construction of square root factor for solution of the lyapunov matrix equation abstract the lyapunov matrix equation is considered in this paper where the solution is a nonnegative definite matrix i e a matrix admitting decomposition in square root factors an algorithm for findings the square root factor without preliminary finding the solution itself is given

an algebraic approximation for the matrix exponential in singularly perturbed boundary value problems a symmetric difference scheme for linear stiff or singularly perturbed boundary value problems of first order with constant coefficients is constructed being based on a stability function containing a matrix square root its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix local damping of errors uniform stability and uniform second order convergence are proved the computation of the specific matrix square root by a well known stable variant of newton s method is discussed a numerical example confirming the results is given

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

halley s method for the matrix sector function the matrix n sector function is a generalization of the matrix sign function it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues it is known that newton s method which can be used for computing the matrix sign function is not globally convergent for the matrix sector function the only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix in this paper we introduce a new algorithm based on halley s generalized iteration formula for solving nonlinear equations it is shown that the iteration has good error propagation properties and high accuracy finally we give two application examples and summarize the results of our numerical experiments comparing newton s the continued fraction and halley s method

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function this paper presents rapidly convergent and more stable recursive algorithms for finding the principal nth root of a complex matrix and the associated matrix sector function the developed algorithms significantly improve the computational aspects of finding the principal nth root of a matrix and the matrix sector function thus the developed algorithms will enhance the capabilities of the existing computational algorithms such as the principal nth root algorithm the matrix sign algorithm and the matrix sector algorithm for developing applications to control system problems

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

separating eigenvalues using the matrix sign function abstract howland has used the matrix sign function to separate the eigenvalues of a given matrix the derived method required complex arithmetic even in the case of real matrices a modification of the method is presented that does not suffer from this drawback moreover it is shown that the problem may be decomposed into smaller subproblems which leads to considerable savings in computational work

robust output tracking control of siso plants with multiple operating points and with parametric and unstructured uncertainties the robust output tracking control problem is considered for single input single output plants with multiple operating points and with both parametric and unstructured uncertainties i e interval parameter uncertainties and norm bounded unmodelled dynamics the parametric uncertainties can involve parameter perturbations as well as degree variations the plant with uncertainties is assumed to be of minimum phase but is not required to have a fixed number of unstable poles a new method is presented by which a linear time invariant robust controller is designed in two steps first a nominal controller is designed to get exact output tracking for the nominal plant then a robust compensator is added to achieve robust properties it is shown that robust stability robust static state property robust transient property and robust tracking property can be achieved simultaneously a computationally tractable design procedure is presented a salient feature of our results shown in the present paper is th

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

considerations on computing real logarithms of matrices hamiltonian logarithms and skew symmetric logarithms abstract the issue of computing a real logarithm of a real matrix is addressed after a brief review of some known methods more attention is paid to three 1 pade approximation techniques 2 newton s method and 3 a series expansion method newton s method has not been previously treated in the literature we address commutativity issues and simplify the algorithmic formulation we also address general structure preserving issues for two applications in which we are interested finding the real hamiltonian logarithm of a symplectic matrix and finding the skew symmetric logarithm of an orthogonal matrix the diagonal pade approximants and the proposed series expansion technique are proven to be structure preserving some algorithmic issues are discussed

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

perturbation bounds for matrix square roots and pythagorean sums abstract perturbation estimates for the square root r a 1 2 and pythagorean sum p i a 2 1 2 of complex matrices are proved we present bounds in the spectral norm for the cases that r and p are accretive i e have positive definite hermitian parts with special attention to the case of a large condition of a the results are based on bounds for the solution of the matrix sylvester equation and for the separation of two matrices

computing the frequency response of linear systems with parametric perturbation abstract the problem of computing the frequency response of linear systems with parametric perturbation is addressed under the assumption that the coefficients of the system transfer function depend on the perturbation parameters in a linear manner we provide results for plotting the frequency response of the perturbed transfer function these results are useful in determining the h norm gain margin and phase margin and in improving the diagonal dominance of multi input multi out put uncertain systems an illustrative example is provided to demonstrate the results

