a combinatorial approach to nonlinear functional expansions an introduction with an example in this paper we present a new approach to causal functionals we introduce combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms this theory parallels the algebraic approach with formal power series in noncommutative variables developed by fliess lamnabhi and lamnabhi lagarrigue this theory makes use of certain combinatorial objects called weighted increasing trees weighted paths and histories we can deduce very efficient algorithms for the computation of the corresponding volterra kernels we present an introduction to our combinatorial theory an example with a nonlinear circuit gives the flavor of our approach the complete proofs and general theory will be discussed in a further paper

on the radius of convergence of interconnected analytic nonlinear input output systems a complete analysis is presented of the radii of convergence of the parallel product cascade and feedback interconnections of analytic nonlinear input output systems represented as fliess operators such operators are described by convergent functional series which are indexed by words over a noncommutative alphabet their generating series are therefore specified in terms of noncommutative formal power series given growth conditions for the coefficients of the generating series for the subsystems the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent in the process of deriving the radius of convergence for the feedback connection it is shown definitively that local convergence is preserved under feedback this had been an open problem in the literature until recently

faa di bruno hopf algebra of the output feedback group for multivariable fliess operators abstract given two nonlinear input output systems written in terms of chen fliess functional expansions i e fliess operators it is known that the feedback interconnected system is always well defined and in the same class an explicit formula for the generating series of a single input single output closed loop system was provided by the first two authors in earlier work via hopf algebra methods this paper is a sequel it has four main innovations first the full multivariable extension of the theory is presented next a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature this grading also facilitates a fully recursive algorithm to compute the antipode of the hopf algebra of the output feedback group and thus the corresponding feedback product can be computed much more efficiently the final innovation is an improved convergence analysis of the antipode operation namely the radius of convergence of the antipode is computed

computable convergence bounds of series expansions for infinite dimensional linear analytic systems and application this paper deals with the convergence of series expansions of trajectories for semi linear infinite dimensional systems which are analytic in state and affine in input a special case of such expansions corresponds to volterra series which are extensively used for the analysis the simulation and the control of weakly nonlinear finite dimensional systems the main results of this paper give computable bounds for both the convergence radius and the truncation error of the series these results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality they are available for distributed and boundary control systems as an illustration these results are applied to an epidemic population dynamic model in this example it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues the relevance of the method is illustrated by simulations

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

interconnection of nonlinear causal systems for nonlinear systems it is not a mathematically obvious fact that the simplest feedback system yields equations which are solvable even when the gain in the loop is

on two definitions of observation spaces this paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls

divergence free preserving high order schemes for magnetohydrodynamics an artificial magnetic resistivity method abstract this paper proposes a new strategy that is very simple divergence free high order accurate yet has an effective discontinuous capturing capability for simulating compressible magnetohydrodynamics mhd the new strategy is to construct artificial diffusion terms in a physically consistent manner such that the artificial terms act as a diffusion term only in the curl of magnetic field to capture numerical discontinuities in the magnetic field while not affecting the divergence field thus maintaining divergence free constraint the physically consistent artificial diffusion terms are built into the induction equations in a conservation law form at a partial differential equation level the proposed method may be viewed as adding an artificial magnetic resistivity to the induction equations and is inherently divergence free both ideal and resistive mhd with and without shock waves and also both inviscid and viscous flows the method is based on finite difference method with co located variable arrangement and we show that any linear finite difference scheme in an arbitrary order of accuracy can be used to discretize the modified governing equations to ensures the divergence free and the global conservation constraints numerically at the discretization level the artificial magnetic resistivity is designed to localize automatically in regions of discontinuity in the curl of magnetic field and vanish wherever the flow is sufficiently smooth with respect to the grid scale thereby maintaining the desirable high order accuracy of the employed discretization scheme in smooth regions two dimensional smooth and non smooth ideal mhd problems are considered to validate the capability of the proposed method

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

algebraic differential equations and rational control systems an equivalence is shown between realizability of input output i o operators by rational control systems and high order algebraic differential equations for i o pairs this generalizes to nonlinear systems the equivalence between autoregressive representations and finite dimensional linear realizability

continuous time hammerstein system identification from sampled data a continuous time hammerstein system driven by a random signal is identified from observations sampled in time the sampling may be uniform or not the a priori information about the system is nonparametric functional forms of both the nonlinear characteristic and the impulse response are completely unknown three kernel algorithms one offline and two semirecursive are presented their convergence to the true characteristic of the nonlinear subsystem is shown the distance between consecutive sampling times must not decrease too fast for the algorithms to converge

