Auxiliary Matrices Research Articles

Numerical Treatment Of The Bounded-Control LQR Problem By Updating The Final Phase Value

A novel approach has been developed for approximately solving the constrained LQR problem, based on updating the final state and costate of an unrestricted related regular problem, and the switching times (when the control meets the constraints). The main result is the expression of a suboptimal control in feedback form by using some corresponding Riccati equation. The gradient method is applied to reduce the cost via explicit algebraic formula for its partial derivatives with respect to the hidden final state/costate of the related regular problem. The numerical method results efficient because it does not involve integrations of states or cost trajectories and reduces the dimension of the relevant unknown parameters. All the relevant objects are calculated from a few auxiliary matrices, which are computed only once and kept in memory. The scheme is here applied to the “cheapest stop of a train” case-study whose optimal solution is already known.

Finite-time H∞ control for one-sided Lipschitz systems with auxiliary matrices

In this paper, finite-time H∞ control is investigated for a class of one-sided Lipschitz systems by using the Finsler׳s lemma. Finite-time boundedness conditions are firstly provided for the one-sided Lipschitz system. The proposed conditions are less conservative since a new Lyapunov function with an additional matrix is constructed, and auxiliary matrices are introduced to make the Lyapunov matrix separate from the system matrix. Then, finite-time H∞ controller is designed by dynamic output feedback. The observer gain and output feedback control gain can be designed in one step optimization. Further discussions and comparisons are also presented in two aspects. At last, numerical simulations are given to verify the validity of the proposed methods.

Analysis and design of delta operator systems with nested actuator saturation

ABSTRACTIn this paper, the problem of estimating the domain of attraction is considered for delta operator systems subject to nested actuator saturation. A set invariance condition is established for the delta operator system with nested actuator saturation in terms of auxiliary feedback matrices. Based on the set invariance condition, an optimisation approach is proposed to estimate the domain of attraction for the delta operator system. Thereby, the partial results of nested actuator saturation for both continuous-time systems and discrete-time systems are extended to delta operator system framework. A numerical example is provided to illustrate the effectiveness of the proposed design techniques.

Extending multivariate- t linear mixed models for multiple longitudinal data with censored responses and heavy tails.

The analysis of complex longitudinal data is challenging due to several inherent features: (i) more than one series of responses are repeatedly collected on each subject at irregularly occasions over a period of time; (ii) censorship due to limits of quantification of responses arises left- and/or right- censoring effects; (iii) outliers or heavy-tailed noises are possibly embodied within multiple response variables. This article formulates the multivariate- t linear mixed model with censored responses (MtLMMC), which allows the analysts to model such data in the presence of the above described features simultaneously. An efficient expectation conditional maximization either (ECME) algorithm is developed to carry out maximum likelihood estimation of model parameters. The implementation of the E-step relies on the mean and covariance matrix of truncated multivariate- t distributions. To enhance the computational efficiency, two auxiliary permutation matrices are incorporated into the procedure to determine the observed and censored parts of each subject. The proposed methodology is demonstrated via a simulation study and a real application on HIV/AIDS data.

Robust model-based clustering via mixtures of skew-t distributions with missing information

Multivariate mixture modeling approach using the skew-t distribution has emerged as a powerful and flexible tool for robust model-based clustering. The occurrence of missing data is a ubiquitous problem in almost every scientific field. In this paper, we offer a computationally flexible EM-type procedure for learning multivariate skew-t mixture models to deal with missing data under missing at random mechanisms. Further, we present an information-based approach to approximating the asymptotic covariance matrix of the maximum likelihood estimators using the outer product of the scores. To assist the development and ease the implementation of our algorithm, two auxiliary permutation matrices are utilized for fast determination of the observed and missing parts of each observation. The practical usefulness of the proposed methodology is illustrated through simulations with varying proportions of artificial missing values and a real data example with genuine missing values.

A simplex method algorithm using a double basis

A simplex method algorithm not requiring explicit LU decomposition updating in the iterations is considered. The solutions obtained with fixed LU factors are corrected using small auxiliary matrices. The results of some numerical experiments are presented.

Optimal scheduling and power allocation in wireless networks with heavy traffic

This paper addresses the problem of joint transmit power allocation and time slot scheduling in a wireless communication system with time varying traffic. The system is handled by a single base station transmitting over time varying channels. This may be the case in practice of a hybrid TDMA-CDMA (Time Division Multiple Access-Code Division Multiple Access) system. The operating time horizon is divided into time slots; a fixed amount of power is available at each time slot. The users share each time slot and the power available at this time slot with the objective of minimizing the expected total queue length. The problem is reformulated, via a heavy traffic approximation, as the optimal control of a reflected diffusion in the positive orthant. We establish a closed form solution for the obtained control problem. The main feature that makes it possible is an astute choice of some auxiliary weighting matrices in the cost rate. The proposed solution relies also on the knowledge of the covariance matrix of the non-standard multi-dimensional Wiener process which is the driving process in the reflected diffusion. We then compute this covariance matrix given the stationary distribution of the multi-dimensional channel process. Further stochastic analysis is provided: the cost variance, and the Fokker–Planck equation for the distribution density of the queue length.

