Modeling and analysis of singular systems via orthogonal triangular functions

This paper presents a new approach to singular system analysis by modeling the system in terms of orthogonal triangular functions (TFs). The proposed method is more accurate compared to block pulse function-based analysis with respect to mean integral square error (MISE). A numerical example involving four states of a singular system is treated and solutions obtained thereof. Four tables and relevant curves are presented to compare the respective coefficients in block pulse function (BPF) domain as well as in TF domain. The percentage error of the samples determined via TF domain are compared with the exact samples of the states. Furthermore, MISE for both BPF and TF analysis are computed and compared to reveal the efficiency of TF-based analysis.     

Optimal control of singular systems via orthogonal functions

In this paper two new recursive algorithms are presented for computing optimal control law of linear time-invariant singular systems with quadratic performance index by using the elegant properties of block-pulse functions (BPFs) and shifted Legendre polynomials (SLPs). Also a unified approach is given to solve the optimal control problem of singular systems via BPFs or SLPs. Two numerical examples are included to demonstrate the validity of the proposed algorithms and approach.     

Sensitivity analysis of linear systems using orthogonal functions

Some useful properties of coefficients for determining the sensitivity functions of linear systems are given. These properties simplify the numerical computations of sensitivity functions with respect to the system parameters.     

Semi-supervised orthogonal discriminant analysis via label propagation

Trace ratio is a natural criterion in discriminant analysis as it directly connects to the Euclidean distances between training data points. This criterion is re-analyzed in this paper and a fast algorithm is developed to find the global optimum for the orthogonal constrained trace ratio problem. Based on this problem, we propose a novel semi-supervised orthogonal discriminant analysis via label propagation. Differing from the existing semi-supervised dimensionality reduction algorithms, our algorithm propagates the label information from the labeled data to the unlabeled data through a specially designed label propagation, and thus the distribution of the unlabeled data can be explored more effectively to learn a better subspace. Extensive experiments on toy examples and real-world applications verify the effectiveness of our algorithm, and demonstrate much improvement over the state-of-the-art algorithms.     

Representation of dynamic output feedback control system variables via orthogonal functions

A general basis for using orthogonal functions to approximate the variables of a dynamic output feedback control system is presented. The use of an operational matrix of integration reduces the problem to a set of linear algebraic equations. The result is also useful for implementing open-loop approximation of derived closed-loop control. The flexible controller structure provides easy adaptation to feedback configurations using state, output, dynamic output or observer output, and feedback compensator. As an example, an approximation analysis of dynamic output feedback control via Fourier series is investigated.     

Numerical solutions of time-varying TS-fuzzy-model-based time-delay dynamic equations via orthogonal functions

In this paper, the approach of orthogonal functions is presented to solve the time-varying Takagi–Sugeno (TS) fuzzy-model-based time-delay dynamic equations (time-varying TSFMTDE). The new method simplifies the procedure of solving the time-varying TSFMTDE into the successive solution of a system of recursive formulae only involving matrix algebra. Based on the presented recursive formulae, an algorithm only including straightforward algebraic computation is also proposed in this paper. The new proposed approach is non-iterative, non-differential, non-integral, straightforward, and well-adapted to computer implementation. Hence, the computational complexity is considerably reduced. The first illustrated example shows that the proposed method, based on the orthogonal functions, can obtain satisfactory results. The second illustrated example, for the pendulum time-delay system with the vibration in the vertical direction on the pivot point having a fuzzy parallel-distributed-compensation controller, is given to demonstrate the application of the proposed approach.     

基于三角函数的正交矩函数的误差预测与分析

提取图像中旋转不变特征是图像处理和模式识别中重要的应用。在极坐标下的正交矩函数则是提取这种特征信息的主要方法。正交矩函数在图像分解和重建过程中的误差是衡量其特征提取精确度的标准。为了提高正交矩函数在图像重建中的性能,提出了一种新的基于三角函数的正交矩函数和一种基于函数误差分析的新的衡量方法,并分别应用新的衡量方法和传统的在大量图像中进行重建误差统计的方法对新的正交矩函数以及另外两种在特征提取方面表现最好的正交矩函数进行了比较。实验结果验证了新的衡量方法的有效性并得到了新的正交矩函数的重建效果更好的结论。     

State estimation using continuous orthogonal functions

A new approach is presented to facilitate research in the state estimation of linear systems using continuous orthogonal functions. The principle of the Luenberger observer is utilized in developing simple algebraic expressions for the estimates of states. This approach has the distinct advantage that the smoothing effect of integration reduces the effect of noise. Hence, this observer gives acceptable estimates of the states in the presence of zero-mean observation noise, even without a filter. Results of simulation indicate that the proposed method works quite well. In addition, the algorithms are recursive and suitable for on-line implementation.     

Analysis of singular systems using orthogonal functions

The use of orthogonal functions to analyze singular systems is investigated. It is shown that the differential-algebraic system equation may be converted to an algebraic generalized Lyapunov equation that can be solved for the coefficients ofx(t)in terms of the orthogonal basis functions. This generalized Lyapunov equation may be considered as a "discrete" equation on the slow subspace of the system, and as a "continuous" equation on its fast subspace. Necessary and sufficient conditions for the existence of a unique solution are given in terms of the relative spectrum of the system. A generalized Bartels/Stewart algorithm based on theQZalgorithm is presented for its efficient solution. Relations are drawn with the invariant subspaces of the system.     

