comments on the drinfeld realization of the quantum affine superalgebra and its hopft algebra structure by generalizing the reshetikhin and semenov tian shansky construction to supersymmetric cases we obtain the drinfeld current realization for the quantum affine superalgebra we find a simple coproduct for the quantum current generators and establish the hopf algebra structure of this super current algebra

observer design for nonlinear control systems abstract this paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities a sufficient liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

vertex operators of uq gl n n and highest weight representations of uq gl 2 2 we determine the exchange relations of the level one q vertex operators of the quantum affine superalgebra uq gl n n we study in detail the level one irreducible highest weight representations of uq gl 2 2 and compute the characters and supercharacters associated with these irreducible modules

finite dimensional irreducible representations of the quantum superalgebra u q gl n 1 it is shown that every finite dimensional irreducible module over the general linear lie superalgebragl n 1 can be deformed to an irreducible module ofu q gl n 1 aq analogue of the universal enveloping algebra ofgl n 1 the results are extended also to all kac modules which in the atypical cases remain indecomposible within each module expressions for the transformations of the gel fand zetlin basis under the action of the algebra generators are written down an analogoue of the poincare birkhoff witt theorem is formulated

algebra of screening operators for the deformed wn algebra we construct a family of intertwining operators screening operators between various fock space modules over the deformed wn algebra they are given as integrals involving a product of screening currents and elliptic theta functions we derive a set of quadratic relations among the screening operators and use them to construct a felder type complex in the case of the deformed w3 algebra

screenings and vertex operators of quantum superalgebra uq sl n 1 we construct the screening currents of the quantum superalgebra uq sl n 1 for an arbitrary level k n 1 we show that these screening currents commute with the superalgebra modulo total difference we propose bosonizations of the vertex operators by using the screening currents we check that these vertex operators are the intertwiners among the fock wakimoto representation and the typical representation for rank n 4

a level one representation of the quantum affine superalgebra uqmn a level one representation of the quantum affine superalgebra uqmn and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields character formulas of level one irreducible highest weight modules of uq are conjectured

approximate high gain observers for non lipschitz observability forms a methodology for the design of continuous practical high gain observers for non linear observable systems is presented only two conditions are required injectivity of the observability map of order n and uniform continuity of its inverse using this map the system is transformed to observability normal form which could have a non lipschitz continuous right hand side and eventually multiplicity of solutions a so called e approximate high gain observer is designed for this form constituting the dynamic part of the observer a uniformly continuous extension of the inverse of the transformation is used as the algebraic part convergence of the error to a neighbourhood of the origin is guaranteed

vertex operators of quantum affine lie algebras u q d n 1 we give an explicit formula for the vertex operators related to the level 1 representations of the quantum affine lie algebrasuq dn 1 in terms of bosons as an application we derive an integral formula for the correlation functions of the vertex models withuq dn 1 symmetry

nonlinear observer design for a general class of discrete time nonlinear systems with real parametric uncertainty this paper is a geometric study of the local observer design for a general class of discrete time nonlinear systems with real parametric uncertainty explicitly we study the observer design problem for a general class of discrete time nonlinear systems with real parametric uncertainty and with an input generator exosystem in this paper we show that for the classical case when the state equilibrium does not change with the parametric uncertainty and when the plant output is purely a function of the state there is no local asymptotic observer for the plant next we show that in sharp contrast to this case for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty there typically exist local exponential observers even when the plant output is purely a function of the state we also present a characterization and construction procedure for local exponential observers for the general class of discrete time nonlinear systems with real parametric uncertainty under some stability assumptions we also show that for the general class of discrete time nonlinear systems considered under some stability assumptions the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs finally we generalize our results to a general class of discrete time nonlinear systems with input generator and with exogenous disturbance

vertex operators for twisted quantum affine algebras we construct explicitly the q vertex operators intertwining operators for the level one modules v i of the classical quantum affine algebras of twisted types using interacting bosons where i 0 1 for a 2 n 1 i 0 for d 3 i 0 n for d 2 1 and i n for a 2 n a perfect crystal graph for d 3 4 is constructed as a by product