qft template generation for time delay plants based on zero inclusion test abstract plant template generation is the key step in applying quantitative feedback theory qft to design robust control for uncertain systems in this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients the main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f t q f q t q q k 0 m 1 q k q k where f q g q h q e j g q and h q are both complex valued affine functions of the m dimensional real vector q and is a fixed frequency based on this condition an efficient algorithm which involves in the worst case evaluation of m algebraic inequalities and solution of m2m 1 one variable quadratic equations is developed for testing the zero inclusion of the value set f t q this zero inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution the proposed template generation technique has a linear computational complexity in resolution and is therefore more efficient than the parameter gridding and interval methods a numerical example illustrating the proposed technique and its computational superiority over the interval method is included

numerically stable improved chebyshev halley type schemes for matrix sign function a general family of iterative methods including a free parameter is derived and proved to be convergent for computing matrix sign function under some restrictions on the parameter several special cases including global convergence behavior are dealt with it is analytically shown that they are asymptotically stable a variety of numerical experiments for matrices with different sizes is considered to show the effectiveness of the proposed members of the family

investigation of robust stability of fractional order multilinear affine systems 2q convex parpolygon approach abstract this paper discusses the robust stability problem of fractional order systems with the multi linear affine uncertainty structure the 2 q convex parpolygon approach has been extended to compute the value set of the fractional order uncertain system and to investigate the robust stability via zero exclusion principle an illustrative example is included for fractional order multi linear affine system to present the advantages of the 2 q convex parpolygon approach over classical value set computation methods in the stability investigation

decoupling of measurable signals via self bounded controlled invariant subspaces minimal unassignable dynamics of feedforward units for prestabilized systems a dynamic feedforward scheme allows measurable signal decoupling to be solved independently of other problems simultaneously present in the design of a control system like e g stabilization robustness insensitivity to disturbances the synthesis procedure based on the properties of self bounded controlled invariant subspaces ensures the minimal complexity of the feedforward unit in terms of minimal unassignable dynamics and minimal dynamic order in the case of left invertible systems and on some specific conditions also in the case of non left invertible systems the output dynamic feedback set up to guarantee stability or more generally robustness or insensitivity properties does not affect the complexity of the feedforward unit since the peculiar layout where the feedback unit receives an additional input from the precompensator preserves in the extended system exactly the same set of unassignable internal eigenvalues of the minimal self bounded controlled invariant subspace as that defined for the original system the overall control structure turns out to be a two degree of freedom controller completely devised in the geometric context

a fast method for computing the principal nth roots of complex matrices abstract based on the generalized continued fraction method for finding the nth roots of real numbers this paper presents a fast computation method for finding the principal nth roots of complex matrices computation algorithms with high convergence rates are developed and their global convergence properties are investigated from the viewpoint of systems theory

numerical stability of the halley iteration for the solution of a system of nonlinear equations let f rq rq and x a simple root in rq of the system of nonlinear equations f x 0 abstract pade approximants apa and abstract rational approximants ara for the operator f have been introduced in 2 and 3 the adjective abstract refers to the use of abstract polynomials 5 for the construction of the rational operators the apa and ara have been used for the solution of a system of nonlinear equations in 4 of particular interest was the following third order iterative procedure a2 xi xi 2 il iai f fi ai2 with f the 1st frechet derivative of f in xi ai f fi the newton correction where fi f xi fi the 2nd frechet derivative of f in xi where fi a2 is the bilinear operator fi evaluated in ai ai and componentwise multiplication and division in rq for q i this technique is known as the halley iteration 6 p 91 in this paper the numerical stability 7 of the halley iteration for the case q i is investigated and illustrated by a numerical example 1 numerical stability of iterations we consider the numerical solution of the equation 1 f x 0 with f rq rq x f x abstract analytic in 0 5 assume that 1 has a simple root x we briefly repeat the definition of condition number given by wozniakowski 7 the condition number should measure the sensitivity of the solution output with respect to changes in the data input we assume that f depends parametrically on a vector d e rp called data vector f x f x d and instead of the exact value f x d we only have the computed value fl f x d in t digit floating point binary arithmetic at best we can expect that fl f x d is the exact value of a slightly perturbed operator at slightly perturbed data 2 fl f x d i af f x ax d ad where i is the q x q unit matrix and 1axii 1 let us now suppose that f x d 0 is solved by an iterative procedure 4 xi f where f can use several fj thejth frechet derivative of f at xi if j i or 2 a single or double prime is used instead of the superscript j if xi is the sequence of successive approximations of x we can at best expect xi to be the representation of a computed value for x llxi x 1 00 is numerically stable if limi c iiii 00 00 lim 1i4 1i 1 we will now prove numerical stability of this iteration under assumptions similar to the assumptions for the newton iteration we will also assume that the divisions in 4 are such that 5 0 iiaiiij kpk 1 2f ji ka fi f7 ai j 0p this content downloaded from 157 55 39 161 on mon 23 may 2016 05 03 48 utc all use subject to http about jstor org terms 174 annie a m cuyt condition 5 takes care of the fact that the denominator of the correction term in 4 does not become too small in comparison with o iai li kpk the assumption of 5 is a natural generalization of the following relations a forq 1 lim a 1l i oco ar f7 fl a72 5a and so 3l e n vi l a i d case 1 k 0 1 0 with d e r in a convergent process 4 lim iix xill 0 and thus lim ai o i e 3m e n v vi m ai 0 p 00 and so vi m a72 o ilaillp also 2 lim ai 0 i e i ooii a ai 2 3n e n dvi n 0 p 5b a 2 a2 and so vi max n m i

