invertibility and inversion of linear periodic systems abstract in this paper the invertibility properties of periodic discrete time systems are studied necessary and sufficient conditions for the existence and a synthesis procedure for constructing left and right inverses are given

disturbance localization with dead beat control for linear periodic discrete time systems in this paper the classical notion of a controlled invariant subspace together with its main properties is extended to the case of linear periodic discrete time systems and used for deriving necessary and sufficient solvability conditions for the disturbance localization problem and a synthesis procedure for the solution moreover the notions of outer and inner controllable subspaces are introduced and studied for the same class of system thus allowing the derivation of necessary and sufficient solvability conditions for the disturbance localization problem with output or state dead beat control and to give synthesis procedures for the solutions

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

robust trajectory tracking for a class of hybrid systems an internal model principle approach this paper deals with the asymptotic tracking of periodic trajectories for hybrid systems having linear dynamics in each operating mode and isolated discrete switching events switching systems parametric uncertainties are considered and the dimension of the state vector is allowed to vary among modes to deal with the hybrid nature of the system and the possible discontinuities of its solutions at switching times a properly amended tracking control problem is defined and a feedback control law based on a discontinuous version of the classical internal model principle is proposed the innovative design of the discrete time dynamics of the compensator guarantees the robust existence of a steady state response giving zero tracking error in the controlled output and local convergence to it

block tridiagonal matrix inversion and fast transmission calculations a method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed with the goal of efficiently determining the matrices involved in the fisher lee relation for the calculation of electron transmission coefficients the new method leads to faster transmission calculations compared to traditional methods as well as freedom in choosing alternate green s function matrix blocks for transmission calculations the new method also lends itself to calculation of the tridiagonal part of the green s function matrix the effect of inaccuracies in the electrode self energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies

a direct method for sparse least squares problems with lower and upper bounds a direct method is developed for solving linear least squares problems mathop min left ax b right 2 limits x wherea is large and sparse and the solution is subject to lower and upper boundsl x u the problem is initially transformed to upper triangular form by a sparseqr factorization an active set algorithm is then used the key step is the stable updating of ther factor associated with the columns ofa corresponding to the free variables when theq factor is not available for this a new method is developed which uses the semi normal equations and iterative refinement

extension and optimization of the find algorithm computing green s and less than green s functions the find algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix such calculation is critical in many applications e g quantum transport in nano devices we extended the algorithm to other matrix inverse related calculations those are required for example to calculate the less than green s function and the current density through the device for a 2d device discretized as an n xxn y mesh the best known algorithms have a running time of o n x 3n y whereas find only requires o n x 2n y even though this complexity has been reduced by an order of magnitude the matrix inverse calculation is still the most time consuming part in the simulation of transport problems we could not reduce the order of complexity but we were able to significantly reduce the constant factor involved in the computation cost by exploiting the sparsity and symmetry the size of the problem beyond which find is faster than other methods typically decreases from a 130x130 2d mesh down to a 40x40 mesh these improvements make the optimized find algorithm even more competitive for real life applications

an approximate framework for quantum transport calculation with model order reduction a new approximate computational framework is proposed for computing the non equilibrium charge density in the context of the non equilibrium green s function negf method for quantum mechanical transport problems the framework consists of a new formulation called the x formulation for single energy density calculation based on the solution of sparse linear systems and a projection based nonlinear model order reduction mor approach to address the large number of energy points required for large applied biases the advantages of the new methods are confirmed by numerical experiments

disturbance localization by measurement feedback for linear periodic discrete time systems abstract the recent notions of inner controllable outer controllable and outer reconstructible subspaces together with the extension of the notion of c a b pair of subspaces to the class of linear periodic discrete time systems are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems by means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases i e with and without an additional requirement of output or state dead beat control

