non iterative computation of infimum in discrete time h optimization and solvability conditions for the discrete time disturbance decoupling problem a non iterative method for the computation of the infimum for a class of discrete time h optimal control problems and the solvability conditions for the general discrete time disturbance decoupling problem are given in this paper the method for the computation of the infimum is applicable to systems where the transfer functions from the disturbance input to the measurement output and from the control input to the controlled output are free of unit circle invariant zeros and satisfy certain geometric conditions the solvability conditions we obtained for the general discrete time disturbance decoupling problem are also necessary and sufficient conditions

on the structure of h sup infinity control systems and related extensions an existence condition of a h sup infinity controller which achieves a prescribed norm bound of the closed loop transfer function is derived in the frequency domain based on a generalization of the notion of j lossless systems this condition is regarded as a frequency domain representation of the well known existence condition in the state space represented in terms of the two algebraic riccati equations a notion of j orthogonal complement which is introduced as a generalization of the usual orthogonal complement play an important role in clarifying the fundamental frequency domain structure of the model matching problem and simplifying the computation of controllers the results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank it is shown that the proper controller achieving the prescribed norm bound exists even in this case in nonstandard cases the controller order can be smaller than the plant order

optimal diagonal scaling for infinity norm optimization abstract a solution is presented for the previously unsolved diagonally scaled multivariable infinity norm optimization problem of minimizing d s a s s x s d 1 s over the set of stable minimum phase diagonal d s and stable x s this problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of structured plant uncertainty the result facilitates the design of feedback controllers which optimize the excess stability margin 3 or equivalently the structured singular value 4 of diagonally perturbed feedback systems

stability and mcmillan degree for rational matrix interpolants abstract given a set of tangential interpolation conditions for a rational matrix function one can ask to find an interpolant which 1 meets a stability or norm side constraint as in nevanlinna pick interpolation or 2 has the minimal possible mcmillan degree among all interpolants with no stability side constraint as in the partial realization problem in this paper a connection between the problems is presented specifically it is shown that a minimal degree solution of a given set of interpolation conditions together with an associated mirror image set of interpolation conditions is automatically stable whenever it is unique this result extends recent results of antoulas and anderson for the scalar version of the problem

inner outer factorizations or right invertible real rational matrices abstract in this paper the problems of doing inner outer factorizations of right invertible real rational matrices are studied by using some new properties of inner matrices the problems can be solved as the general cases a state space algorithm for doing this factorization is also proposed

an alternative solution to the h optimal control problems abstract in this paper we present an alternative solution to the problem min x h n n a bxc where a b rmand c are rational matrices in h n n the solution circumvents the need to extract the matrix inner factors of b and c providing a multivariable extension of sarason s h interpolation theory 1 to the case of matrix valued b s and c s the result has application to the diagonally scaled optimization problem int d a bxc d 1 where the infimum is over d x h n n d diagonal

the robust h sub 2 control problem a worst case design the worst case effect of a disturbance system on the h sub 2 norm of the system is analyzed an explicit expression is given for the worst case h sub 2 norm when the disturbance system is allowed to vary over all nonlinear time varying and possibly noncausal systems with bounded l sub 2 induced operator norm an upper bound for this measure which is equal to the worst case h sub 2 norm if the exogeneous input is scalar is defined some further analysis of this upper bound is done and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given the latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature

an assumed covariant strain based 9 node shell element a new shell element which is free from serious locking problems and which does not possess hourglass modes is proposed solutions obtained with this element exhibit good convergence and are satisfactorily insensitive to mesh distortion the element also exhibits good convergence for plates with variable thickness this element is based on the use of assumed covariant strains which are obtained from the covariant strain field defined with respect to the element natural co ordinate system only the linear version for thin shell cases is considered the element performance is tested by application to several standard plate and shell problems a problem involving variable thickness is also presented

l optimization and hankel approximation an important l infty optimization problem arising in feedback system design is shown to be equivalent to an optimal hankel norm approximation problem this enables the problem to be solved using state space methods based on balanced realizations