convexity inequalities for estimating free energy and relative entropy for quantum systems it is shown that the relative entropy s p q tr p log p tr p log q of two positive semi definite operators p and q satisfies gamma 1 tr p p1 gamma q gamma or s p q or gamma 1 tr p1 gamma qgamma p for 0 gamma or 1 and that these bounds become exact in the limit gamma to phi analogous inequalities hold for states in classical statistical mechanics or information theory with trace replaced by integration or summation furthermore the average of these bounds is in general a better approximation to s p q than either bound alone and the average is amenable to further improvement via repeated richardson extrapolation if p and q are gibbs equilibrium states then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation

a note on mode decoupling of linear time invariant systems using the generalized sign matrix it is shown how to decouple a selected set of modes of linear time invariant systems the decoupling procedure is based on a spectral decomposition of the system matrix using the generalized sign function this decomposition does not require knowledge of eigenvalues and eigenvectors the resulting projectors act as decoupling matrices for a spectral domain of the solution this decomposition is particularly useful in linear time invariant system equations the computational procedure is given within

brief every stabilizing dead time controller has an observer predictor based structure this paper considers the stabilization problem for linear time invariant systems with a single delay h in the feedback loop two parametrizations of all stabilizing controllers are derived in terms of a state space realization of the rational part of the plant these parametrizations have simple structures and clear interpretations the first is in the form of a generalized dead time compensator smith predictor and the second is in the observer predictor form applications of the proposed parametrization to the h 2 optimization and the robust stabilization problems are discussed

computation of the matrix sign function using continued fraction expansion we describe an algorithm which computes the sign function of a complex matrix by using the continued fraction expansion of the inverse of the principal square root function at each step of the iteration we show that the algorithm iteratively computes globally convergent main diagonal pade spl acute approximants the proposed algorithm avoids computing large matrix powers and performs fewer matrix inversions than newton s method the algorithm is multiplication rich and particularly suitable for implementation on vector and parallel computers the stability analysis of the algorithm suggests that the errors introduced during a step are either suppressed or have limited effect on the next step finally we summarize the results of our experiments on computing the sign function of certain matrices

on iterative algorithms for the polar decomposition of a matrix and the matrix sign function in this paper we consider relations between the principal iterations from the pade family of kenney and laub for computing the matrix sign function and the principal iterations from the reciprocal pade family of greco iannazzo and poloni and the dual pade family of zi tak we show global convergence of the principal reciprocal pade iterations and the principal dual pade iterations we adopt the dual pade family of iterations to which the newton method belongs for computing the unitary polar factor of a nonsingular matrix we present numerical experiments with the scaled newton method of higham for computing the unitary polar factor which show how the quality of the computed inverses of matrices in the scaled newton method affects the accuracy of the computed polar factorization of a nonsingular matrix it indicates that assumptions under which the backward stability of the scaled newton method has been proved in the literature cannot be weaker