representation groups for the schur index fix an algebraic number field k and an irreducible complex character x of some finite group g assume k k x contains the values of x so that the simple component a 4 x k of the group algebra kg determined by 1 is central over k it is a basic consequence of the brauer witt theorem that a is a so called cyclotomic algebra see ls we exhibit here a slightly different point of view namely that of constructing a representation group for a similar to the concept used in clifford theory from class field theory it is known that the order of the class a of a in the brauer group br k equals its index m x it suffices to study the primary components of a so fix a prime p and let a denote the pcomponent of a in br k let k be the field obtained from k by adjoining the nth roots of unity if n exp g then by brauer k is a splitting field for a and g we shall see that there is a field l s k with p power degree over k such that

pole placement for linear discrete time systems by periodic output feedbacks abstract necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single input single output linear discrete time systems are established an algorithm for finding a sequence of output feedback gains is presented and illustrated by a simple numerical example

fault detection and isolation of linear discrete time periodic systems using the geometric approach this technical note solves the problem to detect and isolate faults in linear periodic discrete time systems a residual generator is designed through a suitable unknown input observer with the requirement that each residual should be sensitive only to one fault and simultaneously insensitive to the other faults that can affect the system and with the additional requirement of detectability that is the transition matrix of its relative error system has all eigenvalues smaller than in modulus for a given positive p 1 the solution is based on geometric tools and it makes use of the outer observable subspace notion

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

robust stabilization of uncertain systems by periodic feedback an uncertain control system described by a family parametrized by an unknown parameter that takes values in a known constraint set is considered the robust stabilization problem is defined as finding a dynamic output controller that globally stabilizes the uncertain system that is no matter what the parameter value chosen by nature the closed loop system is globally asymptotically stable it is shown that in the class of dynamic periodic controllers the solution to this problem exists under reasonable assumptions given for a general family of abstract non linear models those assumptions in the case of a family of linear finite dimensional models are equivalent to suitable stabilizability and detectability assumptions for this case insensitivity of the design with respect to small errors in the data is proved

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

computing entries of the inverse of a sparse matrix using the find algorithm an accurate and efficient algorithm called fast inverse using nested dissection find for computing non equilibrium green s functions negf for nanoscale transistors has been developed and applied in the simulation of a novel dual gate metal oxide semiconductor field effect transistor mosfet device structure the method is based on the algorithm of nested dissection a graph of the matrix is constructed and decomposed using a tree structure an upward and downward traversal of the tree yields significant performance improvements for both the speed and memory requirements compared to the current state of the art recursive methods for negf this algorithm is quite general and can be applied to any problem where certain entries of the inverse of a sparse matrix e g its diagonal entries the first row or column etc need to be computed as such it is applicable to the calculation of the green s function of partial differential equations find is applicable even when complex boundary conditions are used for example non reflecting boundary conditions

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler groups

fault detection of linear discrete time periodic systems in this note an approach to the design of optimal fault detection systems for linear discrete time periodic systems is proposed which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults the needed computation mainly consists in solving a difference periodic riccati system the proposed approach is finally illustrated by a numerical example

fault detection for linear periodic systems using a geometric approach using the geometric approach a solution is given to the problem of detecting faults in linear periodic discrete time systems the solution consists on developing an unknown input observer the existing conditions are provided in terms of new geometric notion of outer observable subspace that is the dual notion of inner reachable subspace making use of the new notion a residual generator is developed such that the residual is made independent of unknown input disturbance and its output free response goes asymptotically to zero

