blind image deconvolution through bezoutians in this paper we introduce a fast algorithm for computing the univariate gcd of several polynomials not pairwise based on the generalized bezout matrix by using barnett s method this novel approach is devoted to presenting an algorithm that permits to solve the problem of blind image deconvolution by computing greatest common divisors gcd of several polynomials specifically we design a specialized algorithm for computing the gcd of bivariate polynomials of blurred images which correspond to z transforms to recover the original image all algorithms have been implemented in matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach

vibration of rotationally periodic structures abstract the purpose of this paper is to determine eigensolutions of a rotationally periodic structure p of period n in terms of those of the i th substructure s i of p let y i be the generalized displacement vector and f i be the generalized force vector at interface i i where substructures s i and s i 1 are connected transfer function formulation of s i implies that there exists a linear operator g mapping y t and f i at interface i i to y i 1 and f i 1 at interface i i 1 in addition the operator g depends on transfer functions and therefore eigensolutions of s i the periodicity of p then requires that gn be an identity map this results in n self adjoint fredholm integral equations the non trivial solutions of which predict eigensolutions of p as a consequence a periodic structure p with period n will have exactly n eigenvalues lying between two consecutive eigenvalues of the substructure s i if the interface i i contains only one degree of freedom finally three examples are illustrated the first example considers anti plane strain vibration of a linear elastic solid containing four slots of infinitesimal width the second example derives exact eigensolutions of a string mass periodic structure the third example illustrates how eigensolutions of an axisymmetric structure can be recovered under the formulation of rotationally periodic structures

comments on efficient robust constrained model predictive control with a time varying terminal constraint set by wan and kothare abstract we present an algorithm that modifies the original formulation proposed in wan and kothare efficient robust constrained model predictive control with a time varying terminal constraint set systems control lett 48 2003 375 383 the modified algorithm can be proved to be robustly stabilizing and preserves all the advantages of the original algorithm thereby overcoming the limitation pointed out recently by pluymers et al min max feedback mpc using a time varying terminal constraint set and comments on efficient robust constrained model predictive control with a time varying terminal constraint set systems control lett 54 2005 1143 1148

approximate gcd and its application to ill conditioned algebraic equations abstract we describe two algorithms of approximate gcd greatest common divisor for polynomials with coefficients of floating point numbers one is for univariate polynomials and the other for multivariate polynomials the algorithms are careful extensions of the conventional euclidean algorithm and they are applied to solving ill conditioned algebraic equations ill conditioned univariate algebraic equations have multiple or close roots and we can ill conditioned system of algebraic equations which have an approprimate common divisor conventional numeric root finding methods such as newton s method give no satisfactory result for such ill conditioned systems by using an approximate gcd algorithm for multivariate polynomials the system is transformed into a well conditioned problems by a combination of algebraic and numeric methods we call such algorithms hybrid algorithms this paper shows the importance and possible fruitfulness of the hybrid algorithm

modal structure of centrifugal pendulum vibration absorber systems with multiple cyclically symmetric groups of absorbers this work studies the symmetry breaking effects on the vibration mode structure of centrifugal pendulum vibration absorber cpva systems when multiple groups of absorbers are used an absorber group is a set of equally spaced identical absorbers absorbers within a group are cyclically symmetric while the entire system is asymmetric because the groups have no pre defined relative angular spacing one rotational and two translational degrees of freedom for the rotor and a single arclength degree of freedom for each absorber are considered in the planar model the well defined structure of the vibration modes is obtained by analytical and numerical investigations of the associated eigenvalue problem this vibration mode structure is similar to that for cpva systems with equally spaced identical absorbers thus the disrupted symmetry from multiple absorber groups does not destroy the vibration mode structure resulting from the cyclic symmetry within each group the critical speeds and flutter instability of the system are investigated

numerical factorization of multivariate complex polynomials one can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components the importance and nature of this problem however justify a special treatment we exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently certify the decomposition with linear traces and apply interpolation techniques to construct the irreducible factors with a random combination of differentials we lower multiplicities and reduce to the regular case estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision we apply our software to study the singularities of stewart gough platforms