spectral decomposition of a matrix using the generalized sign matrix an algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors a set of eigenprojectors are defined which covers the entire spectrum of a matrix and special attention is given to the projection on the zero eigenvalue some useful applications are discussed in the paper

newton s method for the matrix square root one approach to computing a square root of a matrix a is to apply newton s method to the quadratic matrix equation f x x2 a 0 two widely quoted matrix square root iterations obtained by rewriting this newton iteration are shown to have excellent mathematical convergence properties however by means of a perturbation analysis and supportive numerical examples it is shown that these simplified iterations are numerically unstable a further variant of newton s method for the matrix square root recently proposed in the literature is shown to be for practical purposes numerically stable

residual methods for the large scale matrix pth root and some related problems abstract the problem of finding the p th root of a matrix has received special attention in the last few years standard approaches for this problem include and combine some variations of newton s method which in turn involve matrix factorizations that in general are not suitable for large scale problems motivated by some recently developed low cost iterative schemes for nonlinear problems we consider and analyze specialized residual methods that only require a few matrix matrix products per iteration and hence are suitable for the large scale case as a by product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate preliminary and encouraging numerical results are presented for computing p th roots of large scale symmetric and positive definite matrices for different values of p

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

a parallel algorithm for principal n th roots of matrices an iterative algorithm for computing the principal nth root of a positive definite matrix is presented the algorithm is based on the gauss legendre approximation of a definite integral we present a parallelization in which we use as many processors as the order of the approximation an analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation and thus the number of processors increases we describe the results of our implementation of an eight processor meiko cs 2 comparing the parallel algorithm to the fastest sequential algorithm which is the hoskins walton method

frequency response of uncertain systems with interval plants frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated several simplifications for efficient computation of the envelope of the nyquist plots of the uncertain family are reported stronger computational reduction results are given for the bode magnitude and phase plots of an interval plant controller family of transfer functions these results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design

a basic family of iteration functions for polynomial root finding and its characterizations abstract let p x be a polynomial of degree n 2 with coefficients in a subfield k of the complex numbers for each natural number m 2 let lm x be the m m lower triangular matrix whose diagonal entries are p x and for each j 1 m 1 its jth subdiagonal entries are p j x j for i 1 2 let lmi x be the matrix obtained from lm x by deleting its first i rows and its last i columns l1 1 x 1 then the function bm x x p x det l m 1 1 x det l m 1 x defines a fixed point iteration function having mth order convergence rate for simple roots of p x for m 2 and 3 bm x coincides with newton s and halley s respectively the function bm x is a member of s m m n 2 where for any m m s m m is the set of all rational iteration functions g x k x such that for all roots of p x then g x i mm i x x i with i x k x and well defined at any simple root given g s m m and a simple root of p x gi 0 i 1 m 1 and the asymptotic constant of convergence of the corresponding fixed point iteration is m 1 g m m for bm x we obtain m 1 m det l m 1 2 det l m 1 if all roots of p x are simple bm x is the unique member of s m m n 2 by making use of the identity 0 i 0 n p i x i x i we arrive at two recursive formulas for constructing iteration functions within the s m m family in particular the family of bm x can be generated using one of these formulas moreover the other formula gives a simple scheme for constructing a family of iteration functions credited to euler as well as schroder whose mth order member belongs to s m mn m 2 the iteration functions within s m m can be extended to any arbitrary smooth function f with the uniform replacement of p j with f j in g as well as in m

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

the sensitivity of the algebraic and differential riccati equations in this paper it is shown that the ideas developed by byers in proc summer research conference ams vol 47 contemporary math american mathematical society providence ri 1984 pp 35 49 on the sensitivity of the algebraic riccati equation can be sharpened and extended to norms other than the frobenius norm this extension is crucial from an interpretive point of view because use of the spectral norm allows an identification between the condition number of the algebraic riccati equation and the damping properties of the closed loop dynamical system moreover this approach has the pleasant feature that it carries over to a completely parallel theory for the sensitivity of the differential riccati equation an area that has not been considered previously

