non linear observer based on the euler discretization in this paper we present an estimator based on the euler discretization first we show that the proposed estimator is an exponential observer in the case when the error between trajectories of continuous time and its associated euler discretized system can be neglected otherwise we show that the error of estimation is exponentially attracted by some neighbourhood of the origin which depends only on the sampling time finally numerical simulations using a bioreactor model are given in order to highlight the performance of the proposed estimator

fusion categories arising from semisimple lie algebras using tilting modules we equip certain semisimple categories with a reduced tensor product structure the fusion rules for this tensor product are determined via known character formulas for the involved modules

fast wave computation via fourier integral operators this paper presents a numerical method for time upscaling wave equations i e performing time steps not limited by the courant friedrichs lewy cfl condition the proposed method leverages recent work on fast algorithms for pseudodierential and fourier integral operators fio this algorithmic approach is not asymptotic it is shown how to construct an exact fio propagator by 1 solving hamilton jacobi equations for the phases and 2 sampling rows and columns of low rank matrices at random for the amplitudes the setting of interest is that of scalar waves in two dimensional smooth periodic media of class c 1 over the torus where the bandlimit n of the waves goes to innity in this setting it is demonstrated that the algorithmic complexity for solving the wave equation to xed time t 1 can be as low as o n 2 logn with controlled accuracy numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest

a hybrid redesign of newton observers in the absence of an exact discrete time model abstract we study the newton observer design developed by moraal and grizzle when the exact discrete time model of the sampled data plant is not known analytically we eliminate the dependence on this exact model with a hybrid reconstruction that makes use of continuous time filters to produce the numerical value of the exact model we then implement the newton method with finite difference and secant approximations for the jacobian despite the continuous time filters the proposed hybrid redesign preserves the sampled data characteristic of the newton observer because it only employs discrete time measurements of the output it also offers flexibility to be implemented with nonuniform or event driven sampling we finally study how a line search scheme can be incorporated in this hybrid newton observer to enlarge the region of convergence

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

nonlinear discrete time observer design with linearizable error dynamics a necessary and sufficient condition for the existence of a discrete time nonlinear observer with linearizable error dynamics is provided the result can be applied to any real analytic nonlinear system whose linear part is observable the necessary and sufficient condition is the solvability of a nonlinear functional equation furthermore the well known siegel s theorem on the linearizability of a mapping is naturally reproduced in a corollary the proposed observer design method is constructive and can be applied approximately to any sufficiently smooth linearly observable system yielding a local observer with approximately linear error dynamics

brief paper robust h observer design for sampled data lipschitz nonlinear systems with exact and euler approximate models an lmi approach is proposed for the design of robust h observers for a class of lipschitz nonlinear systems two type of systems are considered lipschitz nonlinear discrete time systems and lipschitz nonlinear sampled data systems with euler approximate discrete time models observer convergence when the exact discrete time model of the system is available is shown then practical convergence of the proposed observer is proved using the euler approximate discrete time model as an additional feature maximizing the admissible lipschitz constant the solution of the proposed lmi optimization problem guaranties robustness against some nonlinear uncertainties the robust h observer synthesis problem is solved for both cases the maximum disturbance attenuation level is achieved through lmi optimization

conformally invariant systems of differential operators abstract kostant s theory of conformally invariant differential operators on certain homogeneous spaces is generalized to cover conformally invariant systems of endomorphism valued differential operators in particular the connection discovered by kostant between conformally invariant operators and highest weight vectors in generalized verma modules is extended

hardy space estimates for the wave equation on compact lie groups abstract let l be the laplacian in this paper we prove that on a compact lie group g of dimension n the multiplier operator e i s l 1 l 2 s 0 1 extends to a bounded operator on the hardy space h p g 0 p if and only if 1 p 1 2 n 1 the result is an analogue of a well known theorem in euclidean space

