Feedforward Design and Optimality Aspects for the Control of Mechatronic Systems

This paper considers the tracking controller design for a mechatronic handling system. The presented approach is based on Port-Hamiltonian systems and it achieves good tracking as well as good disturbance rejection based on a polynomial approach. A PBC controller stabilizes the trajectory and the feedforward part has to fulfill the given limitations and the time optimality. This approach avoids the solution of a boundary value problem and a real-time integration of the internal dynamics. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886–911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.     

Mechatronic Subsystem Studied of Approximate Method Modeled by Hypergraph

The transverse vibrating mechatronic subsystem is considered. Integral parts of this system are: a continuous beam with known boundary conditions and a transducer, extorted by harmonic voltage excitation, to be perfectly bonded to the beam surface. Findings this article are dynamical characteristics of the discussed mechatronic and mechanical system to model them by hypergraphs. Research limitation is that the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered, however for this kind of systems the approach is sufficient. Practical implications of this researches was that global approach is presented, that means in the domain of frequency spectrum analysis. The methods of analysis and obtained results can be base of design and investigation for this type of mechatronic systems. Originality of this paper is that the mechatronic system created from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far because is it relies on application approximate methods of analysis of mechatronic subsystem and modeling the one by hypergraph [1-7]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

Sensitivity of Pareto Solutions in Multiobjective Optimization

The paper presents a sensitivity analysis of Pareto solutions on the basis of the Karush-Kuhn-Tucker (KKT) necessary conditions applied to nonlinear multiobjective programs (MOP) continuously depending on a parameter. Since the KKT conditions are of the first order, the sensitivity properties are considered in the first approximation. An analogue of the shadow prices, well known for scalar linear programs, is obtained for nonlinear MOPs. Two types of sensitivity are investigated: sensitivity in the state space (on the Pareto set) and sensitivity in the cost function space (on the balance set) for a vector cost function. The results obtained can be used in applications for sensitivity computation under small variations of parameters. Illustrative examples are presented.Research of this author was partially supported by Grant BEC2003-09067-C04-03.Research of this author was partially supported by NSERC Grant RGPIN-3492-00.Research of this author was partially supported by Grant BEC2003-09067-C04-02.     

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.     

A Matrix Modification Method for Calculating Approximate Solutions to Systems of Linear Equations

An expensive feature of iterative methods for non-linear problemsmay be the need for frequent solution of systems of linear equations.This paper proposes a method for reducing the amount of suchwork by approximating the inverse of the coefficient matrix.Unlike previously suggested matrix updating techniques, however,the present method is designed to estimate the discrepancy betweenthe exact and approximate solutions. In this way the accuracyof the approximate solutions can be controlled.     

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.     

Equivalence of Balance Points and Pareto Solutions in Multiple-Objective Programming

It is shown that the concept of balance points introduced by Galperin (Ref. 1) is equivalent to the concept of Pareto optimality.     

Approximate Pareto sets of minimal size for multi-objective optimization problems

We are interested in a problem introduced by Vassilvitskii and Yannakakis (2005), the computation of a minimum set of solutions that approximates within an accuracy ε$\epsilon$ the Pareto set of a multi-objective optimization problem. We mainly establish a new 3-approximation algorithm for the bi-objective case. We also propose a study of the greedy algorithm performance for the tri-objective case when the points are given explicitly, answering an open question raised by Koltun and Papadimitriou in (2007).     

深埋隧洞围岩应力的精确解与近似解的对比分析

对不同断面形状的深埋隧洞进行了分析,比较了隧洞围岩应力解析解与通过当量半径方法得到的近似解之间的差别.首先,应用复变函数的基本理论,给出圆形、椭圆、矩形、直墙拱形等几种常见深埋隧洞围岩应力的解析表达式.其次,应用当量半径的折算形式,将其任意形状的边界转化为标准圆形断面,利用Lamé解答得到了各围岩应力分量.最后,考虑隧洞断面形状参数的变化,通过数值算例对精确解和近似解进行了比较,分析了当量半径折算形式的精确度.在此基础上,应用有限元方法验证了复变函数解析解的精确性,以椭圆、矩形和直墙拱形的复变函数解验证当量半径精确度.结果表明,当量半径的折算形式解答与精确解答之间相似程度与隧洞的断面形状和几何参数之间有着密切的关系.     

求时滞Volterra积分系统最优控制问题近似解的一个迭代算法

对一类典型的复杂系统—含多个非线性时滞的Volterra积分系统,给出其控制作用受限(硬约束)最优控制问题近似解的一个迭代算法,证明了该算法是well-defined的,并在一定的条件下证得了算法的收敛性.     

Approximate Solutions of Non-linear Boundary-value Problems

A method for obtaining approximate solutions of non-linear boundaryvalue problems, based on a judicious application of the maximumprinciple, is described. After preliminaries, the method isapplied to generate pointwise bounding approximate solutionsof the Liouville equation. A comparison is also made with themethod based on the existence of complementary variational principles.     

Approximate Solutions of Continuous Dispersion Problems

The problem of positioning p points so as to maximize the minimum distance between them has been studied in both location theory (as the continuous p-dispersion problem) and the design of computer experiments (as the maximin distance design problem). This problem can be formulated as a nonlinear program, either exactly or approximately. We consider formulations of both types and demonstrate that, as p increases, it becomes dramatically more expensive to compute solutions of the exact formulation than to compute solutions of the approximate formulation.     

Mechatronic Design of the 2 DOF PRRRP Parallel Robot

Mechatronics consists in integration of three fields in engineering: mechanics, electronics and information technology. The paradigm of mechatronics is the integration concept, being the first step in the design of complex mechatronic systems. The need for an integrated approach to the design of complex engineering system involving electronic engineering, mechanical engineering and computing has led to the growth of the concept of mechatronics. The paper presents details regarding the integration process for the kinematic axis in mechatronic. The example of the positioning system based on dc motor and screw–nut transmission is analysed. This actuator is further used for driving a 2 DOF PRRRP parallel robot. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calculation of Flexibility of Vibrating Beam as the Subsystem of Mechatronic System by Means the Exact and Approximate Methods

The main aim of this paper is to investigate the transients of characteristics of vibrating beams obtained by the exact and approximate methods and to answer to the question – if the method can be used to nominate the characteristics of mechatronic systems. The approach was to nominate the relevance or irrelevance between the characteristics obtained by considered methods – especially concerning the relevance of the natural frequencies-poles of characteristics of mechanical part of mechatronic system. The main subject of the research is the continuous vibrating beam. Findings of this approach is fact, that approximate solutions fulfill all conditions for vibrating beams and some conditions only, particularly for vibrating beams as the subsystems of mechatronic systems. Practical implications of this paper is the main point is the analysis and the examination of flexibly vibrating discrete-continuous mechatronic systems which characteristics can be nominated with approximate methods only. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation

Approximate stationary solutions of the Ostrovsky equation describing long weakly nonlinear waves in a rotating liquid are constructed. These solutions may be regarded as a periodic sequence of arcs of parabolas containing Korteweg-de Vries solitons at the junctures. Results of numerical computations of the dynamics of the approximate solutions obtained from the nonstationary Ostrovsky equation are presented. It is found that, in the presence of negative dispersion, the shape of a stationary wave is well predicted by the approximate theory, whereas the calculated wave velocity differs slightly from the theoretical value. The stationary solutions in media with positive dispersion are evidently unstable (at least for sufficiently strong rotation), and numerical computations demonstrate a complicated picture of nonstationary destruction.