classical refined and component wise analysis of reinforced shell structures this paper compares early and very recent approaches to the static analysis of reinforced shell wing structures early approaches were those based on the pure semimonocoque theory along with the beam assumptions of the euler bernoulli and timoshenko type the recent approaches are based on a hierarchical one dimensional formulation these are obtained by adopting various polynomial expansions of the displacement field above the cross section of the structure according to the unified formulation which was recently proposed by the first author two classes were developed in the unified formulation framework in the first class taylor expansion models were developed by exploiting n order taylor like polynomials classical beam theories euler bernoulli and timoshenko were obtained as special cases of taylor expansion in the second class lagrange expansion models were built by means of four and nine point lagrange type polynomials over the cross section of the wing the component wise approach was obtai

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

singular j j lossless factorization for strictly proper functions singular j j lossless factorization for strictly proper functions is studied the method of infinite zero compensation derived by copeland and safonov is used to extend the results of kimura to singular systems since j j lossless factorization contains both inner outer factorization and wiener hopf spectral factorization as special cases the results obtained here unify previous results on singular factorization problems

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

families of subcritical spirals in highly counter rotating taylor couette flow a comprehensive numerical exploration of secondary finite amplitude solutions in small gap taylor couette flow for high counter rotating reynolds numbers is provided using newton krylov methods embedded within arclength continuation schemes two different families of rotating waves have been identified short axial wavelength subcritical spirals ascribed to centrifugal mechanisms and large axial scale supercritical spirals and ribbons associated with shear dynamics in the outer linearly stable radial region this study is a first step taken in order to provide the inner structure of the skeleton of equilibria that may be responsible for the intermittent regime usually termed as spiral turbulence that has been reported by many experimentalists in the past

j lossless conjugation and factorization for discrete time systems the j j lossless factorization plays an important role in h control theory just like the wiener hopf factorization does in lqg theory for continuous time systems in this paper the theory of j lossless factorization is extended to discrete time systems the notion j lossless conjugation which is a powerful tool for computing j lossless factorization in continuous time systems is also extended to discrete time systems these extensions are far from trivial the discrete time version j lossless conjugations and factorizations turns out to be much more complicated reflecting the complicated nature of the discrete time riccati equations

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

a generalized chain scattering representation and its algebraic system properties presents a generalization of the chain scattering representation to the case of general plants through the notion of input output consistency the conditions under which the generalized chain scattering representation gcsr and the dual generalized chain scattering representation dgcsr exist are proposed the generalized chain scattering matrices are formulated into a general parameterized form by using the generalized inverse of matrices some algebraic system properties such as the cascade structure property the symmetry duality of the gcsr s and dgcsr s are studied

inner outer factorization for strictly proper functions with j w axis zeros abstract this paper discussed the inner outer factorization for square real rational matrices which may have zeros on the j axis including infinity a factorization method is given in terms of the descriptor form representation of the rational matrix this method uses the solution of a generalized riccati equation we also propose an efficient computation algorithm for solving the generalized riccati equation

the semi stabilizing solution of generalized algebraic riccati equation for descriptor systems this paper considers the generalized algebraic riccati equation gare associated with a non standard j spectral factorization problem ns j sfp for descriptor systems of which system matrices have full column rank we derive necessary and sufficient conditions for the existence of a semi stabilizing solution of the gare based on the hamiltonian matrix pencil approach and gives a parametrization of all the semi stabilizing solutions then we provide a nonstandard j spectral factor ns j sf in terms of a particular semi stabilizing solution of the gare a simple numerical result is also included

parametrization of state feedback h controllers the state feedback h control problem is considered in this paper it is shown that under mild conditions this problem can be solved simply by using the conjugation approach proposed by kimura and kawatani with some extended treatment to the case that t3 s is tall and strictly proper a partial parametrization of state feedback controllers achieving the prescribed norm bound is obtained this is an eminent difference from previous work on the state feedback h control problem where only a static feedback control law the central solution of our result is provided both four block and one block problems are solved in this paper the results of this paper embrace those of doyle et al and petersen

computation of coprime factorizations of rational matrices abstract we propose numerically reliable state space algorithms for computing several coprime factorizations of rational matrices 1 factorizations with factors having poles in a given stability domain 2 factorizations with proper stable factors 3 factorizations with inner and j inner denominators the new algorithms are based on a recursive generalized schur algorithm for pole dislocation they are generally applicable whether the underlying descriptor state space representation is minimal or not and whether it is stabilizable detectable or not the proposed algorithms are useful in solving various computational problems for both standard and descriptor system representations

