Set Of Linear Equality Constraints Research Articles

An Efficient Constrained Weighted Least Squares Method With Bias Reduction for TDOA-Based Localization

This paper addresses the source location problem by using time-difference-of-arrival (TDOA) measurements. The two-stage weighted least squares (TWLS) algorithm has been widely used in the TDOA location. However, the estimation accuracy of the source location is poor and the bias is significant when the measurement noise is large. Owing to the nonlinear nature of the system model, we reformulate the localization problem as a constrained weighted least squares problem and derive the theoretical bias of the source location estimate from the maximum-likelihood (ML) estimation. To reduce the location bias and improve location accuracy, a novel bias-reduced method is developed based on an iterative constrained weighted least squares algorithm. The new method imposes a set of linear equality constraints instead of the quadratic constraints to suppress the bias. Numerical simulations demonstrate the significant performance improvement of the proposed method over the traditional methods. The bias is reduced significantly and the Cramer–Rao lower bound accuracy can also be achieved.

Dynamic Optimal Energy Flow in the Heat and Electricity Integrated Energy System

To retain intact state information of the district heating network (DHN) as possible in the optimization of heat and electricity integrated energy system (HE-IES), this article combines the transient heat flow and steady-state electric power flow to formulate the dynamic optimal energy flow (OEF) model of HE-IES. For efficient and standardized solution, the finite difference method is applied to convert the partial differential equation constraint (introduced by the temperature dynamics in DHN) into a set of linear equality constraints. The structure of applicable difference schemes for system optimization is analyzed, based on which, a scheme with balanced performances in stability, convergence, simulation accuracy, and computation burden is developed. Moreover, with the proposed multi-objective optimization based method to select proper spatial and temporal calculation step sizes, a compromise between model precision and solution complexity can be reached. Case studies validate the feasibility and benefits of our proposed dynamic OEF computing method, which can further provide support for making optimal planning and operating strategies.

Iterative constrained weighted least squares estimator for TDOA and FDOA positioning of multiple disjoint sources in the presence of sensor position and velocity uncertainties

Sensor position and velocity uncertainties are known to be able to degrade the source localization accuracy significantly. This paper focuses on the problem of locating multiple disjoint sources using time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) in the presence of sensor position and velocity errors. First, the explicit Cramér–Rao bound (CRB) expression for joint estimation of source and sensor positions and velocities is derived under the Gaussian noise assumption. Subsequently, we compare the localization accuracy when multiple-source positions and velocities are determined jointly and individually based on the obtained CRB results. The performance gain resulted from multiple-target cooperative positioning is also quantified using the orthogonal projection matrix. Next, the paper proposes a new estimator that formulates the localization problem as a quadratic programming with some indefinite quadratic equality constraints. Due to the non-convex nature of the optimization problem, an iterative constrained weighted least squares (ICWLS) method is developed based on matrix QR decomposition, which can be achieved through some simple and efficient numerical algorithms. The newly proposed iterative method uses a set of linear equality constraints instead of the quadratic constraints to produce a closed-form solution in each iteration. Theoretical analysis demonstrates that the proposed method, if converges, can provide the optimal solution of the formulated non-convex minimization problem. Moreover, its estimation mean-square-error (MSE) is able to reach the corresponding CRB under moderate noise level. Simulations are included to corroborate and support the theoretical development in this paper.

A Low-Cost Robust Distributed Linearly Constrained Beamformer for Wireless Acoustic Sensor Networks With Arbitrary Topology

We propose a new robust distributed linearly constrained beamformer that utilizes a set of linear equality constraints to reduce the cross power spectral density matrix to a block-diagonal form. The proposed beamformer has a convenient objective function for use in arbitrary distributed network topologies while having identical performance to a centralized implementation. Moreover, the new optimization problem is robust to relative acoustic transfer function (RATF) estimation errors and to target activity detection (TAD) errors. Two variants of the proposed beamformer are presented and evaluated in the context of multimicrophone speech enhancement in a wireless acoustic sensor network, and are compared with other state-of-the-art distributed beamformers in terms of communication costs and robustness to RATF estimation errors and TAD errors.