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

brief paper fault tolerant control for a class of nonlinear sampled data systems via a euler approximate observer in this paper a fault tolerant control ftc scheme is developed for a class of nonlinear sampled data systems first a euler approximate discrete model is used to describe the plant under the sampling under this model an observer based fault estimation method is proposed to guarantee the accuracy of both the state and fault estimation values the conditions to make the approximate model consistent with the exact model are presented then an active fault tolerant controller which has a constraint condition for the sampling time is designed to make the faulty system stable finally an inverted pendulum is used to show the efficiency of the proposed method

unknown input observer design for one sided lipschitz discrete time systems subject to time delay in this paper we address the problem of unknown input observer design which simultaneously estimates state and unknown input of a class of nonlinear discrete time systems with time delay a novel approach to the state estimation problem of nonlinear systems where the nonlinearities satisfy the one sided lipschitz and quadratically inner bounded conditions is proposed this approach also allows us to reconstruct the unknown inputs of the systems the nonlinear system is first transformed to a new system which can be decomposed into unknown input free and unknown input dependent subsystems the estimation problem is then reduced to designing observer for the unknown input free subsystem rather than full order observer design in this paper we propose observer design of reduced order which is more practical and cost effective by utilizing several mathematical techniques the time delay issue as well as the bilinear terms which often emerge when designing observers for nonlinear discrete time systems are handled and less conservative observer synthesis conditions are derived in the linear matrix inequalities form two numerical examples are given to show the efficiency and high performance of our results

wave atoms and time upscaling of wave equations we present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying two dimensional time independent periodic media the method consists in representing the time dependent green s function in wave atoms a tight frame of multiscale directional wave packets obeying a precise parabolic balance between oscillations and support size namely wavelength diameter 2 wave atoms offer a uniquely structured representation of the green s function in the sense that the resulting matrix is universally sparse over the class of c coefficients even for large times the matrix has a natural low rank block structure after separation of the spatial indices the parabolic scaling is essential for these properties to hold as a result it becomes realistic to accurately build the full matrix exponential in the wave atom frame using repeated squaring up to some time typically of the form delta t sim sqrt delta x which is bigger than the standard cfl timestep once the expensive precomputation of the green s function has been carried out it can be used to perform unusually large upscaled cheap time steps the algorithm is relatively simple in that it does not require an underlying geometric optics solver we prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks on a n by n grid the expensive precomputation takes somewhere between o n 3log n and o n 4log n steps depending on the separability of the acoustic medium the complexity of upscaled timestepping however beats the o n 3log n bottleneck of pseudospectral methods on an n by n grid for a wide range of physically relevant situations in particular we show that a naive version of the wave atom algorithm provably runs in o n 2 operations for arbitrarily small but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant as a result we get estimates between o n 2 5 log n and o n 3 log n for upscaled timestepping we also show several numerical examples in practice the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions as in reflection seismology in academic examples of accurate propagation of bandlimited wavefronts if the precomputation step is factored out then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3 5 at n 512 and a factor 10 20 at n 1024 for the same accuracy very similar gains are obtained in comparison versus a finite difference method

projective modules in the category _ self duality given a parabolic subalgebra ps of a complex semisimple lie algebra there is a naturally defined category s of g modules which includes all the g modules induced from finite dimensional ps modules this paper treats the question of whether certain projective modules in s are isomorphic to their dual modules the projectives in question are the projective covers of those simple modules occurring in the socles of generalized verma modules their self duality is proved in a number of cases and additional information is obtained on the generalized verma modules

an observer based controller for nonlinear systems a gain scheduling approach abstract this paper presents an observer based controller for discrete time nonlinear systems whose control inputs are subject to a set of equality and or inequality constraints firstly a regularized least square estimator is developed to estimate the states of the nonlinear system then the estimated states are used to generate a feedback control law using the jacobean linearization in conjunction with gain scheduling the stability of the closed loop control system is rigorously investigated applications of the proposed observer based control scheme to two numerical examples are presented then the developed controller is used to control a synchronous machine which is subject to constraints on the control the simulation results show the effectiveness of the proposed observer based control scheme