stability improvement of nonlinear systems by feedback this paper is concerned with the property of stabilizing a nonlinear system to a specified equilibrium point arbitrarily fast by appropriate smooth feedback two closely related forms of this property are explored one refers to the asymptotic transfer to the critical point with exponential decay and it is shown that the systems possessing the above property are precisely those whose corresponding linearization is controllable the second stabilizability form defined here amounts to driving the system back to an arbitrarily small neighborhood of the given point arbitrarily fast by smooth feedback a global stabilization result is also included finding application in hamiltonian systems special emphasis is given to the bilinear case

design of discrete time nonlinear control systems via smooth feedback using the center manifold theory for maps in a systematic way the authors present in this note a necessary condition and sufficient conditions for discrete time nonlinear systems to be stabilizable via smooth state feedback then the authors derive a necessary and sufficient condition characterizing the existence of exponential observers for lyapunov stable discrete time nonlinear systems this in turn leads to a separation principle for the asymptotic stabilization problem as an application the output regulation problem is solved via error feedback

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty this paper is a geometric study of the local observer design for a general class of nonlinear systems with real parametric uncertainty explicitly we study the observer design problem for a general class of nonlinear systems with real parametric uncertainty and with an input generator exosystem in this paper we show that for the classical case when the state equilibrium does not change with the parametric uncertainty and when the plant output is purely a function of the state there is no local asymptotic observer for the plant next we show that in sharp contrast to this case for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty there typically exist local exponential observers even when the plant output is purely a function of the state we also present a characterization and construction procedure for local exponential observers for the general class of nonlinear systems with real parametric uncertainty under some stability assumptions we also show that for the general class of nonlinear systems considered under some stability assumptions the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs finally we generalize our results to a general class of nonlinear systems with input generator and with exogenous disturbance

non linear observer design by observer canonical forms this paper studies the non linear observer design problem by observer canonical forms we present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non linear state coordinate change under which a time variable system is transformed into an observer canonical form these conditions give the multiple versions of li and tao 1986 for the single output case then a comparison is made with the lie algebraic conditions obtained by krener and respondek 1985 and modified by the authors in a recent paper xia and gao 1987

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

observing non linear time variable systems through a canonical form observer abstract bestle and zeitz have introduced a non linear time variable observer canonical form which permits through a non linear state transformation the use of the eigenvalue assignment method to design observers for non linear time variable dynamic systems in this paper a necessary and sufficient condition under which the transformation exists is given it is also proved that the transformation could be determined by integrating single variable functions instead of solving a set of partial differential equations as shown by bestle and zeitz examples are included to show how to do the transformation

growth and roughness of the interface for ballistic deposition in ballistic deposition bd d 1 dimensional particles fall sequentially at random towards an initially flat large but bounded d dimensional surface and each particle sticks to the first point of contact for both lattice and continuum bd a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width sample standard deviation of the height of the interface to constants h t and w t respectively depending on time t we show that h t is asymptotically linear in t while w t 2 grows at least logarithmically in t when d 1 we use duality results showing that w t can be interpreted as the standard deviation of the height for deposition onto a surface growing from a single point

further results on the observer design problem abstract in this paper the observer design problem for nonlinear systems is considered sufficient lyapunov like conditions are presented for the existence of a nonlinear observer the theory we develop considerably improves and extends the results of our recent work 14