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

free vibration analysis of civil engineering structures by component wise models abstract higher order beam models are used in this paper to carry out free vibration analysis of civil engineering structures refined kinematic fields are developed using the carrera unified formulation cuf which allows for the implementation of any order theory without the need for ad hoc formulations the principle of virtual displacements in conjunction with the finite element method fem is used to formulate stiffness and mass matrices in terms of fundamental nuclei the nuclei depend neither on the adopted class of beam theory nor on the fem approximation along the beam axis this paper focuses on a particular class of cuf models that makes use of lagrange polynomials to discretize cross sectional displacement variables this class of models are referred to as component wise cw in recent works according to the cw approach each structural component e g columns walls frame members and floors can be modeled by means of the same 1d formulation a number of typical civil engineering structures e g simple beams arches truss structures and complete industrial and civil buildings are analyzed and cw results are compared to classical beam theories euler bernoulli and timoshenko refined beam models based on taylor like expansions of the displacements on the cross section and classical solid shell fem solutions from the commercial code msc nastran the results highlight the enhanced capabilities of the proposed formulation it is in fact demonstrated that cw models are able to replicate 3d solid results with very low computational efforts

j j lossless factorization for descriptor systems abstract the j j lossless factorization for descriptor systems based on the theory of conjugation is studied the result on j j lossless factorization for proper functions are extended to descriptor systems and those on j lossless factorization for square matrices are extended to nonsquare cases since j j lossless factorization contains both the inner outer factorization and the wiener hopf spectral factorization as special cases and the descriptor system contains the state space system asa special case the results obtained here unify the previous results on factorization problems also j j lossless factorization can be used to solve 4 block h control problems with j axis zeros

a spectral characterization of h optimal feedback performance and its efficient computation abstract we study the spectral properties of a toeplitz hankel operator which arises in the context of the mixed sensitivity h optimization problem and whose largest eigenvalue characterizes the optimal achievable performance 0 the existence of such an operator was first shown by verma and jonckheere 26 who also noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the iteration here we investigate this operator in detail with the objective of efficiency computing its spectrum we define an adjoint linear quadratic problem that involves the same toeplitz hankel operator as shown by jonckheere and silverman 13 16 consequently a finite polynomial algorithm allows 0 to be characterized as simply as the largest root of a polynomial finally a computationally more attractive state space algorithm emerges from the h t 8 lq relationship this algorithm yields a very good accuracy evaluation of the performance 0 by solving just one algebraic riccati equation thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e iteration

a symbolic fourier synthesis of a one point integrated quadrilateral plate element a symbolic fourier analysis is performed on both one point and four point spatially integrated four noded c0 plate elements the analysis shows that the effects of different integration rules is characterized by a single fourier component called the directional averaging operator in addition the conditions for avoiding locking and spurious mechanisms are explicitly identified in a general plate operator expressed in fourier form encouraged by these results a c0 plate element is synthesized employing only one point integrated attributes the stiffness matrix of the new element consists of two parts the usual one point integrated part and the rank compensating part that is identified as the difference between the synthesized and one point integrated plate operators the present element achieves the same accuracy of elements employing the four point spatial integration rule which is not realized in the one point integrated elements using hour glass control techniques preliminary numerical experiments corroborate the present theoretical results and demonstrate the usefulness of the present symbolic analysis and fourier synthesis techniques

linearized pipe flow to reynolds number 10 7 a fourier chebyshev petro galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in an infinite circular pipe our code is unusual in being based on solenoidal velocity variables and in being written in matlab systematic studies are presented of the dependence of eigenvalues transient growth factors and other quantities on the axial and azimuthal wave numbers and the reynolds number r for r ranging from 102 to the idealized physically unrealizable value 107 implications for transition to turbulence are considered in the light of recent theoretical results of s j chapman

