SOLUTION TO A PARAMETRIC EQUATION WITH GENERALIZED BOUNDARY CONDITION IN TRANSPORT THEORY

This paper deals with the solution of a parametric equation with generalized boundary condiiton in transport theory. It gives the distribution of parameter (so called delta-eigenvalue [1]) with which the equation has non-zero solution. A necessary and sufficient condition for the existence of; he control critical eigenvalue delta0 is established.

Two parametric approaches for eigenstructure assignment in second-order linear systems

This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).

An explicit solution to right factorization with applicationin eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix. Applications of this solution to a type of generalized Sylvester matrix equatiorls and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple, and possess desirable structural properties which render the solutions readily implementable. An example demonstrates the effect of the proposed results.

An explicit solution to the matrix equation AV+BW=EVJ

In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.

Eigenstructure assignment in a class of second-order dynamic systems

Eigenstructure assignment using the proportional-plus-derivative feedback controller in a class of secondorder dynamic system is investigated. Simple, general, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established based on two simple Smith form reductions. The approach utilizes directly the original system data and involves manipulations only on n-dimensional matrices. Furthermore, it reveals all the degrees of freedom which can be further utilized to achieve additional system specifications. An example shows the effect of the proposed approach.

Solving the generalized Sylvester matrix equation AV+BW=VF via Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV ＋ BW = VF with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.

SOLUTION TO A PARAMETRIC EQUATION IN TRANSPORT THEORY

In 1976, Ronen et al. raised the question of how to solve the new integrodifferential parameter equation with transport theory. So far there have only been some approximate calculations and numerical analyses about this question. Using functional analysis, this note discusses this question in a strict mathematical way, gives the parameter distribution in L^p space (1≤p≤+∞) with which the

Direct design to stress mapping for cellular structures

This paper aims to instantly predict within any accuracy the stress distribution of cellular structures under parametric design,including the shapes or distributions of the cell geometries,or the magnitudes of external loadings.A classical model reduction technique has to balance the simulation accuracy and interaction speed,and has difficulty achieving this goal.We achieve this by computing offline a design-to-stress mapping that ultimately expresses the stress distribution as an explicit function in terms of its design parameters.The mapping is determined as a solution to an extended finite element analysis problem in a high-dimension space,including both the spatial coordinates and the design parameters.The well-known curse of dimensionality intrinsic to the high-dimension problem is(partly)resolved through a spatial separation using two main techniques.First,the target mapping takes a reduced form as a sum of the products of separated one-variable functions,extending the proper generalized decomposition technique.Second,the simulation problem in a varied computation domain is reformulated as that in a fixed-domain,taking an integration function as the sum of the products of separated one-variable functions,in combination with high-order singular value decomposition.Extensive 2D and 3D examples are shown to demonstrate the approach’s performance.