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

the extended luenberger observer for nonlinear systems abstract for nonlinear single input single output systems dot x f x u y h x u the relationships for a state transformation into the nonlinear observer canonical form are developed it is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation if the transformed nonlinearities are linearized about the reconstructed state with reference to the extended kalman filter algorithm this nonlinear observer design is called the extended luenberger observer

twisted vertex representations of quantum affine algebras recent interests in quantum groups are stimulated by their marvelous relations with quantum yang baxter equations conformal field theory invariants of links and knots and q hypergeometric series besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q deformations or quantum analogues of known structures quantum groups were first defined by drinfeld 2 and jimbo 9 also see 4 as a q deformation of the universal enveloping algebras of the kacmoody algebras in the work of trigonometric solutions of yang baxter equations in the same spirit it was shown in 13 14 that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable kac moody algebras and those of the corresponding quantum groups where both spaces have the same dimension in the case of generic q i e q is not a root of unity moreover one can be very explicit in the case of quantum gl n to write down the irreducible highest weight representations quantum affine algebras are the quantum groups associated to affine lie algebras following drinfeld s realization 3 of q analog of loop algebras the vertex representation of untwisted simply laced quantum affine algebras was constructed in frenkel jing 6 which is a q deformation of frenkel kac 7 and segal 15 construction in the theory of affine lie algebras subsequently the same was done for the quantum affine algebra of type b in 1 in the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the dynkin diagram which generalizes certain important cases in the ordinary twisted vertex operator calculus 5

a note on non linear observers a general non linear exponential observer is presented for non linearly perturbed nonlinear systems this is done by using the non linear variation of constants formula

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

observer design for unknown input nonlinear descriptor systems via convex optimization this paper treats the design problem of full order observers for nonlinear descriptor systems with unknown input ui depending on the available knowledge on the ui dynamics two cases are considered first a ui proportional observer uipo is proposed when the spectral domain of the ui is unknown second a pio is proposed when the spectral domain of the ui is in the low frequency range sufficient conditions for the existence and stability of such observers are given and proved based on the linear matrix inequality lmi approach an algorithm is presented to compute the observer gain matrix that achieves the asymptotic stability objective an example is included to illustrate the method

nonlinear observer design by observer error linearization this paper studies the observer design problem by the observer error linearization approach for nonlinear systems with and without inputs necessary and sufficient conditions for the existence of the linearization transformation are derived for nonlinear systems without inputs the conditions are shown to be corrections to an existing result a computation procedure and a different set of necessary and sufficient conditions based on the computation procedure are presented

state observation of nonlinear uncertain dynamical systems this note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties the design of these observers utilizes techniques related to variable structure systems theory a measure for the rate at which the estimates converge to the actual states is derived

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

on state observers for nonlinear systems abstract for linear systems it is known that there exists a state observer if and only if the system is detectable or in other words asymptotically stabilizable by output injection in this paper it is shown that asymptotic stabilizability by output injections is neither necessary nor sufficient for the construction of a nonlinear observer which may not have an exponential error decay rate the concept of uniform observers is introduced and necessary conditions and sufficient conditions are given for the construction of uniform observers

global finite time stabilization by output feedback for planar systems without observable linearization this note considers the problem of global finite time stabilization by output feedback for a class of planar systems without controllable observable linearization a sufficient condition for the solvability of the problem is established by developing a nonsmooth observer and modifying the adding a power integrator technique we show that an output feedback controller can be explicitly constructed to globally stabilize the systems in finite time as a direct application of the main result global output feedback finite time stabilization is achieved for the double linear integrator systems perturbed by some nonlinear functions which are not necessarily homogeneous

quantum group structure in the fock space resolutions of widehat sl n representations we describe a complex of wakimoto type fock space modules for the affine kac moody algebra widehat sl n the intertwining operators that build the complex are obtained from contour integrals of so called screening operators we show that a quantum group structure underlies the algebra of screening operators this observation greatly facilitates the explicit determination of the intertwiners we conjecture that the complex provides a resolution of an irreducible highest weight module in terms of fock spaces

