Diagonal Dominance Condition Research Articles

The Singularity/Nonsingularity Problem for Matrices Satisfying Diagonal Dominance Conditions in Terms of Directed Graphs

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles.

DECENTRALIZED CONTROL VIA DISTURBANCE ATTENUATION AND EIGENSTRUCTURE ASSIGNMENT

In this paper a simple approach is proposed for decentralized control of linear large-scale systems. Sufficient conditions for diagonal dominance of closed-loop large-scale systems are derived. Based on these conditions, the interactions between the subsystems can be considered as external disturbances for each isolated subsystem. Then a previously proposed approach is used to attenuate disturbances via dynamic output compensators based on complete parametric eigenstructure assignment. Through attenuation of the disturbances, the closed-loop poles of the overall system are assigned to the desirable region, by assigning the eigenstructure of each isolated subsystem appropriately. An example is given to show the effectiveness of the proposed method.

Optimal Robust Control of Robotic Systems

AbstractThere are continuously increasing demands to robotic systems (RS) for faster operation, higher accuracy, and lower energy cost. For the purpose of feedback stabilization, we design decentralized controllers with maximum degree of robustness. The control design criterion is based on a generalized diagonal dominant (GDD) condition proven to be necessary and su.cient for such controllers to be robust against arbitrary, but bounded, disturbances. It is defined by optimal trade‐off relations between the bounds of disturbances, the required accuracy, and the control force limits.

Output feedback decentralized control of large-scale systems using weighted sensitivity functions minimization

This paper considers the problem of achieving stability and desired dynamical transient behavior for linear large-scale systems, by decentralized control. It can be done by making the effects of the interconnections between the subsystems arbitrarily small. Sufficient conditions for stability and diagonal dominance of the closed-loop system are introduced. These conditions are in terms of decentralized subsystems and directly make a constructive H ∞ control design possible. A mixed H ∞ pole region placement is suggested, such that by assigning the closed-loop eigenvalues of the isolated subsystems appropriately, the eigenvalues of the overall closed-loop system are assigned in desirable range. The designs are illustrated by an example.

COMPUTATION OF TRANSIENT NATURAL CONVECTION IN A SQUARE CAVITY BY AN IMPLICIT FINITE-DIFFERENCE SCHEME IN TERMS OF THE STREAM FUNCTION AND TEMPERATURE

In this article we discuss the application of an implicit scheme to the solution by finite differences of transient natural convection in terms of the stream function and temperature. The second-order energy differential and fourth-order momentum equations are discretized according to the well-known alternate direction implicit (ADI) method. Then the temperature field solution is built based on the classic tridiagonal matrix algorithm (TDMA), and the stream function solution is built based on the two original hypotheses proposed here together with the penta diagonal matrix algorithm (PDMA). The time step in the algorithm is obtained analytically as a function of the Prandtl number and the size of the grid imposing the diagonal dominance condition in the pentadiagonal matrix generated by the hypothesis. We verify the hypothesis and the algorithm using the transient natural convection solution in the following parameter ranges: 10 3 r Ra r 10 6 , 10 -2 r Pr r 10 2 , 21 2 21 r grid r 61 2 61. The transient solution to the problem is presented using the average Nusselt number and local Nusselt numbers on the hot wall, CPU time, the temperature and velocity profiles in the y = 0.5 section, and the temperature in the central point of the cavity. The results of the permanent solution are presented using the maximum value of the stream function within the domain and the average Nusselt number on the cold wall as a function of the Rayleigh number, the relaxation parameter, Prandtl number, and the size of the grid. Finally, both the transient and permanent solution results are compared with the results of five other published studies.

Almost decoupling of uncertain multivariable systems

An effective almost decoupling theory using an internal model reference loop is developed for linear uncertain multivariable systems. Based on non-negative matrix theory, a useful design guide is derived to achieve the generalized diagonal dominance condition for the internal model reference loop. Using a Perron root, a sufficient condition is given to guarantee the inverse transfer matrix to be generalized diagonally dominant for all plants in a given family. One of the main features of the proposed scheme is the reduction of uncertainty in the resultant compensated internal loop system, which makes the subsequent design process of the individual channels using standard SISO techniques much simpler.