An Energy-Based Model Encoding Nonlocal Pairwise Pixel Interactions for Multisensor Change Detection

Image change detection (CD) is a challenging problem, particularly when images come from different sensors. In this paper, we present a novel and reliable CD model, which is first based on the estimation of a robust similarity-feature map generated from a pair of bitemporal heterogeneous remote sensing images. This similarity-feature map, which is supposed to represent the difference between the multitemporal multisensor images, is herein defined, by specifying a set of linear equality constraints, expressed for each pair of pixels existing in the before-and-after satellite images acquired through different modalities. An estimation of this overconstrained problem, also formulated as a nonlocal pairwise energy-based model, is then carried out, in the least square sense, by a fast linear-complexity algorithm based on a multidimensional scaling mapping technique. Finally, the fusion of different binary segmentation results, obtained from this similarity-feature map by different automatic thresholding algorithms, allows us to precisely and automatically classify the changed and unchanged regions. The proposed method is tested on satellite data sets acquired by real heterogeneous sensor, and the results obtained demonstrate the robustness of the proposed model compared with the best existing state-of-the-art multimodal CD methods recently proposed in the literature.

Diffusion LMS for Multitask Problems With Local Linear Equality Constraints

We consider distributed multitask learning problems over a network of agents where each agent is interested in estimating its own parameter vector, also called task, and where the tasks at neighboring agents are related according to a set of linear equality constraints. Each agent possesses its own convex cost function of its parameter vector and a set of linear equality constraints involving its own parameter vector and the parameter vectors of its neighboring agents. We propose an adaptive stochastic algorithm based on the projection gradient method and diffusion strategies in order to allow the network to optimize the individual costs subject to all constraints. Although the derivation is carried out for linear equality constraints, the technique can be applied to other forms of convex constraints. We conduct a detailed mean-square-error analysis of the proposed algorithm and derive closed-form expressions to predict its learning behavior. We provide simulations to illustrate the theoretical findings. Finally, the algorithm is employed for solving two problems in a distributed manner: a minimum-cost flow problem over a network and a space-time varying field reconstruction problem.

Almost Asymptotic Regulation of Markovian Jumping Linear Systems in Discrete Time

AbstractThis paper considers the problems of almost asymptotic output regulation for discrete‐time Markovian jumping linear systems. Based on a stochastic Lyapunov‐Krasovskii functional framework, sufficient conditions for the extension of the regulation scheme to such stochastic systems are obtained via state feedback and via error feedback. Relying on a characterization of the feedback controllers, the almost asymptotic regulation is accomplished. The problem of guaranteeing stochastic stability and almost asymptotic tracking is achieved by solving linear matrix inequalities subject to a set of linear equality constraints. In order to ensure relaxed solutions of the regulation equations, a semi‐definite optimization approach via disciplined convex programming is outlined. Simulation results also are given to illustrate the effectiveness of the proposed design approach.

Rank-Constrained Solutions to Linear Matrix Equations Using PowerFactorization

Algorithms to construct/recover low-rank matrices satisfying a set of linear equality constraints have important applications in many signal processing contexts. Recently, theoretical guarantees for minimum-rank matrix recovery have been proven for nuclear norm minimization (NNM), which can be solved using standard convex optimization approaches. While nuclear norm minimization is effective, it can be computationally demanding. In this work, we explore the use of the PowerFactorization (PF) algorithm as a tool for rank-constrained matrix recovery. Empirical results indicate that incremented-rank PF is significantly more successful than NNM at recovering low-rank matrices, in addition to being faster.

An LMI approach to [formula omitted] synthesis subject to almost asymptotic regulation constraints

A multi-objective controller synthesis problem is considered in which an output is to be regulated approximately by assuring a bound on the steady-state peak amplification in response to an infinite-energy disturbance, while also guaranteeing a desired level of performance measured in terms of the worst-case energy gain from a finite-energy input to a performance output. Relying on a characterization of the controllers with which almost asymptotic regulation is accomplished, the problem of guaranteeing the desired level of performance is reduced to solving a system of linear matrix inequalities subject to a set of linear equality constraints. Based on the solution of this system, a procedure is outlined for the construction of a suitable controller whose order is equal to the order of the plant plus the order of the exogenous system.

Quadratic Interpolation and Linear Lifting Design

A quadratic image interpolation method is stated. The formulation is connected to the optimization of lifting steps. This relation triggers the exploration of several interpolation possibilities within the same context, which uses the theory of convex optimization to minimize quadratic functions with linear constraints. The methods consider possible knowledge available from a given application. A set of linear equality constraints that relate wavelet bases and coefficients with the underlying signal is introduced in the formulation. As a consequence, the formulation turns out to be adequate for the design of lifting steps. The resulting steps are related to the prediction minimizing the detail signal energy and to the update minimizing the l2-norm of the approximation signal gradient. Results are reported for the interpolation methods in terms of PSNR and also, coding results are given for the new update lifting steps.