a non linear structure preserving matrix method for the low rank approximation of the sylvester resultant matrix a non linear structure preserving matrix method for the computation of a structured low rank approximation s f g of the sylvester resultant matrix s f g of two inexact polynomials f f y and g g y is considered in this paper it is shown that considerably improved results are obtained when f y and g y are processed prior to the computation of s f g and that these preprocessing operations introduce two parameters these parameters can either be held constant during the computation of s f g which leads to a linear structure preserving matrix method or they can be incremented during the computation of s f g which leads to a non linear structure preserving matrix method it is shown that the non linear method yields a better structured low rank approximation of s f g and that the assignment of f y and g y is important because s f g may be a good structured low rank approximation of s f g but s g f may be a poor structured low rank approximation of s g f because its numerical rank is not defined examples that illustrate the differences between the linear and non linear structure preserving matrix methods and the importance of the assignment of f y and g y are shown

min max feedback mpc using a time varying terminal constraint set and comments on efficient robust constrained model predictive control with a time varying terminal constraint set abstract in this paper a robust mpc scheme using a time varying terminal constraint set for input constrained systems with a polytopic uncertainty description is proposed the new scheme is a refinement of the algorithm published in efficient robust constrained model predictive control with a time varying terminal constraint set z wan m v kothare systems control lett 48 2003 375 383 the original result contains an error in the stability proof which is solved in this paper by introducing within horizon feedback p o m scokaert d q mayne ieee trans automat control 43 8 1998 1136 1142 in the on line algorithm the new scheme is proven to be robustly asymptotically stabilizing but at the cost of an increased computational complexity

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

vibration of rings on a general elastic foundation the free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and galerkin analyses natural frequencies and vibration modes are determined as closed form expressions for a ring having a circumferentially varying foundation of very general description the elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle the foundation stiffnesses vary circumferentially the simple eigensolution expressions explicitly show the parameter dependencies lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation and identify which nodal diameter fourier components contaminate a given n nodal diameter base mode of the free ring discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters nondimensional spring stiffness stiffness angle support angle number of springs and location of the springs the predicted effects of these parameters on the natural frequencies are verified numerically as an application and as the motivating problem for the study the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

real polynomial root finding by means of matrix and polynomial iterations frequently one seeks approximation to all r real roots of a polynomial of degree n with real coefficients which also has nonreal roots we split a polynomial into two factors one of which has degree r and has r real roots we approximate them at a low cost and then decrease the arithmetic time of the known algorithms for this popular problem by roughly a factor of n k if k iterations prepare splitting k is a small integer unless some nonreal roots lie close to the real axis but even if there nonreal roots near the real axis we substantially accelerate the known algorithms we also propose a dual algorithm operating with the associated structured matrices at the price of minor increase of the arithmetic time it facilitates numerical implementation our analysis and tests demonstrate the efficiency of our approach

signal enhancement a composite property mapping algorithm a signal enhancement algorithm is developed that seeks to recover a signal from noise contaminated distorted measurements made on that signal this object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing the measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties conditions for the algorithm s convergence are established in which the primary requirement is that a composite property mapping be closed this is a relatively unrestricted condition in comparison to that required of most existing signal enhancement algorithms

vibration mode structure and simplified modelling of cyclically symmetric or rotationally periodic systems this paper proves the modal vibration properties of general rotating cyclically symmetric or rotationally periodic systems including those with central components to which cyclically symmetric

brief paper decomposition principle in model predictive control for linear systems with bounded disturbances considering a constrained linear system with bounded disturbances this paper proposes a novel approach which aims at enlarging the domain of attraction by combining a set based mpc approach with a decomposition principle the idea of the paper is to extend the pre stabilizing mpc where the mpc control sequence is parameterized as perturbations to a given pre stabilizing feedback gain to the case where the pre stabilizing feedback law is given as the linear combination of a set of feedback gains this procedure leads to a relatively large terminal set and consequently a large domain of attraction even when using short prediction horizons as time evolves by minimizing the nominal performance index the resulting controller reaches the desired optimal controller with a good asymptotic performance compared to the standard pre stabilizing mpc it combines the advantages of having a flexible choice of feedback gains a large domain of attraction and a good asymptotic behavior

constrained robust model predictive control for time delay systems with polytopic description this paper considers an infinite horizon min max model predictive control mpc problem for time delay discrete systems with a polytopic uncertainty description which has rarely been addressed in the literature several new solutions to this problem are proposed which are inter complementary if there is an input delay then the single state feedback solution requires all the system matrices to be stable this deficiency can be removed by adding free control moves before a modified state feedback law if there is no input delay then a computationally efficient off line state feedback solution can be applied in order to obtain desirable region of attraction while at the same time preserve reasonably low on line computational effort one can choose the standard solution simulation results are given to illustrate these new solutions