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

extension of mitc to higher order beam models and shear locking analysis for compact thin walled and composite structures summary a class of mixed interpolated beam elements is introduced in this paper under the framework of the carrera unified formulation cuf to eliminate the detrimental effects due to shear locking the mixed interpolation of tensorial components mitc method is adopted to generate locking free displacement based beam models using general 1d finite elements an assumed distribution of the transverse shear strains is employed for the derivation of the virtual work and the full gauss legendre quadrature is used for the numerical computation of all the components of the stiffness matrix linear quadratic and cubic beam elements are developed using the unified formulation and applied to linear static problems including compact laminated and thin walled structures a comprehensive study of how shear locking affects general beam elements when different classical integration schemes are employed is presented evidencing the outstanding capabilities of the mitc method to overcome this numerical issue refined beam theories based on the expansion of pure and generalized displacement variables are implemented making use of lagrange and legendre polynomials over the cross sectional domain allowing one to capture complex states of stress with a 3d like accuracy the numerical examples are compared to analytic numerical solutions from the literature and commercial software solutions whenever it is possible the efficiency and robustness of the proposed method is demonstrated throughout all the assessments illustrating that mitc elements are the natural choice to avoid shear locking and showing an unprecedent accuracy in the computation of transverse shear stresses for beam formulations this article is protected by copyright all rights reserved

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

on cancellation in h optimal controllers abstract this note clarifies the mechanism of pole zero cancellations that appeared in the derivation of the h optimal controller via a solution of the corresponding model matching problem the cancellable factor is explicitly expressed using chain scattering matrix descriptions a simple state space formula of the sub optimal controller is also presented based on the chain scattering forms and we show that there is a mixed sensitivity problem that can be solved by calculating a solution of only one riccati equation in each iteration

h sup spl infin controller synthesis with time domain constraints for a class of discrete time linear time invariant lti plants we present necessary and sufficient conditions for the existence of a discrete time lti controller which ensures that the h sup spl infin norm of a closed loop map of interest is less than a prespecified level subject to time domain constraints on this closed loop map in cases when such an lti controller does not exist we show that the problem of finding an lti controller that satisfies the input output constraints with the smallest error can be posed as a convex optimization problem based on linear matrix inequalities we also consider briefly other related problems that can be solved completely using a similar approach namely controller synthesis to minimize the optimally scaled h sup spl infin norm and controller synthesis to maximize the guaranteed dissipation

worst case system identification in h sub spl infin validation of a priori information essentially optimal algorithms and error bounds resolves several important open issues pertaining to a worst case control oriented system identification problem known as identification in h sub spl infin first a method is presented for developing confidence that certain a priori information available for identification is not invalid this method requires the solution of a certain nondifferentiable convex program second an essentially optimal identification algorithm is constructed this algorithm is worst case strongly optimal to within a factor of two finally new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm interestingly the development of each of these results draws heavily upon the classical nevanlinna pick interpolation theory as such the authors results establish a clear link between the areas of system identification and optimal interpolation theory both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

minimal indices cancellation and rank revealing factorizations for rational matrix functions we study the problem of eliminating the minimal indices the indices of a minimal polynomial basis of the null space of a rational matrix function by multiplication with a suitably chosen invertible rational matrix function we derive the class of all invertible factors that dislocate the minimal indices to certain zero locations and feature minimal mcmillan degree we impose additional conditions on the factor like being j unitary or j inner either with respect to the imaginary axis or to the unit circle and characterize the classes of solutions en route we extend the well known rank revealing factorization of a constant matrix to rational matrix functions the results are completely general and apply in particular to matrices which are polynomial strictly proper or improper rank deficient with arbitrary poles and zeros including at infinity all characterization are made by using descriptor realizations while the associated computations are performed by employing constant unitary transformations and standard reliable procedures for eigenvalue assignment or for solving riccati equations