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

a quantitative condition for the well posedness of interconnected dynamical systems a sufficient condition for the well posedness of interconnected dynamical systems is reported the condition is stated in terms of the uniform instantaneous gains of subsystems and interconnections and includes vidyasagar s condition 8 as a special case the result is useful for testing the validity of the dynamical model of a large scale system which is obtained by combining the models of subsystems

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

symmetry theorems for the overdetermined eigenvalue problems abstract the well known schiffer conjecture saying that for a smooth bounded domain r n if there exists a positive neumann eigenvalue such that the corresponding neumann eigenfunction u is constant on the boundary of then is a ball in this paper we shall prove that the schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding neumann eigenfunction is constant on the boundary we also prove a similar result to the berenstein conjecture for the overdetermined dirichlet eigenvalue problem

non causal fliess operators and their shuffle algebra fliess operators as a class of non linear operators have been well studied in several respects they have a well developed realization theory and convenient representations in terms of directed infinite products of exponential lie series their interconnection as subsystems has been studied as has their relationship to rational systems they find applications in such diverse areas as discretization methods for controls systems optimal control neural network analysis and the numerical solution of stochastic differential equations one issue concerning fliess operators however that has received little attention is their possible generalization to the non causal case examples of such operators appear implicitly in the literature addressing hilbert adjoints of causal non linear operators and system inversion for the purpose of output tracking but a general systematic treatment of the subject has not appeared in this paper a non causal extension of a fliess operator is developed with the primary focus being on local convergence continuity the associated shuffle algebra and computing adjoint operators

an extended linearization approach to nonlinear system inversion the inversion problem for nonlinear systems is considered from both the internal viewpoint state equation representations and the external viewpoint volterra series representations in order to circumvent difficulties in the known exact inversion theory a method for obtaining what will be called a partial inverse system is proposed for a given nonlinear system a cascade system typically containing differentiators is found such that the overall nonlinear system has a family of constant operating points with inputs and outputs identical furthermore the linearization of the overall system about each of these constant operating points is the identity system

technical communique extending the domain of definition of functional series for nonlinear systems functional series are the basis of much of existing nonlinear system theory the present note extends the well known local series beyond its radius of convergence to describe the entire system trajectory for as long as it exists first steps towards a generalization to systems defined on analytic manifolds are made

generating series for interconnected analytic nonlinear systems given two analytic nonlinear input output systems represented as fliess operators four system interconnections are considered in a unified setting the parallel connection product connection cascade connection and feedback connection in each case the corresponding generating series is produced and conditions for the convergence of the corresponding fliess operator are given in the process an existing notion of a composition product for formal power series has its set of known properties significantly expanded in addition the notion of a feedback product for formal power series is shown to be well defined in a broad context and its basic properties are characterized

a faa di bruno hopf algebra for a group of fliess operators with applications to feedback a faa di bruno type hopf algebra is developed for a group of integral operators known as fliess operators where operator composition is the group product such operators are normally written in terms of generating series over a noncommutative alphabet using a general series expansion for the antipode an explicit formula for the generating series of the compositional inverse operator is derived the result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product that is the generating series for the fliess operator representation of the closed loop system written in terms of the generating series of the fliess operator component systems this formula is employed to provide a proof that local convergence is preserved under feedback

some results on the schiffer conjecture in r 2 abstract let be an open bounded domain in r 2 with connected and c boundary and a solution of 0 1 0 2 n 0 0 3 const 0 for some 0 denoting by 0 1 2 the set of all neumann eigenvalues for the laplacian on we show that 1 if 8 or 2 if is strictly convex and centrally symmetric 13 then must be a disk

preventing numerical errors generated by interface capturing schemes in compressible multi material flows in the present work errors generated in computations of compressible multi material flows using shock capturing schemes are examined specifically pressure oscillations when the specific heats ratio is variable but also temperature spikes and species conservation errors these numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms though temperature errors are irrelevant to solutions to the euler equations it is shown that they have the potential to lead to problems when physical diffusion is included i e for the navier stokes equations these errors are studied analytically and numerically by considering the one dimensional advection of isolated material discontinuities a methodology preventing such errors for weighted essentially non oscillatory weno schemes is presented in which modified weno weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats several multi dimensional problems with various discontinuities shocks material interfaces and contact discontinuities including the single mode richtmyer meshkov instability and turbulence are considered to test the method

a unified approach to generating series for mixed cascades of analytic nonlinear input output systems the primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input output systems in particular it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators fliess operators and analytic functions provided that the composite system is well defined and resides in one of these two classes in each case the radius of convergence of the interconnection is computed

symmetry results for overdetermined boundary value problems of nonlinear elliptic equations abstract we prove the radial symmetry of the solutions of second order nonlinear elliptic equations for overdetermined dirichlet and neumann boundary value problems in addition a global uniqueness theorem of holmgren type is given for nonlinear elliptic equations

