Full Row Rank Research Articles

On utility‐optimised router‐level bandwidth allocation

ABSTRACTKelly's network utility maximisation (NUM) problems and solutions are aimed to maximise the aggregate utility subject to link capacity constraints. They are formulated and solved by using the individual flow rate vector. Because of the architecture of the current networks such as the Internet, the individual flow rates are generally not measurable directly at the routers for the network service provider. However, the aggregate flow rates are more convenient to obtain and to adjust. In this paper, we still study the NUM problems for communication networks but from a router‐level bandwidth allocation standpoint. With the use of the generalised matrix inverse, we propose a general model of utility‐optimised router‐level bandwidth allocation and its solution, where the objective function and the constraints are formulated in terms of the aggregate flow rate vector rather than from the individual flow rate vector as in the usual NUM problem. We find that the new proposed models are equivalent to Kelly's NUM model in the sense that they lead to the same optimum and their solutions satisfy the given routing scheme. We also discuss the special cases where the routing matrix is of full row rank and where there is one single‐hop flow in every link in the network. We suggest a direct application to Internet Protocol‐based virtual private network of the latter case. We present the mathematical models and solution procedures that lead to the utility‐optimised aggregate flow rate vector and further illustrate them by numerical examples. We believe our approach is promising for deployment in communication networks. Copyright © 2012 John Wiley & Sons, Ltd.

Solvability to Robust Output Regulation of Singular Linear Systems

In this paper, robust output regulation problems of singular linear systems are studied. Under appropriate assumptions, it is given that the solvability of robust output regulation problems equals to a specifically row full rank matrix. Moreover, the methods for constructing the feedback regulator are discussed.

Flexible multibody modelling for exact constraint design of compliant mechanisms

In high precision equipment, the use of compliant mechanisms is favourable as elastic joints offer the advantages of low friction and no backlash. If the constraints in a compliant mechanism are not carefully dealt with, even small misalignments can lead to changes in natural frequencies and stiffnesses. Such unwanted behaviour can be avoided by applying exact constraint design, which implies that the mechanism should have exactly the required degrees of freedom and non-redundant constraints so that the system is kinematically and statically determinate. For this purpose, we propose a kinematic analysis using a finite element based multibody modelling approach. In compliant mechanisms, the system’s degrees of freedom are presented clearly from the analysis of a system in which the deformation modes with a low stiffness are free to deform while the deformation modes with a high stiffness are considered rigid. If the Jacobian matrix associated with the dependent coordinates is not full column or row rank, the system is under-constrained or over-constrained. The rank of this matrix is calculated from a singular value decomposition. For an under-constrained system, any motion in the mechanism that is not accounted for by the current set of degrees of freedom is visualised using data from the right singular matrix. For an over-constrained system, a statically indeterminate stress distribution is derived from the left singular matrix and is used to visualise the over-constraints. The analysis is exemplified for the design of a straight guiding mechanism, where under-constrained and over-constrained conditions are visualised clearly.

Optimization approximation solution for regression problem based on extreme learning machine

Extreme learning machine ( ELM) is one of the most popular and important learning algorithms. It comes from single-hidden-layer feedforward neural networks. It has been proved that ELM can achieve better performance than support vector machine ( SVM) in regression and classification. In this paper, mathematically, with regression problem, the step 3 of ELM is studied. First of all, the equation H β = T are reformulated as an optimal model. With the optimality, the necessary conditions of optimal solution are presented. The equation H β = T is replaced by H T H β = H T T . We can prove that the latter must have one solution at least. Second, optimal approximation solution is discussed in cases of H is column full rank, row full rank, neither column nor row full rank. In the last case, the rank-1 and rank-2 methods are used to get optimal approximation solution. In theory, this paper present a better algorithm for ELM.

