two degrees of freedom dead beat control system with robustness this paper deals with the design of a two degrees of freedom tdf dead beat controller with both optimal robustness and minimum settling time based on the parametrization of all stabilizing tdf controllers optimal robustness and minimum time control are achieved simultaneously the minimum time dead beat control is constructed for arbitrary real rational reference input at the same time optimal robustness is achieved under the constraint that the error between reference input and output asymptotically vanishes for any plant perturbation that does not violate the internal stability it is proved that under the same robustness criterion irrespective of its length of settling time the optimal robustness of the tdf system is always superior to that of the one degree of freedom odf system as long as its settling time is finite and they become identical only when the settling time of the odf system goes to infinity a numerical example is given to illustrate the theoretical results

a square root nyquist m filter design for digital communication systems designing matched transmit and receive filters whose combination satisfies the nyquist condition is a classical problem in digital communication systems in this correspondence we propose a novel method for designing such filters the proposed method is based on a cost function whose minimization leads to designs that can strike a balance between the stopband attenuation the residual intersymbol interference isi robust sensitivity to timing jitter and or reduced peak to average power ratio par an iterative algorithm for optimizing the proposed cost function is suggested and its excellent performance is shown by presenting a variety of design examples compared to the published works the proposed method offers the following unique advantages by introducing a symmetry in the filter coefficients filters with reduced computational complexity can be designed we also introduce a design parameter that allows one to strike a balance between the par and other features of the desired filter

ripple free tracking with robustness despite the previous efforts in deadbeat control a need still exists to offer a generic design methodology for both performance and robustness we propose a hybrid two degree of freedom 2dof controller for the fixed order constrained optimization problem addressing performance and robustness specifications the controller is given in terms of the solution of two diophantine equations simple parameterization of these diophantine equations allows for straight forward optimization and constraint satisfaction for both performance and robustness measures

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

on the synthesis of digital transmit filters a simple method for calculating the coefficients of a digital transversal filter when cascaded with analog filters is proposed the optimization criterion takes into account both the spectral characteristics of the signal and the residual amount of intersymbol interference at the decision device input in contrast with standard methods no a priori constraints are applied to the filter in the time or in the frequency domain

finite settling time stabilisation the robust siso case this article deals with the problem of robustness to multiplicative plant perturbations for the case of finite settling time stabilisation fsts of single input single output siso linear discrete time systems fsts is a generalisation of the deadbeat control and as in the case of deadbeat control the main feature of fsts is the placement of all closed loop poles at the origin of the z plane this makes fsts sensitive to plant perturbations hence the need of robust design an efficient robustness index is introduced and the problem is reduced to a finite linear programme where all the benefits of the simplex method such as effectiveness efficiency and ability to provide complete solution to the optimisation problem can be exploited

quiver generalization of a conjecture of king tollu and toumazet abstract stretching the parameters of a littlewood richardson coefficient of value 2 by a factor of n results in a coefficient of value n 1 12 9 19 we give a geometric proof of a generalization for representations of quivers

general solution of robust tracking problem in two degree of freedom control systems in this note we study linear time invariant multivariable systems with two degree of freedom compensators first we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property second utilizing this parametrization we solve a general robust tracking problem including the case of nonsquare plants and show that tracking and robustness can be independently specified in two degree of freedom control systems

double objective optimal multivariable ripple free deadbeat control this paper presents novel results on optimal multivariable deadbeat control given a discrete time stable linear time invariant plant model we give a simple parameterization of all stabilizing ripple free deadbeat controllers of a given order the free parameter is then optimized in the sense that a quadratic index is kept minimal the optimality criterion has the advantage of accounting for both tracking performance and magnitude of the control effort the proposed design procedure is simple to use and allows the tuning of the controller with a scalar weighting factor simulation results are included to illustrate the effectiveness of the proposed design algorithm

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

optimization of binary transversal filters this paper presents a method for optimization with respect to bandwidth efficiency for a binary transversal filter when the input pulse format is nrz the objective is to concentrate as much as possible of the energy in the emitted signal within certain frequency bands the filter coefficients are given by a symmetric eigenvector for a toeplitz matrix and the corresponding eigenvalue equals the optimum value

parametrization of stabilizing controllers for multivariable servo systems with two degrees of freedom a necessary and sufficient condition for the existence of a servo controller is derived using the fractional representation approach for the plant whose measured variables are not necessarily coincident with its controlled outputs an independent parametrization of the servo controllers and the attainable transfer characteristics is proposed the possible classes of all controllers with integrity in the error channel or measured output channel is also clarified using the derived parametrization

a geometric littlewood richardson rule we describe a geometric littlewood richardson rule interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties there are no restrictions on the base eld and all multiplicities arising are 1 this is important for applications this rule should be seen as a generalization of pieri s rule to arbitrary schubert classes by way of explicit homotopies it has straightforward bijections to other littlewood richardson rules such as tableaux and knutson and tao s puzzles this gives the rst geometric proof and interpretation of the littlewood richardson rule geometric consequences are described here and in v2 kv1 kv2 v3 for example the rule also has an interpretation in k theory suggested by buch which gives an extension of puzzles to k theory

quivers long exact sequences and horn type inequalities we give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p groups with fixed isomorphism types this problem is related to some generalized littlewood richardson coefficients that we define in this paper we also show how this problem is related to eigenvalues of hermitian matrices satisfying certain in equalities when m 3 we recover the horn type inequalities that solve the saturation conjecture for littlewood richardson coefficients and horn s conjecture

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

semigroups of sl_3 mathbb c tensor product invariants abstract we find presentations for a family of semigroup algebras related to the problem of decomposing sl 3 c tensor products along the way we find new toric degenerations of the grassmannian variety gr 3 c n which are t invariant for t gl n c the diagonal torus

intersections of schubert varieties and eigenvalue inequalities in an arbitrary finite factor the intersection ring of a complex grassmann manifold is generated by schubert varieties and its structure is governed by the littlewood richardson rule given three schubert varieties s1 s2 s3 with intersection number equal to one we show how to construct an explicit element in their intersection this element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags and is therefore well defined over an arbitrary field of scalars moreover this result also applies to appropriately defined analogues of schubert varieties in the grassmann manifolds associated with a finite von neumann algebra the arguments require the combinatorial structure of honeycombs particularly the structure of the rigid extremal honeycombs it is known that the eigenvalue distributions of self adjoint elements a b c with a b c 0 in the factor r are characterized by a system of inequalities analogous to the classical horn inequalities of linear algebra we prove that these inequalities are in fact true for elements of an arbitrary finite factor in particular if x y z are self adjoint elements of such a factor and x y z 0 then there exist self adjoint a b c r such that a b c 0 and a respectively b c has the same eigenvalue distribution as x respectively y z a complete matricial form of this result is known to imply an affirmative answer to an embedding question formulated by connes the critical point in the proof of this result is the production of elements in the intersection of three schubert varieties when the factor under consideration is the algebra of n n complex matrices our arguments provide new and elementary proofs of the horn inequalities which do not require knowledge of the structure of the cohomology of the grassmann manifolds

on the wiener hopf approach to optimal feedback design abstract a simple solution is given to the problem of designing a proper feedback which minimizes the h 2 norm of the sensitivity function multiplied by a suitable weighting function

a littlewood richardson rule for two step flag varieties this paper studies the geometry of one parameter specializations of subvarieties of grassmannians and two step flag varieties as a consequence we obtain a positive geometric rule for expressing the structure constants of the cohomology of two step flag varieties in terms of their schubert basis a corollary is a positive geometric rule for computing the structure constants of the small quantum cohomology of grassmannians we also obtain a positive geometric rule for computing the classes of subvarieties of grassmannians that arise as the projection of the intersection of two schubert varieties in a partial flag variety these rules have numerous applications to geometry representation theory and the theory of symmetric functions

the symplectic and algebraic geometry of horn s problem one version of horn s problem asks for which does h h h 0 have solutions where h are hermitian matrices with spectra this turns out to be a moment map condition in hamiltonian geometry many of the results around horn s problem proven with great effort by hand are in fact simple consequences of the modern machinery of symplectic geometry and the subtler ones provable via the connection to geometric invariant theory we give an overview of this theory which was not available to horn including all definitions and how it can be used in linear algebra