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

a robust and accurate numerical method for transcritical turbulent flows at supercritical pressure with an arbitrary equation of state this paper addresses issues in high fidelity numerical simulations of transcritical turbulent flows at supercritical pressure the proposed strategy builds on a tabulated look up table method based on refprop database for an accurate estimation of non linear behaviors of thermodynamic and fluid transport properties at the transcritical conditions based on the look up table method we propose a numerical method that satisfies high order spatial accuracy spurious oscillation free property and capability of capturing the abrupt variation in thermodynamic properties across the transcritical contact surface the method introduces artificial mass diffusivity to the continuity and momentum equations in a physically consistent manner in order to capture the steep transcritical thermodynamic variations robustly while maintaining spurious oscillation free property in the velocity field the pressure evolution equation is derived from the full compressible navier stokes equations and solved instead of solving the total energy equation to achieve the spurious pressure oscillation free property with an arbitrary equation of state including the present look up table method flow problems with and without physical diffusion are employed for the numerical tests to validate the robustness accuracy and consistency of the proposed approach

general siso takagi sugeno fuzzy systems with linear rule consequent are universal approximators takagi sugeno ts fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice virtually all the ts fuzzy systems use linear rule consequent at present there exist no results qualitative or quantitative to answer the fundamentally important question that is especially critical to ts fuzzy systems as fuzzy controllers and models are ts fuzzy systems with linear rule consequent universal approximators if the answer is yes then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration in this paper we provide answers to these questions for a general class of single input single output siso fuzzy systems that use any type of continuous input fuzzy sets ts fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case we first constructively prove that this general class of siso ts fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove by utilizing the weierstrass approximation theorem that the general ts fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision furthermore we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound an illustrative numerical example is provided

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

discrete time approximations of fliess operators a convenient way to represent a nonlinear input output system in control theory is via a chen fliess functional expansion or fliess operator the general goal of this paper is to describe how to approximate fliess operators with iterated sums and to provide accurate error estimates for two different scenarios one where the series coefficients are growing at a local convergence rate and the other where they are growing at a global convergence rate in each case it is shown that the error estimates are asymptotically achievable for certain worst case inputs the paper then focuses on the special case where the operators are rational i e they have rational generating series and thus are realizable in terms of bilinear ordinary differential state equations in particular it is shown that a discretization of the state equation via a kind of euler approximation coincides exactly with the discrete time fliess operator approximator of the continuous time rational operator

bilinear system interconnections and generating series of weighted petri nets this paper has three objectives the first objective is to provide a simpler proof concerning a sufficient condition due to ferfera under which bilinearity is preserved for cascade interconnections the next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection the final objective is to show that the well known correspondence between rational series and formal power series recognized by weighted finite state automata can be generalized to produce a correspondence between the generating series of cascaded and feedback connected bilinear systems and a class of weighted petri nets

large eddy simulation of a supersonic boundary layer using an unstructured grid a mach 3 adiabatic flat plate turbulent boundary layer is studied using large eddy simulation les the filtered compressible navier stokes equations are solved on a three dimensional unstructured grid of tetrahedral cells a compressible extension of the rescaling reintroducing process of lund et al is developed to generate the inflow conditions the effect of the subgrid scale motion is incorporated using two approaches namely monotone integrated les miles and the smagorinsky subgrid scale model a detailed grid refinement study is performed

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

hybrid multifluid algorithms extensions of many successful single component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components a new algorithm based on the compressible euler equations for multicomponent fluids augmented by the pressure evolution equation is presented the extended set of equations offers two alternative ways to update the pressure field i using the equation of state or ii using the pressure evolution equation in a numerical implementation these two procedures generally yield different answers the former is a standard conservative update but may produce oscillations near material interfaces the latter is nonconservative but becomes exact near interfaces and automatically maintains pressure equilibrium a hybrid scheme which selects from the two pressure update procedures is presented the scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite very small number of grid cells computed solutions exhibit oscillation free interfaces and have em negligible relative conservation errors in total energy even for very strong shocks the proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend one s favourite scheme to solve multifluid dynamics

left inversion of analytic nonlinear siso systems via formal power series methods given a single input single output siso system f and a function y in the range of f the left inversion problem is to determine an input u such that y f u the goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where f is an analytic mapping in the sense that it has a convergent chen fliess functional expansion and y is a real analytic function in particular it will be shown that given a certain condition on the generating series c of f a corresponding unique analytic u can always be determined via operations on formal power series the condition on c turns out to be equivalent to having a well defined relative degree when f has an input affine analytic state space realization with finite dimension but the method is applicable even when f does not have such a realization the technique is demonstrated on four examples including a continuous stirred chemical reactor