on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

a framework for iterative analysis of non classically damped dynamical systems abstract in this paper we propose a general iterative framework to solve the dynamic problem for linear systems with non classical viscous damping a systematic approach is used to derive families of stationary iterative schemes that as an instance of particular interest decouple the equations of motion for numerical study of the system response for such schemes we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems these techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method we demonstrate that the proposed systematic framework based on a novel application of the homotopy analysis method generalizes iterative schemes previously reported in the literature and importantly provides a unified perspective for the study of iterative solutions of dynamic problems further we establish a connection between our results and the theory of iterative schemes for algebraic linear systems thus providing insights on convergence results and applicability of the method numerical examples illustrate the effectiveness of the approach and indicate future research directions

three dimensional circular formations via set stabilization the paper addresses distributed stabilization of a class of circular formations for steered kinematic particles in three dimensional euclidean space information exchange between the particles is modeled by a bidirectional graph which is assumed to have a spanning tree distributed control design is based on a hierarchical approach that relies on set stabilization and a reduction principle for asymptotic stability of closed sets the approach allows to solve for different formation specifications in a decoupled manner

stability of the second order matrix polynomial this note summarizes some existing stability and instability conditions of the second order matrix polynomial which arises in the formulation of classical mechanics aerodynamics and robotic systems also some sufficient conditions for stability or instability of the second order matrix polynomial are newly developed via the relevant linear matrix equation obtained from the lyapunov theory

stability of matrix second order systems new conditions and perspectives modeling of many dynamic systems results in matrix second order differential equations in the paper the stability issues of matrix second order dynamical systems are discussed in the literature only sufficient conditions of stability and or instability for a system in matrix second order form are available in this paper necessary and sufficient conditions of asymptotic stability for time invariant systems in matrix second order form under different types of dynamic loadings conservative nonconservative are derived and a physical interpretation is carried out the stability conditions in the sense of lyapunov the jw axis behavior of eigenvalues are also analyzed as the conditions are gained directly in terms of physical parameters of the system the effect of different loadings on the system stability is made transparent by dealing with the stability issues directly in matrix second order form

a sufficient condition for the stability of interval matrix polynomials the root location of sets of scalar polynomials whose coefficients are confined to intervals and the associated extension to eigenvalues of sets of constant matrices whose coefficients are contained in intervals are reviewed a central result for complex scalar interval polynomials is a theorem developed by v l kharatonov 1978 which states that each member of a set of such polynomials is stable if and only if eight special polynomials from the set are stable the case of interval matrix polynomials is examined and a kharitonov like result for their strong stability is provided this in turn yields a sufficient condition for stability of a set of interval matrix polynomials

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

the extreme eigenvalues and stability of real symmetric interval matrices a novel khoritonov like algorithm for computing the minimal and maximal eigenvalues of n n dimensional symmetric interval matrices is presented it is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2 sup n 1 symmetric vertex matrices whereas its minimal eigenvalue coincides with the minimal of another special set of 2 sup n 1 symmetric vertex matrices as immediate corollaries of this algorithm weak necessary and sufficient conditions for testing the hurwitz and schur stability of symmetric interval matrices where one has to test the stability of 2 sup n 1 and 2 sup n symmetric vertex matrices respectively are obtained

maximal unidirectional perturbation bounds for stability of polynomials and matrices abstract given a stable polynomial or matrix we consider a class of unidirectional perturbations in this note we provide a closed form for the maximal perturbation bounds under which stability is preserved the results are then used to derive a closed form for the maximal stability box around the coefficients of a nominally stable polynomial

necessary and sufficient conditions for the output feedback regularization of descriptor systems the problem of output feedback regularization of descriptor systems is studied necessary and sufficient conditions are given to regulate the descriptor systems by derivative and proportional output feedback into a closed loop regular system which is of index at most one

generalized matrix euclidean algorithms for solving diophantine equations and associated problems abstract based on the matrix fraction descriptions of multi variable control systems this paper presents the generalized matrix euclidean algorithms for solving diophantine equations and associated problems a chain rule is developed for finding the greatest common right left divisor of two square non square polynomial matrices and the irreducible right left matrix fraction description from the reducible or irreducible left right matrix fraction description also the development of multi variable pole assignment controllers are discussed for engineering applications

