Jacobi Iterative Algorithm Research Articles

Iterative algorithm to the generalized coupled Sylvester‐transpose matrix equations with application in robust and minimum norm observer design of linear systems

AbstractThis article consider a class of generalized coupled Sylvester‐transpose matrix equations which play an important role in control and systems theory. Based on the Jacobi iterative algorithm, the full‐row rank accelerated Jacobi gradient based iterative (RRAJGI) algorithm and the full‐column rank accelerated Jacobi gradient based iterative (CRAJGI) algorithm are proposed. By using the Frobenius norm of matrix and the trace function of matrix, the convergence of the algorithms are proved. The results show that the new algorithms are convergent for arbitrary initial matrices under the convergence number satisfies appropriate conditions. Numerical examples show that RRAJGI algorithm and CRAJGI algorithm have the advantages of faster convergence speed and higher convergence accuracy than other existing algorithms. Finally, an application example for robust and minimum norm observer design of linear systems is given.

Parallel design and implementation of Jacobi iterative algorithm based on ternary optical computer

Parallel design and implementation of Jacobi iterative algorithm based on ternary optical computer

Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system

This paper is dedicated to solving the iterative solution to the discrete-time periodic Sylvester matrix equations. Inspired by Jacobi iterative algorithm and hierarchical identification principle, the Jacobi gradient based iterative (JGI) algorithm and the accelerated Jacobi gradient based iterative (AJGI) algorithm are proposed. It is verified that the proposed algorithms are convergent for any initial matrix when the parameter factor μ satisfies certain condition. The necessary and sufficient conditions are provided for the presented new algorithms. Moreover, we also apply the JGI algorithm and AJGI algorithm to study a more generalized discrete-time periodic matrix equations and give the convergence conditions of the algorithms. Finally, two numerical examples are given to illustrate the effectiveness, accuracy and superiority of the proposed algorithms.

Resilient distributed MPC for systems under synchronous round-robin scheduling

This paper is concerned with the resilient dynamic output-feedback (DOF) distributed model predictive control (DMPC) problem for discrete-time polytopic uncertain systems under synchronous Round-Robin (RR) scheduling. In order to alleviate the computation burden and improve the system robustness against uncertainties, the global system is decomposed into several subsystems, where each subsystem under synchronous RR scheduling communicates with each other via a network. The RR scheduling is adopted to avoid data collisions, however the updating information at each time instant is unfortunately reduced, and the underlying RR scheduling of subsystems are deeply coupled. The main purpose of this paper is to design a set of resilient DOF-based DMPC controllers for systems under the consideration of polytopic uncertainties and synchronous RR scheduling, such that the desirable performance can be obtained at a low cost of computational time. A novel distributed performance index dependent of the synchronous RR scheduling is constructed, where the last iteration information from the neighbor subsystems is used to deal with various couplings. Then, by resorting to the distributed RR-dependent Lyapunov-like approach and inequality analysis technique, a certain upper bound of the objective is put forward to establish a solvable auxiliary optimization problem (AOP). Moreover, by using the Jacobi iteration algorithm to solve such a problem online, the distributed feedback gains are directly obtained to guarantee the convergence of system states. Finally, two examples including a distillation process example and a numerical example are employed to show the effectiveness of the proposed resilient DMPC strategy.

