Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.

Optimal Estimation of High-Dimensional Covariance Matrices with Missing and Noisy Data

The estimation of covariance matrices is very important in many fields, such as statistics. In real applications, data are frequently influenced by high dimensions and noise. However, most relevant studies are based on complete data. This paper studies the optimal estimation of high-dimensional covariance matrices based on missing and noisy sample under the norm. First, the model with sub-Gaussian additive noise is presented. The generalized sample covariance is then modified to define a hard thresholding estimator , and the minimax upper bound is derived. After that, the minimax lower bound is derived, and it is concluded that the estimator presented in this article is rate-optimal. Finally, numerical simulation analysis is performed. The result shows that for missing samples with sub-Gaussian noise, if the true covariance matrix is sparse, the hard thresholding estimator outperforms the traditional estimate method.

Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation

In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .

On the Solutions of the Matrix Equations in Optimal Stochastic Control

In this paper, the matrix algebraic equations involved in the optimal control problem of time-invariant linear Ito stochastic systems, named Riccati- Ito equations in the paper, are investigated. The necessary and sufficient condition for the existence of positive definite solutions of the Riccati- Ito equations is obtained and an iterative solution to the Riccati- Ito equations is also given in the paper thus a complete solution to the basic problem of optimal control of time-invariant linear Ito stochastic systems is then obtained. An example is given at the end of the paper to illustrate the application of the result of the paper.

Optimal strategy of searching FPD weights scanning matrix using GA-PSO

This paper discusses a kind of optimal method used for searching flat panel display （FPD） scanning matrix. The method adopts bionic algorithm： genetic algorithm （GA） and particle swarm optimization （PSO） algorithm. The method using single GA is more time-consuming, and the search efficiency is low in later evolution; the PSO algorithm is easily falling into the local optimal solution and appears the premature convergent phenomenon. Hence, a hybrid approach of GAPSO is found to optimize the search for high grayscale weights scanning matrix. Finally in the acceptable time, it finds a weight scanning matrix （WSM） of 256 gray scales with Matlab, whose scanning efficiency reaches 94.73% and the linearity is very good.

Optimizing Memory Access Efficiency in CUDA Kernel via Data Layout Technique

Over the past decade, Graphics Processing Units (GPUs) have revolutionized high-performance computing, playing pivotal roles in advancing fields like IoT, autonomous vehicles, and exascale computing. Despite these advancements, efficiently programming GPUs remains a daunting challenge, often relying on trial-and-error optimization methods. This paper introduces an optimization technique for CUDA programs through a novel Data Layout strategy, aimed at restructuring memory data arrangement to significantly enhance data access locality. Focusing on the dynamic programming algorithm for chained matrix multiplication—a critical operation across various domains including artificial intelligence (AI), high-performance computing (HPC), and the Internet of Things (IoT)—this technique facilitates more localized access. We specifically illustrate the importance of efficient matrix multiplication in these areas, underscoring the technique’s broader applicability and its potential to address some of the most pressing computational challenges in GPU-accelerated applications. Our findings reveal a remarkable reduction in memory consumption and a substantial 50% decrease in execution time for CUDA programs utilizing this technique, thereby setting a new benchmark for optimization in GPU computing.

Least-Squares Solutions of the Matrix Equation A^TXA=B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.

Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.

Optimization of AZ80 magnesium alloy squeeze cast process parameters using morphological matrix

The squeeze cast process parameters of AZ80 magnesium alloy were optimized by morphological matrix. Experiments were conducted by varying squeeze pressure, die pre-heat temperature and pressure duration using L9（33） orthogonal array of Taguchi method. In Taguchi method, a 3-level orthogonal array was used to determine the signal/noise ratio. Analysis of variance was used to determine the most significant process parameters affecting the mechanical properties. Mechanical properties such as ultimate tensile strength, elongation and hardness of the components were ascertained using multi variable linear regression analysis. Optimal squeeze cast process parameters were obtained.

Optimization of the biphasic release of indomethacin from HPMC/pectin/calcium chloride matrix tablet by response surface methodology

We studied the effect of two independent variables, the pectin/calcium chloride weight ratio and the overall matrix weight in HPMC/pectin/calcium matrix tablet, on the release of indomethacin. A two-factor 5-level central composite experimental design was employed. Responses of the Peppas correlation parameters n and K and the 10% release time （T0.1） were optimized by response surface methodology. Significant effect of the independent variables on the biphasic release parameters, n and K, was observed. N, K and T0.1 were well fitted with the second-order quadratic equations rather than linear equations. Moreover, the mathematic fitting and the response surfaces showed significant cross-interaction between the pectin/calcium chloride ratio and the overall matrix weight. The optimal formulation with larger n, longer T0.1 and smaller K consisted of medium pectin/calcium chloride ratio around 1.0 and medium matrix weight around 200 mg. Validation studies on the optimal formulations showed good predictability of the n, K and T0.1 values with biases within the range of-7.33% and 6.26%. Our results support that central composite design can be used to optimize drug release from HPMC/pectin/calcium matrix tablet with high predictability.

