the quantum heisenberg group h 1 q the structure of the quantum heisenberg group is studied in the two different frameworks of the lie algebra deformations and of the quantum matrix pseudogroups the r matrix connecting the two approaches together with its classical limit r are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed some unusual properties of the quantum enveloping heisenberg algebra are shown

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

frequency domain approach to optimally robust residual generation and evaluation for model based fault diagnosis methods of residual generation and evaluation for model based fault diagnosis in uncertain linear dynamic systems are investigated with the aid of frequency domain approaches and h optimization techniques based on the factorization technique the construction of residual generators in terms of a parametrization concept is proposed with its help the fault detection and isolation problem with perfect model uncertainty decoupling is studied and moreover by its optimal approximation the problem of optimally robust residual generation is treated the optimization problem is solved by h techniques to increase the robustness of residual evaluation a frequency domain residual evaluation index is introduced and optimal input adaptive fault thresholds are derived with respect to the frequency domain evaluation index the results are illustrated in terms of an inverted pendulum example

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

inner outer factorization for strictly proper transfer matrices considers inner outer factorization for strictly proper transfer matrices we provide characterizations to the solution of this particular factorization problem and develop a computational algorithm to solve it a numerical example is used to illustrate the proposed theory and algorithm

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

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

on inner outer factorization abstract the calculation of an inner outer factorization of a proper rational matrix by state space methods is considered particular attention is given to the extraction of the inner factor so that a minimal realization is obtained for this factor

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

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

a simple state space design of an interactor for a non square system via system matrix pencil approach abstract the construction of an interactor cancelling the infinite zeros of a non square proper transfer matrix is discussed in this paper along the line of approach of b r copeland m g safonov int j robust nonlinear control 2 1992 139 in this paper some new properties of the infinite eigenstructure of the system matrix pencil of the transfer matrix are presented a simple state space design of the interactor is proposed by using these properties then some new features about state space relations among the original system the interactor and the compensated system are presented also several conditions about the invariance of the stabilizability or controllability between the state space realizations of the original system and the compensated system are shown furthermore the proposed method of designing the interactor in this paper can be applied to a tall or fat transfer matrix with full normal rank

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

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

the extended j spectral factorization for descriptor systems in this paper we study an extended j spectral factorization problem i e j spectral factorization problem for descriptor systems which may not have full column normal rank and may exhibit poles and zeros on the extended imaginary axis we present necessary and sufficient solvability conditions and develop a numerically reliable method for the underlying problem our method is implemented by using only orthogonal transformations and has an acceptable computational complexity

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

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

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

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

eigenvalue eigenvector sensitivity analysis of linear time invariant singular systems this note refers to the eigenvalue eigenvector sensitivity analysis of linear time invariant multivariable singular systems procedures that can be easily programmed on a digital computer are presented for the determination of the deviation of the eigenvalues and the corresponding right and left eigenvectors which are caused by the system s parameter changes about their nominal values the method is illustrated by two examples

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

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

j factorizations of a general discrete time system abstract the problems of extending the j lossless conjugation the j j spectral factorization and the j j lossless factorization to a completely general discrete time system are considered existence conditions together with a numerically sound prototype algorithm for computing the factors are provided for a system having any type of singularity including arbitrary normal rank poles and zeros at infinity at zero or on the unit circle

a numerical design of interactors for general descriptor systems in this technical note we develop a numerically reliable method to design an interactor for a general descriptor system relative to a stability region cg and an offending zero set our main result is based on a condensed form this condensed form is independent of the offending zeros and is computed using only orthogonal transformations and hence is numerically backward stable

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

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

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

path integrals for quantum algebras and the classical limit coherent states path integral formalism for the simplest quantum algebras q oscillator suq 2 and suq 1 1 is introduced in the classical limit canonical structure is derived with a modified symplectic and riemannian metric non constant deformation induced curvature for the phase spaces is obtained

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

conditions for minimizing the norm sensitivity of characteristic roots the sensitivity of a characteristic root of an n by n real matrix is measured by the euclidean norm of the root s n2derivatives with respect to the elements of the matrix let denote a real root and sigma j omega a complex root conditions for minimizing the sensitivity norms based on lambda sigma omega and sigma j omega are obtained since the conditions apply for all n and involve simple algebraic properties of the matrix they may have useful applications

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

generalized lqr control and kalman filtering with relations to computations of inner outer and spectral factorizations we investigate the generalized linear quadratic regulator lqr control where the dimension of the control input is strictly greater than the dimension of the controlled output and the weighting matrix on the control signal is singular the dual problem is the generalized kalman filtering where the dimension of the input noise process is strictly smaller than the dimension of the output measurement and the covariance of the observation noise is singular these two problems are intimately related to inner outer factorizations for nonsquare stable transfer matrices with square inners of the smaller size such inner outer factorizations are in turn related to spectral factorizations for power spectral density psd matrices whose normal ranks are not full we propose iterative algorithms and establish their convergence for inner outer and spectral factorizations which in turn solve the generalized lqr control and kalman filtering

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

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

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

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

quantum heisenberg group and algebra contraction left and right regular representations it is shown that the quantum heisenberg group hq 1 can be obtained by means of contraction from the quantum suq 2 group its dual hopf algebra is the quantum heisenberg algebra uq h 1 left and right regular representations for uq h 1 as acting on its dual hq 1 are derived by imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state by duality left and right regular representations for quantum heisenberg group with the quantum heisenberg algebra as a representation module are also constructed as before the reduction of left representations leads to finite dimensional irreducible ones for which the intertwining operator is also investigated

computation of the general j j prime lossless factorization the j j lossless factorization problem for a generalized state space system whose transfer matrix is completely general is investigated the main novelty is that we remove all restrictive hypotheses and allow for arbitrary normal rank poles and zeros on the imaginary axis or at infinity the numerically sound formulas for the solution bear the same nice expressions and striking simplicity of the inner outer factorization and may be equally well applied for the j j spectral factorization

a qualitative bound in robustness of stabilization by state feedback it is proved that placing the poles of a linear time invariant system arbitrarily far to the left of the imaginary axis is not possible if small perturbations in the model coefficients are taken into account given a nominal controllable system a sub 0 b sub 0 with one input and at least two states and an open ball around b sub 0 no matter how small there exists a real number gamma and a perturbation b within that ball such that for any feedback matrix k placing the eigenvalues of a sub 0 b sub 0 k to the left of res gamma there is an eigenvalue of a sub 0 bk with real part not less than gamma

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

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

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

q difference intertwining operators for a lorentz quantum algebra representations r r of a lorentz quantum algebra u are constructed they are labeled by two complex numbers r r and act in the space of formal power series of two noncommuting variables these variables are built from elements of the matrix lorentz quantum group l which is dual to u the conditions for reducibility of r r are given q difference intertwining operators in which realize partial equivalences of the representations r r are constructed explicitly the whole construction is a generalization of a known procedure for q 1 the case when q is a root of unity is also considered in detail

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

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

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

induced representations of quantum 1 1 galilei algebras we develop a systematic method to construct induced representations of quantum algebras the procedure makes use of two hopf algebras with a nondegenerate pairing and a pair of dual bases for them we apply the method on three different quantum deformations of the galilei algebra in 1 1 dimensions we obtain several families of induced representations including some results already known

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

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