higher order mitc general shell elements two mixed interpolated general shell finite elements for non linear analysis a 9 node element and a 16 node element are presented the elements are based on the mixed interpolation of tensorial components mitc approach in which the covariant strain component fields for the in plane and shear actions are interpolated and tied to the also interpolated displacement field both the 9 node element referred to as the mitc9 element and the 16 node element referred to as the mitc16 element are tested numerically and found to have high predictive capabilities

computation of general inner outer and spectral factorizations in this paper we solve two problems in linear systems theory the computation of the inner outer and spectral factorizations of a continuous time system considered in the most general setting we show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic riccati equation of order usually much smaller than the mcmillan degree of the transfer function matrix of the system the proposed procedures are completely general being applicable for a polynomial proper improper system whose transfer function matrix could be rank deficient and could have poles zeros on the imaginary axis or at infinity as an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix numerical refinements and examples illustrating the proposed approach are discussed in detail

two sided tangential interpolation with real rational units in h the problem of computing units in rh that satisfy ball gohberg rodman two sided tangential interpolation conditions over the open right half complex plane is considered we first show that every unit in rh can be written up to left multiplication by an orthogonal matrix as a finite product of positive real functions in rh with invertible value at this factorization is used to recast interpolation conditions on units as coupled positive real interpolation problems for the factors an algebraic necessary and sufficient condition for the existence of solutions and a parameterization of all solutions when one exists are given we introduce a multivariable parity interlacing property for left interpolation data and present an algorithm to compute units the algorithm involves a line search and is guaranteed to terminate in a finite number of operations when the data set has parity interlacing property as an application strong stabilization of lti plants is discussed a procedure to construct a stable stabilizing controller is given the results are illustrated with a simple mimo example a necessary and sufficient condition characterizing all real rational units that satisfy tangential interpolation conditions is given an algorithm to compute units for left interpolation is presented

j j lossless factorization based on conjugation this paper is concerned with a derivation of the state space form of the j j lossless factorization which contains both the inner outer factorization and the spectral factorization of positive matrices as special cases also the j j lossless factorization gives a unified framework of h control theory we use the method of conjugation which makes the derivation much simpler than the previous literature most of which used the technique of j j spectral factorization a necessary and sufficient condition is represented in terms of two riccati equations one of which is degenerated

j lossless and extended j lossless factorizations approach for domain h control the paper addresses problems of numerical conditioning of the discrete time h x control design based on j lossless factorizations of the chain scattering representation of a plant it is demonstrated that for a sufficiently small sampling period forward shift operator techniques may become ill conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so called i operator form of the source problem state space models of all i domain controllers are given the solutions are obtained by considering two coupled i domain algebraic riccati equations that can be solved by applying a generalized eigenproblem formulation numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number an extended j lossless factorization for generalized plants with zeros on the stability boundary circle is also given a so called zero compensator technique adapted to the i domain models is utilized to cancel suc

on the linear stability of spiral flow between rotating cylinders a numerical study is made of the effects of both axisymmetric and non axisymmetric disturbances on the stability of spiral flow between rotating cylinders if we let omega 1 and omega 2 be the angular speeds of the inner and outer cylinders and r 1 and r 2 be their respective radii then for fixed values of eta r 1 r 2 and mu omega 2 omega 1 the onset of instability depends on both the taylor number t and the axial reynolds number r here r is based on the gap width between the cylinders and the average axial velocity of the basic flow while t is based on the average angular speeds of the cylinders using the compound matrix method we have computed the complete stability boundary in the r t plane for axisymmetric disturbances with eta 0 95 and mu 0 we find that for sufficiently high reynolds numbers there are two distinct axisymmetric modes corresponding to the usual shear and rotational instabilities we have also obtained the stability boundaries for non axisymmetric disturbances for r leqslant 6000 for eta 0 95 and 0 77 with mu 0 these last results are found to be in substantial agreement with the experimental observations of snyder 1962 1965 nagih 1972 and mavec 1973 in the low and moderate axial reynolds number regimes

