on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

stability analysis of second order sliding mode control systems with input delay using poincare map this note discusses the stability of dynamical systems with input delay under the second order sliding mode control algorithm poincare map is constructed to analyze the switching dynamics and to derive the stability conditions different parameter setting options are given for ensuring stability simulation examples are presented to verify the theoretical results

brief paper oscillations and transfer properties of relay feedback systems with time delay linear plants oscillations and transfer properties of relay feedback systems with linear plants having a dead time time delay are considered poincare maps are found explicitly in terms of the system parameters the oscillatory and transfer properties are presented in the format of the locus of a perturbed relay system lprs which represents the continuum of all possible periodic solutions and transfer properties of the system with a given linear plant conditions of local orbital asymptotic stability are found in terms of the system parameters the proposed approach is illustrated by examples

multirate and composite control of two time scale discrete time systems the design of stabilizing feedback control of singularly perturbed diserete time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control composite control strategies are developed for the case of single rate measurements all variables are measured at the same rate as well as for the case of multirate measurements slow variables are measured at a rate slower than that of fast variables

truncated predictor control of lipschitz nonlinear systems with time varying input delay this note deals with control design for lipschitz nonlinear systems with time varying input delay based on a truncated prediction of the system state over the delay period both a state and an output feedback control law are constructed within the framework of lyapunov krasovskii functionals a set of conditions are identified under which the closed loop system under either the state feedback or the output feedback law is globally asymptotically stable at the origin a numerical example is included to demonstrate the effectiveness of the proposed designs

analysis of a second order sliding mode algorithm in presence of input delays in this note a double integrator system under the action of a second order sliding mode control algorithm is considered and the resulting closed loop behavior in presence of an input delay is analysed due to the delay in the limit the system trajectories are periodic whenever the control modulus is chosen to be constant the amplitude and period of the resulting oscillations are fixed for any initial value if the control behaves asymmetrically it is shown that this is no more true since the overall dynamical system can admit diverse limit cycles

maclane homology and topological hochschild homology the aim of this paper is to show that the topological hochschild homology of a discrete ring r in the sense of 2 and the maclane homology of r see lo or s are isomorphic the method is to show that they are both isomorphic to a certain kind of homology of the category of finitely generated projective rmodules with coefficients in the bifunctor horn that the latter agrees with maclane homology was shown in 8 and that it agrees with topological hochschild homology is the main result of this paper in the appended last section we describe a related spectral sequence

second order integral sliding mode control for uncertain systems with control input time delay based on singular perturbation approach this note proposes a robust integral sliding mode rism manifold and a corresponding design method for stabilization control for uncertain systems with control input time delay delay independent sufficient condition for the existence of the rism surface is given in terms of lmi an improved sliding mode control smc which is delay dependent and suitable for small input time delay keeps system stay on the neighborhood of the rism surface in finite time with uncertainties and disturbances the reaching condition for the neighborhood of the rism is satisfied furthermore the control makes second order sliding mode based on a numerical example the proposed method is verified to be efficient and feasible

global stability of relay feedback systems for a large class of relay feedback systems rfs there will be limit cycle oscillations conditions to check existence and local stability of limit cycles for these systems are well known global stability conditions however are practically nonexistent the paper presents conditions in the form of linear matrix inequalities lmis that when satisfied guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time invariant lti stable systems the analysis consists in finding quadratic surface lyapunov functions for poincare maps associated with rfs these results are based on the discovery that a typical poincare map induced by an lti flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state moreover level sets of this function are convex subsets of linear manifolds the search for quadratic lyapunov functions on switching surfaces is done by solving a set of lmis although this analysis methodology yields only a sufficient criterion of stability it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle in fact it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology examples analyzed include minimum phase systems systems of relative degree larger than one and of high dimension such results lead us to believe that globally stable limit cycles of rfs frequently have quadratic surface lyapunov functions

e structure for q r maclane homology of a ringr is the hochschild homology of the so called cubical construction q r which is a chain algebra if we take a commutative ring r the dixmier product onq r is no longer commutative the main result of this paper is that it is commutative up to higher homotopies i e that it is an e1 algebra the e1 operad which acts onq r is constructed by using the analogue of theq complex in the context of finite sets for the precise notation of an operad action on these complexes the definition of an e1 monoidal functor is introduced

