The Relation between the Stabilization Problem for Discrete Event Systems Modeled with Timed Petri Nets via Lyapunov Methods and Max-Plus Algebra

A discrete event system is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Timed Petri nets are a graphical and mathematical modeling tool applicable to discrete event systems in order to represent its states evolution where the timing at which the state changes is taken into consideration. One of the most important performance issues to be considered in a discrete event system is its stability. Lyapunov theory provides the required tools needed to aboard the stability and stabilization problems for discrete event systems modeled with timed Petri nets whose mathematical model is given in terms of difference equations. By proving stability one guarantees a bound on the discrete event systems state dynamics. When the system is unstable, a sufficient condition to stabilize the system is given. It is shown that it is possible to restrict the discrete event systems state space in such a way that boundedness is achieved. However, the restriction is not numerically precisely known. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.

基于Max-plus代数法的市域铁路快慢车运行特性

目的:为编制市域铁路快慢车模式的运行计划和检验快慢车方案的鲁棒性,针对快慢车运行特点,提出一种基于Max-plus代数法的市域铁路快慢车运行系统的闭环模型。方法:将列车运行系统作为典型的离散事件动态系统,定义其模型变量与参数,同时定义系统约束规则;基于Max-plus代数法建立列车运行系统Max-plus开环线性模型,并进行了快慢车模式下开环线性模型的变换;基于Max-plus代数法建立列车运行系统Max-plus闭环线性模型,并以一段计划开行快慢车的市域铁路作为算例,对该算例建立闭环模型,并求解与分析输出演化过程,最后通过状态转移变量矩阵的求解结果生成列车运行时刻表。结果及结论:该算例的快慢车运行系统稳定,一个周期系统的缓冲时间为291 s;通过单参数摄动情形下的鲁棒性分析获得了使快慢车运行系统保持运行一致均衡性的摄动元取值区间;首班车在始发站的出发时刻不具备鲁棒性,当第4列列车为快车时,其越行后成为第3列列车,该列车在越行站越行时刻不具备鲁棒性。

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

Stochastic stabilization of Markovian jump cloud control systems based on max-plus algebra

In this paper, stochastic stabilization is investigated by max-plus algebra for a Markovian jump cloud control system with a reference signal. For the Markovian jump cloud control system, there exists framework adjustment whose evolution is satisfied with a Markov chain. Using max-plus algebra, a maxplus stochastic system is used to describe the Markovian jump cloud control system. A causal feedback matrix is obtained by exponential stability analysis for a causal feedback controller of the Markovian jump cloud control system. A sufficient condition is given to ensure existence on the causal feedback matrix of the causal feedback controller. Based on the causal feedback controller, stochastic stabilization in probability is analyzed for the Markovian jump cloud control system with a reference signal.Simulation results are given to show effectiveness of the causal feedback controller for the Markovian jump cloud control system.

Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

Extended Fan's Algebraic Method and Its Application to KdV and Variant Boussinesq Equations

An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.

CONVERGENCE ANALYSIS OF RUNGE-KUTTA METHODS FOR A CLASS OF RETARDED DIFFERENTIAL ALGEBRAIC SYSTEMS

This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.

Studies on heteronuclear diatomic molecular dissociation energies using algebraic energy method

The algebraic energy method （AEM） is applied to the study of molecular dissociation energy De for 11 heteronuclear diatomic electronic states： a^3∑＋ state of NaK, X^2∑＋ state of XeBr, X^2∑＋ state of HgI, X^1∑＋ state of LiH, A3∏（1） state of IC1, X^1∑＋ state of CsH, A（3∏1） and B0＋（3∏） states of CIF, 21∏ state of KRb, X^1∑＋ state of CO, and c^3∑＋ state of NaK molecule. The results show that the values of De computed by using the AEM are satisfactorily accurate compared with experimental ones. The AEM can serve as an economic and useful tool to generate a reliable De within an allowed experimental error for the electronic states whose molecular dissociation energies are unavailable from the existing literature

Algebraic Method‑Based Point‑to‑Point Trajectory Planning of an Under‑Constrained Cable‑Suspended Parallel Robot with Variable Angle and Height Cable Mast

To avoid impacts and vibrations during the processes of acceleration and deceleration while possessing flexible working ways for cable-suspended parallel robots(CSPRs),point-to-point trajectory planning demands an under-constrained cable-suspended parallel robot(UCPR)with variable angle and height cable mast as described in this paper.The end-effector of the UCPR with three cables can achieve three translational degrees of freedom(DOFs).The inverse kinematic and dynamic modeling of the UCPR considering the angle and height of cable mast are completed.The motion trajectory of the end-effector comprising six segments is given.The connection points of the trajectory segments(except for point P3 in the X direction)are devised to have zero instantaneous velocities,which ensure that the acceleration has continuity and the planned acceleration curve achieves smooth transition.The trajectory is respectively planned using three algebraic methods,including fifth degree polynomial,cycloid trajectory,and double-S velocity curve.The results indicate that the trajectory planned by fifth degree polynomial method is much closer to the given trajectory of the end-effector.Numerical simulation and experiments are accomplished for the given trajectory based on fifth degree polynomial planning.At the points where the velocity suddenly changes,the length and tension variation curves of the planned and unplanned three cables are compared and analyzed.The OptiTrack motion capture system is adopted to track the end-effector of the UCPR during the experiment.The effectiveness and feasibility of fifth degree polynomial planning are validated.