closed form solutions to a class of h optimization problems a closed form solution is presented to the h sub optimal control problem via state or output feedback the problem formulation is general and does not impose any restrictions on feedthrough terms between the control input and the controlled output or between the disturbance input and the measurement output variables it is however required that both the transfer function from the disturbance input to the measurement output and the transfer function from the control input to the controlled output have no invariant zeros on the jw axis and that they satisfy certain geometric conditions for the same class of systems the conditions under which the h optimal control problem via state feedback has a solution are given moreover explicit expressions for the optimal solutions are provided finally the pole zero cancellations in the closed loop system resulting from the h optimal or sub optimal state or output feedback are examined

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

computation of inner outer factorization of rational matrices we propose a new numerically reliable computational approach to determine the inner outer factorization of a rational transfer matrix g of a linear descriptor system in contrast to existing computationally involved one shot methods which require the solution of riccati or generalized riccati equations the new approach relies on an efficient recursive zeros dislocation technique the resulting inner and outer factors always have minimal order descriptor representations the approach proposed is completely general being applicable whenever g is proper strictly proper or not or of full column row rank or not

h sub spl infin compensator design with minimal order observers a method based on a bounded real lemma is presented in the form of sufficient conditions for designing observer based proper reduced order h sub spl infin compensators the proposed compensators are shown to include the strictly proper full order central h sub spl infin compensators as a special class a compensator design algorithm is included to illustrate the use of the proposed compensator design equations

matrix valued nevanlinna pick interpolation with complexity constraint an optimization approach over the last several years a new theory of nevanlinna pick interpolation with complexity constraint has been developed for scalar interpolants in this paper we generalize this theory to the matrix valued case also allowing for multiple interpolation points we parameterize a class of interpolants consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros this is a complete parameterization and for each choice of interpolant we provide a convex optimization problem for determining it this is derived in the context of duality theory of mathematical programming to solve the convex optimization problem we employ a homotopy continuation technique previously developed for the scalar case these results can be applied to many classes of engineering problems and to illustrate this we provide some examples in particular we apply our method to a benchmark problem in multivariate robust control by constructing a controller satisfying all design specifications but having only half the mcmillan degree of conventional h sup spl infin controllers we demonstrate the advantage of the proposed method

exact computation of infimum for a class of continuous time h optimal control problem with a nonzero direct feedthrough term from the disturbance input to the controlled output a noniterative method for the computation of infimum for a class of continuous time h optimal control problem is considered in this paper the problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems the method is applicable to systems where i the transfer function from the disturbance input to the measurement output is free of imaginary axis invariant zeros and left invertible and ii the transfer function from the control input to the controlled output of the given system is free of imaginary axis invariant zeros and right invertible the result presented in this paper is a continuation of the previous work of the author and his co workers chen et al 1992 in which the direct feedthrough term from the disturbance input to the measurement output of the given system are required to be zero

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

characterization of the sub layers for 2 block h optimal control problem an analysis of the sub layers for the 2 block h optimal control problem is made in this paper the sub layers are characterized by a 2 block h control problem through the concept of the diagonalizing matrix pair this matrix pair is constructed from the modified 1 block h optimal control problem and its maximal schmidt pair the optimal h norm of the sub layers which can possibly be attained is independent of any different optimal solution

on extended j j lossless factorization abstract the extended j j lossless factorization for discrete time rational matrix functions with zeros on the unit circle is considered a necessary and sufficient condition for the existence of the extended j j lossless factorization is obtained the condition is given in terms of the original system parameters in the form of a generalized eigenvalue problem and a discrete time algebraic riccati equation state space representations for the extended j j lossless factorization are provided

on the use of hadamard weightings in the partial decoupling of h based designs abstract in this paper a solution to the hadamard weighted h hwh 1 block problem is proposed the solution is based on a method by safonov and le 1988 the robust stability of designs based on the hwh method is also discussed

