Matrix Quadratic Form Research Articles

Quasi-Degradation Probability of Two-User NOMA Over Rician Fading Channels

Non-orthogonal multiple access (NOMA) has a great potential to offer a higher spectral efficiency of multi-user wireless networks than orthogonal multiples access (OMA). Previous work has established the condition, referred to quasi-degradation (QD) probability, under which NOMA has no performance loss compared to the capacity-achieving dirty paper coding for the two-user case. Existing results assume Rayleigh fading channels without line-of-sight (LOS). In many practical scenarios, the channel LOS component is critical to the link quality where the channel gain follows a Rician distribution instead of a Rayleigh distribution. In this work, we analyze the QD probability over multi-input and single-output (MISO) channels subject to Rician fading. The QD probability heavily depends on the angle between two user channels, which involves a matrix quadratic form in random vectors and a stochastic matrix. With the deterministic LOS component, the distribution of the matrix quadratic form is non-central that dramatically complicates the derivation of the QD probability. To remedy this difficulty, a series of approximations is proposed that yields a closed-form expression for the QD probability over MISO Rician channels. Numerical results are presented to assess the analysis accuracy and get insights into the optimality of NOMA over Rician fading channels.

Electric Vehicle Battery SOC Estimation based on GNL Model Adaptive Kalman Filter

With the efficient development of the electric vehicle, it is urgent to recycle and utilize the decommissioned power battery which increases rapidly in quantity year by year. Accurate and reliable state of charge (SOC) estimation of the battery is the key technology to realize the battery cascade utilization. The traditional estimation methods do not take the self-discharge factors into account which affect the aging battery to a great extent. This research adopts the GNL circuit equivalent model, which considers the self-discharge factor and discretizes its state space equation by matrix quadratic form. The adaptive unscented Kalman filter algorithm (AUKF) is used to estimate and update the SOC in time. The experimental comparison verifies the effectiveness of AUKF for aging batteries. The results show that the proposed method can obtain less error of the state estimated value and fast following features which meets the actual demand of SOC estimation.

A study for the moment of Wishart distribution

The skewness of a matrix quadratic form XX 0 is obtained using the expectation of stochastic matrix and applying the properties of commutation matrices, where X Np;n (0; ;In).

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X(⁎)AX, when A=A(⁎) is a positive semidefinite matrix, X has a matrix multivariate elliptical distribution and X(⁎) denotes the usual conjugate transpose of X. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.

Parity Space-Based Fault Estimation for Linear Discrete Time-Varying Systems

This note is concerned with the problem of parity space-based fault estimation for linear discrete time-varying systems. The design of parity space-based fault estimator is formulated as to find a minimum for a matrix quadratic form. It is shown that the minimum provides a unified solution to the fault estimation problem with different design criteria, such as the performance indices of maximum singular value and Frobenius-norm. A necessary and sufficient condition for the minimum is derived and an analytic solution to the parity space-based fault estimator is obtained. Numerical examples are given to show the effectiveness of the proposed method.

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n × p multivariate normal random matrix with general covariance Σ Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y ′ WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.

Robust balancing for rotating machines

The balancing of rotors divides broadly into two categories: balancing in situ and balancing in a balancing machine. In the latter case, the most common practice is to arrange balance corrections on the rotor such that the net excitations of each of the four in-plane rigid-body modes of the free rotor is zero by deploying balance corrections on two independent planes. In a small proportion of cases, the net excitations of the first pair of flexural modes are also zeroed using a third correction plane. This paper proposes that, when rotors are balanced in a balancing machine (not similar to the machine stator), substantially more utility can be gained from the balancing operation by combining a suitably weighted account of the specific balancing requirements of the machine with knowledge of the expected machine characteristics than can be achieved by ignoring this knowledge. A single cost function is established based on a numerical model of the machine. Then, depending on circumstances, either the expected value of this cost function or its worst possible value can be minimized. The methods proposed require that relatively detailed knowledge of the distribution of residual unbalance be obtained experimentally. The paper briefly discusses some practical methods for how such information might be extracted. The definition of the cost function as a matrix quadratic form provides potentially valuable information about the necessary number and the optimal location of balance planes on a given rotor, and methods for determining an optimal set of balance planes are outlined.

