the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

static fundamental solutions for a bi material full space in this paper the complete static response of two joined dissimilar half spaces due to an arbitrary interior point load is derived by means of a method of displacement potentials and integral transforms a dual format of the solution in the form of a hankel integral representation and algebraic closed form expressions is presented as illustrations its analytical degeneration to benchmark solutions for a homogeneous medium as well as its variation under general geometric and material conditions are shown the importance of the dual format of the bi material solution in connection with the method of asymptotic decomposition to the development of a rigorous treatment of its dynamic counterpart is also demonstrated

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

complete results for a class of state feedback disturbance attenuation problems the author considers a problem of disturbance attenuation using full state feedback the particular class of linear systems under consideration is closely related to the one block problem in h sup infinity control the author gives a complete solution to this problem in terms of a certain algebraic riccati equation

distributed computation of linear matrix equations an optimization perspective this paper investigates the distributed computation of the well known linear matrix equation in the form of axb f with the matrices a b x and f of appropriate dimensions over multiagent networks from an optimization perspective in this paper we consider the standard distributed matrix information structures where each agent of the considered multiagent network has access to one of the subblock matrices of a b and f to be specific we first propose different decomposition methods to reformulate the matrix equations in standard structures as distributed constrained optimization problems by introducing substitutional variables we show that the solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations and we design distributed continuous time algorithms for the constrained optimization problems even by using augmented matrices and a derivative feedback technique moreover we prove the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition

mathematical boundary integral equation analysis of an embedded shell under dynamic excitations a boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half space under axisymmetric excitations by virtue of a set of ring load green s functions for the shell and a group of dynamic fundamental solutions for the semi infinite medium the structure medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations through the analysis of an auxiliary pair of cauchy integral equations the singularities of the contact stress distributions arc rendered explicit with a direct incorporation of such analytical features into the formulation an effective computational procedure is developed which involves an interpolation of regular functions only typical results for the dynamic contact load distributions displacements and complex compliance functions are included as illustrations in addition to furnishing quantities of direct engineering interest this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods

exact computation of the infimum in h sub infinity optimization via output feedback the authors present a simple and noniterative procedure for the computation of the exact value of the infimum in the standard h sub infinity optimal control with output feedback the problem formulation is general and does not place any restrictions on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables the method is applicable to systems that satisfy the following conditions 1 the transfer function from the control input to the controlled output is right invertible and has no invariant zeros on the j omega axis and 2 the transfer function from the disturbance to the measurement output is left invertible and has no invariant zeros on the j omega axis

on positive definite solutions to the algebraic riccati equation abstract complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic riccati equation are given it is also shown that when a positive definite solution exists it is either unique or else there are uncountably many such solutions

a non iterative method for computing the infimum in h optimization this paper presents a simple and non iterative procedure for the computation of the exact value of the infimum in the singular h optimization problem and is an extension of our earlier work the problem formulation is general and does not place any restriction on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables our method is applicable to a class of singular h optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output have no invariant zeros on the jw axis and also satisfy certain geometric conditions the computation of the infimum in our method involves solving two well defined riccati and two lyapunov 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

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

three dimensional green s function for time harmonic dynamics in a viscoelastic layer abstract this paper presents green s function for time harmonic elastodynamic problems for a single layer domain three dimensional region bounded by two parallel planes with traction free boundary conditions the semi analytic solution is built in three steps a potential displacement representation b angular fourier series c radial hankel transform reflection matrices are presented for the plate domain kernels are integrally split into a singular closed form term the static half space solution plus an incremental solution in order to compute the inverse hankel transform for displacements and stress components a modified complex integration path is required theoretical considerations allow an adequate delimitation of such a complex path a specific treatment is proposed for low excitation frequencies where asymmetric lamb waves play a major role a series of numerical benchmarks are presented to validate the implementation of the functions displacements and tractions