Application of Fussy Decision-making in Selecting the Optimal Variety from City Flower Candidates

Fuzzy similar priority ratio is adopted to select the optimal variety from 6 city flower candidates in a certain city,i.e.Nelumbo nucifera x1,Prunus persica Batsch.var.duplex Rehd.x2,Rosa chinensis Jacq.x3,Dendranthema morifolium x4,Jassminum nudiflorum Lindl.x5 and Prunus mume x6.The results show that the priority sequence of 6 candidates was x1,x6,x5,x3,x4 and x2.

Marginal optimization method to improve the inconsistent comparison matrix in the analytic hierarchy process

To improve the inconsistency in the analytic hierarchy process(AHP), a new method based on marginal optimization theory is proposed. During the improving process, this paper regards the reduction of consistency ratio(CR) as benefit, and the maximum modification compared to the original pairwise comparison matrix(PCM) as cost, then the improvement of consistency is transformed to a benefit/cost analysis problem. According to the maximal marginal effect principle, the elements of PCM are modified by a fixed increment(or decrement) step by step till the consistency ratio becomes acceptable, which can ensure minimum adjustment to the original PCM so that the decision makers’ judgment is preserved as much as possible. The correctness of the proposed method is proved mathematically by theorem. Firstly, the marginal benefit/cost ratio is calculated for each single element of the PCM when it has been modified by a fixed increment(or decrement).Then, modification to the element with the maximum marginal benefit/cost ratio is accepted. Next, the marginal benefit/cost ratio is calculated again upon the revised matrix, and followed by choosing the modification to the element with the maximum marginal benefit/cost ratio. The process of calculating marginal effect and choosing the best modified element is repeated for each revised matrix till acceptable consistency is reached, i.e., CR<0.1. Finally,illustrative examples show the proposed method is more effective and better in preserving the original comparison information than existing methods.

Optimal Load Balancing Leveling Method for Multi-leg Flexible Platforms

The working platforms supported with multiple extensible legs must be leveled before they come into operation.Although the supporting stiffness and reliability of the platform are improved with the increasing number of the supporting legs,the increased overdetermination of the multi-leg platform systems leads to leveling coupling problem among legs and virtual leg problem in which some of the supporting legs bear zero or quasi zero loads.These problems make it quite complex and time consuming to level such a multi-leg platform.Based on rigid body kinematics,an approximate equation is formulated to rapidly calculate the leg extension for leveling a rigid platform,then a proportional speed control strategy is proposed to reduce the unexpected platform distortion and leveling coupling between supporting legs.Taking both the load coupling between supporting legs and the elastic flexibility of the working platform into consideration,an optimal balancing legs’ loads(OBLL) model is firstly put forward to deal with the traditional virtual leg problem.By taking advantage of the concept of supporting stiffness matrix,a coupling extension method(CEM) is developed to solve this OBLL problem for multi-leg flexible platform.At the end,with the concept of supporting stiffness matrix and static transmissibility matrix,an optimal load balancing leveling method is proposed to achieve geometric leveling and legs’ loads balancing simultaneously.Three numerical examples are given out to illustrate the performance of proposed methods.This paper proposes a method which can effectively quantify all of the legs’ extension at the same time,achieve geometric leveling and legs’ loads balancing simultaneously.By using the proposed methods,the stability,precision and efficiency of auto-leveling control process can be improved.

Automatic Generation of Optimally Rigid Formations Using Decentralized Methods

In this paper, decentralized methods of optimally rigid graphs generation for formation control are researched. The notion of optimally rigid graph is first defined in this paper to describe a special kind of rigid graphs. The optimally rigid graphs can be used to decrease the topology complexity of graphs while maintaining their shapes. To minimize the communication complexity of formations, we study the theory of optimally rigid formation generation. First, four important propositions are presented to demonstrate the feasibility of using a decentralized method to generate optimally rigid graphs. Then, a formation algorithm for multi-agent systems based on these propositions is proposed. At last, some simulation examples are given to show the efficiency of the proposed algorithm.

