Regular Pencil Research Articles

Descriptor fractional linear systems with regular pencils

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

On Modifications to the Spectral Dichotomy Algorithm

We introduce modifications of an existing spectral dichotomy algorithm [6] with respect to a circle for general regular matrix pencils. The modifications result in a cost-effective algorithm, mathematically equivalent to the original one, but six times less expensive. The algorithm approximates iteratively the spectral projector onto the right deflating subspace associated with the eigenvalues inside/outside the circle as well as a criterion to ascertain its numerical quality. We also show how to approximate the left projector from the algorithm without any extra work. Effective stopping criteria for the algorithm are discussed. Numerical examples which illustrate the theory are presented.

Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils

The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.

SINGULAR FRACTIONAL CONTINUOUS-TIME AND DISCRETE-TIME LINEAR SYSTEMS

Abstract New classes of singular fractional continuous-time and discrete-time linear systems are introduced. Electrical circuits are example of singular fractional continuous-time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and Laplace transformation the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources or at least one node with branches with supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of singular fractional discrete-time linear systems is derived. The considerations are illustrated by numerical examples.

About canonical forms of regular matrix pencil

The algorithm of local reduction of the regular matrix pencil to canonical form is constructed in the article.

Stability of descriptor positive linear systems

Purpose – The purpose of the paper is to propose a new method for checking the positivity and asymptotic stability of descriptor continuous‐time and discrete‐time linear systems.Design/methodology/approach – The proposed method is based on the elementary operations applied to descriptor linear systems with regular pencils.Findings – The descriptor systems with regular pencils are transformed to equivalent standard systems which are used for checking of the asymptotic stability of descriptor systems.Originality/value – The paper proposes a new method for investigation of the positivity and asymptotic stability of descriptor continuous‐time and discrete‐time linear systems.

Positive stable realizations for fractional descriptor continuous-time linear systems

Abstract A method for computation of positive asymptotically stable realizations of fractional descriptor continuous-time linear systems with regular pencil is proposed. The method is based on the decomposition of the improper transfer matrix into strictly proper matrix and a polynomial matrix. A procedure for decomposition of a positive asymptotically stable realization is given and illustrated by a numerical example.

Descriptor Fractional Linear Systems with Regular Pencils

AbstractNew classes of descriptor fractional continuous‐time and discrete‐time linear systems with regular pencils are introduced. Electrical circuits are an example of descriptor fractional continuous‐time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and Laplace transformation the solution to the state equation of descriptor fractional linear systems is derived. It is shown that every electrical circuit is a descriptor fractional systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources, or at least one node with branches containing supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of descriptor fractional discrete‐time linear systems is derived. A method for decomposition of the descriptor fractional linear systems with regular pencils into dynamic and static parts is proposed. The considerations are illustrated by numerical examples.

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.

Positivity of descriptor linear systems with regular pencils

Positivity of descriptor linear systems with regular pencilsThe positivity of descriptor continuous-time and discrete-time linear systems with regular pencils are addressed. Such systems can be reduced to standard linear systems and can be decomposed into dynamical and static parts. Two definitions of the positive systems are proposed. It is shown that the definitions are not equivalent. Conditions for the positivity of the systems and the relationship between two classes of positive systems are established. The considerations are illustrated by examples of electrical circuits and numerical examples.

A generalization of matrix inversion with application to linear differential-algebraic systems

A new generalized inversion for square matrices based on projections is introduced. It includes as special cases known generalized inverses such as the Moore-Penrose and the Drazin inverses. When associated with a regular matrix pencil, it can be expressed by a contour integral formula and can be used, in particular, to write down an explicit representation of the solutions of linear differential algebraic systems. The representation can further be simplified when a well chosen expansion is used for the exponential function. An illustration is given with the expansion in Laguerre functions.

Frequency compensation for a class of DAE's arising in electrical circuits

AbstractStructured perturbations of regular pencils of the form $sE-A,E,A \in {\rm I\!R}^{n\times n},\; s \in {\rm I\!\!\!C}$, are considered which model the addition of a capacitance c in an electrical circuit in order to improve the frequency response. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Singular fractional linear systems and electrical circuits

Singular fractional linear systems and electrical circuitsA new class of singular fractional linear systems and electrical circuits is introduced. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and the Laplace transformation, the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional system if it contains at least one mesh consisting of branches only with an ideal supercapacitor and voltage sources or at least one node with branches with supercoils.