relations between least squares and least rank solutions of the matrix equation axb c a matrix x is called a least squares solution of the matrix equation axb c if it minimizes the f norm of c axb a least rank solution of axb c if it minimizes the rank of c axb these two types of solution are not necessarily the same in this paper we establish necessary and sufficient conditions for the two types of solutions to coincide by using some matrix rank formulas and nested decompositions of matrices

paper adaptive controllers for discrete time systems with arbitrary zeros an overview the paper presents a comprehensive review of the problem of designing discrete time adaptive controllers for systems with arbitrary zeros minimum or non minimum phase systems being particular cases the authors have attempted to give a unified presentation of the various existing schemes by stripping away the technical details and highlighting their close interrelation through the use of a common framework both algorithmic and theoretical aspects are discussed with particular emphasis on the stability of the schemes

analytic resolution of time domain half space green s functions for internal loads by a displacement potential laplace hankel cagniard transform method a refined yet compact analytical formulation is presented for the time domain elastodynamic response of a three dimensional half space subject to an arbitrary internal or surface force distribution

elastodynamic green s functions for a smoothly heterogeneous half space in this paper the response of a vertically heterogeneous elastic half space with a smooth modulus variation under a set of time harmonic ring and point sources is derived analytically a method of evaluation via asymptotic decomposition for the singular green s functions is presented in the technique the green s functions are decomposed into an analytical part and a residual component capturing the corresponding singular behavior the analytical parts of the ring and point load green s functions are expressible in terms of the elliptic integrals and algebraic functions respectively the residual integrals which are regular can be evaluated by numerical contour integration to obtain correct results one must note and take into account the existence of multiple poles along the formal path of the inversion integrals the details of which are discussed in the paper to highlight the various aspects of the physical problem a set of illustrative numerical results is included

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

lower bounds on the infima in some cal h _ infty optimization problems we consider three optimization problems system inversion model matching with probably unstable plant and full information control a common theme in numerical solution of these problems is that the infimal performance among solutions has to be approximated before a suboptimal solution can be found recently a sequence of lower bounds that converges monotonously towards the exact infimum has been established for the system inversion problem the goal of this technical note is twofold first we show that except for some rare cases these lower bounds converge at least root exponentially fast i e we establish the good naturedness of this approximation method second we show how the approximation method can be extended such that also arbitrarily good lower bounds on the infima in the model matching problem and the full information control problem can be obtained

mappings of the finite and the infinite zero structures and invertibility structures of general linear multivariable systems under the bilinear transformation this paper presents a comprehensive picture of the mapping of structural properties associated with general linear multivariable systems under bilinear transformation while the mapping of poles of linear multivariable systems under such a transformation is well known the question of how the structural invariant properties of a given system are mapped remains unanswered this paper addresses this question more specifically we investigate how the finite and infinite zero structures as well as invertibility structures of a general continuous time discrete time linear time invariant multivariable system are mapped to those of its discrete time continuous time counterpart under the bilinear inverse bilinear transformation we demonstrate that the structural invariant indices lists i2 and i3 of morse remain invariant under the bilinear transformation while the structural invariant indices lists i1 and i4 of morse are in general changed

on solutions of matrix equation axb cyd f in this paper the matrix equation with two unknown matrices x y of form axb cyd f is discussed by applying the canonical correlation decomposition ccd of matrix pairs we obtain expressions of the least squares solutions of the matrix equation and sufficient and necessary conditions for the existence and uniqueness of the solutions we also derive a general form of the solutions we also study the least squares hermitian skew hermitian solutions of equation axah cych f