Determination of di (hydrogenated tallow alkyl) dimethyl ammonium compounds in textile auxiliaries by ultra-high performance liquid chromatography-tandem mass spectrometry

A method has been developed for the determination of di (hydrogenated tallow alkyl) dimethyl ammonium compounds (DHTDMAC) in textile auxiliaries by ultra-high performance liquid chromatography-tandem mass spectrometry. (UPLC-MS/MS). The samples were extracted and diluted with acidified methanol by 5% (v/v) formic acid under ultrasonic assistance. The separation was performed on an Eclipse Plus C18 column (50 mm x 2.1 mm, 1.8 microm) using 0.1% (v/v) formic acid solution and methanol as the mobile phases. Identification and quantification were achieved by UPLC-MS/MS with electrospray ionization (ESI) source in positive ion mode and multiple reaction monitoring (MRM) mode. The results indicated that the calibration curve of DHTDMAC showed good linear relationship between peak area and mass concentration in the range of 10-280 microg/L with the correlation coefficient (r2) of 0.9991. The limit of detection (LOD, S/N=3) and the limit of quantification (LOQ, S/N= 10) of this method were 3 mg/kg and 10 mg/kg, respectively. The average recoveries from three typical textile auxiliary matrices including dispersant, antistatic agent and fabric softener, at three spiked levels were in the range of 97.2%-108.3% with the relative standard deviations (RSDs) of 1.5%-4.6%. The method is sensitive, accurate, simple and effective for the analysis of DHTDMAC in textile auxiliaries.

Distributed optimization for multi-task learning via nuclear-norm approximation

We exploit a variational characterization of the nuclear norm to extend the framework of distributed convex optimization to machine learning problems that focus on the sparsity of the aggregate solution. We propose two distributed dynamics that can be used for multi-task feature learning and recommender systems in scenarios with more tasks or users than features. Our first dynamics tackles a convex minimization on local decision variables subject to agreement on a set of local auxiliary matrices. Our second dynamics employs a saddle-point reformulation through Fenchel conjugation of quadratic forms, avoiding the computation of the inverse of the local matrices. We show the correctness of both coordination algorithms using a general analytical framework developed in our previous work that combines distributed optimization and subgradient methods for saddle-point problems.

Positive real control for discrete-time singular systems with affine parameter dependence

This paper presents a less conservative and numerically tractable solution to the static state feedback positive real control problem for affinely parameter dependent discrete-time singular systems. Relied on the use of auxiliary matrices and a positive scalar decision variable, a novel necessary and sufficient condition of positive realness is first derived in terms of a strict matrix inequality for linear time-invariant discrete-time singular systems. This characterization leads to a numerically efficient and reliable way for the controller design synthesis. Then, the results are further expanded to parameter dependent singular systems whose coefficient matrices are affine functions of a time-invariant uncertain parameter vector. Both robust analysis and robust controller synthesis are addressed. Numerical examples are included to illustrate the effectiveness of the present results.

A reproducing kernel Hilbert space constructive approximation for a special class of integral equations

A reproducing kernel Hilbert space approach is proposed to study a class of integral equations with Toeplitz and Hankel kernel functions. The existence property and approximate representations of the solutions are given by constructing appropriate auxiliary operators and positive definite matrices within a reproducing kernel Hilbert space framework. Moreover, conditions for the boundedness and uniqueness of the solution are also obtained.

Perspective on natural orbital functional theory

The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top‐down and bottom‐up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two‐particle reduced density matrix (2‐RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrices, which are constrained to certain bounds due to the N‐representabilty conditions of the 2‐RDM. Appropriate forms of these matrices lead to different implementations known in the literature as PNOFi (i = 1–5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field. © 2014 Wiley Periodicals, Inc.

Robust stabilization and l 2-gain control of uncertain discrete-time constrained piecewise-affine systems

In this paper, the robust stabilization and l 2 gain control are investigated for a class of uncertain discrete-time piecewise-affine systems with input saturation and state constraints. By introducing some auxiliary feedback matrices, the LMI-based local stabilization condition is firstly proposed, where the designed controller can be saturated, and meanwhile, the domain of attraction is well estimated. Under energy bounded disturbance, the disturbance tolerance condition is proposed under which all trajectories starting from the origin will remain inside a bounded state region, then the l 2-gain controller is designed in the restricted region. Finally, numerical examples and simulations illustrate the effectiveness and the reduced conservativeness of the proposed results.