grey box modelling and identification using physical knowledge and bayesian techniques abstract advanced control design requires a model that describes process behaviour adequately such a model can be constructed using physical modelling or statistical identification techniques both have their disadvantages physical models are rigorous and thus often expensive to construct while black box model structures are not necessarily compatible with physical reality since both approaches have undeniable merits as well their combination seems to be attractive and rewarding this paper discusses an approach to statistical estimation where a best linearly parametrized dynamic regression model is identified which is also consistent with specified knowledge about process responses such characteristics are forced upon the black box models yielding a grey box model set it will be shown that crucial physical knowledge such as process stability and sign of stationary gains can be translated into linear inequality constraints on the black box model parameters in order to select best estimators in the grey box class a bayesian approach is adopted given a prior distribution associated with the physical knowledge and given the data likelihood a posterior distribution is constructed maximum and average a posteriori estimators are analysed explicit solutions are given for special cases of gaussian likelihood and a prior which is uniformly or piece wise linearly distributed on a linearly constrained grey box model class finally simulation results and an application to a distillation process show the advantage of the contrained estimates under realistic experiment conditions considerable variance reductions at the cost of a somewhat larger bias can be achieved indicating the potential for e g adaptive control applications

technical communique maximal perturbation bounds for robust α stability of matrix second order systems with one parameter perturbations in this paper the robust a stability problem of matrix second order systems with perturbations in the form of a one parameter family of matrices is investigated all the system matrices including the second order differential coefficient matrices are assumed to have such perturbations based on the kronecker product a necessary and sufficient condition for the robust a stability problem is presented by transforming such a problem into checking the nonsingularity of a class of uncertain matrices then a closed form for the maximal perturbation bounds for preserving the a stability is given finally illustrative examples are given to show that our results are effective and less conservative than the results obtained by other researchers

the canonical diophantine equations with applications a fundamental relationship between appropriate pairs of polynomial matrices is presented this relationship termed canonical diophantine equations can be used to resolve a number of standard polynomial matrix problems here the general diophantine equation is constructively resolved in a unique minimal way in addition prime canonical factorizations of a system transfer matrix are derived from knowledge of any dual factorization

application of kharitonov s theorem to mechanical systems kharitonov s theorem is used to derive a robust stability condition for proportional integral derivative pid controlled multi degree of freedom mechanical systems the characteristic equation of such a system is given as a determinant of a third order polynomial with matrix coefficients from which a scalar interval polynomial is obtained a simple procedure for designing pid controllers for these mechanical systems is described and a new kharitonov like result which states roughly that a controller designed for an upper bounding inertia matrix results in stable set point regulation for all other inertias is proved

diagonal dominance of damping and the decoupling approximation in linear vibratory systems a common approximation in the analysis of non classically damped systems is to ignore the off diagonal elements of the modal damping matrix this procedure is termed the decoupling approximation it is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant in addition the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant it is shown numerically in this paper that over a finite range errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off diagonal elements decreasing continuously in magnitude an explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients small off diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation any error criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate

a sufficient condition for the stability of matrix polynomials a sufficient condition is presented for the stability of the matrix polynomials based on algebraic properties of the matrix coefficients the stability condition is derived from the lyapunov theory by a multivariable feedback system that is associated to the matrix polynomial illustrative examples are given a block schwarz form related to the matrix polynomial is obtained directly from the given realization algorithm

stability analysis of interval matrices improved bounds new sufficient conditions for the stability of interval matrices are presented based on the lyapunov stability approach as well as on the frequency domain approach the conditions help to overcome the conservatism of the existing criteria for the stability of interval matrices based on a matrix approach several numerical examples are given to demonstrate the new conditions and to compare them with results previously reported

stability of a polytope of matrices counterexamples while there have been significant breakthroughs for the stability of a polytope of polynomials since v l kharitonov s 1978 seminal result on interval polynomials for a polytope of matrices the stability problem is considered far from completely resolved counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case these counterexamples illustrate the fundamental differences between polynomial stability and matrix stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail

sufficient condition for asymptotic stability of discrete interval systems abstract a counter example is given to a recent result on the stability analysis of interval matrices by jiang 1988 a sufficient condition is then presented for the asymptotic stability of the discrete time interval system x k 1 a i x k by testing the norms of extreme matrices of the interval matrix a i such that they are all less than one

