Parametric Solutions to the Generalized Discrete Yakubovich‐Transpose Matrix Equation

This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich‐transpose matrix equation X − AX^TB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich‐transpose matrix equation, which is called the Karm‐Yakubovich‐transpose matrix equation, is considered. The explicit solutions to the Karm‐Yakubovich‐transpose matrix equation are also presented by the so‐called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.     

Solutions to the matrix equation

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BY, with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B), generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.     

A symmetric preserving iterative method for generalized Sylvester equation

The generalized Sylvester matrix equation AX + YB = C is encountered in many systems and control applications, and also has several applications relating to the problem of image restoration, and the numerical solution of implicit ordinary differential equations. In this paper, we construct a symmetric preserving iterative method, basing on the classic Conjugate Gradient Least Squares (CGLS) method, for AX + YB = C with the unknown matrices X, Y having symmetric structures. With this method, for any arbitrary initial symmetric matrix pair, a desired solution can be obtained within finitely iterate steps. The unique optimal (least norm) solution can also be obtained by choosing a special kind of initial matrix. We also consider the matrix nearness problem. Some numerical results confirm the efficiency of these algorithms. It is more important that some numerical stability analysis on the matrix nearness problem is given combined with numerical examples, which is not given in the earlier papers. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Unsupervised image segmentation using triplet Markov fields

Hidden Markov fields (HMF) models are widely applied to various problems arising in image processing. In these models, the hidden process of interest X is a Markov field and must be estimated from its observable noisy version Y. The success of HMF is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed one remains Markovian, which facilitates different processing strategies such as Bayesian restoration. HMF have been recently generalized to “pairwise” Markov fields (PMF), which offer similar processing advantages and superior modeling capabilities. In PMF one directly assumes the Markovianity of the pair (X, Y). Afterwards, “triplet” Markov fields (TMF), in which the distribution of the pair (X, Y) is the marginal distribution of a Markov field (X, U, Y), where U is an auxiliary process, have been proposed and still allow restoration processing. The aim of this paper is to propose a new parameter estimation method adapted to TMF, and to study the corresponding unsupervised image segmentation methods. The latter are validated via experiments and real image processing.     

Generalized Fourier series for non-linear systems: application to the study of limit cycles in second-order approximation

The approximate solution of non-linear differential equations is studied to second-order using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Liénard's ordinary differential equation, [Xdot] + g(X) = f(X)[Xdot] are presented. In the generalized van der Pol equation with f(X)= ε(1 ? X²) and g(X) = AX + 2BX₃ a very good second-order approximation is given that depends on the values of A/B and ε.     

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

The inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ² + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.     

PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.     

Fixed rank solutions of the matrix equation with statistical applications

By applying the canonical correlation decomposition of matrix pairs, the general fixed rank least square solutions of matrix equation Xβ=Y are derived. As statistical applications, an algorithm for computing the least square estimator of the multivariate reduced rank regression model Y=Xβ+?, r(β)=t is given.     

An algorithm for solving the matrix equation X = FXF T + S

A now algorithm for solving the matrix equation X = FXF ^T + S, which is important in the control system design, is presented in this paper. The algorithm is based on the QR algorithm for finding the eigenvalues of a matrix and works efficiently for large dimensional problems. A simple example is given to illustrate the algorithm. The method is also applicable to other types of equations such as the Lyapunov equation A ^T X + XA + B = 0.     

Improved diagnosis of lossy resonator bandpass filters using Y‐parameters

This article presents a generalized coupling matrix (CM) extraction technique directly from the measured or electromagnetic simulated admittance parameter (Y‐parameters) of a narrow band cross‐coupled resonator bandpass filter with losses. The characteristic polynomials corresponding to the Y‐parameters rather than the S‐parameters are derived and then determined by the Cauchy method. Once determining the characteristic polynomials, the complex poles and residues of Y‐parameters are also determined. Using these complex poles and residues, the complex CM can be obtained by a sequence of similarity transformations, and then the desired extracted CM and unloaded Q for each resonator are obtained from this complex CM. To verify the effectiveness of the proposed method, three filter CM extraction examples are demonstrated. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:807–814, 2015.     