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

a central limit theorem with applications to percolation epidemics and boolean models suppose x x x x zd is a white noise process and h b defined for finite subsets b of z d is determined in a stationary way by the restriction of x to b using a martingale approach we prove a central limit theorem clt for h as b becomes large subject to h satisfying a stabilization condition the effect of changing x x at a single site needs to be local this clt is then applied to component counts for percolation and boolean models to the size of the big cluster for percolation on a box and to the final size of a spatial epidemic

mathematics of random growing interfaces we establish a thermodynamic limit and gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces the results hold for the standard ballistic deposition model as well as the surface relaxation model in the off lattice setting the results are proved with the aid of general limit theorems for stabilizing functionals of marked poisson point processes

design of asymptotic estimators an approach based on neural networks and nonlinear programming a methodology to design state estimators for a class of nonlinear continuous time dynamic systems that is based on neural networks and nonlinear programming is proposed the estimator has the structure of a luenberger observer with a linear gain and a parameterized in general nonlinear function whose argument is an innovation term representing the difference between the current measurement and its prediction the problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error toward this end if a neural network is used to take on this function the parameters i e the neural weights are chosen together with the gain by constraining the derivative of a quadratic lyapunov function for the estimation error to be negative definite on a given compact set it is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points the gain is determined by solving a lyapunov equation the neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid point constraints that are not satisfied techniques based on low discrepancy sequences are applied to deal with a small number of sampling points and hence to reduce the computational burden required to optimize the parameters numerical results are reported and comparisons with those obtained by the extended kalman filter are made

on exponential observers for nonlinear systems abstract the aim of the paper is to identify the class of nonlinear systems that have exponential observers a concept introduced by previous authors it is shown that a necessary condition for the existence of an exponential observer for a nonlinear system is that the corresponding linearized system is detectable and for local exponential stabilization problems the condition is also sufficient this paper gives also a theorem on the separation property for the exponential design problem and it enables us to tell exactly to what extent the classical local linearization approach is applicable

comparative study of non linear state observation techniques abstract this paper contains a comparative study of four techniques for observing the slates of non linear systems the first technique examined is inspired by bestle and zeitz and icrener and respondek in this method a non linear transformation is found that brings the system into a canonical form from where observer design is facilitated the second technique is attributable to thau in this method the error between the system s true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid the third technique is due to baumann and rugh in this method extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant finally techniques from variable structure systems are utilized to design an observer that yields an exponentially decaying error like thau s observer but unl

on nonlinear observers in this note we make some remarks about the general observer problem for nonlinear systems we show how the existence of a partial observer without stability requirements is locally equivalent to the existence of a conditioned invariant distribution we argue that by considering not only identity observers we can put the stability issue in a broader perspective

further results on the uniform observability of discrete time nonlinear systems in this technical note the relation between notions of uniform observability of discrete time nonlinear systems based on injectivity of an observation map window the full rankness of its jacobian and a k function is investigated it is proven that a system is uniformly observable in the sense of injectivity of the observation map together with the full rankness of its jacobian if and only if it is so in some k function senses

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

uq sl 2 1 vertex operators screen currents and correlation functions at an arbitrary level bosonized q vertex operators related to the four dimensional evaluation modules of the quantum affine superalgebra uq sl 2 1 are constructed for arbitrary level k where 0 1 is a complex parameter appearing in the four dimensional evaluation representations they are intertwiners among the level highest weight fock wakimoto modules screen currents which commute with the action of uq sl 2 1 up to total differences are presented integral formulas for n point functions of type i and type ii q vertex operators are proposed

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

the extended kalman filter as an exponential observer for nonlinear systems we analyze the behavior of the extended kalman filter as a state estimator for nonlinear deterministic systems using the direct method of lyapunov we prove that under certain conditions the extended kalman filter is an exponential observer i e the dynamics of the estimation error is exponentially stable furthermore we discuss a generalization of the kalman filter with exponential data weighting to nonlinear systems

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

level one representations and vertex operators of quantum affine superalgebra uq gl n n level one representations of the quantum affine superalgebra uq gl n n associated with the appropriate nonstandard system of simple roots and q vertex operators intertwining operators associated with the level one modules are constructed explicitly in terms of free bosonic fields