two polynomial matrix operations this note presents an algorithm to carry out the division of two polynomial matrices and an algorithm to transform a polynomial matrix into a row reduced one these algorithms are useful in the polynomial matrix fraction description of multivariable systems

simplified sufficient conditions for the asymptotic stability of interval matrices it will be shown that the sufficient conditions for the asymptotic stability of interval matrices proved in mansour 1988 can be further simplified by considering only a part of the extreme matrices

the generalized matrix continued fraction descriptions in the second cauer form this note presents two generalized matrix continued fraction descriptions in the second cauer form obtained from a nonsingular rational matrix and or polynomial matrix fraction descriptions computational algorithms are developed for the expansion and inversion of the generalized matrix continued fraction descriptions which are useful in developing applications to multivariable control systems

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

a division algorithm for polynomial matrices a new algorithm is presented for directly determining both the quotient and the remainder associated with the division of one polynomial matrix by another the procedure requires that the denominator matrix be column proper with nonzero column degrees however unlike earlier algorithms only real matrix multiplications are employed i e there is no need for matrix inversions either explicit or implicit

stability analysis of interval matrices continuous and discrete systems this paper considers the stability properties of dynamical interval systems two new methods of determining an upper bound on interval perturbations are presented one of the methods is applicable to both continuous and discrete systems

eigenvalues of linear viscoelastic systems the calculation of eigenvalues of single and multiple degree of freedom linear viscoelastic systems is considered the assumed viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions current methods to solve this type of problem normally use the state space approach involving additional internal variables such approaches often increase the size of the eigenvalue problem to be solved and can become computationally expensive for large systems here an approximate non state space based approach is proposed for this type of problem the proposed approximations are based on certain physical assumptions which simplify the underlying characteristic equation to be solved closed form approximate expressions of the complex and real eigenvalues of the system are derived these approximate expressions are obtained as functions of the elastic eigenvalues only this enables one to approximately calculate the eigenvalues of complex viscoelastic systems by simple post processing of the elastic undamped eigenvalues representative numerical examples are given to verify the accuracy of the derived expressions

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

stabilisation of matrix polynomials a state feedback is proposed to analyse the stability of a matrix polynomial in closed loop first it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable following the ideas of the routh hurwitz stability procedure for scalar polynomials certain continued fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the euclid s division algorithm which leads to an extension of the well known routh hurwitz stability criteria but this time in terms of matrix coefficients after that stability of the closed loop matrix polynomial is guaranteed based on a corollary of a lyapunov theorem the sufficient stability conditions are i the matrices of one column of the presented array must be symmetric and positive definite and ii the matrices of the cascade realisation must satisfy a commutative condition these stability conditions are also necessary for matrix polynomial of second order

a criterion for flatness of hodge bundles over curves and geometric applications for the definition of the kodaira dimension as well as for a discussion of the conjecture cn m i refer to the bourbaki talk by esnault 7 here i only want to say that c n and cn 1 are true and that for applications to albanese mappings a stronger version of cn m is needed which in 7 is called conjecture c m if m 1 it merely says that in addition to the validity of cn in the special case when y is an elliptic curve and k x f 0 there should exist a surjective morphism y y such that the fibre product x x y is birational to a product y x f for u n 2 m 1 this last requirement follows from a result of arakelov 1 theorem 1 1 which implies that deg f tox r 0 and equality holds if and only if f is isotrivial i e after a finite covering of y the fibre product is birational to a product here cox r is the relative dualizing sheaf see sect 5 1 kawamata s proof of c as presented in 13 depends on a result of fujita 11 about line bundle quotients of f t o x r d i m y l very roughly fujita s result says that arakelov s inequality generalizes to the case when dimx is arbitrary but still dim y 1 here i want to look more closely to what happens if deg f o2x r 0 the natural framework for this question is griffith s theory of variations of hodge structure our main result is the following