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

a feedback theory of two degree of freedom optimal wiener hopf design the design of linear two degree of freedom stabilizing controllers is treated in a quadratic cost setting the class of all such controllers which give finite cost is established and the tradeoff possible between optimum performance tracking cost sensitivity and stability margins is discussed

mmse design of time and color multiplexing codes illumination multiplexing has proven itself to be a valuable tool in image quality improvement for many computer vision and graphics applications provided that its limitations on photon noise and saturation are properly tackled currently multiplexing codes are constructed according to the maximum signal to noise ratio snr and they are primarily employed for time multiplexing a technique that requires the number of measurements m to be equal to the number of illumination sources n in this work we propose an illumination multiplexing method based on minimum mean square errors mmses for the more general setting of both time and color multiplexing performed together with km n where k is the number of color channels under the umbrella of the proposed mmse formulation the conventional maximum snr approach can be thought of as a special case of the mmse design the formulated mmse problem is a difficult non convex problem but it can be approximated by sequential semi definite convex programming and a 1 d exhaustive search the proposed formulation and algorithm can be readily specialized to max snr and or time multiplexing designs thereby giving the optimized codes a much broader scope of application computer simulations show that the conventional max snr design is suboptimal to the proposed mmse design though both see significant quality improvements as m increases experiments also demonstrate the effectiveness and superiority of the proposed method in illuminating various objects

approximate iterations for structured matrices important matrix valued functions f a are e g the inverse a 1 the square root sqrt a and the sign function their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices a and f a often f a possesses this structure only approximately however intermediate matrices arising during the evaluation may lose the structure of the initial matrix this would make the computations inefficient and even infeasible however the main result of this paper is that an iterative fixed point like process for the evaluation of f a can be transformed under certain general assumptions into another process which preserves the convergence rate and benefits from the underlying structure it is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique we demonstrate our results by verifying all requirements in the case of the iterative computation of a 1 and sqrt a

the pade iterations for the matrix sign function and their reciprocals are optimal abstract it is proved that among the rational iterations locally converging with order s 1 to the sign function the pade iterations and their reciprocals are the unique with the lowest sum of the degrees of numerator and denominator

computing matrix valued nevanlinna pick interpolation abstract we describe a computational method known as the nevanlinna algorithm for the matrix valued nevanlinna pick interpolation the original interpolation problem formulated using the caratheodory class of matrix valued rational functions is first converted to an equivalent setting using the schur class of rational functions as a result the necessary and sufficient pick s condition for the interpolation becomes consistent with the scalar valued formulation so that some efficient techniques developed for the scalar valued interpolation can be employed or modified for the matrix valued case we give a brief yet sufficiently clear derivation and a detailed arithmetic complexity analysis for the algorithm we show that an n point matrix valued nevanlinna pick interpolation using the new algorithm requires approximately 95 nm 3 complex arithmetic operations where m is the matrix dimension

an adaptive angular sweep algorithm for value set construction the generalized mapping theorem provides a means to synthesize convex hull generator points for uncertain polynomials and transfer functions however its application requires maximizing a projection onto a unit ray over a uniform grid of angles which is inefficient and ignores the errors associated with a finite grid the algorithm developed in the paper addresses both of these issues by characterizing errors and synthesizing generator points only when needed to meet a given convergence tolerance

frequency domain criteria for l sup p robust stability of continuous linear systems necessary and sufficient criteria in the frequency domain for robust stability are given under the assumption that coefficients of a characteristic polynomial belong to a transformed l sup p ball three cases are considered in detail p infinity interval uncertainty p 2 ellipsoidal uncertainty and p 1 octahedral uncertainty it is shown that frequency domain stability criteria are efficient when one deals with robust stability problems the frequency domain approach considered can be extended to various problems of robust stability

a fast convergent numerical method for matrix sign function with application in sdes in this article a new matrix iterative method for computing the sign of a matrix is offered convergence along with asymptotical stability of the presented method is studied an accelerated form of the iteration will further be constructed finally the application of the given approach in numerical solution of stochastic differential equations and in solving algebraic riccati equations is pointed out