variable projection methods for approximate greatest common divisor computations we consider the problem of finding for a given n tuple of polynomials real or complex the closest n tuple that has a common divisor of degree at least d extended weighted euclidean seminorm of the coefficients is used as a measure of closeness two equivalent representations of the problem are considered i direct parameterization over the common divisors and quotients image representation and ii sylvester low rank approximation kernel representation we use the duality between least squares and least norm problems to show that i and ii are closely related to mosaic hankel low rank approximation this allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic hankel low rank approximation we develop optimization methods based on the variable projection principle both for image and kernel representation these methods have linear complexity in the degrees of the polynomials for small and large d we provide a software implementation of the developed methods which is based on a software package for structured low rank approximation

nonsmooth neural network for convex time dependent constraint satisfaction problems this paper introduces a nonsmooth ns neural network that is able to operate in a time dependent td context and is potentially useful for solving some classes of ns td problems the proposed network is named nonsmooth time dependent network ntn and is an extension to a td setting of a previous ns neural network for programming problems suppose c t t ge 0 is a nonempty td convex feasibility set defined by td inequality constraints the constraints are in general ns nondifferentiable functions of the state variables and time ntn is described by the subdifferential with respect to the state variables of an ns td barrier function and a vector field corresponding to the unconstrained dynamics this paper shows that for suitable values of the penalty parameter the ntn dynamics displays two main phases in the first phase any solution of ntn not starting in c 0 at t 0 is able to reach the moving set c cdot in finite time t h whereas in the second phase the solution tracks the moving set i e it stays within c t for all subsequent times t ge t h ntn is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times this new and peculiar dynamics displayed by ntn is potentially useful for addressing some significant td signal processing tasks as an illustration this paper discusses a number of examples where ntn is applied to the solution of ns td convex feasibility problems

matrix representation of the shifting operation and numerical properties of the eres method for computing the greatest common divisor of sets of many polynomials the extended row equivalence and shifting eres method is a matrix based method developed for the computation of the greatest common divisor gcd of sets of many polynomials in this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the eres method by introducing its complete algebraic representation then we analyse in depth its overall numerical stability in finite precision arithmetic numerical examples and comparison with other methods are also presented

efficient robust constrained model predictive control with a time varying terminal constraint set abstract an efficient robust constrained model predictive control algorithm with a time varying terminal constraint set is developed for systems with model uncertainty and input constraints the approach is novel in that it off line constructs a continuum of terminal constraint sets and on line achieves robust stability by using a relatively short control horizon even n 0 with a time varying terminal constraint set this algorithm not only dramatically reduces the on line computation but also significantly enlarges the size of the allowable set of initial conditions moreover this control scheme retains the unconstrained optimal performance in the neighborhood of the equilibrium the controller design is illustrated through a benchmark problem

method for finding multiple roots of polynomials conventional numerical methods for finding multiple roots of polynomials are inaccurate the accuracy is unsatisfactory because the derivatives of the polynomial in the intermediate steps of the associated root finding procedures are eliminated engineering applications require that this problem be solved this work presents an easy to implement method that theoretically completely resolves the multiple root issue the proposed method adopts the euclidean algorithm to obtain the greatest common divisor gcd of a polynomial and its fast derivative the gcd may be approximate because of computational inaccuracy the multiple roots are then deflated into simple ones and then determined by conventional root finding methods the multiplicities of the roots are accordingly calculated a detailed derivation and test examples are provided to demonstrate the efficiency of this method

structured total least norm and approximate gcds of inexact polynomials the determination of an approximate greatest common divisor gcd of two inexact polynomials f f y and g g y arises in several applications including signal processing and control this approximate gcd can be obtained by computing a structured low rank approximation s f g of the sylvester resultant matrix s f g in this paper the method of structured total least norm stln is used to compute a low rank approximation of s f g and it is shown that important issues that have a considerable effect on the approximate gcd have not been considered for example the established works only yield one matrix s f g and therefore one approximate gcd but it is shown in this paper that a family of structured low rank approximations can be computed each member of which yields a different approximate gcd examples that illustrate the importance of these and other issues are presented