The N-th moment of matrix quadratic form

We have obtained a formula for vec( ⊗ i = 1 N B i ). Using this formula and the 2 Nth moment of the matrix normal distribution, we obtain the Nth moment of matrix quadratic form. Furthermore, we have obtained the kurtosis of Wishart distribution.

Convergence analysis of binary relation inference networks

Two methods for studying the consistency problems of a class of binary relation inference networks are described. One method is derived using the mathematical concepts of energy function (Et) and delta energy function (ΔEt), where both functions have closely related geometrical interpretations. By properly formulating ΔEt as matrix quadratic form, network convergence is shown to be directly related to the matrix property of negative semidefiniteness. The other method, which can be applied in either a discrete-time or continuous-time framework, is based on studying the eigenvalue problem for an associated state-space model of the inference network. The merits and limitations of the proposed methods are discussed, with reference to several specific examples.

Third moment of matrix quadratic form

The third moment of a matrix quadratic form is obtained using the sixth moment of the matrix normal distribution and applying the properties of commutation matrices and Kronecker products.

Spline-Based Adaptive Control of Distributed Parameter Systems

In this paper spline functions are used for designing an alternative - more practical - approach to modeling and to control of a class of continuous-time linear distributed parameter systems. The key step or submitted approach is the simultaneous spline approximation of both the signals and of the kernels of integral operator describing the system. This approximation under acceptable assumptions permits to create simple, finite-dimensi onal and parametric models easily lmplementabl e for adaptive control purposes. The control synthesis is based on the Integral continuous-time and continuous-space loss functional. Choosing suitable spline approximation the functional can be converted to the matrix quadratic form. The minimization is then performed via quadratic programming met hod. The control signals are continuous-time functions and are generated by the controller in the form of polynomial splines of defined dogree. The restrictions on amplitude and speed changes of the signals are introduced. The results of simulation experiments are given.

The variance matrix of a matrix quadratic form %81¡ under normality assumptions

In order to obtain the first and second moments of a matrix quadratic form under normality assumptions its moment generating function will be derived and then differentiated. Use is being made of matrix differential calculus as developed by the author

On the dispersion matrix of a matrix quadratic form connected with the noncentral Wishart distribution

Recently Magnus and Neudecker [3] derived the dispersion matrix of vec X′ X, when X′ is a p × n random matrix ( n > p) and vec X′ has the distribution N np(vec M′, I n ⊗ V) . This note is concerned with the matrix quadratic form X′ AX, where X′ is a defined above and A is a nonrandom (not necessarily symmetric) matrix. The dispersion matrix of vec X′ AX is then derived by applying results of Magnus and Neudecker [3] and Neudecker and Wansbeek [4]. This generalizes an earlier result of Giguère and Styan [2] which assumes a symmetric A.

A computational procedure to stabilize nonlinear sampled-data systems

In this paper, a sufficient condition for the stability of a class of nonlinear sampled-data systems, expressed in a matrix quadratic form, is derived. A programmed computational algorithm is constructed to implement the stabilization scheme. Application to third-order control problem is presented in order to demonstrate the developed procedure.

Some Matrix Results and Extensions of Cochran’s Theorem

This paper gives some matrix results and some necessary and sufficient conditions for a matrix quadratic form in normal variables to be distributed as a difference of two independent Wishart distributions. Some extensions of Cochran’s theorem are then derived concerning differences of independent Wishart distributions.

Asymptotic solutions to the matrix Riccati equation

This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation. This method is fast, efficient and gives all possible solutions to the matrix quadratic form. Matrix sign functions are used to find the asymptotic solutions.

On the derivative of the trace of a matrix quadratic form

In the optimisation of linear systems with a random plant gain, the problem of computing the derivative of the trace of a matrix quadratic form is encountered. A derivation of this derivative is presented.

Erratum: On the derivative of the trace of a matrix quadratic form

Erratum: On the derivative of the trace of a matrix quadratic form