Overview of Precoding Techniques for Massive MIMO

Massive multiple-input multiple-output (MIMO) is playing a crucial role in the fifth generation (5G) and beyond 5G (B5G) communication systems. Unfortunately, the complexity of massive MIMO systems is tremendously increased when a large number of antennas and radio frequency chains (RF) are utilized. Therefore, a plethora of research efforts has been conducted to find the optimal precoding algorithm with lowest complexity. The main aim of this paper is to provide insights on such precoding algorithms to a generalist of wireless communications. The added value of this paper is that the classification of massive MIMO precoding algorithms is provided with easily distinguishable classes of precoding solutions. This paper covers linear precoding algorithms starting with precoders based on approximate matrix inversion methods such as the truncated polynomial expansion (TPE), the Neumann series approximation (NSA), the Newton iteration (NI), and the Chebyshev iteration (CI) algorithms. The paper also presents the fixed-point iteration-based linear precoding algorithms such as the Gauss-Seidel (GS) algorithm, the successive over relaxation (SOR) algorithm, the conjugate gradient (CG) algorithm, and the Jacobi iteration (JI) algorithm. In addition, the paper reviews the direct matrix decomposition based linear precoding algorithms such as the QR decomposition and Cholesky decomposition (CD). The non-linear precoders are also presented which include the dirty-paper coding (DPC), Tomlinson-Harashima (TH), vector perturbation (VP), and lattice reduction aided (LR) algorithms. Due to the necessity to deal with a high consuming power by the base station (BS) with a large number of antennas in massive MIMO systems, a special subsection is included to describe the characteristics of the peak-to-average power ratio precoding (PAPR) algorithms such as the constant envelope (CE) algorithm, approximate message passing (AMP), and quantized precoding (QP) algorithms. This paper also reviews the machine learning role in precoding techniques. Although many precoding techniques are essentially proposed for a small-scale MIMO, they have been exploited in massive MIMO networks. Therefore, this paper presents the application of small-scale MIMO precoding techniques for massive MIMO. This paper demonstrates the precoding schemes in promising multiple antenna technologies such as the cell-free massive MIMO (CF-M-MIMO), beamspace massive MIMO, and intelligent reflecting surfaces (IRSs). In-depth discussion on the pros and cons, performance-complexity profile, and implementation solidity is provided. This paper also provides a discussion on the channel estimation and energy efficiency. This paper also presents potential future directions in massive MIMO precoding algorithms.

COMPARISON OF GPU AND CPU EFFICIENCY WHILE SOLVING HEAT CONDUCTION PROBLEMS

Overview of GPU usage while solving different engineering problems, comparison between CPU and GPU computations and overview of the heat conduction problem are provided in this paper. The Jacobi iterative algorithm was implemented by using Python, TensorFlow GPU library and NVIDIA CUDA technology. Numerical experiments were conducted with 6 CPUs and 4 GPUs. The fastest used GPU completed the calculations 19 times faster than the slowest CPU. On average, GPU was from 9 to 11 times faster than CPU. Significant relative speed-up in GPU calculations starts when the matrix contains at least 4002 floating-point numbers.

基于雅克比迭代算法室内温度分布问题的研究——封闭空间内点热源的散热情况

基于雅克比迭代算法室内温度分布问题的研究——封闭空间内点热源的散热情况

Mathematical Modelling Time Dependent for Heat Transfer Process in Homogenous Composite ZnO/Activated CNF

A theoretical study of heat transfer on composite Zinc Oxide (ZnO)/Activated Carbon Nanofibers (CNF) by using mathematical model from diffusion equation for stationary composite with various heating time was studied. The equation and boundary conditions are interpreted by using Taylor series expansion to obtain a steady-state numerical solution from the Jacobi iteration algorithm. The composites where put inside the furnace model by filaments on the two sides of the walls as the source of heat (general model for furnace). By using this model, the position of sample to receive the homogeneous temperature determined. We compared the steady-state and unsteady-state conditions to identify the temperature distribution at real-time. The results show that the heat distribution will reach an asymptotic state by performing a numerical analysis is 200 K and 300 K for 25 minute time of heating. In this study, shows the efficient way for annealing homogenous materials to identify the position inside the furnace which can be used as a guide to receive the homogenous temperature during the annealing processes.

An Improved Jacobi-Based Detector for Massive MIMO Systems

Massive multiple-input-multiple-output (MIMO) is one of the key technologies in the fifth generation (5G) cellular communication systems. For uplink massive MIMO systems, the typical linear detection such as minimum mean square error (MMSE) presents a near-optimal performance. Due to the required direct matrix inverse, however, the MMSE detection algorithm becomes computationally very expensive, especially when the number of users is large. For achieving the high detection accuracy as well as reducing the computational complexity in massive MIMO systems, we propose an improved Jacobi iterative algorithm by accelerating the convergence rate in the signal detection process.Specifically, the steepest descent (SD) method is utilized to achieve an efficient searching direction. Then, the whole-correction method is applied to update the iterative process. As the result, the fast convergence and the low computationally complexity of the proposed Jacobi-based algorithm are obtained and proved. Simulation results also demonstrate that the proposed algorithm performs better than the conventional algorithms in terms of the bit error rate (BER) and achieves a near-optimal detection accuracy as the typical MMSE detector, but utilizing a small number of iterations.