division of generalized polynomials using the comrade matrix abstract let a and b be two polynomials expressed in generalized form i e relative to a given orthogonal basis it is shown how their greatest common divisor d the quotient a d and the quotient and remainder on division of a by b can be determined simultaneously without any conversion into standard power form this is done by applying elementary row operations to the matrix b a where a is the comrade matrix associated with a

factorization results for left polynomials in some associative real algebras state of the art applications and open questions abstract we discuss existence of factorizations with linear factors for left polynomials over certain associative real involutive algebras most notably over clifford algebras because of their relevance to kinematics and mechanism science we put particular emphasis on factorization results for quaternion dual quaternion and split quaternion polynomials a general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non division rings we explain the current state of the art present some new results and provide examples and counter examples

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

the computation of multiple roots of a polynomial this paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise and whose exact form contains multiple roots the addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots but the algorithms presented in this paper preserve the multiplicities of the roots in particular even though the given polynomial is corrupted by noise and all computations are performed on these inexact coefficients the algorithms sew together the simple roots that originate from the same multiple root thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial the algorithms described in this paper do not require that the noise level imposed on the coefficients be known and all parameters are calculated from the given inexact coefficients examples that demonstrate the theory are presented

a geometrical approach to finding multivariate approximate lcms and gcds abstract in this article we present a new approach to compute an approximate least common multiple lcm and an approximate greatest common divisor gcd of two multivariate polynomials this approach uses the geometrical notion of principal angles whereas the main computational tools are the implicitly restarted arnoldi method and sparse qr decomposition upper and lower bounds are derived for the largest and smallest singular values of the highly structured macaulay matrix this leads to an upper bound on its condition number and an upper bound on the 2 norm of the product of two multivariate polynomials numerical examples are provided

the amended dsesc power method for polynomial root finding cardinal s matrix version of the sebastiao e silva polynomial root finder rapidly approximates the roots as the eigenvalues of the associated frobenius matrix we preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters as in cardinal s algorithm we repeatedly square the frobenius matrix in nearly linear arithmetic time per squaring which yields dramatic speedup versus the recent effective polynomial root finder based on the application of the inverse power method to the frobenius matrix

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

constrained linear mpc with time varying terminal cost using convex combinations recent papers ieee transactions on automatic control 48 6 2003 1092 1096 automatica 38 2002 1061 1068 systems and control letters 48 2003 375 383 have introduced dual mode mpc algorithms using a time varying terminal cost and or constraint the advantage of these methods is the enlargement of the admissible set of initial states without sacrificing local optimality of the controller but this comes at the cost of a higher computational complexity this paper delivers two main contributions in this area first a new mpc algorithm with a time varying terminal cost and constraint is introduced the algorithm uses convex combinations of off line computed ellipsoidal terminal constraint sets and uses the associated cost as a terminal cost in this way a significant on line computational advantage is obtained the second main contribution is the introduction of a general stability theorem proving stability of both the new mpc algorithm and several existing mpc schemes ieee transactions on automatic control 48 6 2003 1092 1096 automatica 38 2002 1061 1068 this allows a theoretical comparison to be made between the different algorithms the new algorithm using convex combinations is illustrated and compared with other methods on the example of an inverted pendulum

model predictive control for linear parameter varying constrained systems using ellipsoidal set prediction this paper proposes a new model predictive control mpc method for linear parameter varying systems with bounded parameter variation subject to input constraints the method adopts closed loop prediction and constructs ellipsoidal sets to predict the future states with reasonable computational effort then the information on the parameter variation rate is exploited to improve the accuracy of the prediction furthermore a relaxed terminal condition which guarantees the stability for infinite horizon is introduced to enlarge the stabilizable region it is shown that the feasibility of the mpc problem at the initial step ensures the stability of the closed loop system finally a simulation result illustrates the effectiveness of the proposed method