analysis of closed loop performance and frequency domain design of compensating filters for sliding mode control systems for a sliding mode sm system containing parasitic dynamics a model of chattering and a model of the averaged motions of the order equal to the original order of the system are obtained via the use of the locus of a perturbed relay system lprs method the latter model proves that the averaged motions in an sm system differ from those in the reduced order model in particular such phenomena as the nonideal disturbance rejection and the nonideal input tracking are observed it is shown that the non reduced order model transforms into the reduced one when the parasitic dynamics are removed yet the chattering and the averaged motion being the two components of the motion in the sm system belong to two different frequency ranges spectra as a result a frequency characteristic shaping aimed at achieving the desired frequency of chattering and enhancing the closed loop performance becomes possible this opportunity is explored in this paper examples that illustrate the proposed approach are provided

a decentralized model reference adaptive variable structure controller for large scale time varying delay systems in this note the problem of decentralized model reference adaptive variable structure control for a class of perturbed large scale systems with varying time delay interconnections is investigated based on the lyapunov stability theorem an adaptive variable structure control strategy for solving the robust tracking problem without the knowledge of upper bound of perturbations is developed the use of adaptive technique is to adapt the unknown upper bound of perturbations so that the objective of globally asymptotical stability is achieved once the system enters the sliding manifold the dynamics of controlled systems are insensitive to matching perturbations finally an example is given to demonstrate the feasibility of the proposed control scheme

the topological hochschild homology of the gaussian integers the calculation in this paper gives the 2 torsion in the topological hochschild homology of rings of integers in quadratic extensions of the rationals in general it is not hard to deduce the p torsion in the topological hochschild homology of rings of integers which do not ramify at p from the corresponding torsion for the integers thus for the ring of gaussian integers and also for the integers with the square root of 2 or 2 adjoined this paper completes the calculation of the topological hochschild homology spectrum which is a priori known to be a product of eilenberg maclane spectra precisely because of this a priori knowledge the homotopy groups can be found by doing a homology calculation a spectral sequence arising from the filtration of topological hochschild homology by simplicial skeleta converges to the desired homology if one reduces the rings in question modulo 2 the spectral sequence collapses at its second term this comparison bounds the nontriviality of higher differentials in the spectral sequence for the original ring it is explicitly demonstrated using simplicial calculations that the higher differentials are as nontrivial as they could be given this bound 0 introduction this paper gives an explicit calculation of the 2 torsion in the topological hochschild homology thh of rings of integers in quadratic extensions of the rationals which are ramified at the prime 2 theorem 1 14 the calculation of the torsion in thh of rings of integers in extensions unramified at p can be deduced from bokstedt s calculation of this invariant for the integers see 2 2 this means that given the results of this paper the 2 torsion of thh is known for any quadratic number ring it also means that for rings of integers in extensions which ramify only at the prime 2 specifically the gaussian integers z a 2 and z 2 the homotopy type of the thh spectrum is completely known and given in

topological hochschild homology of number rings we calculate an explicit formula for the topological hochschild homology of a discrete valuation ring of characteristic zero with finite residue field from this we deduce the topological hochschild homology of global number rings ams subject classification 19d55 16e40 19d50 55t99

detecting k theory by cyclic homology we discuss which part of the rationalized algebraic k theory of a group ring is detected via trace maps to hochschild homology cyclic homology periodic cyclic or negative cyclic homology

decoupling transformation for weakly coupled linear systems the non singular transformation that completely decouples a weakly coupled linear system under non restrictive conditions is defined the transformation matrices are obtained from two algebraic matrix equations algorithms that efficiently generate solutions to these equations are derived the proposed transformation also completely decouples the corresponding lyapunov matrix differential equation

laguerre polynomials and the inverse laplace transform using discrete data abstract we consider the problem of finding a function defined on 0 from a countable set of values of its laplace transform the problem is severely ill posed we shall use the expansion of the function in a series of laguerre polynomials to convert the problem in an analytic interpolation problem then using the coefficients of lagrange polynomials we shall construct a stable approximation solution error estimate is given numerical results are produced

output tracking control of uncertain nonlinear second order systems abstract the solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum time optimal control problem two different types of uncertainty are considered in the paper in the first case a constant bound on the uncertain dynamics is assumed to be known in the second case the bound is a function of both the measurable and the unmeasurable state variable of the system in both cases the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features the resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one or to a vicinity of this latter is proved also in the uncertain case

