a sufficient condition for stability of a polytope of matrices abstract in this paper we study the stability of interval dynamical systems a sufficient condition for the stability of a polytope of matrices which is shown to be necessary and sufficient for a certain class of matrices is obtained the results developed can also be applied to the stability of a positive cone of matrices and sufficient conditions for the stability of interval dynamical systems are obtained a relationship between real parts of eigenvalues and matrix measures is also presented

work by robert kalaba on automated sensitivity analysis this paper reviews work by robert kalaba on the development of a program nasa for the automated sensitivity analysis of parametrized nonlinear systems over parameter intervals nasa incorporates a fast and efficient algorithm feed for higher order partial derivative evaluation as well as an adaptive homotopy continuation procedure for obtaining all required initial conditions applications are envisioned for fields such as economics and the biological sciences where models tend to be complex and closed form solutions are difficult to obtain

resonances of characteristic modes for perfectly conducting objects resonances i e extrema of the eigenvalues of characteristic modes for closed perfectly conducting objects are investigated the characteristic modal solutions based on the electric magnetic and combined field integral operators efio mfio and cfio are studied and compared with analytical solutions for a sphere all these formulations are found to capture both external radiating and internal cavity resonances predicted by the analytical expressions at the internal resonances the eigenvalues obtained with the efio and mfio based approaches are not correct and the corresponding modes are nonunique these solutions also exhibit a strong duality between the electric tm and magnetic te type modes a connection is found between the external and internal resonances and the condition numbers of the matrices the modal expansion of the cfio based solution is correct even though it also experiences the nonuniqueness of the efio and mfio based solutions

computing bounds to real eigenvalues of real interval matrices in this study a new method with algorithms for computing bounds to real eigenvalues of real interval matrices is developed the algorithms are based on the properties of continuous functions the method can provide the tightest eigenvalue bounds and improve some former research results numerical examples illustrate the applicability and effectiveness of the new method copyright 2007 john wiley sons ltd

optimal antenna currents for q superdirectivity and radiation patterns using convex optimization the high q factor low bandwidth and low efficiency make the design of small antennas challenging here convex optimization is used to determine current distributions that provide upper bounds on the antenna performance optimization formulations for maximal gain q factor quotient minimal q factor for superdirectivity and minimum q for given far fields are presented the effects of antennas embedded in structures are also discussed the results are illustrated for planar geometries

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

counter example to a recent result on the stability of interval matrices by s bialas abstract in bialas 1983 a condition is given which is supposedly necessary and sufficient for the stability of interval matrices we demonstrate via a counter example that the condition given is in fact not sufficient

automatic differentiation of functions of derivatives in a recent paper an algorithm feed was introduced for the systematic exact evaluation of higher order partial derivatives of functions of many variables the present paper demonstrates that feed can be applied to a much broader class of functions than envisioned previously specifically feed can be used to evaluate the higher order partial derivatives of functions which are defined in terms of the derivatives of other functions a task required in many applications

comment on on sufficient conditions for the stability of interval matrices abstract some new sufficient conditions for the stability of interval matrices were given by argoun 1986 these conditions however are incorrect

minimization of antenna quality factor optimal currents on arbitrarily shaped radiators with respect to the minimum quality factor are found using a simple and efficient procedure the solution starts with a reformulation of the problem of minimizing quality factor q as an alternative so called dual problem taking advantage of modal decomposition and group theory it is shown that the dual problem can easily be solved and always results in minimal quality factor q moreover the optimization procedure is generalized to minimize quality factor q for embedded antennas with respect to the arbitrarily weighted radiation patterns or with prescribed magnitude of the electric and magnetic near fields the obtained numerical results are compatible with previous results based on the composition of modal currents convex optimization and quasi static approximations however using the methodology in this paper the class of solvable problems is significantly extended

linear system stability under parameter uncertainties a sufficient condition for linear system stability under parameter variations is presented it yields parameter bounds which are not conservative and the test is easily implemented on a digital computer

robust stability of state space models with structured uncertainties a method for robust eigenvalue location analysis of linear state space models affected by structured real parametric perturbations is proposed the approach based on algebraic matrix properties deals with state space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters a method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed the method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space this problem which can be reduced to finding the real solutions of a system of polynomial equations simplifies considerably when cases with one or two uncertain parameters are considered for these cases necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems

antenna bandwidth optimization with single frequency simulation a method to compute antenna q using an electromagnetic simulation at a single frequency is described this method can easily be integrated into global optimization algorithms in this way the optimization time of some antenna parameters e g bandwidth may be significantly reduced the method is validated by direct comparison with the physical bound of the analyzed structure numerical examples for rectangular antennas and antennas with a rectangular ground plane illustrate the integration of the method into a genetic algorithm the results predicted by optimization agree very well with those obtained using a commercial electromagnetic solver these results suggest that the method can be used to yield antennas with q factors within 20 of their corresponding physical bound

stability analysis of dynamic interval systems abstract the stability analysis of interval matrices with certain stability margins is considered by means of similarity transformation and gershgorin s theorem new criteria are developed and illustrated by examples we also demonstrate a counter example to a recent result by argoun 1986

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

sufficient conditions for stability of interval matrices extremely simple sufficient conditions are presented for the stability of interval matrices these complement the necessary and sufficient conditions developed by bialas 1983 which become unwieldy for large matrices 4 4 and above

robustness analysis of interval matrices based on kharitonov s theorem this paper deals with the robustness analysis problem of interval matrices expansion of det a b is used to get characteristic polynomials of two corresponding systems kharitonov s theorem is applied to test hurwitz properties of the two polynomials a new sufficient condition for all eigenvalues of the interval matrix to lie in a damping cone on complex plan is derived illustrative examples are given

reduced conservatism of eigenvalue estimation provides new techniques to give simple estimates of the eigenvalues of a matrix in some cases the new results theorems 1 and 2 improve those obtained by gershgorin s theorem the additional use of theorems 3 and 4 give less conservative estimates than those using gershgorin s theorem alone

solving nonlinear equations by adaptive homotopy continuation standard homotopy continuation methods for solving systems of nonlinear equations require the continuation parameter to move from 0 to 1 along the real line difficulties can occur however if a point of singularity is encountered during the course of the integration to ameliorate these difficulties this paper proposes extending the continuation parameter to complex values and adaptively computing a continuation path in the complex plane that avoids points giving rise to singularities specifically it is proposed that the continuation parameter move from 0 0i to 1 0i along a spider web grid centered at 1 0i in the complex plane the actual path through the grid is determined step by step in accordance with two objectives short path length and avoidance of singular points a two phase homotopy continuation is used to study the implementation of this idea numerical examples are presented which indicate the effectiveness of the approach

sufficient conditions for the stability of interval matrices the stability of interval dynamical systems is studied sufficient conditions for the polytope of interval matrices are examined and some of the proofs are greatly simplified more generally sufficient conditions are obtained and the approach taken in the new proofs has the potential for further generalizations of the result obtained in this paper

a necessary and sufficient condition for the stability of interval matrices consider the linear dynamical system described by the equation xdot t ax t where a is a constant real matrix with dimensions n n and the elements of matrix a are not known precisely we know only the upper and lower bounds of the intervals in which the elements of matrix a are confined in this paper there is proved a necessary and sufficient condition for the stability of solutions of eqn 1 with incomplete information about the elements of matrix a

new robustness bounds for discrete systems with random perturbations some new stochastic stability robustness bounds for state space models are reported nominally exponentially stable discrete time systems are assumed to be subject to random parameter perturbations and novel bounds are obtained on the maximum variances of these random perturbations to maintain stability robustness the methods employed in the analysis are the existing ones used in the deterministic framework after transforming the stochastic robustness problem to a deterministic one the results are compared with each other and with the exact stability region in an example

antenna synthesis and optimization using generalized characteristic modes a set of modes called generalized characteristic modes is described previously developed characteristic modes exhibit orthogonality properties over an obstacle surface as well as the complete sphere at infinity the generalized characteristic modes described here also exhibit orthogonality properties over the obstacle surface but rather than showing orthogonality over the sphere at infinity they are orthogonal over any region of space similar to generalized inagaki modes theories developed for antenna pattern synthesis and directivity optimization using generalized inagaki modes can be applied in an entirely parallel fashion using generalized characteristic modes comparisons are made between generalized characteristic modes and generalized inagaki modes through three examples

on the analysis of eigenvalue assignment robustness the robust eigenvalue assignment of systems subject to parameter perturbations is addressed based on some essential properties of induced norms and matrix measures the authors derive some sufficient conditions which ensure the assignment of the system s eigenvalues in the specified region irrespective of the system perturbations the robustness bounds for eigenvalue assignment are obtained without the need to solve the lyapunov equation an example is given to illustrate the effectiveness and ease of the proposed analysis methods