computing the polynomial remainder sequence via bezout matrices in this paper we address the task of computing the polynomial remainder sequence appearing in the euclidean algorithm applied to two polynomials u x and v x of degree n and m respectively m n via renewed approach based on the approximate block diagonalization of a b ezout matrix b u v more specifically this algorithm consists on using the e cient schur complementation method given in terms of a bmatrix b u v associated with the input polynomials we also compare the proposed approach to the approximate block diagonalization of a hankel matrix h u v in which the successive transformation matrices are upper triangular toeplitz matrices all algorithms have been implemented in matlab numerical experiments performed with a wide variety of test problems show the e ectiveness of this algorithm in terms of e cience stability and robustness be two polynomials with deg u x n deg v x m and m n the classical euclidean algorithm applied to u x and v x returns a sequence of quotients qk x and remainders rk x such that r 1 x u x r0 x v x rk 2 x rk 1 x qk x rk x k 1 k 2 where rk x is the polynomial remainder of the division of rk 2 x and rk 1 x and rk x is the greatest common divisor gcd of u x and v x the question of computing the quotients and the remainders is beginning to be studied by collins and brown see 11 9 10 via e cient euclidean based methods naturally the classical algorithm turns out to be unstable when applied to approximate polynomials one might think of designing stabilized numerical versions of the euclidean algorithm to the approximate case see 25 23 several papers have been devoted to describe the natural relation between the euclidean algorithm and the block lu factorization of the hankel and b

resolution des systemes d equations algebriques abstract let f1 fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field counting the common zeros at infinity an algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation which degree is the number of common zeros this algorithm gives the whole algebraic and geometric structure of the set of zeros multiplicities conjugate zeros when all the polynomials have the same degree the complexity of this algorithm is polynomial relatively to the generic number of solutions

root finding by expansion with independent constraints elimination methods are highly effective for the solution of linear and nonlinear systems of equations but reversal of the elimination principle can be beneficial as well competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms that is their convergence from the start a well known example is the dual linear and nonlinear programming which enhances the power of optimization algorithms we believe that this is just an ad hoc application of general principle of expansion with independent constraints it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization at the end of this paper we comment on other applications and extensions of this principle presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros such as weierstrass durand kerner s root finder which reduces its root finding task to viete s vieta s system of n polynomial equations with n unknowns we adjust this root finder to the approximation of a single zero of p preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear together with computing a zero of a polynomial p the algorithm deflates this polynomial as by product and then could be reapplied to the quotient to approximate the next zero of p alternatively by using m processors that exchange no data one can concurrently approximate up to m zeros of p our tests confirm the efficiency of the proposed algorithms technically our root finding boils down to computations with structured matrices polynomials and partial fraction decompositions our study of these links can be of independent interest e g as by product we express the inverse of a sylvester matrix via its last column thus extending the celebrated result of gohberg and sementsul 1972 22 from toeplitz to sylvester matrix inverses

a linear programming approach to constrained robust predictive control a receding horizon predictive control algorithm for systems with model uncertainty and input constraints is developed the proposed algorithm adopts the receding horizon dual mode i e free control moves and invariant set paradigm the approach is novel in that it provides a convenient way of combining predictions of control moves which are optimal in the sense of worst case performance with large target invariant sets thus the proposed algorithm has a large stabilizable set of states corresponding to a cautious state feedback law while enjoying the good performance of a tightly tuned but robust control law unlike earlier approaches which are based on qp or semidefinite programming here computational complexity is reduced through the use of lp

a structured rank revealing method for sylvester matrix we propose a fast algorithm for computing the numeric ranks of sylvester matrices let s denote the sylvester matrix and h denote the hankel like sylvester matrix the algorithm is based on a fast cholesky factorization of s ts or h th and relies on a stabilized version of the generalized schur algorithm for matrices with displacement structure all computations can be done in o r n m where n m and r denote the size and the numerical rank of the sylvester matrix respectively

the computation of the degree of the greatest common divisor of three bernstein basis polynomials abstract this paper considers the application of the sylvester resultant matrix to the computation of the degree of the greatest common divisor gcd of three bernstein basis polynomials f y g y and h y it is shown that the governing equations can be written in two forms which lead to different sylvester matrices the first form requires that the polynomials be considered in pairs but different pairs of polynomials may yield different computational answers for example the solution of the computations gcd f g and gcd g h may differ from the solution of the computations gcd f g and gcd f h depending on f y g y and h y this problem does not arise when the second form is considered which requires that the three polynomials be considered simultaneously complications arise in both forms because of the combinatorial terms in the bernstein basis functions which cause the entries of the matrices to span several orders of magnitude even if the coefficients of the polynomials are of the same order of magnitude it is shown that the adverse effects of this wide range of magnitudes can be mitigated by postmultiplying both forms of the sylvester matrix by a diagonal matrix of combinatorial terms and preprocessing f y g y and h y by three operations results of gcd computations from the two forms of the sylvester matrix when f y g y and h y are perturbed by noise and with the omission and inclusion of the preprocessing operations are shown