relay based identification of a class of nonminimum phase siso processes a set of general expressions is derived from a single symmetrical relay feedback test for identification of a class of process transfer functions using these expressions the parameters of open loop stable nonminimum phase transfer function models may be obtained from simple measurements made on the limit cycle for comparison the conventional describing function based identification formulae are presented fourier series based curve fitting with the options of nonlinear least squares method and trust region algorithm is used to measure limit cycle parameters in the presence of measurement noise examples are given to illustrate the value of the proposed method

general linear and functor cohomology over finite fields in recent years there has been considerable success in computing extgroups of modular representations associated to the general linear group by relating this problem to one of computing ext groups in functor categories f l s f s in this paper we extend our ability to make such ext group calculations by establishing several fundamental results throughout this paper we work over fields of positive characteristic p the reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of ext groups for example the existence of nonzero ext groups of positive degree is equivalent to the existence of objects which are not direct sums of simple objects indeed a knowledge of ext groups provides considerable knowledge of compound objects in the study of modular representation theory of finite chevalley groups such as gln fq ext groups play an even more central role it has been shown in cps that a knowledge of certain ext groups is sufficient to prove lusztig s conjecture concerning the dimension and characters of irreducible representations we consider two different categories of functors the category f fq of all functors from finite dimensional fq vector spaces to fq vector spaces where fq is the finite field of cardinality q and the category p fq of strict polynomial functors of finite degree as defined in f s the category p fq presents several advantages over the category f fq from the point of view of computing extgroups these are the accessibility of injectives and projectives the existence of a base change and an even easier access to ext groups of tensor products this explains the usefulness of our comparison in theorem 3 10 of ext groups in the category p fq with ext groups in the category f fq weaker forms of this theorem have been known to us since 1995 and to s betley independently

numerical methods to design the reaching phase of output feedback variable structure control abstract two numerical methods are developed to design a variable structure control that satisfies the reaching condition using static output feedback the design is formulated as a nonsmooth convex optimization problem for which existing algorithms are available it is shown how the resulting control law can be modified to be robust in the presence of parameter uncertainty or a disturbance numerical examples successfully demonstrate the developed techniques

on the discrete time integral sliding mode control a new discrete time integral sliding mode control dismc scheme is proposed for sampled data systems the new control scheme is characterized by a discrete time integral sliding manifold which inherits the desired properties of the continuous time integral sliding manifold such as full order sliding manifold with pole assignment and elimination of the reaching phase in particular comparing with existing discrete time sliding mode control the new scheme is able to achieve more precise tracking performance it will be shown in this work that the new control scheme achieves o t2 steady state error for state regulation with the widely adopted delay based disturbance estimation another desirable feature is the proposed dismc prevents the generation of overlarge control actions due to deadbeat response which is usually inevitable due to the existence of poles at the origin for a reduced order sliding manifold designed for sampled data systems both the theoretical analysis and illustrative example demonstrate the validity of the proposed scheme

the inverse laplace transform and analytic pseudo differential operators abstract by comparing the laplace transform l with the differential operatord we obtain a formula for the inverse laplace transform l 1 1 v 1 cos d v l wherevis a unitary transformation operator this helps us obtain an explicit spectral representation of l some applications of the above relation are discussed

componentwise ultimate bounds for positive discrete time delay systems perturbed by interval disturbances this paper presents a method to derive componentwise ultimate upper bounds and componentwise ultimate lower bounds for linear positive systems with time varying delays and bounded disturbances the disturbance vector is assumed to vary within a known interval whose lower bound may be different from zero we first derive a sufficient condition for the existence of componentwise ultimate bounds this condition is given in terms of the spectral radius of the system matrices which is easy to check and allows us to compute directly both the smallest componentwise ultimate upper bound and the largest componentwise ultimate lower bound then by using the comparison method we extend the obtained result to a class of nonlinear time delay systems which has linear positive bounds two numerical examples are given to illustrate the effectiveness of the obtained results

independent of delay stabilization using implicit lyapunov function method nonlinear implicit lyapunov function based control algorithms with delay in control and or measurement channel are presented the proposed control algorithms provide global asymptotic stability for all delays less than a certain threshold value and global asymptotic stability with respect to a compact set containing the origin for any delay over the threshold value the algorithms are also applicable for the fast varying delay and sampled data cases the theoretical results are supported by numerical simulations and comparison