On risk concentration for convex combinations of linear estimators

We consider the estimation problem for an unknown vector β ∈ Rp in a linear model Y = Xβ + σξ, where ξ ∈ Rⁿ is a standard discrete white Gaussian noise and X is a known n × p matrix with n ≥ p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate β in this situation, we use a family of spectral regularizations of the maximum likelihood method β^α(Y) = H ^α(X ^T X) β ^?(Y), α ∈ R⁺, where β ^?(Y) is the maximum likelihood estimate for β and {H ^α(·): R⁺ → [0, 1], α ∈ R⁺} is a given ordered family of functions indexed by a regularization parameter α. The final estimate for β is constructed as a convex combination (in α) of the estimates β^α(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method.     

Composite State Feedback of the Finite Frequency H∞ Control for Discrete‐Time Singularly Perturbed Systems

This paper mainly is concerned with the finite frequency H_∞ control for the discrete‐time singularly perturbed systems. A state feedback controller is designed to stabilize the whole system and to satisfy the desired design specifications. The generalized Kalman–Yakubovich–Popov (GKYP) lemma is used to convert the related frequency domain inequalities in finite frequency ranges to feasible linear matrix inequalities. Based on the Lyapunov stability method, stable conditions are obtained for discrete‐time singularly perturbed systems. A bounded real lemma then is derived, which characterizes the H_∞ norm performance in specific frequency ranges. Furthermore, the approach for the design of a composite state feedback controller is put forward combined with the unique frequency characteristics of singularly perturbed systems. Detailed analysis of the performance achieved by the piecewise composite controller is provided when it is applied to the original system, and the effectiveness and merits of the proposed controller are illustrated with a numerical result.     

A wide view auto‐stereoscopic 3D display with an eye‐tracking system for enhanced horizontal viewing position and viewing distance

This paper describes the development of auto‐stereoscopic three‐dimensional (3D) display with an eye‐tracking system for not only the X‐axis (right–left) and Y‐axis (up–down) plane directions but also the Z‐axis (forward–backward) direction. In the past, the eye‐tracking 3D system for the XY‐axes plane directions that we had developed had a narrow 3D viewing space in the Z‐axis direction because of occurrence of 3D crosstalk variation on screen. The 3D crosstalk variation on screen was occurred when the viewer's eye position moved back and forth along the Z‐axis direction. The reason was that the liquid crystal (LC) barrier pitch was fixed and the LC barrier was able to control the only barrier aperture position. To solve this problem, we developed the LC barrier that is able to control the barrier pitch as well as the barrier aperture position in real time, corresponding to the viewer's eye position. As a result, the 3D viewing space has achieved to expand up to 320–1016 mm from the display surface in the Z‐axis direction and within a range of ±267 mm in the X‐axis direction. In terms of the Y‐axis direction, the viewing space is not necessary to be considered, because of a stripe‐shaped parallax barrier.     

Morphogenetic approach to system identification

In this paper, we propose a novel approach to system identification based on morphogenetic theory (MT). Given a context H defined by a set of M objects, each described by a set of N attributes, and a vector X of desired outputs for each object, MT combines notions from formal concept analysis and tensor calculus so as to generate a morphogenetic system (MS). The MS is defined by a set of weights s¹, …, s^N, one for each attribute. Given H and X, weights are computed so as to generate the projection Y of X on the space of the attributes with the minimum distance between Y and X. An MS can be represented as a neuron, morphogenetic neuron, with a number of synapses equal to the number of attributes and synaptic weights equal to s¹, …, s^N. Unlike traditional neural network paradigm, which adopts an iterative process to determine synaptic weights, in MT, weights are computed at once. We introduce a method to generate a morphogenetic neural network (MNN) for identification problems. The method is based on extending appropriately and iteratively the attribute space so as to reduce the error between desired output and computed output. By using four well‐known datasets, we show that an MNN can identify an unknown system with a precision comparable with classical multilayer perceptron with complexity similar to the MNN but reducing drastically the time needed to generate the neural network. Furthermore, the structure of the MNN is generated automatically by the method and does not require a trial‐and‐error approach often applied in classical neural networks. © 2009 Wiley Periodicals, Inc.     