Application of material-mesh algebraic collapsing acceleration technique in method of characteristics——based neutron transport code

The algebraic collapsing acceleration(ACA)technique maximizes the use of geometric flexibility of the method of characteristics(MOC).The spatial grids for loworder ACA are the same as the high-order transport,which makes the numerical solution of ACA equations costly,especially for large-size problems.To speed-up the MOC transport iterations effectively for general geometry,a coarse-mesh ACA method that involves selectively merging fine-mesh cells with identical materials,called material-mesh ACA(MMACA),is presented.The energy group batching(EGB)strategy in the tracing process is proposed to increase the parallel efficiency for microscopic crosssection problems.Microscopic and macroscopic crosssection benchmark problems are used to validate and analyse the accuracy and efficiency of the MMACA method.The maximum errors in the multiplication factor and pin power distributions are from the VERA-4 B-2 D case with silver-indium-cadmium(AIC)control rods inserted and are 104 pcm and 1.97%,respectively.Compared with the single-thread ACA solution,the maximum speed-up ratio reached 25 on 12 CPU cores for microscopic cross-section VERA-4-2 D problem.For the C5 G7-2 D and LRA-2 D benchmarks,the MMACA method can reduce the computation time by approximately one half.The present work proposes the MMACA method and demonstrates its ability to effectively accelerate MOC transport iterations.

ON THE METHODS FOR FINDING ROOTS OF ALGEBRAIC EQUATIONS WITH WEIERSTRASS' CORRECTIONS

This is a study of the Durand-Kerner and Nourein methods for finding the roots of a given algebraic equation simultaneously. We consider the conditions under which the iterative methods fail. The numerical example is presented.

Concurrence of Quantum States: Algebraic Dynamical Method Study XXX Models in a Time-Depending Random External Field

Based on algebraic dynamics and the concept of the concurrence of the entanglement, we investigate the evolutive properties of the two-qubit entanglement that formed by Heisenberg XXX models under a time-depending external field. For this system, the property of the concurrence that is only dependent on the coupling constant J and total values of the external field is proved. Furthermore, we found that the thermal concurrence of the system under a static random external field is a function of the coupling constant J, temperature T, and the magnitude of external field.

BCS Ground State and XXZ Antiferromagnetic Model as SU（2）,SU（1,1） Coherent States：AN Algebraic Diagonalization Method

An algebraic diagonalization method is proposed. As two examples, the Hamiltonians of BCS ground stateunder mean-field approximation and XXZ antiferromagnetic model in linear spin-wave frame have been diagonalized byusing SU(2), SU(1,1) Lie algebraic method, respectively. Meanwhile, the eigenstates of the above two models are revealedto be SU(2), SU(1,1) coherent states, respectively. The relation between the usual Bogoliubov Valatin transformationand the algebraic method in a special case is also discussed.

A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic method.The results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.This article fairly clarifies the behaviors of surface waves in shallow waters.In the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great importance.Thus,this study will serve as a reference to guide scientists working in this field.

ALGEBRAIC MULTI-GRID METHOD IN TWO-DIMENSION ELECTRICALLY LARGE PROBLEMS

In this paper the algebraic multi-grid principle is applied to the multilevel moment method, which makes the new multilevel method easier to implement and more adaptive to structure. Moreover, the error spectrum is analyzed, and the reason why conjugate gradient iteration is not a good relaxation scheme for multi-grid algorithm is explored. The numerical results show that our algebraic block Gauss Seidel multi-grid algorithm is very effective.

Parallel Rosenbrock Methods for DifferentialAlgebraic Equations

A class of parallel Rosenbrock methods for differential algebraic equations are presented in this paper. The local truncation errors are defined and the order conditions are established by using the DA-trees and DA-series. The paper also deals with the convergence of the parallel Rosenbrock methods for h -> 0 and states the bounds for the global errors of the methods. Some particular methods are obtained by solving the order equations and a numerical example is given, from which the theoretical orders are actually observed.

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.

PARALLEL MULTISPLITTING AOR METHOD FOR SOLVING A CLASS OF SYSTEM OF NONLINEAR ALGEBRAIC EQUATIONS

A class of parallel multisplitting AOR method for solving large scale system of nonlinear equations A φ(x)+Bψ(x)=b was proposed. Under certain conditions, the existence and uniqueness of the solution of this system of nonlinear equations were proved, and the global convergence theory of the new method was set up.

BCS Ground State and XXZ Antiferromagnetic Model as SU(2), SU(1,1) Coherent States: An Algebraic Diagonalization Method

An algebraic diagonalization method is proposed. As two examples, the Hamiltonians of BCS ground stateunder mean-field approximation and XXZ antiferromagnetic model in linear spin-wave frame have been diagonalized byusing SU(2), SU(1,1) Lie algebraic method, respectively. Meanwhile, the eigenstates of the above two models are revealedto be SU(2), SU(1,1) coherent states, respectively. The relation between the usual Bogoliubov Valatin transformationand the algebraic method in a special case is also discussed.