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

axisymmetric stress transfers from an embedded elastic cylindrical shell to a half space within the framework of three dimensional classical elastostatics and thin shell theories a rigorous mathematical formulation is presented for the torsionless axisymmetric stress transfer problem of a cylindrical shell of finite length embedded in a semi infinite solid by virtue of a set of ring load green s functions for the shell and a group of fundamental solutions for the half space the mechanical interaction problem is shown to be reducible to a pair of fredholm integral equations through the analysis of an auxiliary set of cauchy integral equations the singularities of the resultant contact stress distributions are rendered explicit the results of which are incorporated in a numerical procedure typical solutions for the axial and radial load transfers contact stress distributions as well as other related responses are included as illustrations in addition to furnishing results of direct relevance to a number of engineering applications the present treatment is apt to be useful as a basis of assessment for various approximate methods for this class of contact problems

weight selection in feedback design with degree constraints we present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint performance is cast as an interpolation problem with bounded analytic functions minimizers of a certain weighted entropy functional provide interpolants having degree less than the number of constraints the choice of weight parameterizes all such bounded degree solutions however the relationship between the weights and the shape of corresponding transfer functions is not direct thus in this paper we develop a formalism that guides weight selection

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

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

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

seismic soil structure interaction analysis by direct boundary element methods abstract upon establishing the mathematical framework necessary for a proper understanding of the analytical theory a regularized form of the conventional direct boundary integral equation formulation for three dimensional elastodynamics is presented for a general anisotropic medium founded on the basis of a full decomposition of the greens functions into regular and singular parts the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as cauchy principal values extended to deal with general seismic soil structure interaction problems in semi infinite media the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi layered viscoelastic half space greens functions and interfacial boundary tractions arising in typical soil structure foundation configurations a set of new benchmark numerical results are included

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

squaring down of non strictly proper systems in an earlier work a procedure for squaring down proper systems by static and dynamic compensators has been developed the compensators designed there have the following properties a they are asymptotically stable b the additional finite zeros induced by them are assignable to the open left half complex plane and c they preserve the fundamental properties such as stabilizability detectability infinite zero structure and minimum phase nature of the original system the earlier design procedure is extended to non strictly proper systems while retaining all the above mentioned properties

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

h2 optimal control of linear systems with tracking disturbance rejection constraints this paper considers the design of two degree of freedom linear multivariable controllers for four block matrix transfer functions with objectives pertaining to both random and deterministic signals the former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances the cost functional is set up by adding quadratic terms to the usual lqg wiener hopf cost in such a way as to ensure the existence of unique solutions even when the signals to be tracked rejected as well as models have poles on the imaginary axis the solutions of the servomechanism problem are explicitly parametrized by free stable rational and proper parameters under two sets of assumptions one which is necessary and sufficient for the existence of solutions and another stronger set which avoids kronecker products and the associated higher dimensional factorizations involved in

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

maximum rank matrix completion abstract the maximum rank completion problem is the problem of given a partial matrix that is a matrix where we are only given some of the entries filling in the unknown entries in such a way as to maximize the rank applications include bipartite matching and matroid intersection for linearly represented matroids we describe an algorithm that finds a maximum rank completion by perturbing an arbitrary completion in a greedy way

completing triangular block matrices with maximal and minimal ranks abstract this paper considers two completion problems on block matrices with maximal and minimal ranks let a a ij be an n n block matrix where a ij n i j 1 is given and a ij 1 i is a variant block entry then determine all these variant block entries such that a a ij has maximal and minimal possible ranks respectively by making use of the theory of generalized inverses of matrices we present complete solutions to these two problems as applications we also determine maximal and minimal ranks of the matrix expression a bxc when x is a variant triangular block matrix and then present a necessary and sufficient condition for the matrix equation bxc a to have a triangular block solution

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

entropy minimizing and risk sensitive control rules abstract glover and doyle 4 have shown that the entropy minimising criterion is identical with the infinite horizon form of the leqg optimality criterion leqg linear exponential of quadratic gaussian whittle and kuhn 11 have given a hamiltonian formulation which provides a natural treatment of leqg optimisation this paper contains a direct proof that the control yielded by these hamiltonian methods is entropy minimising the conclusion would follow from the two assertions above but a simple adjoining of the proofs of these two assertions yields an argument so circuitous that a direct proof is of interest