A note on AMP algoritm and its extension

The ABS methods are of direct finite algorithms which solve a linear systems of equations at most in number of equations. Amini, Mahdavi-Amiri and Peyghami in [3] present an ABS type algorithm so that two new equations are satisfied in every iteration when coefficient matrix is a full row rank matrix. Their article has some problems. In this paper we first point to them and then we extend their algorithm so that it is applicable to any arbitrary rank of coefficient matrix.

FINITE FLIESS FUNCTIONAL EXPANSION AND ITS APPLICATION TO FLIGHT CONTROL OF MISSILES

This paper addresses output tracking of nonlinear systems via state feedback and Fliess expansion. Under the full row rank assumption on the “decoupling matrix,� it is shown that a suitable state feedback can be designed such that the Fliess expansion of the closed-loop system contains only finite terms. A significant advantage of this form is that it avoids the truncation error in numerical calculation of Fliess expansions. As an application, this expansion is invoked to solve the problem of attitude control of missiles. Piecewise constant control is designed on each sampling interval to realize tracking optimization. Numerical simulations demonstrate the efficacy of the proposed control scheme.

Spread of fuzzy variable and expectation-spread model for fuzzy portfolio optimization problem

Using Lebesgue-Stieltjes (L-S) integral, we first define the nth moment of a fuzzy variable, and derive some useful moment formulas for common possibility distributions. Particularly, the second moment, called spread, can be represented as quadratic convex functions with respect to fuzzy parameters. Then, we take the spread as a new risk criteria in fuzzy decision systems, combine it with the expectation of fuzzy variable, and develop three classes of fuzzy expectation-spread models for portfolio optimization problems. In the case when the returns are trapezoidal and triangular fuzzy variables, we employ the parametric representations of spreads to turn the proposed expectation-spread models into their equivalent parametric programming problems. In the case when the return parametric matrix is full row rank, the equivalent parametric programming problems become convex programming ones that can be solved by conventional optimization methods or general purpose softwares. Finally, we demonstrate the developed modeling ideas via three numerical examples, and also compare the expectation-spread method with traditional expectation-variance method via a number of numerical experiments.

Robust Controllability of T–S Fuzzy-Model-Based Control Systems With Parametric Uncertainties

The robust controllability problem for the Takagi-Sugeno (T-S) fuzzy-model-based control systems is studied in this paper. Under the assumption that the nominal T-S fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal T-S fuzzy-model-based control systems has a full row rank for its controllability matrix), a sufficient condition is proposed to preserve the assumed property when the parameter uncertainties are added into the nominal T-S fuzzy-model-based control systems. The proposed sufficient condition can provide the explicit relationship of the bounds on parameter uncertainties to preserve the assumed property. Besides, a robustly global controllability condition and the related robustly global stabilizability condition of the uncertain T-S fuzzy-model-based control systems are also presented in this paper. A nonlinear mass-spring-damper mechanical system with parameter uncertainties is given as an example to illustrate the application of the proposed sufficient conditions.

Notes on factor prime factorizations for n-D polynomial matrices

Multidimensional (n?D) polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. This note is a continuation of the papers in Wang (Circuits Syst I 54: 1398---1405, 2007 and Circuits Syst II 55: 61---64, 2008). A complete constructive criterion is given for a matrix of 2 × m full row rank to be FLP. We also present a reduction method for dealing with the factorization problems with cases of non-regular divisors.

Modal parameter identification based on ARMAV and state–space approaches

An accurate prediction for the response of civil and mechanical engineering structures subject to ambient excitation requires the information of dynamic properties of these structures including natural frequencies, damping ratios and mode shapes. Since the excitation force is not available as a measured signal, we need to develop techniques which are capable of accurately extracting the modal parameters from output-only data. This article presents the results of modal parameter identification using two time-domain methods as follows: the autoregressive moving average vector (ARMAV) method and the state–space method. These methods directly work with the recorded time signals and allow the analysis of structures where only the output is measured, while the input is unmeasured and unknown. The equivalence between ARMAV and state–space approaches for the problem of modal parameter identification of vibrating systems is shown in the article. Using only the singular value decomposition of a block Hankel matrix of sample covariances, it is shown that these two approaches give identical modal parameters in the case where the block Hankel matrix has full row rank. The time-domain modal identification algorithms have a serious problem of model order determination: when extracting structural modes these algorithms always generate spurious modes. A modal indicator to differentiate spurious and structural modes is presented. Numerical and experimental examples are given to show the effectiveness of the ARMAV or state–space approaches in modal parameter identification using response data only.