complete comparative static differential equations equations having form 1 arise as first order conditions in microeconomic constrained optimization models as defining characterizations for general equilibrium and macroeconomic models and as steady state solution characterizations for descriptive and optimal growth models see silberberg 12 hansen 7 cass and shell 3 and brock l if y r r is continuously differentiable over a neighborhood of a point x0 a in r n l for which y xo lx 0 and iy x0 or 0 then the implicit function theorem guarantees the existence of a continuously differentiable function x n cr r over some open neighborhood n cro of cx in r such that dx z ix yx x x a lyjx a x g e n cp 2

allowable coefficient perturbations with preserved stability of a hurwitz polynomial given a polynomial p s ansn an 11sn 1 a1s a0 which satisfies the routh hurwitz criterion a procedure is given for finding non conservative upper bounds on the allowable variations of the coefficients such that the perturbed polynomial maintains the stability property

the adiabatic analogue of the margolus levitin theorem in this letter we derive the minimum time needed for any state of a given quantum system to evolve adiabatically into a distinct orthogonal state this problem is relevant to deriving physical limits in quantum computation and quantum information processing our result represents an adiabatic analogue to the margolus levitin theorem margolus and levitin 1998 physica d 120 188

adiabatic quantum gates and boolean functions we discuss the logical implementation of quantum gates and boolean functions in the framework of quantum adiabatic method which uses the language of ground states spectral gaps and hamiltonians instead of the standard unitary transformation language

eigenvalue crossing avoidance in characteristic modes the theory of characteristic modes tcm provides a set of eigenvalues and eigenvectors that are determined entirely by a structure s geometry at a particular frequency when examining a design over a range of frequencies consistent interpretation of the modal data is crucial here we present a way of interpreting crossing avoidances observed when tracking characteristic mode eigenvalue traces over frequency using the framework of coupled mode theory a method for decomposing a pair of characteristic modes into auxiliary modes is developed the auxiliary modes have a degeneracy eigenvalue crossing and maintain continuity of modal current distributions through the coupling region of the original characteristic modes we demonstrate several examples using our method on simple objects furthermore we explore the effect of multiplicative noise and lumped reactive loading on induced crossing avoidance

wideband characteristic mode tracking a method is proposed by which a large number of characteristic modes of an arbitrary structure are tracked over a very wide frequency range it is necessary for any serious wideband modal analysis of complex structures the method can track a large number of modes using the modal eigenvectors from a generalized eigenvalue problem parameterized by frequency and has been successfully applied to over one hundred antenna geometries the method is detailed and applied to two distinct geometries and the results discussed

eigenvalue tracking due to parameter variation this note describes an efficient eigenvalue tracking algorithm which is applicable in many engineering systems where the effect of parameter variation on system stability is to be determined starting with the original system eigenvalues the algorithm uses determinants to compute first order eigenvalue sensitivities which are used in an iterative method that converges rapidly to the new eigenvalues corresponding to the parameter change the algorithm tracks all the system eigenvalues starting from a given base case it is shown to be less costly than repeated eigenvalue evaluation and is illustrated by means of a simple power system example

a finite algorithm for the exact evaluation of higher order partial derivatives of functions of many variables abstract a formal algorithm is given for the systematic exact evaluation of higher order partial derivatives of functions of many variables the algorithm improves upon wengert s method in two key respects applications are envisioned wherever gradients jacobians hessians and power series expansions could be employed

technical communique robust stabilization for uncertain discrete singular systems this paper considers the problem of robust stabilization for uncertain discrete singular systems subject to time invariant norm bounded unstructured uncertainty in the state matrix the problem we address is the design of state feedback controllers such that the resulting closed loop system is regular causal as well as stable for all admissible uncertainties in this connection the concept of generalized quadratic stabilizability is defined in terms of matrix inequalities a necessary and sufficient condition for generalized quadratic stabilizability is derived under certain assumptions when this condition holds the explicit formula of a state feedback controller solving the problem is also given the results can be viewed as extensions of existing results on robust stabilization of uncertain discrete standard state space systems

on the relationship between the spectral radius and stability radius for discrete systems the relationship between the spectral radius and the stability radius is established via elementary calculation the result gives a discrete counterpart of the existing result for continuous systems and provides an upper bound for the stability radius the main result of the calculation is summarized in the form of a theorem