Optimal guaranteed cost control for fuzzy descriptor systems with time-varying delay

Based on the delay-independent rule, the problem of optimal guaranteed cost control for a class of Takagi-Sugeno （T-S） fuzzy descriptor systems with time-varying delay is studied. A linear quadratic cost function is considered as the performance index of the closed-loop system. Sufficient conditions for the existence of guaranteed cost controllers via state feedback are given in terms of linear matrix inequalities （LMIs）, and the design of an optimal guaranteed cost controller can be reduced to a convex optimization problem. It is shown that the designed controller not only guarantees the asymptotic stability of the closed-loop fuzzy descriptor delay system, but also provides an optimized upper bound of the guaranteed cost. At last, a numerical example is given to illustrate the effectiveness of the proposed method and the perfect performance of the optimal guaranteed cost controller.

Optimal observation configuration of UAVs based on angle and range measurements and cooperative target tracking in three-dimensional space

This article investigates the optimal observation configuration of unmanned aerial vehicles(UAVs) based on angle and range measurements, and generalizes predecessors' researches in two dimensions into three dimensions. The relative geometry of the UAVs-target will significantly affect the state estimation performance of the target, the cost function based on the Fisher information matrix(FIM) is used to derive the FIM determinant of UAVs' observation in three-dimensional space, and the optimal observation geometric configuration that maximizes the determinant of the FIM is obtained. It is shown that the optimal observation configuration of the UAVs-target is usually not unique, and the optimal observation configuration is proved for two UAVs and three UAVs in three-dimension. The long-range over-the-horizon target tracking is simulated and analyzed based on the analysis of optimal observation configuration for two UAVs. The simulation results show that the theoretical analysis and control algorithm can effectively improve the positioning accuracy of the target. It can provide a helpful reference for the design of over-the-horizon target localization based on UAVs.

Applying Analytical Derivative and Sparse Matrix Techniques to Large-Scale Process Optimization Problems

The performance of analytical derivative and sparse matrix techniques applied to a traditional dense sequential quadratic programming (SQP) is studied, and the strategy utilizing those techniques is also presented.Computational results on two typical chemical optimization problems demonstrate significant enhancement in efficiency, which shows this strategy is promising and suitable for large-scale process optimization problems.

Sensing Matrix Optimization for Multi-Target Localization Using Compressed Sensing in Wireless Sensor Network

In the multi-target localization based on Compressed Sensing(CS),the sensing matrix's characteristic is significant to the localization accuracy.To improve the CS-based localization approach's performance,we propose a sensing matrix optimization method in this paper,which considers the optimization under the guidance of the t%-averaged mutual coherence.First,we study sensing matrix optimization and model it as a constrained combinatorial optimization problem.Second,the t%-averaged mutual coherence is adopted as the optimality index to evaluate the quality of different sensing matrixes,where the threshold t is derived through the K-means clustering.With the settled optimality index,a hybrid metaheuristic algorithm named Genetic Algorithm-Tabu Local Search(GA-TLS)is proposed to address the combinatorial optimization problem to obtain the final optimized sensing matrix.Extensive simulation results reveal that the CS localization approaches using different recovery algorithms benefit from the proposed sensing matrix optimization method,with much less localization error compared to the traditional sensing matrix optimization methods.

Multi-Model Based PSO Method for Burden Distribution Matrix Optimization With Expected Burden Distribution Output Behaviors

Burden distribution is one of the most important operations, and also an important upper regulation in blast furnace(BF) iron-making process. Burden distribution output behaviors(BDOB) at the throat of BF is a 3-dimensional spatial distribution produced by burden distribution matrix(BDM),including burden surface output shape(BSOS) and material layer initial thickness distribution(MLITD). Due to the lack of effective model to describe the complex input-output relations,BDM optimization and adjustment is carried out by experienced foremen. Focusing on this practical challenge, this work studies complex burden distribution input-output relations, and gives a description of expected MLITD under specific integral constraint on the basis of engineering practice. Furthermore, according to the decision variables in different number fields, this work studies optimization of BDM with expected MLITD, and proposes a multi-mode based particle swarm optimization(PSO) procedure for optimization of decision variables. Finally, experiments using industrial data show that the proposed model is effective, and optimized BDM calculated by this multi-model based PSO method can be used for expected distribution tracking.

Optimization Algorithm for Reduction the Size of Dixon Resultant Matrix:A Case Study on Mechanical Application

In the process of eliminating variables in a symbolic polynomial system,the extraneous factors are referred to the unwanted parameters of resulting polynomial.This paper aims at reducing the number of these factors via optimizing the size of Dixon matrix.An optimal configuration of Dixon matrix would lead to the enhancement of the process of computing the resultant which uses for solving polynomial systems.To do so,an optimization algorithm along with a number of new polynomials is introduced to replace the polynomials and implement a complexity analysis.Moreover,the monomial multipliers are optimally positioned to multiply each of the polynomials.Furthermore,through practical implementation and considering standard and mechanical examples the efficiency of the method is evaluated.