computation of approximate polynomial gcds and an extension computation of approximate polynomial greatest common divisors gcds is important both theoretically and due to its applications to control linear systems network theory and computer aided design we study two approaches to the solution so far omitted by the researchers despite intensive recent work in this area correlation to numerical pade approximation enabled us to improve computations for both problems gcds and pade reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the gcd problem and to devise effective solution algorithms in particular unlike the known algorithms we estimate the degree of approximate gcds at a low computational cost and this enables us to obtain certified correct solution for a large class of input polynomials we also restate the problem in terms of the norm of the perturbation of the zeros rather than the coefficients of the input polynomials which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

euclidean remainders for generalized polynomials abstract given two polynomials a and b expressed in generalized form i e relative to a given orthogonal basis it is shown how the complete set of euclidean remainders can be determined also in generalized form the computational scheme involves reducing the matrix b a where a is the comrade matrix associated with a by elementary row operations to a quasitriangular form as a by product solutions to a set of linear congruences are also obtained

invariance properties and characterization of the greatest common divisor of a set of polynomials on m sets of polynomials from r s of degree d x1d4c5 m d the notion of extended r equivalence emd under x1d4c5 m d is characterized by a complete invariant and a canonical form it is shown that the greatest common divisor g c d of x1d4c5 m d is an invariant of x1d4c5 m d for proper sets x1d4c5 m d it is also shown that the g c d is invariant under the combined action of e r e transformations and shifting operations finally a systematic procedure for the computation of the g c d of x1d4c5 mdis given this procedure is based on the use of elementary row operations on real matrices and shifting operations

gpgcd an iterative method for calculating approximate gcd of univariate polynomials we present an iterative algorithm for calculating approximate greatest common divisor gcd of univariate polynomials with the real or the complex coefficients for a given pair of polynomials and a degree our algorithm finds a pair of polynomials which has a gcd of the given degree and whose coefficients are perturbed from those in the original inputs making the perturbations as small as possible along with the gcd the problem of approximate gcd is transferred to a constrained minimization problem then solved with the so called modified newton method which is a generalization of the gradient projection method by searching the solution iteratively we demonstrate that in some test cases our algorithm calculates approximate gcd with perturbations as small as those calculated by a method based on the structured total least norm stln method and the uvgcd method while our method runs significantly faster than theirs by approximately up to 30 or 10 times respectively compared with their implementation we also show that our algorithm properly handles some ill conditioned polynomials which have a gcd with small or large leading coefficient

constrained robust feedback model predictive control for uncertain systems with polytopic description for the constrained uncertain systems with polytopic description a new design method of robust feedback model predictive control is proposed by using a sequence of feedback control laws and designing the parameter dependent lyapunov function for each model the design method provides more degrees of freedom and then can improve the control performance and enlarge the feasible region based on the characteristic property of the design method an offline design algorithm is developed to reduce the mpc controller s online computation burden the numerical examples verify the effectiveness of the results presented in this article

a subdivision method for computing nearest gcd with certification a new subdivision method for computing the nearest univariate gcd is described and analyzed it is based on an exclusion test and an inclusion test the exclusion test in a cell exploits taylor expansion of the polynomial at the center of the cell the inclusion test uses smale s a theorems to certify the existence and unicity of a solution in a cell under the condition of simple roots for the distance minimization problem we analyze the complexity of the algorithm in terms of a condition number which is the inverse of the distance to the set of degenerate systems we report on some experimentation on representative examples to illustrate the behavior of the algorithm

constrained rhc for lpv systems with bounded rates of parameter variations this paper studies a stabilization problem of polytopically uncertain linear parameter varying systems with input constraints and bounded rates of parameter variations in the framework of finite receding horizon control rhc a system containing parameter uncertainties is modified into a system with parameter incremental uncertainties within each horizon for the system modified in this manner a robust rhc is designed by solving an optimization problem at each time instant based on the feasibility of the problem and the optimality of its solution the closed loop system stability is guaranteed a numerical example is included to illustrate the validity of the results