Some sets of orthogonal polynomial kernel functions

Kernel methods provide high performance in a variety of machine learning tasks. However, the success of kernel methods is heavily dependent on the selection of the right kernel function and proper setting of its parameters. Several sets of kernel functions based on orthogonal polynomials have been proposed recently. Besides their good performance in the error rate, these kernel functions have only one parameter chosen from a small set of integers, and it facilitates kernel selection greatly. Two sets of orthogonal polynomial kernel functions, namely the triangularly modified Chebyshev kernels and the triangularly modified Legendre kernels, are proposed in this study. Furthermore, we compare the construction methods of some orthogonal polynomial kernels and highlight the similarities and differences among them. Experiments on 32 data sets are performed for better illustration and comparison of these kernel functions in classification and regression scenarios. In general, there is difference among these orthogonal polynomial kernels in terms of accuracy, and most orthogonal polynomial kernels can match the commonly used kernels, such as the polynomial kernel, the Gaussian kernel and the wavelet kernel. Compared with these universal kernels, the orthogonal polynomial kernels each have a unique easily optimized parameter, and they store statistically significantly less support vectors in support vector classification. New presented kernels can obtain better generalization performance both for classification tasks and regression tasks.     

Prolongations and stability analysis via lyapunov functions of dynamical polysystems

The purpose of this paper is to explore asymptotic stability properties of dynamical polysystems in the vicinity of an arbitrary set by means of suitably defined Lyapunov functions. Both necessary and sufficient conditions are established using higher-order prolongations and transitizing extensions of the reachable map. For smooth systems in particular, the main sufficient conditions derived can be tested without prior knowledge of the system trajectories.     

Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.     

Properties of the magnitude terms of orthogonal scaling functions

The spectrum of the convolution of two continuous functions can be determined as the continuous Fourier transform of the cross-correlation function. The same can be said about the spectrum of the convolution of two infinite discrete sequences, which can be determined as the discrete time Fourier transform of the cross-correlation function of the two sequences. In current digital signal processing, the spectrum of the continuous Fourier transform and the discrete time Fourier transform are approximately determined by numerical integration or by densely taking the discrete Fourier transform. It has been shown that all three transforms share many analogous properties. In this paper we will show another useful property of determining the spectrum terms of the convolution of two finite length sequences by determining the discrete Fourier transform of the modified cross-correlation function. In addition, two properties of the magnitude terms of orthogonal wavelet scaling functions are developed. These properties are used as constraints for an exhaustive search to determine a robust lower bound on conjoint localization of orthogonal scaling functions.     

Computing orthogonal rational functions with poles near the boundary

Computing orthogonal rational functions is a far from trivial problem, especially for poles close to the boundary of the support of the orthogonality measure. In this paper we analyze some of the difficulties involved and present two different approaches for solving this problem.     

Solution of variational problems via general orthogonal polynomials

The general orthogonal polynomials approximation is employed to solve variational problems. The operational matrix of integration is applied to reduce an integral equation to an algebraic equation with expansion coefficients. A simple and straightforward algorithm is then developed to calculate the expansion coefficients of the general orthogonal polynomials. The proposed method is general and various classical orthogonal polynomial approximations of the same problem can be obtained as a special case of the derived results.     

Solution of integral equations via generalized orthogonal polynomials

A new approximation method using a generalized orthogonal polynomial (GOP) is employed for solving integral equations. The integration operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomial, is developed. The dependent variables in the integral equation are assumed to be expressed by a GOP series. A set of algebraic equations is obtained from the integral equation. The calculation of coefficients is straightforward and easy. Examples are given, and the results obtained from individual orthogonal polynomial approximations are compared with each other. It is found that nearly all individual orthogonal polynomials, except Hermite polynomials, offer excellent results.     

Discrete pulse orthogonal functions for the analysis,parameter estimation and optimal control of linear digital systems

Discrete pulse orthogonal functions (DPOFs) are introduced, and their operational matrix is proposed to solve various control problems. The applications of DPOFs in digital control systems are analogous to those of block pulse functions (BPFs) in continuous systems. Applying the DPOFs, the approximate solutions of digital time-invariant systems can be easily obtained by a convenient algorithm. Three examples are presented to demonstrate the applications of DPOFs.     

Regularized orthogonal linear discriminant analysis

In this paper the regularized orthogonal linear discriminant analysis (ROLDA) is studied. The major issue of the regularized linear discriminant analysis is to choose an appropriate regularization parameter. In existing regularized linear discriminant analysis methods, they all select the “best” regularization parameter from a given parameter candidate set by using cross-validation for classification. An obvious limitation of such regularized linear discriminant analysis methods is that it is not clear how to choose an appropriate candidate set. Therefore, up to now, there is no concrete mathematical theory available in selecting an appropriate regularization parameter in practical applications of the regularized linear discriminant analysis. The present work is to fill this gap. Here we derive the mathematical relationship between orthogonal linear discriminant analysis and the regularized orthogonal linear discriminant analysis first, and then by means of this relationship we find a mathematical criterion for selecting the regularization parameter in ROLDA and consequently we develop a new regularized orthogonal linear discriminant analysis method, in which no candidate set of regularization parameter is needed. The effectiveness of our proposed regularized orthogonal linear discriminant analysis is illustrated by some real-world data sets.