a framework for nonlinear sampled data observer design via approximate discrete time models and emulation we study observer design for sampled data nonlinear systems using two approaches i the observer is designed via an approximate discrete time model of the plant ii the observer is designed based on the continuous time plant model and then discretized for sampled data implementation emulation we investigate under what conditions and in what sense these designs achieve convergence for the unknown exact discrete time model we present examples which show that designs that violate our conditions may indeed lead to instability when implemented on the exact model

decentralized sampled data fuzzy observer design for nonlinear interconnected systems this paper presents the decentralized sampled data fuzzy observer design techniques for nonlinear interconnected systems which are assumed to be composed of fuzzy subsystems and unknown interconnections to design the decentralized fuzzy observer the estimation error dynamics is obtained and the performance function is defined based on the estimation error dynamics and the performance function the continuous time decentralized fuzzy observer is first proposed to minimize the interconnection bound to attenuation degree ratio in addition the decentralized sampled data fuzzy observers are designed by using the approximate discretization and the exact discrete time design approaches respectively each proposed observer design technique is formulated into the optimal problems with the linear matrix inequalities finally several simulation examples show the validity and superiority of the proposed observer design techniques by comparing with the conventional techniques

the homomorphisms between scalar generalized verma modules associated to maximal parabolic subalgebras let g be a finite dimensional simple lie algebra over c we classify the homomorphisms between g modules induced from one dimensional modules of maximal parabolic subalgebras 1

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

observer design for a class of nonlinear systems a viable design methodology to construct observers for a class of nonlinear systems is developed the proposed technique is based on the off line solution of a riccati equation and can be solved using commercially available software packages for globally valid results the class of systems considered is characterized by globally lipschitz nonlinearities local results relax this assumption to only a local requirement for a more general description of nonlinear systems the methodology yields approximate observers locally the proposed theory is used to design an observer for a single link flexible joint robot this observer estimates the robot link variables based on the joint measurements

a seeger sogge stein theorem for bilinear fourier integral operators abstract we establish the regularity of bilinear fourier integral operators with bilinear amplitudes in s 1 0 m n 2 and non degenerate phase functions from l p l q l r under the assumptions that m n 1 1 p 1 2 1 q 1 2 and 1 p 1 q 1 r this is a bilinear version of the classical theorem of seeger sogge stein concerning the l p boundedness of linear fourier integral operators moreover our result goes beyond the aforementioned theorem in that it also includes the case of quasi banach target spaces

regularity of fourier integral operators with caustics the caustics of fourier integral operators are defined as caustics of the corresponding schwartz kernels lagrangian distributions on x y the caustic set c of the canonical relation is characterized as the set of points where the rank of the projection c x x y is smaller than its maximal value dim x x y 1 we derive the l p y l q x estimates on fourier integral operators with caustics of corank 1 such as caustics of type a m 1 m n for the values of p and q outside of a certain neighborhood of the line of duality q p the l p l q estimates are proved to be caustics insensitive we apply our results to the analysis of the blow up of the estimates on the half wave operator just before the geodesic flow forms caustics

backstepping designs for the stabilisation of nonlinear sampled data systems via approximate discrete time model the problems of backstepping adaptive backstepping and reduced order observer based output feedback control of sampled data nonlinear systems in strict feedback form are considered controller design methods based on the euler approximate model are presented for these problems the controllers are designed to compensate the effects of the discrepancy between the euler approximate model and exact discrete time model parameter estimation error in adaptive control and observer error in output feedback control which behave as disturbance it is shown that the obtained controllers semi globally practically asymptotically stabilise the plant model under standard assumptions then numerical examples are given to illustrate the design methods