new predictive scheme for the control of lti systems with input delay and unknown disturbances a new predictive scheme is proposed for the control of linear time invariant lti systems with a constant and known delay in the input and unknown disturbances it has been achieved to include disturbances effect in the prediction even though there are completely unknown the artstein reduction is then revisited thanks to the computation of this new prediction an extensive comparison with the standard scheme is presented throughout the article it is proved that the new scheme leads to feedback controllers that are able to reject perfectly constant disturbances for time varying ones a better attenuation is achieved for a wide range of perturbations and for both linear and nonlinear controllers a criterion is given to characterize this class of perturbations finally some simulations illustrate the results

singularly perturbed analysis of chattering in relay control systems for sliding mode control systems with fast actuators sufficient conditions for the exponential decreasing of the amplitude of chattering and unlimited growth of frequency are found the connection between the stability of actuators and the stability of the plant on the one hand and the stability of the sliding mode system as the whole on the other hand is investigated the algorithm for correction of sliding mode equations is suggested for taking into account the presence of fast actuators

an lmi approach for second order sliding set design using piecewise lyapunov functions abstract this paper pursues recent results on second order sliding set design for uncertain linear systems with matched uncertainties a methodology that allows the designer to use a lesser number of derivatives it is based on the direct lyapunov method and its conditions are expressed as linear matrix inequalities which are efficiently solved via convex optimization techniques in contrast with former reports on this matter the class of lyapunov functions in this work is enlarged to the set of those which are piecewise c 1 which naturally leads to design advantages as well as to a more natural adaptation of the lyapunov function to the discontinuities arising in the sliding set design methodology simulation examples are included to show the effectiveness of the proposed approach

global asymptotic stabilization of feedforward systems with delay in the input the problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved no limitation on the size of the delay is imposed explicit expressions of bounded control laws are determined

euler s discretization effect on a twisting algorithm based sliding mode control in this paper the euler s discretization of the second order sliding mode control system with the twisting algorithm is studied it shows that for all values of discretization steps initial conditions and allowable parameter setting the system trajectories are always bounded it also shows that for the uncertain systems under a mild condition the system trajectories are also bounded further the periodic behaviors and limit cycles of the system trajectories are explored with conditions for the existence of periodic orbits formulated extensive simulation examples are given to show typical trajectories and comparison between the first order and second order sliding mode control systems

static sliding mode control of systems with arbitrary relative degree by using artificial delay static output feedback stabilization of systems with relative degree n with matched disturbances is considered assuming that the system is controllable a static output feedback sliding mode controller is designed where the output derivatives up to the order n 1 are approximated by using the current and the delayed values of the output numerical examples illustrate the efficiency of the method

complex variable and regularization methods of inversion of the laplace transform in this paper three methods are derived for approximating f given its laplace transform g on 0 oo i e fo f t exp st dt g s assuming that g e l 0 oo the first method is based on a sinc like rational approximation of 9 the second on a sinc solution of the integral equation fo0 f t exp st dt g s via standard regularization and the third method is based on first converting fo0 f t exp st dt g s to a convolution integral over r and then finding a sinc approximation to f via the application of a special regularization procedure to solve the fourier transform problem we also obtain bounds on the error of approximation which depend on both the method of approximation and the regularization parameter

accuracy of homogeneous sliding modes in the presence of fast actuators it is shown that the higher the order r of the homogeneous sliding mode is the less sensitive it is to the presence of unaccounted for fast stable actuators with being an infinitesimal actuator time constant the sliding variable magnitude is proved to be proportional to r the system chattering is shown to be not amplified in the presence of actuators

global asymptotic stabilization for chains of integrators with a delay in the input the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved no limitation on the size of the delay is imposed to validate the approach a third order example is presented

output feedback sliding mode control for sampled data systems we address the output feedback sliding mode control problem for a sampled data linear system with external disturbances of the matching type by taking into account the disturbance compensation a deadbeat high gain output feedback control strategy with additional dynamics is able to attenuate the disturbances in the framework of singular perturbation analysis it is shown that the closed loop system exhibits good robustness against exogenous disturbances