An Efficient FPGA Parallel Implementation for 2-D MUSIC Algorithm

The multi signal classification (MUSIC) algorithm is a classical one in the direction estimation of radar. An highly efficient implementation scheme is proposed in this paper. In order to solve the problem that computing two-dimensional(2-D) MUSIC algorithm is not fast enough in engineering implementation and the resource consumption is large, and we have completed the simulation implementation of the algorithm on field programmable gate array (FPGA). Through the analysis of the steps of Jacobi transformation, a high efficient serial control scheme is proposed to realize Jacobi transformation, and the characteristic value decomposition is realized by using cyclic Jacobi iterative algorithm. This design avoids finding the most value through threshold comparison, reduces the hardware complexity and improves the decomposition speed. In spectral peak search, a parallel computing method is presented, which realizes parallel 2-D spectral peak search through spatial division, this method can effectively improve the effectiveness of search, and also can achieve the estimation of the target at the junction of airspace. Compared with the traditional implementation scheme, the design scheme has the advantages of high precision, high real time and saving hardware resources. Theoretical analysis and MATLAB simulation test show that this method and its FPGA design have good performance and have high application value in the practical process.

Mathematical model and numerical solutions for the coupled gas–solid heat transfer process in moving packed beds

A theoretical study was performed into the coupled gas–solid heat transfer process in a moving cooling packed clinker bed. A mathematical model for the clinker cooling process including effects from clinker movement is proposed. The aim is to predict through numerical methods the temperature distribution in the clinker layer and determine the optimal operating conditions. The equations and boundary conditions are discretized using the Taylor series expansion. The Jacobi iteration algorithm is employed to solve the difference equations to obtain temperature distributions for the clinker cooling process. The results are validated using industrial data. The relative errors are 7.0% and 1.1% for secondary and tertiary air temperatures, respectively. An analysis of the temperature difference distribution between clinker and cooling air confirmed the need to apply thermal non-equilibrium conditions in packed bed modeling. The different clinker speeds and thicknesses are also calculated. The results show that, with the same thickness of clinker layer, the most effective speed of the clinker is 0.008 m/s and ensures clinker cooling and heat recovery requirements. With the same clinker mass flow rate, operating with a thicker clinker layer can improve heat recovery and decrease the clinker outlet temperature; both can be used as a guiding framework in real cement production.

Distributed model predictive control with saturated inputs by a saturation‐dependent Lyapunov function

SummaryIn this paper, distributed model predictive control (MPC) problems are considered for input‐saturated polytopic uncertain systems by a saturation‐dependent Lyapunov function approach. The actuator saturation is processed by the transformation into the linear convex combination form. By the decomposition of the control input, distributed MPC controllers are designed in parallel for each subsystems. The Lyapunov Function we select is saturation dependent, which is less conservative than the general Lyapunov Function approach. An invariant set condition is provided and min–max distributed MPC is proposed based on the invariant set. The robust distributed MPC controllers are determined by solving a linear matrix inequality (LMI) optimization problem. To reduce the conservatism, we present a robust distributed MPC algorithm, which is not only saturation dependent but also parameter dependent. A Jacobi iterative algorithm is developed to coordinate the distributed MPC controllers. A simulation example with multi‐subsystem is carried out to demonstrate the effectiveness of the proposed distributed MPC algorithms. Copyright © 2016 John Wiley & Sons, Ltd.

Constrained robust distributed model predictive control for uncertain discrete-time Markovian jump linear system

This paper is concerned with the robust distributed model predictive control (MPC) problem for a class of uncertain discrete-time Markovian jump linear systems (MJLSs), subject to constraints on the inputs and states. Polytopic uncertainties both in system matrices and transition probability matrices of Markov process are taken into consideration. The global system is decomposed into several subsystems, and in this way these subsystems are able to exchange information with each other via internet. Furthermore, by constructing a novel Lyapunov functional, a sufficient condition is derived to guarantee the robust stability in the mean square sense for each subsystem with admissible constraints and uncertainties. In terms of the Cauchy–Schwarz inequality, the constrained problem of minimizing an upper bound on the worst-case infinite horizon cost function is transformed into a convex optimization problem involving linear matrix inequalities (LMIs). By solving a series of LMIs, a novel Jacobi iterative algorithm is proposed to design a distributed mode-dependent state-feedback controller, which ensures the local optimality at each sampling instant. Finally, compared with centralized and decentralized control schemes, two numerical simulation examples are employed to show the effectiveness of the proposed distributed algorithm.