Distributed Computation for Linear Programming Problems Satisfying a Certain Diagonal Dominance Condition

A block-coordinate ascent method for a class of linear programming problems whose constraints satisfy a certain diagonal dominance condition is proposed. This method partitions the original linear program into subprograms where each subprogram corresponds uniquely to a subset of the decision variables of the problem. At each iteration, one of the subprograms is solved by adjusting its corresponding variables, while the other variables are held constant. The algorithmic mapping underlying this method is shown to be monotone and contractive. Using the contractive property, the method is shown to converge even when implemented in an asynchronous, distributed manner, and that its rate of convergence can be estimated from the synchronization parameter.

Multigrid solution and grid redistribution for convection–diffusion

AbstractWe examine multigrid solution for convection–diffusion problems in which convective effects are significant. In particular, we consider the case where the fine grid satisfies the associated cell or diagonal dominance condition but coarse grid levels violate this condition. Related numerical experiments are conducted. We also introduce an alternative local elliptic projection technique and compare this with interpolating the error corrections from coarse to fine grids. In addition, we demonstrate the use of grid redistribution and optimization techniques in the multilevel context.

A simple stability criterion for a linear system of neutral integro-differential equations

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the asymptotic stability of the trivial solution of a class of linear neutral integro-delay-differential equations of the formwhere ẋ denotes the right derivative, and then generalize the result to a vector system. Asymptotic stability of the trivial solution of the form (1·1) and several of its variants has been considered by many authors (see the discussion at the end). Also, there exists a well developed fundamental theory for neutral delay-differential equations (e.g. existence, uniqueness, continuous dependence etc. of solutions; see for instance the survey article by Akhmerov[1] and the references in it); however there exists no ‘easily verifiable’ sufficient condition for the asymptotic stability of the trivial solution of (1·1). By the term ‘easily verifiable’ we mean a verification which is as easy as in the case of Routh-Hurwitz criterion or the diagonal dominance condition of a matrix. A number of results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [8] that a linear neutral differential equation can have unbounded solutions even though the related characteristic equation has only purely imaginary roots (see also Gromova[7], Datko[3], Snow [16], Brumley [2]); such behaviour is not possible in the case of ordinary or (non-neutral) delay-differential equations. It is known (theorem 6·1 of Henry [10]) that if the characteristic equation associated with a only with negative real parts and if all the roots are uniformly bounded away from the imaginary axis, then asymptotic stability of the trivial solution can be asserted.

Preconditioning By Incomplete Block Cyclic Reduction

Iterative methods for solving linear systems arising from the discretization of elliptic/parabolic partial differential equations require the use of preconditioners to gain increased rates of convergence. Preconditioners arising from incomplete factorizations have been shown to be very effective. However, the recursiveness of these methods can offset these gains somewhat on a vector processor. In this paper, an incomplete factorization based on block cyclic reduction is developed. It is shown that under block diagonal dominance conditions the off-diagonal terms decay quadratically, yielding more effective algorithms.

Composite matrix inverses and generalized Gershgorin sets

In computational matrix algebra, Gershgorin's Theorem [8] and Ostrowski's Theorem[14] play key roles in establishing regularity conditions for complexn×nmatrices. As a result of this importance as well as possible applications in other areas, several extensions of these results have been made (Kotelyanski[9], Ky Fan [5], Brauer[2], Nwokah[10]), for point set matrices. More recently, further extensions have been made to include regularity conditions for composite matrices (matrices partitioned into blocks) (Brenner [3], Van der Sluis[16], Feingold and Varga[6], Cook [4]). Cook's result was motivated by engineering applications but is rather restrictive since, like Feingold and Varga, it requires that the matrices under consideration be diagonally dominant. In this paper, the diagonal dominance condition is replaced by a more general condition (theH-matrix condition). Estimates for the diagonal blocks of block-partitioned matrix are then obtained. The results on inclusion regions for eigenvalues unifiex the previous results of Feingolg and Varga [6], kotelyanski [9], Brauer [2], and generalizes the earlier results of Nwokah [10], from point matrices to block-partitioned matrices. In a sense, these results complement those of brenner [3] Van der Sluis [16], but like those of Cook [4], have a more immediate direct application to control engineering in the area of computer-aided design of composite feedback control systems (Nwokah [12]), which will be briefly reviewed towards the end of the paper.

Preconditioning by incomplete block cyclic reduction

Iterative methods for solving linear systems arising from the discretization of elliptic/parabolic partial differential equations require the use of preconditioners to gain increased rates of convergence. Preconditioners arising from incomplete factorizations have been shown to be very effective. However, the recursiveness of these methods can offset these gains somewhat on a vector processor. In this paper, an incomplete factorization based on block cyclic reduction is developed. It is shown that under block diagonal dominance conditions the off-diagonal terms decay quadratically, yielding more effective algorithms.