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

tuning of the natural frequency spectrum of a circular plate by in plate stress abstract initial stresses are often introduced deliberately into centrally clamped peripherally free circular plates to increase the plate natural frequencies increased natural frequencies in the critical modes extend the range of stable rotation speed current axisymmetric initial stressing schemes increase the natural frequencies of the zero nodal circle two and higher nodal diameter vibration modes but the zero and one nodal diameter natural frequencies decrease for sufficiently large initial stress the zero nodal diameter natural frequency vanishes and the plate experiences divergence buckling thereby limiting the initial stress that can be induced in this paper a new initial stressing method is presented whereby the natural frequencies of all zero nodal circle modes increase simultaneously thus avoiding buckling of the plate through excessive initial stress the stress field is induced by applying uniform pressure to the boundaries of eccentric interior holes through proper location of the holes the critical natural frequencies increase monotonically with increased pressure and the magnitudes of the frequency shifts increase with the introduction of additional stressed holes the initial stressing schemes can be directed towards optimal increases in specific vibration modes calculations predict natural frequency increases between 35 and 90 depending on the vibration mode experimental measurements are consistent with the model predictions

constrained mpc for uncertain linear systems with ellipsoidal target sets abstract robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain time varying linear systems with input constraints under fixed state feedback control laws with the introduction of extra degrees of freedom in the form of perturbed control laws these stabilizable regions can be enlarged this was done in lee and kouvaritakis automatica 36 2000 1497 1504 in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets we also extend the analysis to take into account both polytopic and unstructured bounded disturbances as well as unstructured uncertainties

technical communique a synthesis approach of on line constrained robust model predictive control a new method to address constrained robust model predictive control mpc for systems with polytopic description rpc spd is proposed this is motivated by two previous techniques one is the well known robust mpc that parameterizes the infinite horizon control moves into a single linear state feedback law the other is a nominal on line mpc that splits the infinite horizon control moves into a set of free control moves over a fixed horizon and a state feedback law in the terminal region the basic idea here is to adopt the rationale of the second technique to improve feasibility and optimality of the first one a parameter dependent lyapunov function is developed for establishing the closed loop stability the performance of the controller is demonstrated by an example

overdetermined weierstrass iteration and the nearest consistent system we propose a generalization of the weierstrass iteration for overdetermined systems of equations and we prove that the proposed method is the gauss newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure in the univariate case we show the connection of our method to the optimization problem formulated by karmarkar and lakshman for the nearest gcd in the multivariate case we generalize the expressions of karmarkar and lakshman and give explicitly several iteration functions to compute the optimum the arithmetic complexity of the iterations is detailed

technical communique improving off line approach to robust mpc based on nominal performance cost this paper gives two alternative off line synthesis approaches to robust model predictive control rmpc for systems with polytopic description in each approach a sequence of explicit control laws that correspond to a sequence of nested asymptotically invariant ellipsoids are constructed off line in order to accommodate a wider class of systems nominal performance cost is chosen to substitute the worst case one in an existing technique in the design of control law for a larger ellipsoid the second approach further incorporates the knowledge of control laws associated with all smaller ellipsoids so as to further improve feasibility and optimality the effectiveness of the alternative approaches is illustrated by a simulation example

an elementary proof of barnett s theorem about the greatest common divisor of several univarlate polynomials abstract this articule provides a new proof of barnett s theorem giving the degree of the greatest common divisor of several univariate polynomials with coefficients in a field in terms of the rank of a well defined matrix the new proof is elementary and self contained no use of jordan form or invariant factors and it is based on some easy to state properties of subresultants moreover this proof allows one to generalize barnett s results to the case when the considered polynomials have their coefficients in an integral domain

bernstein bezoutian matrices several computational and structural properties of bezoutian matrices expressed with respect to the bernstein polynomial basis are shown the exploitation of such properties allows the design of fast algorithms for the solution of bernstein bezoutian linear systems without never making use of potentially ill conditioned reductions to the monomial basis in particular we devise an algorithm for the computation of the greatest common divisor gcd of two polynomials in bernstein form a series of numerical tests are reported and discussed which indicate that bernstein bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the bernstein and the power basis