Preconditioned generalized mixed-type splitting iterative method for solving weighted least-squares problems

Recently, Shen et al. [Preconditioned iterative methods for solving weighted linear least squares problems, Appl. Math. Mech.-Engl. Ed. 33(3) (2012), pp. 375–384] have considered four kinds of preconditioned generalized accelerated overrelaxation (GAOR) methods for solving the linear systems based on a class of weighted least-squares problems and examined their convergence rates. More recently, Yun [Comparison results on the preconditioned GAOR method for generalized least squares problems, Int. J. Comput. Math. 89 (2012), pp. 2094–2105] has focused on the same problem and suggested three different types of preconditioned GAOR methods and studied their convergence properties. In this paper, we first introduce the generalized mixed-type splitting (GMTS) iterative method for solving the linear systems corresponding to the weighted least-squares problems. The GMTS iterative method exploits auxiliary matrices L1 and D1 which gratify certain conditions. In order to improve the convergence rate of the GMTS method, different types of preconditioners are applied. In addition, the convergence of the (preconditioned) GMTS iterative methods is discussed. It is both theoretically and experimentally demonstrated that by appropriate choices of auxiliary matrices L1 and D1, the (preconditioned) GMTS method outperforms the (preconditioned) GAOR method.

On State Feedback H∞ Control for Discrete-Time Singular Systems

This technical note deals with the state feedback H∞ control problem for linear time-invariant discrete-time singular systems. Relied on the use of auxiliary matrices and a positive scalar, a novel necessary and sufficient condition for the bounded real lemma is derived for discrete-time singular systems. The characterization is reduced to a strict linear matrix inequality (LMI) when the scalar is fixed, and the resulting LMI is non-conservative as long as the scalar is chosen sufficiently large. Moreover, the result is further expanded to H∞ controller design, and a numerically efficient and reliable design procedure is given. Since no particular restriction is imposed on the auxiliary matrices, the proposed result outperforms the existing methods in the literature. Numerical examples are included to illustrate the effectiveness of the present result.

Optimal saturated feedback laws for LQR problems with bounded controls

Optimal and suboptimal strategies are substantiated and illustrated for linear-quadratic problems with penalized endpoints, when bounds in control values are imposed. The optimal solution for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to some unrestricted problem that has the same coefficients, except for the final penalization matrix \(S\), and starts at a generally different initial state. This result reduces the searching span for the solution: from the infinite-dimensional set of admissible control trajectories to the finite-dimensional space of symmetric positive semi-definite symmetric matrices \(\hat{S}\) and initial states \(\hat{x}_{0}\). An efficient scheme is also proposed to approximate (and eventually to find) the optimal feedback strategy on-line, based on the updating of \(\hat{S}\) at successive sampling times \(t_{k},\) and on the possibility to generate the corresponding Riccati matrix \(P(t,T,\hat{S})\) for \(t_{k}<t\le t_{k+1}\) from auxiliary matrices stored in memory. Numerical simulations are provided, compared, and checked against the analytical solutions of two classical case-studies.

Learning from incomplete data via parameterized [formula omitted] mixture models through eigenvalue decomposition

A framework of using t mixture models with fourteen eigen-decomposed covariance structures for the unsupervised learning of heterogeneous multivariate data with possible missing values is designed and implemented. Computationally flexible EM-type algorithms are developed for parameter estimation of these models under a missing at random (MAR) mechanism. For ease of computation and theoretical developments, two auxiliary indicator matrices are incorporated into the estimating procedure for exactly extracting the location of observed and missing components of each observation. Computational aspects related to the specification of starting values, convergence assessment and model choice are also discussed. The practical usefulness of the proposed methodology is illustrated with real data examples and a simulation study with varying proportions of missing values.

Mixtures of common factor analyzers for high-dimensional data with missing information

Mixtures of common factor analyzers (MCFA), thought of as a parsimonious extension of mixture factor analyzers (MFA), have recently been developed as a novel approach to analyzing high-dimensional data, where the number of observations n is not very large relative to their dimension p. The key idea behind MCFA is to reduce further the number of parameters in the specification of the component-covariance matrices. An attractive and important feature of MCFA is to allow visualizing data in lower dimensions. The occurrence of missing data persists in many scientific investigations and often complicates data analysis. In this paper, we establish a computationally flexible EM-type algorithm for parameter estimation of the MCFA model with partially observed data. To facilitate the implementation, two auxiliary permutation matrices are incorporated into the estimating procedure for exactly extracting the location of observed and missing components of each observation. Practical issues related to the specification of initial values, model-based clustering and discriminant procedure are also discussed. Our methodology is illustrated through real and simulated examples.

Entanglement and separability of two-qubit states on Minkowski space with compact support

The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.

Robust statistical modelling using the multivariate skew t distribution with complete and incomplete data

Missing data is inevitable in many situations that could hamper data analysis for scientific investigations. We establish flexible analytical tools for multivariate skew t models when fat-tailed, asymmetric and missing observations simultaneously occur in the input data. For the ease of computation and theoretical developments, two auxiliary indicator matrices are incorporated into the model for the determination of observed and missing components of each observation that can effectively reduce the computational complexity. Under the missing at random assumption, we present a Monte Carlo version of the expectation conditional maximization algorithm, which is performed to estimate the parameters and retrieve each missing observation with a single value. Additionally, a Metropolis–Hastings within Gibbs sampler with data augmentation is developed to account for the uncertainty of parameters as well as missing outcomes. The methodology is illustrated through two real data sets.