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

block computation and representation of a sparse nullspace basis of a rectangular matrix abstract in this paper we propose a new method to efficiently compute a representation of an orthogonal basis of the nullspace of a sparse matrix operator b t with b r n m n m we assume that b has full rank i e rank b m it is well known that the last n m columns of the orthogonal matrix q in a qr factorization b qr form such a desired null basis the orthogonal matrix q can be represented either explicitly as a matrix or implicitly as a matrix h of householder vectors typically the matrix h represents the orthogonal factor much more compactly than q we will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity this new algorithm may e g be used to construct a null space basis of the discrete divergence operator in the finite element context and we will provide numerical results for this particular application

trajectory tracking for a particle in elliptical billiards an infinitely rigid unitary mass particle is considered moving on a planar region delimited by a rigid elliptical barrier elliptical billiards under the action of proper control forces a class of periodic trajectories involving an infinite sequence of non smooth impacts between the mass and the barrier at fixed times is found by using an lmis based procedure the jumps in the velocities at the impact times render difficult if not impossible to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour hence the tracking control problem is properly stated using notions similar to the quasi stability concept in v lakshmikantham d d bainov and p s simeonov theory of impulsive differential equations 6 world scientific 1989 a controller whose state is subject to discontinuities based on the internal model principle is shown to solve the proposed tracking problem giving rise to control forces that are piecewise continuous fun

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

a geometric approach to nonlinear fault detection and isolation we present a differential geometric approach to the problem of fault detection and isolation for nonlinear systems a necessary condition for the problem to be solvable is derived in terms of an unobservability distribution which is computable by means of suitable algorithms the existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an observable quotient subsystem unaffected by all fault signals but one for this subsystem a fault detection filter is designed

linear function dead beat observers with disturbance localization for linear periodic discrete time systems abstract in this paper the classical notion of a conditioned invariant subspace is extended to the case of linear periodic discrete time systems properly characterized and suitably specialized to the notion of an outer reconstructible subspace these ideas are used in order to find necessary and sufficient conditions for the existence of a linear function dead beat observer for such systems under unknown and unmeasurable disturbances and a synthesis procedure for such an observer

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

a hybrid method for the parallel computation of green s functions quantum transport models for nanodevices using the non equilibrium green s function method require the repeated calculation of the block tridiagonal part of the green s and lesser green s function matrices this problem is related to the calculation of the inverse of a sparse matrix because of the large number of times this calculation needs to be performed this is computationally very expensive even on supercomputers the classical approach is based on recurrence formulas which cannot be efficiently parallelized this practically prevents the solution of large problems with hundreds of thousands of atoms we propose new recurrences for a general class of sparse matrices to calculate green s and lesser green s function matrices which extend formulas derived by takahashi and others we show that these recurrences may lead to a dramatically reduced computational cost because they only require computing a small number of entries of the inverse matrix then we propose a parallelization strategy for block tridiagonal matrices which involves a combination of schur complement calculations and cyclic reduction it achieves good scalability even on problems of modest size

schur indices and schur groups w x schur groups were introduced in 8 as a certain kind of smallest groups admitting characters with nontrivial schur index indeed every finite group g having an irreducible character x with m x 1 involves such schur groups they occur as terminal steps by repeated application of the brauer witt theorem over local fields of characteristic zero there is a connection with the schur subgroup s k of the brauer group br k of such a field k this s k has been studied in detail by w x w x yamada 12 and quite recently by adem and milgram 1 from the character theoretic point of view it appears to be of interest to give explicit generators for s k in terms of schur groups see section 6 in the course w x of this investigation the authors noticed that there must be a gap in 8 concerning the exceptional dyadic schur groups yielding nontrivial 2 lo cal index it turned out that the definition of the schur groups of type qd q was too restrictive the correct definition will be given in section 5 below we take the opportunity to discuss these terminal q reduc2 tions once again and the class of 2 groups involved theorem 4 and 5 the schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups for the sake of clarity we begin our present work by making some general comments on schur indices in group extensions this should be compared with turull w x 11 our approach is perhaps more elementary and we just demonstrate w x how to use the results on stable clifford theory 7 more details may be w x found in the recent thesis 5 of the first named author

a lifting based approach to observer based fault detection of linear periodic systems in this note an approach based on lifting is introduced for the observer based fault detection fd of linear discrete time periodic systems the main contribution of this note is to give an analytic expression to transform the linear time invariant lti observer based residual generator designed based on the lifted lti reformulation of the periodic system into a periodic observer based residual generator to reduce detection delay the result enables a systematic transfer of the existing observer based fd methods for discrete lti systems to discrete time periodic systems

on the calculation of local invariants of irreducible characters abstract let k be a finite extension of the field q p of p adic numbers let be an irreducible character of a finite group whose values are in k associated with and k is an element of the brauer group of k and therefore by standard results a local invariant in q z the paper defines some classes of finite groups that the paper calls p basic groups and gives formulas to calculate the local invariant of their characters it also shows how one can calculate the local invariant of any irreducible character of any finite group that has values in k by reducing the problem to the case where the groups are p basic and using the explicit formulas given in the paper

asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts an infinitely rigid mass is considered which moves on a plane under the action of proper control forces within a closed region delimited by an infinitely massive and rigid circular barrier a suitably amended tracking problem for a class of periodic trajectories involving an infinite number of nonsmooth impacts between the mass and the barrier is formally stated and solved

a fast algorithm for sparse matrix computations related to inversion we have developed a fast algorithm for computing certain entries of the inverse of a sparse matrix such computations are critical to many applications such as the calculation of non equilibrium green s functions g r and g for nano devices the find fast inverse using nested dissection algorithm is optimal in the big o sense however in practice find suffers from two problems due to the width 2 separators used by its partitioning scheme one problem is the presence of a large constant factor in the computational cost of find the other problem is that the partitioning scheme used by find is incompatible with most existing partitioning methods and libraries for nested dissection which all use width 1 separators our new algorithm resolves these problems by thoroughly decomposing the computation process such that width 1 separators can be used resulting in a significant speedup over find for realistic devices up to twelve fold in simulation the new algorithm also has the added advantage that desired off diagonal entries can be computed for free consequently our algorithm is faster than the current state of the art recursive methods for meshes of any size furthermore the framework used in the analysis of our algorithm is the first attempt to explicitly apply the widely used relationship between mesh nodes and matrix computations to the problem of multiple eliminations with reuse of intermediate results this framework makes our algorithm easier to generalize and also easier to compare against other methods related to elimination trees finally our accuracy analysis shows that the algorithms that require back substitution are subject to significant extra round off errors which become extremely large even for some well conditioned matrices or matrices with only moderately large condition numbers when compared to these back substitution algorithms our algorithm is generally a few orders of magnitude more accurate and our produced round off errors stay at a reasonable level

geometric tools for solving the fdi problem for linear periodic discrete time systems this paper studies the problem of detecting and isolating faults in linear periodic discrete time systems the aim is to design an observer based residual generator where each residual is sensitive to one fault whilst remaining insensitive to the other faults that can affect the system making use of the geometric tools and in particular of the outer observable subspace notion the fault detection and isolation fdi problem is formulated and necessary and solvability conditions are given an algorithmic procedure is described to determine the solution of the fdi problem

on row and column orderings for sparse least squares problems let p 1 and p 2 be respectively m times m and n times n permutation matrices with m geqq n suppose the m times n sparse matrix bar a p 1 ap 2 is reduced to upper trapezoidal form left begin array 20 c r 0 end array right through the application of givens rotations sequentially to the rows of bar a it is well known that the sparsity of r depends only on the choice of p 2 but the choice of p 1 can drastically affect the arithmetic required to compute r in this paper we provide a mechanism for studying the connection between good row and good column orderings along with a modified nested dissection algorithm for finding a good p 2 which automatically induces a good p 1 an analysis for a model problem is given along with some experimental results

failure detection and identification using the geometric concept of an unobservability subspace a solution is given to the problem of detecting and identifying control system component failures in linear time invariant systems conditions are developed for the existence of a causal linear time invariant processor that can detect and uniquely identify a component failure first for the case where components can fail simultaneously and then for the case where they fail only one at a time explicit design algorithms are provided when these conditions are satisfied in addition to time domain solvability conditions frequency domain interpretations of the results are given and connections are drawn with results already available in the literature