Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

In the present paper, we consider large scale nonsymmetric differential matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability and others. We show how to apply Krylov-type methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as Backward Differentiation Formula (BDF) or Rosenbrok method. We also show how these technique could be easily used to solve some problems from the well known transport equation. Some numerical experiments are given to illustrate the application of the proposed methods to large-scale problem

Some open problems in matrix theory arising in linear systems and control

Control theory has long provided a rich source of motivation for developments in matrix theory. Accordingly, we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control. The problems discussed include robust stability, matrix exponentials, induced norms, stabilizability and pole assignability, and nonstandard matrix equations. A substantial number of references are included to acquaint matrix theorists with problems and trends in this application area.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30236/1/0000630.pd

Solution of arbitrary fully fuzzy matrix equations and pair fully fuzzy matrix equations

Control system theory often involved the application of matrix equations and pair matrix equations where there are possibilities that uncertainty conditions can exist. In this case, the classical matrix equations and pair matrix equations are not well equipped to handle these conditions. Even though there are some previous studies in solving the matrix equations and pair matrix equations with uncertainty conditions, there are some limitations that include the fuzzy arithmetic operations, the type of fuzzy coefficients and the singularity of matrix coefficients. Therefore, this study aims to construct new methods for solving matrix equations and pair matrix equations with all the coefficients of the matrix equations are arbitrary left-right triangular fuzzy numbers (LR-TFN), which either positive, negative or near-zero. In constructing these methods, some modifications on the existing fuzzy subtraction and multiplication arithmetic operators are necessary. By modifying the existing fuzzy arithmetic operators, the constructed methods exceed the positive restriction to allow the negative and near-zero LR-TFN as the coefficients of the equations. The constructed methods also utilized the Kronecker product and Vec-operator in transforming the fully fuzzy matrix equations and pair fully fuzzy matrix equations to a simpler form of equations. On top of that, new associated linear systems are developed based on the modified fuzzy multiplication arithmetic operators. The constructed methods are verified by presenting some numerical examples. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary fully fuzzy matrix equations and pair fully fuzzy matrix equations, with minimum complexity of the fuzzy operations. The constructed methods are applicable for singular and non-singular matrices regardless of the size of the matrix. With that, the constructed methods are considered as a new contribution to the application of control system theory

Ergodic backward stochastic difference equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem

Computer algebra for solving dynamics problems of piezoelectric robots with large number of joints

The application of general control theory to complex mechanical systems represents an extremely difficult problem. If industrial piezoelectric robots have large number of joints, development of new control algorithms is unavoidable in order to achieve high positioning accuracy. The efficiency of computer algebra application was compared with the most popular methods of forming the dynamic equations of robots in real time. To this end, a computer algebra system VIBRAN was used. Expressions for the generalized inertia matrix of the robots have been derived by means of the computer algebra technique with the following automatic program code generation. As shown in the paper, such application could drastically reduce the number of floating point product operations that are required for efficient numerical simulation of piezoelectric robots

On A Set Of Classical Canonical Forms For An NxN Matrix

The term matrix was first used in 1850 by J. J. Sylvester. 1858 Arthur Cay ley began the systematic development of the theory of Today, matrix theory has application in such diverse fields as inventory control in factories, quantum theory in physics cost In matrices. analysis in transportations and other industries, deployment problems in military operations, and data analysis in sociology and psychology. This paper concerns itself with the set of necessary and sufficient conditions which will insure certain or particular canonical forms of matrices. Canonical forms of matrices are an important phenomena in the study of Differential Equations and other mathematical courses. In this paper, the development of the canonical form of a matrix under similarity transformation shall be presented. Letting A be a n-square matrix with elements in the field F, where F is a real number field, and let P be a non-singular matrix with elements in F. The set of all matrices P -1 AP constitutes a class of matrices similar to A. These matrices are known as canonical forms. In other words, a matrix of order nxn will be transformed to a canonical form of a matrix. If A and B denote two n-square matrices with elements in The necessary matrix. F such that P_1AP = B, A and B are said to be similar, and sufficient conditions that A and B be similar are that the X-matrices A- Xl and B- XI have the same invariant factor and the same elementary divisors. Chapter I includes Notations and Definitions. Chapter II includes a set of Basic Theorems of matrices to be used in the development of the canonical forms. In Chapter III five canonical forms of matrices of order nxn shall be developed. These forms are Diagonal, Triangular, Rational, Jordon, and Smith Normal Form. In Chapter IV some examples of the canonical forms in Chapter III will be presente

Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis