet models and it is shown that the dynamical features of chaotic systems can be captured by the identified models providing the wavelet basis functions are properly selected. Two basis selection algorithms, orthogonal least squares and a new matching pursuit orthogonal least squares, are considered and compared. Several examples are included to illustrate the results.     

Iterative algorithm for minimal norm least squares solution to general linear matrix equations

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory.