A method for the calculation of eigenfunction expansions

In this paper, a method is presented for the calculation of the coefficients of the series expansion of a function f(t), in the base orthonormal set made up by the eigenfunctions of the self-adjoint operator $\text{L: L(x(t)) = (}\text{d}\text{dt}\text{)( p(t)(}\text{dx(t)}\text{dt}\text{))?q(t)x(t)}$. We show that the values of the numbers $\text{t}\text{}$x_k> can be obtained by solving the differential equation L + λ) y(t) = Kf(t), in the interval of definition, for each of the eigenvalues λ of L and by using as initial conditions those which determine one of its associated orthonormal functions. This makes the method specially interesting for its implementation on a hybrid computer: One advantage of the proposed method is that the analysis of f(t) does not require the simultaneous presence of the functions of the base set and the problem signal, thus eliminating both the problems of the synchronized generation of signals and the need for storing it in memory.     

How to Compute Interval Inclusions of Geodetic Coordinates from Interval Inclusions of Cartesian Coordinates

In satellite geodesy the position of a point P is usually determined by computing its coordinate vector x with respect to an earth-fixed Cartesian coordinate system S. S is chosen such that a rotational ellipsoid E, closely approximating the surface of the earth, has normal form with respect to S. Since the geodetic coordinates of P with respect to E (ellipsoidal latitude , ellipsoidal longitude , and ellipsoidal height h) describe the location of P with respect to the surface of the earth much better than x, a frequently appearing problem is to compute , , and h from x.In practice this problem is solved by iterative methods, the convergence properties of which are not analyzed in detail but for which fast (numerical) convergence is observed for points near to E.In the present paper a theoretically well founded new method is developed, working for all P having a unique nearest point in E.In addition it will be shown that the new method can be implemented such that interval inclusions for , , and h can be computed from interval inclusions of the components of x.     

Dyadic output feedback laws generated as Kronecker products : optimal and suboptimal solutions

The problem of eigenvalue assignment in the system dx/dt equals; Ax + Bu, y = Dx, using the dyadic output feedback law u = u₀ + q · p^Ty is considered via a formulation developed earlier by the author, in which p and q occur in the Kronecker product vector p?q. The equations governing the values of p and q which give an optimum approximation to a prescribed spectrum of eigenvalues are derived, and a special case is solved. Various facets of the problem of generating suboptimal solutions are discussed.     

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝ^n×n for the general case where the matrix elements a_ij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λ_μ, x^(μ)), μ = 1, ..., n, is suggested for the case where λ_μ is a simple eigenvalue. It is based on the use of an affine interval approximation of a_ij(p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.     

Computing complex eigenvalues of large non-Hermitian matrices

The generalized eigenvalue problem Ax=λBx with a non-symmetric matrix A is solved by means of inverse vector iteration. The algorithm makes use of the band structure of the matrices, thus allowing quite large dimensions (d≤3742). In the application all complex eigenvalues for the resistive Alfvén modes are successively computed.     

Stability regions in the parameter space: D-decomposition revisited

The challenging problem in linear control theory is to describe the total set of parameters (controller coefficients or plant characteristics) which provide stability of a system. For the case of one complex or two real parameters and SISO system (with a characteristic polynomial depending linearly on these parameters) the problem can be solved graphically by use of the so-called D-decomposition. Our goal is to extend the technique and to link it with general M-Δ framework. In this way we investigate the geometry of D-decomposition for polynomials and estimate the number of root invariant regions. Several examples verify that these estimates are tight. We also extend D-decomposition for the matrix case, i.e. for MIMO systems. For instance, we partition real axis or complex plane of the parameter k into regions with invariant number of stable eigenvalues of the matrix A+kB. Similar technique can be applied to double-input double-output systems with two parameters.     

Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems

In this paper, a sufficient and necessary condition for the asymptotic stability of the discrete linear interval system X(k + 1) = A₁X(k) is presented; i.e. all the eigenvalues of each matrix A?A₁, (an interval matrix) have magnitudes less than 1 if and only if all the eigenvalues of every ‘extreme matrix’ of the interval matrix A₁, have magnitudes less than 1.     

Mappings of linear vector spaces induced by a transformation matrix and its applications†

The projections of a vector Ax along a certain orthonormal basis are chosen as the n components of a vector ρ. As the vector x sweeps the n–dimensional space, the locus of the point vector ρ is a closed surface, called the ρ locus. Some of the properties of the ρ locus are similar to those of the eigenvalues of the A matrix. These properties are exploited in the analysis of the unforced, linear time-invariant, dynamical system given by [xdot] = Ax and for the forced systems of the [xdot] = Ax + Bu, where u represents an impulse, step, ramp or sinusoidal input.     

Feedback decoupling-controller design of 3-D systems in state space

This paper solves the input-output decoupling problem of three-dimensional (3-D) systems formulated in state-space representation. The control policy adopted is of the static state feedback type u=Kx+Nw where K, N are appropriate matrices to be determined, x is the system state vector, and w is the new input vector assumed equidimensional to the actual input vector u. The procedure derived determines K and N such that the resulting closed-loop system has a diagonal and nonsingular transfer-function matrix. The case, where only partial input-output decoupling is possible, is also considered, and the corresponding state-feedback matrices K and N are determined. The results are illustrated by simple numerical examples. The required 3-D generalization of the well known Cayley-Hamilton theorem is provided.     

The eigenvalue problem (A − λB)x = 0 for symmetric matrices of high order

Vibrational problems of complex structures treated by the method of finite elements lead to the general eigenvalue problem $\text{(A ? λB)x = 0}$, where $\text{A}$ and $\text{B}$ are symmetric and sparse matrices of high order. The smallest eigenvalues and corresponding eigenvectors of interest are usually computed by a variant of the inverse vector iteration. Instead of this, the smallest eigenvalue can be computed as the minimum of the corresponding Rayleigh quotient for instance by the method of the coordinate relaxation of Faddejew/Faddejewa. The slow convergence of this simple algorithm can however be sped up considerably in analogy to the successive overrelaxation method by a systematic overrelaxation. Numerical experiments indicate indeed a rate of convergence of this coordinate overrelaxation as a function of the relaxation parameter which is comparable to that of the usual seccessive overrelaxation for linear equations. In comparison with known procedures for the solution of the general eigenvalue problem there result some important computational advantages with regard to the amount of work. Finally, the higher eigenvalues can be computed successively by minimizing the Rayleigh quotient of a modified eigenvalue problem based on a deflation process.     

A new metric for L–R fuzzy numbers and its application in fuzzy linear systems

In this paper, a metric based on modified Euclidean metric on interval numbers, for L–R fuzzy numbers with fixed $L(\cdot)$ and $R(\cdot)$ is introduced. Then, it is applied for solving L–R fuzzy linear system (L–R-FLS) with fuzzy right-hand side, so that L–R-FLS is transformed to the minimization problem. The solution of the mentioned non-linear programming problem is our favorite fuzzy number vector solution. Two constructive Algorithms are proposed in detail and the method is illustrated by solving several numerical examples.     

Interval operators of a function of which the Lipschitz matrix is an interval M-matrix

There are several interval iterations by applying interval operators which supply an interval sequence including all solutions of an equationf(x)=0 in a given intervalX ₀ and which require the knowledge of an interval Lipschitz matrixL off. In this paper statements are made about existence and convergence in case thatL is an intervalM-matrix.     

Determination of eigenvalues in Marshall's model reduction

This paper is concerned with the problem of determining eigenvalues, retained in Marshall's reduced order model, which have the largest effect on the retaining state vector or give the best approximate reduced system in the sense of ISE criterion. The main objective is to show that the usual choice of eigenvalues in Marshall's model reduction technique (decomposition of eigenvalues into fast modes and slow modes) does not necessarily give a well approximated reduced order model. Here the measure of approximation is defined and calculated by means of ISE criterion. The effectiveness of this method is explained by two illustrative examples.     

Optimal servo problems with incomplete state feedback

A method of designing optimal dynamic compensators for a multi-input multi-output system with polynomial types of desired outputs assuming that the outputs alone are available for feedback is presented in this paper. A performance index quadratic in the state-variable vector, x and a certain derivative of the input to the plant, u is chosen as the criterion for optimality. The problem is solved by first augmenting the plant by a ‘pr’th order compensator and then reducing the augmented system to a regulator system.     

Power Domination in Circular-Arc Graphs

A set S?V is a power dominating set (PDS) of a graph G=(V,E) if every vertex and every edge in G can be observed based on the observation rules of power system monitoring. The power domination problem involves minimizing the cardinality of a PDS of a graph. We consider this combinatorial optimization problem and present a linear time algorithm for finding the minimum PDS of an interval graph if the interval ordering of the graph is provided. In addition, we show that the algorithm, which runs in Θ(nlogn) time, where n is the number of intervals, is asymptotically optimal if the interval ordering is not given. We also show that the results hold for the class of circular-arc graphs.     

Stochastic H 2 /H ∞-control for a dynamical system with internal noises multiplicative with respect to state, control, and external disturbance

We consider an optimal control problem for a dynamical system under the influence of disturbances of both deterministic and stochastic nature. The system is defined on a finite time interval, and its diffusion coefficient depends on the control signal. The controller in the feedback circuit is assumed to be static, nonstationary, linear in the state vector, and satisfying the condition ‖L‖_∞ < γ that bounds the norm of operator L: v ?z for the transition of external disturbance to the controllable output signal. Solving the optimization H ₂/H _∞-control problem, we get three matrix functions satisfying a system of two differential equations of Riccati type and one matrix algebraic equation. In the special case of a stochastic system whose diffusion coefficient does not depend on the control signal, the system is reduced to two related Riccati equations.     

On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Algorithms are presented to solve the special eigenvalue problem $\text{AZ}\text{=}\text{Zλ}$, where $\text{A}$ is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.     

On higher order centered forms

If the real-valued mappingf has a representation of the formf(x)=f(c)+(x-c) ⁿ h(x), xεX, then we introduce an interval expression which approximates the range of values off over the compact intervalX with ordern+1. The well known centered form is the special casen=1 of this result.