Fault‐Tolerant Finite Frequency H∞ Control for Uncertain Mechanical System with Input Delay and Constraint

In this paper, a novel fault‐tolerant finite frequency H_∞ controller (FFHC) is developed for uncertain mechanical system with input delay and constraint. First, the mathematical model of uncertain mechanical system is derived, where the uncertainties occur in mass, damping and stiffness matrices, respectively. Then, in view of the fact that the dominant resonance energies are caused by low‐order vibration modes of mechanical system, the finite frequency control algorithm is investigated to suppress these low‐order resonances peaks. By virtue of Lyapunov‐Krasovskii functional (LKF) and generalized Kalman‐Yakubovich‐Popov (GKYP) lemma, the desirable fault‐tolerant controller can be obtained by convex optimization. Numerical simulations verify the improvements and advantages of proposed cotroller in disturbance rejection when compared with the classic entire frequency H_∞ controller (EFHC).     

Cooperative downconversion in Y3Al5O12 : RE3+,Yb3+ (RE = Tb,Tm, and Pr) nanophosphors

Abstract— Efficient near‐infrared (NIR) quantum cutting (QC) through cooperative downconversion in Y₃Al₅O₁₂ : RE³⁺,Yb³⁺ (RE = Tb, Tm, and Pr) nanophosphors has been demonstrated, which involves the conversion of the visible photon from RE³⁺ into the NIR emission of Yb³⁺ with the optimal quantum efficiency approaching 200%. The authors have analyzed the measured luminescence spectra and decay lifetimes and propose a mechanism to rationalize the NIR QC effect. The results indicate the potential of developing RE³⁺‐Yb³ dual‐ion‐based nanophosphors for the downconversion of high‐energy photons to multiple photons with an energy greater than the bandgap of silicon‐based‐photonics display devices.     

Rare‐earth materials for use in the dark

Abstract— Recently, it was found that some materials doped with rare‐earth ions show bright and long‐lasting phosphorescence. They do not include radioactive elements and can be safely used as luminous paints for use in the dark. Some of them are better than the traditional zinc sulfide doped with copper (ZnS:Cu). The most important rare‐earth materials with long‐lasting phosphorescence are aluminates such as alkaline‐earth aluminates MAl₂O₄:Eu²⁺, Dy³⁺ (M = Sr, Ca) and garnets Y₃Ga₅O₁₂:Tb³⁺, Gd₃Ga₅O₁₂:Tb³⁺, Cd₃Al₂Ge₃O₁₂:Tb³⁺, Cd₃M₂Ge₃O₁₂:Pr³⁺ (M = Al, Ge), Y₃Al_5?xGa_xO₁₂:Ce³⁺ (x = 3, 3.5). Some oxides such as InBO₃:Tb³⁺, Ba₂SiO₄:Dy³⁺ also show long‐lasting phosphorescence properties. Other sulfide materials include ZnS:Eu, Ca_xSr_1?x S:Bi, Tm, Cu or Ca_xSr_1?xS:Eu. Alkaline‐earth aluminates MAl₂O₄:Eu²⁺ (M = Mg, Ca, Sr, Ba) codoped with RE³⁺ (RE = Y, La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) were synthesized by using homogeneous precipitation method.     

Fault estimation observer design for a class of nonlinear multiagent systems in finite‐frequency domain

This paper focuses on the fault estimation observer design problem in the finite‐frequency domain for a class of Lipschitz nonlinear multiagent systems subject to system components or actuator fault. First, the relative output estimation error is defined based on the directed communication topology of multiagent systems, and an observer error system is obtained by connecting adaptive fault estimation observer and the state equation of the original system. Then, sufficient conditions for the existence of the fault estimation observer are obtained by using a generalized Kalman‐Yakubovich‐Popov lemma and properties of the matrix trace, which guarantee that the observer error system satisfies robustness performance in the finite‐frequency domain. Meanwhile, the pole assignment method is used to configure the poles of the observer error system in a certain area. Finally, the simulation results are presented to illustrate the effectiveness of the proposed method.