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

output feedback variable structure control design this paper describes a procedure for variable structure output feedback control vsofc for a class of multivariable linear time invariant systems the equivalent control method of utkin in the output feedback mode yields a reduced order system exhibiting output feedback equivalent dynamics using the kimura davison sufficient conditions for pole placement using output feedback a sufficient condition is derived to assign the eigenvalues of the reduced order system the observability and controllability of the reduced system is discussed two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques a control design is discussed as well

sliding mode control in the presence of input delay a singular perturbation approach sliding mode control smc in the presence of small unavoidable input delay as may be present in controller implementation is studied linear systems with bounded matched disturbances and uncertain system matrices are considered where input delay in the smc will produce oscillations or potentially even unbounded solutions without a priori knowledge of the bounds on the state dependent terms as required by existing methods the design objective is to achieve ultimate boundedness of the closed loop system with a bound proportional to the delay and disturbance bounds this is a non trivial problem because the relay gain depends on the state bound whereas the latter bound depends on the relay gain a controller with linear gain proportional to the scalar email protected is proposed which for small enough m 0 produces a closed loop singularly perturbed system and yields the desired ultimate bound a constructive linear matrix inequality lmi based solution for evaluation of both the design parameters and the ultimate bound is derived the superiority of the proposed control over existing methodologies that ignore input delay within the design is demonstrated through an example

singular perturbation analysis of cheap control problem for sampled data systems this paper studies the discrete time cheap control problem for sampled data systems using the theory of singular perturbations it is shown by using the two time scale property of singularly perturbed systems that the problem can be solved in terms of two reduced order subproblems for which computations can be done in parallel thus increasing the computational speed similarly to the continuous time case the singular perturbation approach enables the decomposition of the algebraic riccati equation into two reduced order pure slow and pure fast continuous time algebraic equations

finite time global stabilization by means of time varying distributed delay feedback the paper contains certain results concerning the finite time global stabilization for triangular control systems described by retarded functional differential equations by means of time varying distributed delay feedback these results enable us to present solutions to feedback stabilization problems for systems with delayed input the results are obtained by using the backstepping technique

dynamics of relay relaxation oscillators relaxation oscillators can usually be represented as a feedback system with hysteresis the relay relaxation oscillator consists of relay hysteresis and a linear system in feedback the objective of this work is to study the existence of periodic orbits and the dynamics of coupled relay oscillators in particular we give a complete analysis for the case of unimodal periodic orbits and illustrate the presence of degenerate and asymmetric orbits we also discuss how complex orbits can arise from bifurcation of unimodal orbits finally we focus on oscillators with an integrator as the linear component and study the entrainment under external forcing and phase locking when such oscillators are coupled in a ring

higher equivariant excision abstract we develop a theory of goodwillie calculus for functors between g equivariant homotopy theories where g is a finite group we construct j excisive approximations for any finite g set j these combine into a poset the goodwillie tree that extends the classical goodwillie tower we prove convergence results for the tree of a functor on pointed g spaces that commutes with fixed points and we reinterpret the tom dieck splitting as an instance of a more general splitting phenomenon that occurs for the fixed points of the equivariant derivatives of these functors as our main example we describe the layers of the tree of the identity functor in terms of the equivariant spanier whitehead duals of the partition complexes

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

analysis of delayed sliding mode control systems under zero order holder discretization zero order holder discretization effects in sliding mode control systems with an input delay are studied stability conditions are formulated conditions for the existence of periodic solutions are derived and the existence of periodic steady states is investigated the influence of the discretization step and the delay on the period and the amplitude of steady state oscillations is discussed structure of basins of attraction of periodic orbits with different switching patterns is studied simulation results are presented to illustrate various behaviors

robust semiglobal stabilization of the second order system by relay feedback with an uncertain variable time delay we present sufficient conditions for robust relay delayed semiglobal stabilization of second order systems which relate the upper bound to an uncertain time delay and the parameters of the plant we also suggest an algorithm of delayed relay control gain adaptation for semiglobal stabilization which is based on delayed information about the sign of the controlled variable only the proposed algorithm suppresses bounded uncertainties in the time delay that is being designed for the upper bound of uncertainty in the time delay the control law ensures semiglobal stabilization independently of any variable time delay obeying the given upper bound

the bad truth about laplace s transform inverting the laplace transform is a paradigm for exponentially ill posed problems for a class of operators including the laplace transform we give forward and inverse formulae that have fast implementations using the fast fourier transform these formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the laplace transform