approximate decoupling of the equations of motion of damped linear systems abstract one common procedure in solving a normalized damped linear system with non zero off diagonal damping elements is to replace the normalized damping matrix by a selected diagonal matrix the extent of approximation introduced by this method of decoupling the system is evaluated and tight upper bounds on the norm of errors are derived moreover when the normalized damping matrix is diagonally dominant it is shown that decoupling the system by neglecting the off diagonal elements indeed minimizes the error upper bound the results expounded in this paper are applicable to any damped linear dynamical system the coefficient matrices of which are symmetric and positive definite

exact unidirectional perturbation bounds for robustness of uncertain generalized state space systems continuous time cases abstract the robustness of continuous time generalized state space systems with unidirectional perturbations is investigated in this paper by transforming the original problem to a robust rank problem we provide a simple method to compute the exact bounds on such perturbations so that the important features of regularity impulse immunity and stability of nominal systems are all preserved this approach can be viewed as a generalization of fu and barmish s result to generalized state space setups it also provides a tractable solution to many practical robustness problems this is clearly explained by a simple circuit example 1997 elsevier science ltd

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

a less conservative lmi condition for robust cal d stability of polynomial matrix polytopes a projection approach this technical note investigates the problem of checking robust d stability of a polytope of polynomial matrices finsler s projection lemma is taken to derive a new sufficient condition in terms of a linear matrix inequality lmi feasibility problem the basic idea behind this condition is to lift the existing stability condition into the one of larger space by means of finsler s lemma and based on it to introduce additional decision variables examples are given to show that the proposed condition can yield less conservative results than the previous one

semi infinite arrays of isotropic point scatterers a unified approach we solve the two dimensional problem of acoustic scattering by a semi infinite pe riodic array of identical isotropic point scatterers i e objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions this model is appropriate for scatterers on which dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity the problem is also relevant to the scattering of an e polarized electromagnetic wave by an array of highly conducting wires the actual geometry of each scatterer is characterized by a single parameter in the equations related to the single body scattering problem and determined from a harmonic boundary value problem using a mixture of analytical and numerical techniques we confirm that a number of phenomena reported for specific geometries are in fact present in the general case such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions we show that the semi infinite array problem is equivalent to that of inverting an infinite toeplitz matrix which in turn can be formulated as a discrete wiener hopf problem numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes

modal control of large flexible space structures using collocated actuators and sensors the authors consider the problem of assigning the eigenvalues associated with critical modes of a large flexible space structure into a specified region in the left half plane via direct velocity feedback control dvfc using collocated actuators and sensors conditions for the existence of a dvfc using collocated actuators and sensors that can achieve the eigenvalue assignment are derived when there exist nonunique feasible dvfcs the one with least frobenius norm feedback gain is determined an experimental four bay truss is used to illustrate the results

the design of linear controllers with symbolic algebra abstract modern transfer function methods use quite a lot of abstract algebra in the design of control systems in computing the final results symbolic algebra turns out to be an important tool in this paper symbolic computation algorithms will be developed to enable computations directly in the theoretically useful transfer function rings a symbolic software package called hinf will be developed and used both for educational and theoretical purposes

electromagnetic wave diffraction by semi infinite strip grating a nonlinear operator equation in terms of the operator of reflection by a plane periodic semi infinite grating of thin metal strips is obtained for the h polarized electromagnetic wave incidence this equation is solved numerically for an incident plane wave the algorithm created allows the solution of a nonlinear operator equation in the fourier amplitude of the reflected field with the method of successive iterations the initial approach chosen is substantiated and the solution convergence is numerically verified the field reflected by a semi infinite grating is calculated and analyzed for different numerical values of the exciting wave angle of incidence the strip width and the incident field wavelength

new class of control laws for robotic manipulators part 1 non adaptive case a new class of exponentially stabilizing control laws for joint level control of robot arms is introduced it has recently been recognized that the non linear dynamics associated with robotic manip

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

wave diffraction by semi infinite venetian blind type grating the h polarized electromagnetic wave diffraction by semi infinite periodic venetian blind type grating is considered the grating consists of strips placed in the parallel planes the problem is reduced to a nonlinear operator equation with respect to the unknown reflection operator of a structure for its solution a complex approach which consists in using the successive over relaxation method and the gauss seidel method is suggested the near and far fields scattered by the grating are calculated the dependences of the reflection coefficient and diffraction patterns are analyzed for different incidence angles width of strips and distance between them