Distributed space-time trellis code for asynchronous cooperative communications under frequency-selective channels

In most cooperative communication works, perfect synchronization among relay nodes is assumed in order to achieve the cooperative diversity. However, this assumption is not realistic due to the distributed nature of each relay node. In this paper, we propose a family of distributed space-time trellis code (DSTTC) that does not require the synchronization assumption. It is shown that the proposed DSTTC has minimum memory order, and the construction of such DSTTCs is equivalent to using the generator matrices that have full row rank, regardless of sub-matrices shifting problem. Here, a sub-matrix corresponds to the generator matrix of a relay node under frequency-selective channels. We derive sufficient conditions on the code design such that the full cooperative and multipath diversities can be achieved. By further studying the diversity product, we design the DSTTCs which can achieve full diversity and the maximum coding gain through the exhaustive computer search. The newly proposed codes exhibit good properties, e.g., high energy efficiency and low synchronization cost, and can be applied to distributed wireless networks. Finally, various numerical examples are provided to corroborate the analytical studies.

Limited-Shift-Full-Rank Matrices With Applications in Asynchronous Cooperative Communications

Shift-full-rank (SFR) matrices are matrices that have full row rank no matter how their rows are shifted. SFR matrices have been used lately as generator matrices for a family of space–time trellis codes to achieve full diversity in asynchronous cooperative communications, where the numbers of columns of the SFR matrices correspond to the memory sizes of the trellis codes. A systematic construction of SFR matrices, including the shortest (square) SFR (SSFR) matrices, has been also previously proposed. In this paper, we study a variation of SFR matrices with a relaxed condition: limited-shift-full-rank (LT-SFR) matrices, i.e., the matrices that have full row rank no matter how their rows are shifted as long as the shifts are within some range called delay tolerance. As the generator matrices for the previously proposed space–time trellis codes, LT-SFR matrices can guarantee asynchronous full diversity of the corresponding codes when the timing errors are within the delay tolerance. Therefore, due to the relaxed condition imposed on LT-SFR matrices, more eligible generator matrices than SFR matrices become available.

Relationship between state-space and ARMAV approaches to modal parameter identification

In this paper we study the relationship between state-space and autoregressive moving average vector (ARMAV) approaches for the problem of estimating the modal parameters of a vibrating system. Using only the singular value decomposition of the block Hankel matrix of covariances, it is shown that these two methods give identical modal parameters in the cases where this block Hankel matrix has full row rank.

Left invertibility and duality for linear systems

Left and right invertibilities, as well as inversion procedures and algorithms, have been widely studied for years. We consider here, for linear time invariant (LTI) systems, such types of one side invertibilities but in the time domain, which is more demanding than just in the frequency domain, i.e. for the Transfer Function Matrix (TFM). New geometric conditions are given for Time Domain (TD) Left Invertibility, as well as structural conditions which require not only full column rank but also the absence of some types of finite zeros, namely the “observable” ones. For general monic (full column rank) systems having “observable” zeros, we enhance a cascade decomposition which allows for the TD Left Inversion of the “pole” factor, and for the TFM Left Inversion of the “zero” factor. Duality in the classical TFM setting just relies on transposition. In the present TD context, this is not relevant. We give necessary and sufficient conditions for TD Right Invertibility, that show that the underlying duality is indeed the one between zeros and poles, which is a posteriori natural. Full row rank LTI systems can be TD Right Inverted if and only if they have no finite observable poles.