nonlocal automated sensitivity analysis previously a complete ordinary differential equation ode system was developed for tracking the solution x of a parameterized system of nonlinear equations 0 x over an interval 0 this paper develops a two phase complex homotopy continuation method for obtaining the required initial conditions at t an initial short artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function which can ameliorate the problem of artificially induced singularities also the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short minimal number of integration steps and numerically stable avoidance of singular points a fortran program nasa has been developed for solving the complete ode system starting with a two phase initialization the program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives numerical experiments are reported which illustrate the program s effectiveness in ref 1 an analytically complete system of ordinary differential equations odes was developed for finding a solution vector x for a parameterized system of nonlinear equations 0 p x the well known ode dx d x was supplemented with odes for the adjoint a and the determinant 6 of the jacobian matrix vx x this differential system in principle permits nonlocal sensitivity analysis i e the tracking of the solution vector x and the sensitivity vector dx d together with the adjoint a and the determinant i over any closed interval 0 where the determinant i remains nonzero a fortran program was used successfully in ref 1 to solve this differential system for several illustrative examples subsequently the program was modified to incorporate a fast and efficient algorithm feed for the automatic evaluation of higher order partial derivatives 2 nevertheless to make the program a truly practical tool a simple and reliable procedure was still needed for obtaining the initial conditions x a and c required by the differential system at the initial parameter point ct this paper develops a simple but effective homotopy continuation method for obtaining these initial conditions which differs in two principal respects from standard homotopy continuation methods first the proposed continuation essentially proceeds through the function rather than through an artificial construct letting f x d x t a standard continuation method for finding a solu tion vector x x to the system of equations 0 f x would generally involve solving a system of equations of the form

correcting argoun s approach for the stability of interval matrices recently xu and shao 1989 pointed out that the approach of argoun 1986 for the stability of interval matrices is incorrect this note corrects his approach so that it can be used to check the stability of linear continuous time systems with perturbations

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

a method for the evaluation of radiation q based on modal approach a new formula for the evaluation of the modal radiation q factor is derived the total q of selected structures is to be calculated from the set of eigenmodes with associated eigen energies and eigen powers thanks to the analytical expression of these quantities the procedure is highly accurate respecting arbitrary current densities flowing along the radiating device the electric field integral equation delaunay triangulation method of moments rao wilton glisson basis function and the theory of characteristic modes constitute the underlying theoretical background in terms of the modal radiation q all necessary relations are presented and the essential points of implementation are discussed calculation of the modal energies and q factors enable us to study the effect of the radiating shape separately to the feeding this approach can be very helpful in antenna design a few examples are given including a thin strip dipole two coupled dipoles a bowtie antenna and an electrically small meander folded dipole results are compared with prior estimates and some observations are discussed good agreement is observed for different methods

individual tracking of an eigenvalue and eigenvector of a parameterized matrix let m a be an n x n complex matrix valued differentiable function of a parameter ci varying over some simply connected region co of the complex plane c in a previous paper l a complete initial value system of differential equations was developed for both the eigenvalues and right and left eigenvectors of m a assuming m a has iz distinct eigenvalues for each c in co in the present paper it is shown that an initial value system can also be developed for tracking a single eigenvalue of m a together with one of its corresponding right or left eigenvectors the computational feasibility of the initial value system is illustrated by numerical example

local and nonlocal comparative static analysis of economic systems a complete differential equation system has been developed for the local and nonlocal comparative static analysis of general parametrized economic systems j x a 0 the differential equations allow both qualitative and quantitative sensitivity studies at parameter points or over parameter intervals a fortran program comstat is now available for the numerical integration of the differential equations the only input requirements are j and initial conditions initialization is rendered routine by a two stage application of comstat derivatives are automatically evaluated via a generalized wengert method freeing the user from ever having to form analytical derivatives comstat is applied to a problem in optimal economic growth to show the sensitivity of the steady state capital labor ratio to changes in the time preference parameter

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

antenna synthesis and optimization using weighted inagaki modes inagaki modes represent a new modal formalism for problems of radiation and scattering from arbitrary discrete and continuous structures these modes can be generalized to be orthogonal over any sector of the sphere at infinity as well as the source region itself theory and application of these modes to problems of pattern synthesis and array optimization including constraints are presented examples are given for a line source an array of printed dipoles and a dipole array in free space