Multi-Plet Two-Channel Perfect Reconstruction FIR and Causal Stable IIR Filter Banks

This paper proposes a new class of two-channel structural perfect reconstruction (PR) finite-impulse response / infinite-impulse response (FIR/IIR) filter banks (FBs) called the multi-plet FBs. It generalizes the structural PR FBs proposed by Phoong and the conventional triplet FB in that it employs multiple lifting steps similar to the conventional lifting structure. Apart from the important structural PR property, the multi-plet FB can be systematically designed to meet a given specification on the passband/stopband ripples and transition bandwidth: a low-order prototype PR FB with a much wider transition band is first designed using nonlinear optimization in order to obtain a symmetric frequency response and prescribed passband and stopband ripples. As only a few variables are required, its performances can easily be controlled. A subfilter is then designed using second-order cone programming (SOCP) so that the prototype FB can be warped by means of frequency transformation to meet the desired transition bandwidth, while preserving the PR condition, passband/stopband ripples and lifting structure of the prototype FB. The design procedure is very general and it applies to both linear-phase/low-delay FIR and IIR multi-plet FBs. By formulating the K-regularity conditions as a set of linear equality constraints on the subfilter coefficients, multi-plet-based wavelet bases can easily be designed under the SOCP framework. Design examples show that the proposed approach offers more flexibility in controlling the frequency characteristics of the PR FBs and lower design complexity than conventional methods

On the Design and Multiplierless Realization of Perfect Reconstruction Triplet-Based FIR Filter Banks and Wavelet Bases

This paper proposes new methods for the efficient design and realization of perfect reconstruction (PR) two-channel finite-impulse response (FIR) triplet filter banks (FBs) and wavelet bases. It extends the linear-phase FIR triplet FBs of Ansari et al. to include FIR triplet FBs with lower system delay and a prescribed order of K regularity. The design problem using either the minimax error or least-squares criteria is formulated as a semidefinite programming problem, which is a very flexible framework to incorporate linear and convex quadratic constraints. The K regularity conditions are also expressed as a set of linear equality constraints in the variables to be optimized and they are structurally imposed into the design problem by eliminating the redundant variables. The design method is applicable to linear-phase as well as low-delay triplet FBs. Design examples are given to demonstrate the effectiveness of the proposed method. Furthermore, it was found that the analysis and synthesis filters of the triplet FB have a more symmetric frequency responses. This property is exploited to construct a class of PR M-channel uniform FBs and wavelets with M=2/sup L/, where L is a positive integer, using a particular tree structure. The filter lengths of the two-channel FBs down the tree are approximately reduced by a factor of two at each level or stage, while the transition bandwidths are successively increased by the same factor. Because of the downsampling operations, the frequency responses of the final analysis filters closely resemble those in a uniform FB with identical transition bandwidth. This triplet-based uniform M-channel FB has very low design complexity and the PR condition and K regularity conditions are structurally imposed. Furthermore, it has considerably lower arithmetic complexity and system delay than conventional tree structure using identical FB at all levels. The multiplierless realization of these FBs using sum-of-power-of-two (SOPOT) coefficients and multiplier block is also studied.

New Design and Realization Techniques for a Class of Perfect Reconstruction Two-Channel FIR Filterbanks and Wavelets Bases

This paper proposes two new methods for designing a class of two-channel perfect reconstruction (PR) finite impulse response (FIR) filterbanks (FBs) and wavelets with K-regularity of high order and studies its multiplier-less implementation. It is based on the two-channel structural PR FB proposed by Phoong et al (1995). The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved, respectively, as a quadratic programming problem with linear equality constraints (QPLC) and a semidefinite programming (SDP) problem. We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two (SOPOT) coefficients while preserving the regularity constraints using Bernstein polynomials. However, this implementation usually requires long coefficient wordlength and another direct-form implementation, which can realize multiplier-less wavelets with K-regularity condition up to fifth order, is proposed. Several design examples are given to demonstrate the effectiveness of the proposed methods.

The Design of a Class of Perfect Reconstruction Two-Channel FIR Linear-Phase Filterbanks and Wavelets Bases Using Semidefinite Programming

This paper proposes a new method for designing a class of two-channel perfect reconstruction (PR) linear-phase FIR filterbanks (FBs) and wavelets previously proposed by Phoong et al. By expressing the given K-regularity constraints as a set of linear equality constraints in the design variables, the design problem using the minimax error criterion can be solved using semidefinite programming (SDP). Design examples show that the proposed method is very effective and it yields equiripple stopband response while satisfying the given K-regularity condition.