Screening for minimal hepatic encephalopathy in asymptomatic drivers with liver cirrhosis

Background and study aims Minimal hepatic encephalopathy (MHE) represents a part of the spectrum of hepatic encephalopathy (HE). It can have a far-reaching impact on quality and ability to function in daily life and may progress to overt HE. This study was designed to screen for MHE in drivers with liver cirrhosis in Mansoura, a city in the Nile delta in Egypt. Patients and methods A total of 174 consecutive drivers with positive serology for viral markers and cirrhosis were screened for MHE. Questionnaires and standard psychometric tests and well-informed consent were performed at the same setting. The diagnosis of MHE was made when one or both symbol digit test (SDT) and number connection test (NCT) appeared abnormal. Beck’s inventory and Mini Mental State Examination questionnaires were performed for those diagnosed as MHE. After overnight fasting, venous blood samples were taken for haematologic tests and routine liver function tests by conventional methods. Arterial ammonia was also measured. Results A total of 66 patients showed evidence for MHE out of 139 patients who fulfilled the inclusion criteria. No significant differences were present, apart from a significantly elevated arterial ammonia level ( p-value <0.001) and a bad self-reported driving history ( p < 0.05) in the MHE-positive group when compared with the MHE-negative group. Multivariate logistic regression revealed that advanced Child–Pugh grade ( p < 0.001), hepatitis B virus (HBV)-related aetiology ( p < 0.001) and smoking are significant risk factors for MHE. MHE is significantly commoner among Child–Pugh C patients ( p < 0.05) when compared with the other Child–Pugh grades. Conclusion Our data revealed a high prevalence of MHE (47%) among Egyptian drivers with liver cirrhosis. It is hence recommended to include the driving history as well as regular pencil–paper standard psychometric testing in evaluating those at risk, especially in the outpatient setting, for early detection and proper management.

On generalized inverses of singular matrix pencils

On generalized inverses of singular matrix pencilsLinear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.

Construction of an asymptotic solution of one optimal control problem

We construct an asymptotic solution of a singularly perturbed optimal control problem in the case of a regular pencil of boundary matrices.

An isoperimetric type problem for Bézier curves of degree n

We present an efficient algorithm, which solves an isoperimetric type problem in the class of Bézier curves of degree n > 2 . This algorithm requires to compute the maximal characteristic value and the corresponding principal vector of a regular pencil problem with block square matrices of dimension 2 n − 2 . The coordinates of the principal vector determine the control points of the extremal curve, while the characteristic value gives the maximal area bounded by this curve.

Zero assignment of matrix pencils by additive structured transformations

Matrix pencil models are natural descriptions of linear networks and systems. Changing the values of elements of networks, that is redesigning them, implies changes in the zero structure of the associated pencil and this is achieved by structured additive transformations. The paper examines the problem of zero assignment of regular matrix pencils by a special type of structured additive transformations. For a certain family of network redesign problems the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem has certain common features with the pole assignment of linear systems by structured static compensators and thus the new powerful methodology of global linearization [J. Leventides, N. Karcanias, Sufficient conditions for arbitrary Pole assignment by constant decentralised output feedback, Mathematics of Control for Signals and Systems 8 (1995) 222–240; J. Leventides, N. Karcanias, Global asymptotic linearisation of the pole placement map: A closed form solution for the constant output feedback problem, Automatica 31 (1995) 1303–1309] can be used. For regular pencils with infinite zeros, families of structured degenerate additive transformations are defined and parameterized and this lead to the derivation of conditions for zero structure assignment, as well as methodology for computing such solutions. The case of regular pencils with no infinite zeros is also considered and conditions of zero assignment are developed. The results here provide the means for studying problems of linear network redesign by modification of the non-dynamic elements.

Similarity class of a matrix with a prescribed submatrix

In this article we study the possible similarity classes of a square matrix when an arbitrary submatrix is prescribed. As the main result, we solve the even more general problem of describing the possible strict equivalence classes of a regular pencil when a subpencil is prescribed. This result improves the result by [Cabral and Silva, Similarity invariants of completions of submatrices, Lin. Alg. Appl. 169 (1992), 151–161.], since an explicit solution is obtained without any existential quantifiers involved, over an algebraically closed field. In fact, the sufficiency of the conditions obtained by [Gohberg, Kaashoek and Van Schangen, Eigenvalues of completions of submatrices, Lin. Multilin. Alg. 25 (1989), 55–70.] is proved. In the proof, we use various results and techniques including matrix pencils completions, Kronecker canonical form, Littlewood–Richardson coefficients, Young diagrams, the solution of the Carlson problem and localization techniques.