Bounds for the relative change of singular values of a real matrix

This paper amis to derive bounds for the relative change in the singular values of a real matrix. Both the full column rank and the full row rank cases are considered. Two numerical experiments show that the introduced bounds are confirmed and their values are dependent on the element-wise perturbation of the matrix. Mathematics Subject Classification: Primary 65F15; Secondary 65G05

Extended reduced rank two Abaffian update schemes in the ABS-type methods

The ABS methods, introduced by Abaffy, Broyden and Spedicato, are direct iteration methods for solving a linear system where the ith iterate satisfies the first i equations, therefore a system of m equations is solved in at most m steps. Recently, we have presented a new approach to devise a class of ABS-type methods for solving full row rank systems [K. Amini, N. Mahdavi-Amiri, M. R. Peyghami, ABS-type methods for solving full row rank linear systems using a new rank two update, Bulletin of the Australian Mathematical Society 69 (2004) 17–31], the ith iterate of which solves the first 2 i equations. Here, to reduce the space and computation time, we use a new extended rank two update formula for the Abaffian matrix so that the number of rows of the Abaffian matrix is reduced by two in every iteration. This extension along with the reduction offer more flexibility for the definition of the Abaffian matrix.

Interval iterative methods for computing Moore–Penrose inverse

In this paper, we import interval method to the iteration for computing Moore–Penrose inverse of the full row (or column) rank matrix. Through modifying the classical Newton iteration by interval method, we can get better numerical results. The convergence of the interval iteration is proven. We also give some numerical examples to compare interval iteration with classical Newton iteration.

Shift-full-rank matrices and applications in space-time trellis codes for relay networks with asynchronous cooperative diversity

To achieve full cooperative diversity in a relay network, most of the existing space-time coding schemes require the synchronization between terminals. A family of space-time trellis codes that achieve full cooperative diversity order without the assumption of synchronization has been recently proposed. The family is based on the stack construction by Hammons and El Gamal and its generalizations by Lu and Kumar. It has been shown that the construction of such a family is equivalent to the construction of binary matrices that have full row rank no matter how their rows are shifted, where a row corresponds to a terminal (or transmit antenna) and its length corresponds to the memory size of the trellis code on that terminal. We call such matrices as shift-full-rank (SFR) matrices. A family of SFR matrices has been also constructed, but the memory sizes of the corresponding space-time trellis codes (the number of columns of SFR matrices) grow exponentially in terms of the number of terminals (the number of rows of SFR matrices), which may cause a high decoding complexity when the number of terminals is not small. In this paper, we systematically study and construct SFR matrices of any sizes for any number of terminals. Furthermore, we construct shortest (square) SFR (SSFR) matrices that correspond to space-time trellis codes with the smallest memory sizes and asynchronous full cooperative diversity. We also present some simulation results to illustrate the performances of the space-time trellis codes associated with SFR matrices in asynchronous cooperative communications.

To Solving Multiparameter Problems of Algebra. 5. The ∇V-q Factorization Algorithm and its Applications

The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles.

A Gaussian mixture model for underdetermined independent component analysis

This paper proposes a Bayesian Independent Component Analysis based on the Gaussian mixture model for underdetermined blind source separation. The proposed algorithm follows a hierarchical learning and alternative estimations for sources and mixing matrix. The independent sources are estimated from their a posteriori means and the mixing matrix is estimated by Maximum Likelihood (ML). Both estimations require the a posteriori correlations of sources which exist in the underdetermined model with full row rank in general. The correlations are approximated with the help of linear response theory and factorized approximation to the true posterior. Under this framework, each source prior is modeled as a mixture of Gaussians. This mixture model provides us a flexibility that it can deal with the hybrid mixtures of both sparse and non-sparse sources, while most algorithms for underdetermined model only assume sparse prior for the sources. Simulations by using synthetic data validate the effectiveness of the learning algorithm.