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

directional interpolation via all pass transfer function matrices and its application in hankel norm approximations a new approach to the problem of multivariable interpolation via all pass transfer function matrices that are not necessarily stable is presented the approach applies to both state space and classical function theoretic arguments and obtains a very simple expression for the all pass matrix that satisfies the interpolation requirement unlike the solution that is obtained by the generalized nevanlinna pick algorithm this expression is derived in closed form explicitly in terms of the interpolation parameters it allows a detailed investigation of the structure of the all pass solution and it is readily used in hankel norm approximations of linear multivariable systems

controller design with real parametric uncertainty a number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty in particular the complex synthesis procedure has been successfully applied to a number of engineering problems however the presence of real parametric uncertainty in the problem description substantially complicates matters so that standard complex synthesis techniques are no longer adequate in this paper we develop a procedure to tackle the mixed real and complex synthesis problem this procedure involves a d g k iteration between computing the mixed upper bound and solving an h optimal control problem and has guaranteed convergence to a local minimum of the nonconvex problem the procedure has been implemented in software and several controller designs are compared with the corresponding complex synthesis designs

direct computation of infimum in discrete time h optimization using measurement feedback a direct and non iterative method for the computation of the infimum for a class of discrete time h optimal control problem is considered in this paper the problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems the method is applicable to systems where i the transfer function from the disturbance input to the measurement output is free of unit circle invariant zeros and left invertible and ii the transfer function from the control input to the controlled output of the given system is free of the unit circle invariant zeros and right invertible

optimal constant i o similarity scaling for full information and state feedback control problems abstract in robust control problems capturing all robustness and performance objectives in a single h norm cost function is impossible an alternative approach still untilizing the h norm involves diagonal similarity scaling of certain closed loop transfer functions the set of allowable diagonal scalings is problem dependent and reflects assumptions about the uncertainty and desired performance objectives the scaling set considered here is a prescribed convex set of positive definite matrices we consider the optimal constant scaling problem for the full information h control problem the solution is obtained by transforming the original problem into a convex feasibility problem specifically a structured linear matrix inequality in special cases solvability of the full information problem is equivalent to solvability of the state feedback problem

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

more on optimal stress points reduced integration element distortions and error estimation the presented work addresses the relationships between optimal sampling points reduced integration and geometric distortion with the objective of estimating errors in terms of those considerations isoparametric quadratic plane and solid elements are used as a vehicle for the study geometric distortion measures and evaluation conditions based on convergence requirements are first defined in terms of the polynomial orders of the geometry and applied strain using these the concept of optimal stress sampling already established for undistorted elements is extended to distorted geometry and shown to be effective over a range of geometries and strains errors in the strain displacement relationship and numerical integration of the strains are used to estimate the total response error and to rationalize the connection between optimal stress points and reduced integration enhanced convergence by extension to the representation of linear strains in elements with quadratic geometry is identified as the main advantage of reduced integration the applicability of the proposed and other distortion parameters to vetting of element geometry and error prediction is discussed

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

h control problem with jω axis zeros abstract in this paper we discuss the so called one block h control problem in the case where the plant has j axis including infinity poles and zeros first we derive a necessary and sufficient condition for the h model matching problem to have a solution in the transfer function setting second we give a solvability condition which enables us to check the solvability by i examining matrix norm conditions related to j axis zeros and ii solving two generalized riccati equations finally we extend our result to the standard h problem

conjugation and h control of discrete time systems we establish conjugation notion in discrete time systems first introduced into the h control theory of continuous time systems by kimura 1989 in discrete time systems conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle with the aid of j lossless conjugation conjugation by a j lossless system it is shown that the parametrization of sub optimal solutions of h model matching problems is reduced to a lyapunov type equation the parametrization of all solutions is given in an extremely simple way it is further proved that the j lossless conjugation of the h model matching problem is a natural state space representation of classical interpolation in discrete time systems

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

the singular minimum entropy h control problem abstract in this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has h norm less than some a priori given bound we find an explicit expression for the infimum moreover we give a characterization when the infimum is attained contrary to the regular case for the singular minimum entropy h control problem the infimum is not always attained