The application of generalized diagonal dominance to linear system stability theory

Abstract This paper shows that the well-known diagonal dominance condition associated with Rosenbrock's stability theorems is unnecessary and can be replaced by a less restrictive generalized diagonal dominance condition. Similar results are derived for generalized block diagonally dominant systems. Whenever possible graphical interpretations are given to facilitate the implementation of these ideas in computer aided design packages.

A quotient-difference algorithm for the determination of eigenvalues of periodic tridiagonal matrices

A generalised sparse factorisation of periodic tridiagonal matrices was introduced in Evans and Okolie [1]. By applying a similar factorisation strategy in a similarity transformation we obtain a sparse form extension of the Rutishauser's [2] L.R. and Q.D. schemes to determine the eigenvalues of a wide class of symmetric periodic tridiagonal matrices under certain diagonal dominance conditions. The method proposed is recommended for use in practical applications such as in the solution of the self-adjoint periodic characteristic Sturm Liouville problem.

LU decomposition of M-matrices by elimination without pivoting

It is shown that if A or − A is a singular M-matrix satisfying the generalized diagonal dominance condition y T A⩾0 for some vector y⪢ 0, then A can be factored into A = LU by a certain elimination algorithm, where L is a lower triangular M-matrix with unit diagonal and U is an upper triangular M-matrix. The existence of LU decomposition of symmetric permutations of A and for irreducible M-matrices and symmetric M-matrices follow as colollaries. This work is motivated by applications to the solution of homogeneous systems of linear equations Ax = 0, where A or − A is an M-matrix. These applications arise, e.g., in the analysis of Markov chains, input-output economic models, and compartmental systems. A converse of the theorem metioned above can be established by considering the reduced normal form of A.

A connective stability result for interconnected passive systems

When a large scale system is formed from linear interconnections of a number of passive subsystems, a diagonal dominance condition on the interconnection matrix is known to ensure asymptotic stability. It is shown that the same condition ensures that stability is retained when a large class of nonlinearities, including nonlinearities with memory, is allowed in the interconnections. In particular, arbitrary distributed delays may be inserted without loss of stability.

A diagonal dominance criterion for exponential dichotomy

Roughly speaking, a system of linear differential equations has an exponential dichotomy if it has a subspace of solutions shrinking exponentially and a complementary subspace of solutions growing exponentially. In the case of constant coefficients, this happens if and only if the eigenvalues of the coefficient matrix have nonzero real parts. In the general case, Lazer has shown that if the coefficient matrix function is bounded and satisfies a diagonal dominance condition (which, in the constant case, is a sufficient but not necessary condition that the eigenvalues have nonzero real parts) then the system has an exponential dichotomy. In this paper we prove the same result with a weaker diagonal dominance condition, thus generalizing a theorem of Nakajima.

Compartmental system analysis: Realization of a class of linear systems with physical constraints

This paper considers the compartmental system analysis from the system theoretic point of view. The specific problem treated here is that of realization of rational transfer function in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x} (t) = Ax(t)+</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bu(t), y(t)= c' x(t)</tex> . This problem, however, differs from the usual one in that physical consideration of the compartmental system naturally leads to a set of constraints on the realization. Specifically the matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is required to satisfy a preassigned sign pattern as well as a diagonal-dominant condition. Realizability conditions and the characterization of minimal realization are discussed in detail. Counter examples are constructed to show that the minimal dimension does not coincide with the McMillan degree of the transfer function. Necessary and sufficient conditions for realizability are obtained when i) the degree of the transfer function is equal to or less than three or ii) the graphical structure of the compartment system is of noncyclic tree type.

A note on irreducibility for linear operators on partially ordered finite dimensional vector spaces

Many of the important applications of the Perron-Frobenius theory of nonnegative matrices assume that certain matrices are irreducible. The purpose of this note is to introduce a weaker condition which can be used in place of irreducibility, even in the more general setting of linear operators on a partially ordered finite dimensional vector space. Applications to convergence theorems, comparison results, and generalized diagonal dominance conditions are given.

A Gerschgorin theorem for difference equations—II

It is shown that the bounded solutions of the difference equation x( m + 1) = A( m + 1) x( m) arise as fixed points of a contraction mapping when A( m) satisfies a diagonal dominance condition on {0, 1, 2,…}.