an algorithm for reducing complexity in parametric predictive control multi parametric quadratic programming solutions to predictive control can require large numbers of regions which would make practical implementation difficult this paper demonstrates how an underexplored prediction class can be combined with parametric programming to give far simpler solutions and at little cost to optimality the key proposal is to explore how one can give more flexibility in the shape of the predictions without increasing the required numbers of degrees of freedom this flexibility allows terminal regions to be enlarged

a matrix pencil based numerical method for the computation of the gcd of polynomials the paper presents a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s p sub m d of maximal degree d it is based on a previously proposed theoretical procedure karcanias 1989 that characterizes the gcd of p sub m d as the output decoupling zero polynomial of a linear system s a spl circ c spl circ that may be associated with p sub m d the computation of the gcd is thus reduced to finding the finite zeros of the pencil sw aw where w is the unobservable subspace of s a spl circ c spl circ if k dim w the gcd is determined as any nonzero entry of the kth compound c sub k sw a spl circ w the method defines the exact degree of gcd works satisfactorily with any number of polynomials and evaluates successfully approximate solutions

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

computing greatest common divisors and squarefree decompositions through matrix methods the parametric and approximate cases abstract barnett s method through bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way two different uses of this method in computer algebra are introduced here firstly we describe an algorithm for parameterizing the greatest common divisor of several polynomials in k x y being x a parameter taking values in an real field k secondly we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following corless et al r m corless p m gianni b m trager s watt the singular value decomposition for polynomial systems in acm international symposium on symbolic and algebraic computation 1995 pp 195 207 given a family of polynomials whose coefficients are imperfectly known we describe an algorithm for computing their approximate greatest common divisor by using as main tools barnett s method and singular value decomposition computations furthermore we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients

a framework for multiple robust explicit mpc controllers for linear systems this article presents a multi mode explicit controller for constrained linear systems with bounded disturbances using a switching strategy based on model predictive control mpc in the proposed approach the system switches among several mpc controllers having different performance levels the switching is done so as to achieve increasing levels of performance as time evolves reaching the desired controller in finite time steps the conditions needed for switching and robust convergence of the multi mode mpc controllers are provided compared with standard robust explicit mpc implementations the proposed approach has the flexibility of having a large domain of attraction a good asymptotic behaviour and a small number of partitions

an ellipsoidal off line mpc scheme for uncertain polytopic discrete time systems an off line model predictive control mpc method based on ellipsoidal calculus and viability theory is described in order to address control problems in the presence of state and input constraints for uncertain polytopic linear plants subject to persistent disturbances in order to reduce the computational burdens and conservativeness of traditional polytopic mpc schemes the present approach carries out off line most of the computations and it makes use of closed loop predictions to improve the control performance this is done by recursively pre computing suitable ellipsoidal inner approximations of the exact controllable sets and solving on line a simple and numerically low demanding optimization problem subject to a set membership constraint comparisons with three other recent off line mpc approaches are also provided in the final example

technical communique min max predictive control strategies for input saturated polytopic uncertain systems a novel robust predictive controller for input saturated ltv discrete time systems with polytopic model uncertainties is presented the solution is based on the minimization at each time instant of an upper bound of the worst case infinite horizon quadratic cost under the constraint of steering the set valued future state evolutions emanating from the current state into a feasible and positive invariant set a condition on the initial state is given that suffices to ensure problem solvability for all subsequent time instants under the latter the proposed predictive controller is proved to robustly asymptotically stabilize the input saturated ltv polytopic system

fast parallel computation of the polynomial remainder sequence via bezout and hankel matrices if u x and v x are polynomials of degree n and m respectively m n all the coefficients of the polynomials generated by the euclidean scheme applied to u x and v x can be computed by using o log 3 n parallel arithmetic steps and n 2 log n processors over any field of characteristic 0 supporting fft fast fourier transform if the field does not support fft the number of processors is increased by a factor of log log n if the field does not allow division by n the number of processors is increased by a factor of n this result is obtained by reducing the euclidean scheme to computing the block triangular factorization of the bezout matrix associated with u x and v x this approach is also extended to the evaluation of polynomial gcd greatest common divisor over any field of constants in o log 2 n steps with the same number of processors

brief an improved approach for constrained robust model predictive control in this paper we present a new technique to address constrained robust model predictive control the main advantage of this new approach with respect to other well known techniques is the reduced conservativeness specifically the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several lyapunov functions each one corresponding to a different vertex of the uncertainty s polytope