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

a spectral factorization algorithm for discrete time descriptor systems via generalized eigenproblems abstract this paper considers a factorization of discrete time improper spectral matrices using the generalized eigenproblem gep approach we first derive a factorization algorithm for the spectral matrix formed by an rh transfer matrix based on the gep associated with a descriptor representation of the spectral matrix and a related linear equation we then extend the factorization algorithm to the case where the spectral matrix is formed by an improper rl matrix a simple example is included to show the applicability of the present algorithm

the structure of all pass matrices that satisfy directional interpolation requirements all pass matrices are considered which are permitted to have right half plane rhp poles less than some specified number this is an extension of the inner interpolation problem where no rhp poles are permitted all pass matrices which satisfy the directional interpolation requirements are obtained in closed form the simple derivation provides an insight into the mechanism by which the interpolation requirements impose the infinity norm restriction under the number condition of half plane poles

on the takagi interpolation problem in this paper we approach the interpolation problem of takagi from the perspective of time series modeling the framework of exact identification and the notion of mpum are used in order to generalize some results on the nevanlinna interpolation problem these results are of independent interest and concern the relation between the roots of the determinant of the denominator of an unfalsified model and the signature of the pick matrix of the data

optimal hankel norm model reductions and weiner hopf factorization i the canonical case we consider the problem of approximating a given stable rational matrix function g s of mcmillan degree n by a function s f s where has mcmillan degree l n and f is antistable it is known that the minimum possible l norm of the error g f l or equivalently the minimum possible hankel norm of the stable part of the error g h is equal to the l 1 st hankel singular value l 1 g s of g we give an explicit linear fractional map parametrization for the class of all functions s f s as above which satisfy g f l l 1 g s the coefficients of the linear fractional map are completely determined by the matrices a b and c in a realization g s c si a 1b for g s and the unique solutions of a pair of lyapunov equations involving these matrices note that without loss of generality g 0 the basic idea is to use the approach of ball and helton to reduce the problem to a symmetric wiener hopf factorization problem which in turn can be solved by applying a result of kaashoek and ran the results obtained here are equivalent to the results of glover but our analysis gives an alternative more geometric approach

new strategy for assumed stresses for 4 node hybrid stress membrane element an oblique skew co ordinate system defined at the origin of the isoparametric co ordinates is introduced as the reference co ordinate system for assumed stresses in the formulation of a 4 node hybrid membrane element the initially uncoupled stresses in these reference co ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of pian and wu7 when suitable incompatible displacements are introduced such strategy allows the element stiffness to be formulated by using either the hellinger reissner or the complementary energy functional numerical results show that the performances of three elements constructed by using different sets of incompatible displacements in this paper are practically identical and are all of slightly higher accuracy than that of the pian sumihara element6

reduced integration and the shear flexible beam element the timoshenko beam element is studied in an investigation of shear locking in the development of c 0 continuous elements using shear flexible or penalty function type formulations a penalty type argument is used to degenerate thick elements to thin elements and the various approximations of the shear related energy terms act as different types of constraints depending on the formulation two types of constraints may emerge which are classified as true or spurious formulations that ensure only true constraints in the extreme penalty limit cases display superior performance in the thick element situation the argument is extended to a shallow curved timoshenko beam and another mechanism inplane locking arises if all energy terms corresponding to the membrane energy are exactly integrated

gain perturbation tolerance in h state feedback control the gain perturbation tolerance property of h optimal state feedback hisf systems is considered the following two problems are discussed 1 for what gain perturbations would the perturbed closed loop system remain stable 2 for what gain perturbations would the perturbed closed loop transfer function satisfy the specified h norm performance in each case we establish general sufficient conditions in the frequency domain in particular under a certain orthogonality condition the hisf systems are shown to possess an infinite gain increase tolerance for stability the result obtained also provides insight into the gain reduction tolerance problem furthermore we show that under certain conditions the specified h norm performance of the hisf systems is preserved even with an infinite gain increase concerning closed loop performance robustness against gain perturbations we see remarkable differences between the case where controlled outputs include a direct transmission of controls and the c