Convergence of Price-Based Resource Allocation Algorithms in Multicellular Multicarrier Systems

Inter-cell interference mitigation is a key challenge in the heterogeneous wireless networks which are expected to use an aggressive frequency reuse factor and a high-density access point deployment to improve coverage and spectral efficiency. In this paper, the problem of resources allocation in multicell Orthogonal Frequency Division Multiple Access wireless system is considered with universal frequency reuse and target of Weighted Sum-Rate Maximization. We address multi cell modified iterative water filling as an iterative power allocation algorithm. Also, a new extension of fixed point implementation of Successive Convex Approximation for Low complExity (SCALE) algorithm to multicellular system [referred to as Multi Cell Fixed point SCALE (MCF-SCALE)] is presented and it has been shown both of them resulted to the same convergence point. It is also demonstrated that using Lagrangian multiplier instead of noise variance in Standard Yates framework (as has been used in some previous papers) is not a suitable method for proving convergence and all the previous results based on this pattern need to be revised. Finally, a new framework is presented for proving the convergence of MCF-SCALE algorithm based on Jacobi iterative algorithm. Moreover, some previous convergence criteria are shown to be interpreted as a special case of this condition.

Coverage Optimization and Spatial Load Balancing by Robotic Sensor Networks

This technical note studies robotic sensor networks performing static coverage optimization with area constraints. Given a density function describing the probability of events happening and a performance function measuring the cost to service a location, the objective is to position sensors in the environment so as to minimize the expected servicing cost. Moreover, because of load balancing considerations, the area of the region assigned to each robot is constrained to be a pre-specified amount. We characterize the optimal configurations as center generalized Voronoi configurations. The generalized Voronoi partition depends on a set of weights, one per robot, assigned to the network. We design a Jacobi iterative algorithm to find the weight assignment whose corresponding generalized Voronoi partition satisfies the area constraints. This algorithm is distributed over the generalized Delaunay graph. We also design the ?move-to-center-and-compute-weight? strategy to steer the robotic network towards the set of center generalized Voronoi configurations while monotonically optimizing coverage.

Distributed traffic control system based on model predictive control

The paper investigates a distributed control system scheme for urban road traffic management. The control algorithm is based on model predictive control (MPC) involving Jacobi iteration algorithm to solve constrained and nonlinear programming problem. The signal controllers of traffic network constitute a network of computers. They can distribute their computation realizing an efficient traffic control without any central management. However the optimal control inputs can be also calculated by a single traffic controller if the traffic network contains few intersections. The control aim is to relieve traffic congestion, reduce travel time and improve homogenous traffic flow in urban traffic area using distributed control architecture. The MPC based control strategy can be implemented in any urban transportation network but adequate measurement system and modern traffic controllers are needed. Theory, realization possibilities and simulation of the control method are also presented. The simulation results show that the system is able to ameliorate the network efficiency and reduce travel time. The distributed MPC based traffic control strategy proves the effectiveness by realizing a dependable control operation and creating optimal flow in the network subjected to control input constraints.

Application of Jacobi algorithm for ISI channels

Here, the Jacobi iterative algorithm is applied to combat intersymbol interference (ISI) caused by frequency-selective channels. The performance bound of the equaliser is analysed in order to gain an insight into its asymptotic behaviour. Because of the error propagation problem, the potential of this algorithm is not reached in an uncoded system. However, its extension to a coded system with the application of the turbo-processing principle results in a new turbo equalisation algorithm, which demonstrates comparable performance with reduced complexity compared with some existing filter-based turbo equalisation schemes; and superior performance compared with some frequency domain solutions, such as orthogonal frequency division multiplexing and single-carrier frequency domain equalisation.

Techniques for compiling programs on distributed memory multicomputers

It is widely accepted that distributed memory parallel computers will play an important role in solving computation-intensive problems. However, the design of an algorithm in a distributed memory system is time-consuming and error-prone, because a programmer is forced to manage both parallelism and communication. In this paper, we present techniques for compiling programs on distributed memory parallel computers. We will study the storage management of data arrays and the execution schedule arrangement of Do-loop programs on distributed memory parallel computers. First, we introduce formulas for representing data distribution of specific data arrays across processors. Then, we define communication cost for some message-passing communication operations. Next, we derive a dynamic programming algorithm for data distribution. After that, we show how to improve the communication time by pipelining data, and illustrate how to use data-dependence information for pipelining data. Jacobi's iterative algorithm and the Gauss elimination algorithm for linear systems are used to illustrate our method. We also present experimental results on a 32-node nCUBE-2 computer.