regions of convergence of a pade family of iterations for the matrix sector function and the matrix pth root in this paper we prove a conjecture on a common region of a convergence of pade iterations for the matrix sector function for this purpose we show that all pade approximants to a special case of hypergeometric function have a power series expansion with positive coefficients using a sharpened version of schwarz s lemma we also demonstrate a better estimate of the convergence speed our results are also applicable to a family of rational iterations for computing the matrix pth root

parallel algorithm for solving some spectral problems of linear algebra abstract an algorithm is described for computing the deflating subspaces of a regular linear matrix pencil b a more precisely the algorithm is intended to compute the projection matrices p and i p onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle this algorithm can be considered as an orthogonal version of the matrix sign function method it possesses a quadratic convergence rate and simultaneously with the projection operators operators computes a condition number of the problem we point out the extent to which this algorithm can be parallelized applications are made to solving the riccati equation

a power method for computing square roots of complex matrices abstract in this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived these methods are globally convergent and are based on eigenvalue shifting and powering specifically it is shown for each positive integerr 2 a convergent method of orderrcan be developed these algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues

computation of the frequency response of systems with uncertain parameters a simplification abstract computation of all possible frequency responses of a linear system that depends on m uncertain parameters is the subject of this paper using a brute force approach this computation could be carried out using an m l dimensional parameter sweep i e there is the usual frequency sweep plus one sweep for each uncertain parameter a much simpler computational alternative is possible if the uncertain system s transfer function has a linear dependence on the parameters and if the set of possible parameter values has a polytopic structure for this class of uncertain systems it has previously been shown that for a single fixed frequency all possible responses can be determined using several 1 dimensional sweeps thus for a range of frequencies the obvious extension would involve several 2 dimensional sweeps however this obvious extension is not the simplest way to carry out the computation indeed this paper will show that all possible responses for a range of frequencies can be determined

robust stability under structured and unstructured perturbations the problem of robust stability for linear time invariant single output control systems subject to both structured parametric and unstructured h sub infinity perturbations is studied a generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time invariant dynamic system under perturbations of mixed type is presented the solution involves calculating the h sub infinity norm of a finite number of extremal plants the problem of calculating the exact structured and unstructured stability margins is then constructively solved a feedback control system containing a linear time invariant plant which is subject to both structured and unstructured perturbations is considered the case where the system to be controlled is interval is treated and a nonconservative easily verifiable necessary and sufficient condition for robust stability is given the solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered

determination of a matrix function using the divided difference method of newton and the interpolation technique of hermite computing a function f a of an n by n matrix a is a frequently occurring problem in control theory and other applications in this paper we introduce an effective approach for the determination of matrix function f a we propose a new technique which is based on the extension of newton divided difference and the interpolation technique of hermite and using the eigenvalues of the given matrix a the new algorithm is tested on several problems to show the efficiency of the presented method finally the application of this method in control theory is highlighted

positive and negative solution of dual riccati equations by matrix sign function iteration abstract for a matrix algebraic riccati equation it is shown that the positive definite stabilizing solution the negative definite anti stabilizing solution and the positive and negative solutions of the corresponding dual equation may all be computed from a single matrix sign function iteration this is potentially far more efficient than four separate schur vector solutions moreover necessary and sufficient conditions are given for the existence of unique stabilizing or antistabilizing solutions of an algebraic riccati equation in terms of the matrix sign of an associated hamiltonian matrix

structured matrices and newton s iteration unified approach abstract recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices we recall some fundamental techniques for such a unification and then specify it in full details for newton s iteration which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step the iteration is particularly suitable for n n structured matrices represented with o n entries of their short generators rather than with their own n 2 entries based on such a representation matrix operations are performed much more rapidly and use much less memory space a major problem is to control the length of the generators which tends to grow quite rapidly in the iterative process two known methods solve this problem for toeplitz like and cauchy like matrices we extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity some novel techniques are introduced in this study in particular for the estimation of the norms of the inverse displacement operators

palindromic matrix polynomials matrix functions and integral representations abstract we study the properties of palindromic quadratic matrix polynomials z p qz pz 2 i e quadratic polynomials where the coefficients p and q are square matrices and where the constant and the leading coefficients are equal we show that for suitable choices of the matrix coefficients p and q it is possible to characterize by means of z well known matrix functions namely the matrix square root the matrix polar factor the matrix sign and the geometric mean of two matrices finally we provide some integral representations of these matrix functions