Performance bounds for queueing networks and scheduling policies

Except for the class of queueing networks and scheduling policies admitting a product form solution for the steady-state distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. In most applications such as manufacturing systems, however, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper the authors introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steady state for general quadratic forms, the authors obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an augmenting set of linear equality and inequality constraints. Together, these allow the authors to bound the performance, either above or below, by solving a linear program. The authors illustrate this technique on several typical problems of interest in manufacturing systems. For an open re-entrant line modeling a semiconductor plant, the authors plot a bound on the mean delay (called cycle-time) as a function of line loading. It is shown that the last buffer first serve policy is almost optimal in light traffic. For another such line, it is shown that it dominates the first buffer first serve policy. For a set of open queueing networks, the authors compare their lower bounds with those obtained by another method of Ou and Wein (1992). For a closed queueing network, the authors bracket the performance of all buffer priority policies, including the suggested priority policy of Harrison and Wein (1990). The authors also study the asymptotic heavy traffic limits of the lower and upper bounds. For a manufacturing system with machine failures, it is shown how the performance changes with failure and repair rates. For systems with finite buffers, the authors show how to bound the throughput. Finally, the authors illustrate the application of their method to GI/GI/1 queues. The authors obtain analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

LSEQIEQ: a FORTRAN IV subroutine package for the analysis of multiple linear regression problems with possibly deficient pseudorank and linear equality and inequality constraints

An algorithm and FORTRAN IV subroutine package are presented which solve the general linear least-squares problem: minimize∥AX−B∥ minimize∥X∥ where A is a p by n data matrix of values preceding the n unknown coefficients symbolized by the vector X and B is a p-vector of dependent variables corresponding to the p rows of the data matrix A, subject to an arbitrary set of linear equality constraints and inequality bounds on the coefficients of the vector X. The solution is effected by orthogonal decomposition of the data matrix and the code includes a provision for estimating the true rank (actual number of independent variables) of the problem and extracting a solution vector which incorporates this intervariable correlation. The computer storage requirements of the subroutines are kept to a minimum by implementing a sequential accumulation algorithm which alleviates the need to store the data matrix in memory at any time. Thus, an effectively infinite number of data points can be processed in a small fixed storage space. A detailed description of subroutine usage is given and three applications of the code are presented which deal with rank deficient data matrices which are: (1) large and overdetermined, (2) small, but whose unknown coefficients are bounded by inequality constraints, and (3) underdetermined, with bounded coefficients subject to linear equality constraints. The first problem is concerned with the application of thermodynamics to the solution properties of silicate liquids where a strongly rank deficient data matrix is analyzed. The second application of the algorithm involves a rapid method of estimating the precious metal content of mineral phases in an ore body. Lastly, a complex underdetermined problem, from the solution chemistry of clay minerals, is investigated. A solution is obtained only after careful and judicious construction of the inequality constraint matrix.These applications stem from typical least-squares problems encountered in igneous petrology, economic geology, and geochemistry and emphasize the advantages of the present approach over standard regression techniques.

Optimal Control of Linear Econometric Systems with Linear Equality Constraints on the Control Variables

with at a (n x 1) target vector and Kt a known (n x n) symmetric, positive semidefinite matrix. The optimal control problem normally treated is to determine that set of values of the control variables U1 ,..., UT which minimizes the welfare function (1.2), subject to the constraint of the model (1.1), and given the initial condition xo. In this paper the optimal control problem will be extended to problems in which the control variables must satisfy a set of linear equality constraints. It turns out that for this modified problem the optimal control rule for ut, t=1,..., T is similar in form to the normal control problem. An important example in which this control problem arises is when the components of the (m x 1) control vector u, are the amount of money allocated from a central authority to each of rn local authorities and it is required that their sum at time t be equal to the total financial resources available (as in macroeconomic management). A description of this example is given in Section 4. In Section 2 the modified control problem is formally stated and the solution derived. Section 3 demonstrates that the principle of certainty equivalence and the separation theorem hold.

An algorithm for linearly constrained adaptive array processing

A constrained least mean-squares algorithm has been derived which is capable of adjusting an array of sensors in real time to respond to a signal coming from a desired direction while discriminating against noises coming from other directions. Analysis and computer simulations confirm that the algorithm is able to iteratively adapt variable weights on the taps of the sensor array to minimize noise power in the array output. A set of linear equality constraints on the weights maintains a chosen frequency characteristic for the array in the direction of interest. The array problem would be a classical constrained least-mean-squares problem except that the signal and noise statistics are assumed unknown a priori. A geometrical presentation shows that the algorithm is able to maintain the constraints and prevent the accumulation of quantization errors in a digital implementation.