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

discrete time nonlinear observer design using functional equations abstract the present research work proposes a new approach to the discrete time nonlinear observer design problem based on the early ideas that influenced the development of the linear luenberger observer the proposed approach develops a nonlinear analogue the formulation of the discrete time nonlinear observer design problem is realized via a system of first order linear nonhomogeneous functional equations and a rather general set of necessary and sufficient conditions for solvability is derived using results from functional equations theory the solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package

on the observer design in discrete time abstract the paper deals with the equivalence under coordinates change and output transformation of a discrete time nonlinear system to observer canonical forms necessary and sufficient conditions for local equivalence to these forms are given

two families of semiglobal state observers for analytic discrete time systems two families of observers for discrete time nonlinear systems are presented in this paper whose design is based on the taylor approximation of the inverse of the observation map semiglobal convergence results are provided under the assumption that the system observation map is a globally analytic diffeomorphism the performances of the observers in the two families are compared both from theoretical and practical points of view

leading weight vectors and homomorphisms between generalized verma modules abstract kostant introduced leading weight vectors in his paper on verma modules and quasi invariant differential operators we extend his idea to study homomorphisms between generalized verma modules we classify all first order leading weight vectors and thus determine corresponding hom spaces between generalized verma modules moreover in light of the correspondence between leading weight vectors and invariant differential operators we obtain a new and practical approach to the existence of first order invariant differential operators

controlling linear systems with nonlinear perturbations under sampled data output feedback digital redesign approach this article develops a digital redesign dr technique for sampled data observer based output feedback control of a continuous time linear system with nonlinear perturbation it is assumed that the nonlinear perturbation is a locally lipschitz function to deal with the discrete time modelling error in nonlinear systems as opposed to the previous approach the dr problem is configured as a stabilisation one for error dynamics between the closed loop system of nominal linear model under an analogue state feedback controller and that of the linear system with the nonlinear perturbation under a sampled data output feedback controller a constructive dr condition is formulated in the format of linear matrix inequalities the stability of the actual sampled data control system is guaranteed within the dr procedure the effectiveness of the proposed dr methodology is demonstrated through a numerical simulation

conformally invariant differential operators on minkowski space and their curved analogues this article describes the construction of a natural family of conformally invariant differential operators on a four dimensional pseudo riemannian manifold included in this family are the usual massless field equations for arbitrary helicity but there are many more besides the article begins by classifying the invariant operators on flat space this is a fairly straightforward task in representation theory best solved through the theory of verma modules the method generates conformally invariant operators in the curved case by means of penrose s local twistor transport

bernstein gelfand gelfand sequences the bernstein gelfand gelfand sequences extend the complexes of homogeneous vector bundles to curved cartan geometries

observer based stabilisation of linear systems with parameter uncertainties by using enhanced lmi conditions this paper deals with the problem of observer based stabilisation for linear systems with structured norm bounded parameter uncertainties a new design methodology is established thanks to a judicious use of some mathematical artefacts such as the well known young inequality and various matrix decompositions the proposed method allows one to compute simultaneously the observer and controller gains by solving a single bilinear matrix inequality bmi which becomes a linear matrix inequality lmi by freezing some scalars furthermore we show that some existing and elegant results reported in the literature can be regarded as particular cases of the stability conditions presented here numerical examples and evaluations of the conservatism are provided to show the effectiveness of the proposed design methodology

homomorphisms between generalized verma modules let g be a finite dimensional complex semisimple lie algebra and p a parabolic subalgebra the first result is a necessary and sufficient condition in the spirit of the bernstein gelfand gelfand theorem on verma modules for lepowsky s standard map between two generalized verma modules for g to be zero the main result gives a complete description of all homomorphisms between the generalized verma modules induced from one dimensional p modules in the hermitian symmetric situation

brief paper a generalized framework for robust nonlinear h filtering of lipschitz descriptor systems with parametric and nonlinear uncertainties in this paper a generalized robust nonlinear h filtering method is proposed for a class of lipschitz descriptor systems in which the nonlinearities appear both in the state and measured output equations the system is assumed to have norm bounded uncertainties in the realization matrices as well as nonlinear uncertainties we synthesize the h filter through semidefinite programming and strict lmis the admissible lipschitz constants of the nonlinear functions are maximized through lmi optimization the resulting h filter guarantees asymptotic stability of the estimation error dynamics with prespecified disturbance attenuation level and is robust against time varying parametric uncertainties as well as lipschitz nonlinear additive uncertainty explicit bound on the tolerable nonlinear uncertainty is derived based on a norm wise robustness analysis