fast and stable algorithms for computing the principal square root of a complex matrix this note presents recursive algorithms that are rapidly convergent and more stable for finding the principal square root of a complex matrix also the developed algorithms are utilized to derive the fast and stable matrix sign algorithms which are useful in developing applications to control system problems

diagonal dominance for robust stability of mimo interval systems this paper provides sufficient conditions for robust stability of a multivariable interval feedback system the result is based on the robustness of the well known diagonal dominance of the system return difference matrix which allows for a simple application of the nyquist criterion to a multi input multi output mimo system

discrete generalized algebraic riccati equations and polynomial matrix factorization abstract this paper presents an algorithm for solving discrete generalized algebraic riccati equations with the help of an orthogonal projector a generalization of the procedure of forming and correcting the orthogonal projector is considered and also that of correcting the proper solution by the newton raphson scheme the possibility to use the discrete generalized riccati equations for polynomial matrix factorization with respect to the unit circle is demonstrated a numerical example is given

bezout factors and l sup 1 optimal controllers for delay systems using a two parameter compensator scheme the authors consider the simultaneous problem of optimal robust stabilization and optimal tracking for single input single output siso systems in an l sup spl infin setting using a two parameter compensator scheme optimal robustness is linked to the work done by georgiou and smith 1992 in the l sup 2 setting optimal tracking involves the resolution of l sup 1 optimization problems the authors consider in particular the robust control of delay systems they determine explicit expressions of the bezout factors for general delay systems which are in the callier desoer class spl bscr spl circ 0 finally they solve several general l sup 1 optimization problems and give an algorithm to solve the optimal robust control problem for a large class of delay systems

the sign matrix and the separation of matrix eigenvalues abstract the sign matrices uniquely associated with the matrices m j i 2 where j are the corners of a rectangle oriented at 4 to the axes of a cartesian coordinate system may be used to compute the number of eigenvalues of the arbitrarily chosen matrix m which lie within the rectangle and to determine the left and right invariant subspaces of m associated with these eigenvalues this paper is concerned with the proof of this statement and with the details of the computation of the required sign matrices

mapping frequency response bounds to the time domain a rigorous method of converting a transfer function description with any shaped uncertainty region in the complex frequency domain to input specific guaranteed time domain bounds is derived periodic input signals are considered first followed by an extension to aperiodic input such as a step the results can be used with confidence in controller design techniques that make use of the complex frequency plane and require translation to the time domain

the dual pade families of iterations for the matrix pth root and the matrix p sector function in this paper we consider the pade family of iterations b laszkiewicz k email protected tak a pade family of iterations for the matrix sector function and the matrix pth root numer linear algebra appl 16 2009 951 970 and a new dual pade family of iterations for computing the principal pth root of a matrix including the newton and halley methods as particular cases we prove convergence of iterations of these families in certain regions we also propose a new dual pade family of iterations for computing the matrix p sector function and we determine a certain region of convergence for this purpose we study properties of the pade approximants to the function 1 z 1 p we show a connection of the series expansion with respect to b of the iterates generated by iterations of the dual pade family for computing the matrix pth root i b 1 p with binomial scalar expansion of 1 b 1 p

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration

technical communique a note on absolute stability of uncertain systems the absolute stability of nonlinear systems is an important robustness issue which has been studied intensively since its first formulation by lur e recently extensions of the solutions to different versions of this problem have been developed for cases with either structured or unstructured uncertainties in the linear part in this note the circle criterion and the popov criterion are recast in an even more general setting one which accommodates both types of perturbations in the linear plant numerical examples are provided to illustrate the application of these frequency domain tools

computation of the frequency response of multilinear affine systems deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form p s q l sub o q l sub 1 q s spl middot spl middot spl middot l sub n q s sup n whose coefficients depend linearly on q q sub 1 q sub 2 q sub q sup t and the uncertainty box is q q q sub i spl epsiv q sub i q sub i i 1 2 q using the geometric structure of the value set of p s q a powerful edge elimination procedure is proposed for computing the bode nyquist and nichols envelopes of these uncertain systems a numerical example is included to illustrate the benefit of the method presented

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed