Grigorios I. Kalogeropoulos

A compound matrix algorithm for the computation of the Smith form of a polynomial matrix

In the present paper is presented a numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations. Using the notion of compound matrices, the Smith canonical form of a polynomial matrixM(s)∈ℝnxn[s] is calculated directly from its definition, requiring only the construction of all thep-compound matricesCp(M(s)) ofM(s), 1<p≤n. This technique produces a stable and accurate numerical algorithm working satisfactorily for any polynomial matrix of any degree.

Discretizing LTI Descriptor (Regular) Differential Input Systems with Consistent Initial Conditions

A technique for discretizing efficiently the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is introduced and fully discussed. Additionally, an upper bound for the error that derives from the procedure of discretization is also provided. Practically speaking, we are interested in such kind of systems, since they are inherent in many physical, economical and engineering phenomena.

The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

In the present paper, we study the derivation of the Weierstrass Canonical Form (WCF) of a regular matrix pencil. In order to compute the WCF, we use two important computational tools: a) the QZ algorithm to specify the required root range of the pencil and b) the updating technique to compute the index of annihilation. The proposed updating technique takes advantages of the already computed rank of the sequences of matrices that appears during our procedure reducing significantly the required floating-point operations. The algorithm is implemented in a numerical stable manner, giving efficient results. Error analysis and the required complexity of the algorithm are included.

Normalizability and feedback stabilization for second-order linear descriptor differential systems

ln this paper we investigate the normalizability of second-order linear descriptor differential system and simultaneously the relocation of its poles. The whole procedure is divided into three main algorithmic steps. Firstly, we normalize the descriptor system. Afterwards, we solve a linear and a multi-linear sub-problem. The proposed method computes a reduced set of quadratic Plucker relations which describes completely the specific Grassmann variety. Finally, using these relations the feedback matrices are fully determined which provide the solution to the pole assignment problem. An illustrative example of the algorithmic procedure is also discussed.

The dynamic response of homogeneous LTI descriptor differential systems under perturbations of the right matrix coefficients

This paper is concerned with the dynamic response of a general class of linear time invariant differential systems, the right parameter of which undergoes step perturbations. We solve both systems using the complex Weierstrass canonical form (powerful tool of matrix pencil theory). After that, we calculate and compare the relationship between the two solutions. This comparison is of considerable importance in numerical analysis since it has a direct bearing upon the accuracy of any particular method used to construct the solution of the base system. A numerical example is also provided.

Simulate the State Changing of a Descriptor System in (Almost) Zero Time Using the Normal Probability Distribution

In number of control applications, the ability of manipulate the state vector from the input is more than vital. Thus, in the present paper, we develop analytically a methodology for the state changing of a linear control descriptor differential system based also on a linear combination of Dirac δ-function and its derivatives. Using linear algebra techniques and the generalized inverse theory, the input’s coefficients are determined. In our practical numerical application, the Dirac distribution is approximated by the normal probability distribution.

Investigating the solution properties of symmetric/skew-symmetric LTI homogeneous matrix descriptor discrete-time systems with consistent initial conditions and constant delay period

In this paper, the Thompsons canonical form for a regular and singular matrix pencil of complex matrices with symmetric and skew symmetric structural properties is introduced for the solution of linear and time invariant (LTI) matrix homogeneous descriptor discrete time system with consistent initial conditions and time delay. Under this approach, the main equation is divided into several sub-systems whose solutions are derived. Note that the regularity or singularity of matrix pencil predetermines the number of sub-systems.

Error Analysis of the Complex Kronecker Canonical Form

In some interesting applications in control and system theory, i.e. in engineering, in ecology (Leslie population model), in financial/actuarial (Leontief multi input - multi output) science, linear descriptor (singular) differential/difference equations with time-invariant coefficients and (non-) consistent initial conditions have been extensively used. The solution properties of those systems are based on the Kronecker canonical form, which is an important component of the Matrix Pencil Theory. In this paper, we present some preliminary results for the error analysis of the complex Kronecker canonical form based on the Euclidean norm. Finally, under some weak assumptions an interesting new necessary condition is also derived.

An Efficient Algorithm for Approximating Impulse Inputs and Transferring Instantly the State of Linear Matrix Control Systems

Last decades, the control and system theory has been beneficed more by the new technological trends. Thus, in numerous (computational) applications, for instant in electronics, in computers, in engineering, as well as, in financial issues, some elements of the control and system theory have been introduced. In this paper, we are interested to change instantly the state of linear matrix differential systems into zero time by using very appropriate inputs. Thus, our desired inputs are chosen to be sequences of Dirac function and its derivatives. A new algorithmic method for the approximation of the Dirac sequence and the calculations of the relative matrix coefficients is introduced. This approach extends further our knowledge to this issue.

Transferring Instantly the State of a Linear Singular Descriptor Differential System

In numerous computational applications in mechanics, in engineering, as well as, in financial issues, the ability of manipulating instantly the state vector from the input is more than significant. Thus, in this paper, we extend a method for the instantly state transferring of linear singular descriptor differential systems, which is based on impulsive distributions. Using linear algebra techniques and the generalized inverse theory, the input’s coefficients are determined.