an observer based anti windup scheme for non linear systems with input constraints this article addresses the problem of anti windup design for feedback linearizable non linear systems with input constraints subject to dynamic controllers with linear dynamics and non linear gains a method for designing the controller gains and a non linear observer based anti windup modification is proposed which allows reducing the behaviour of the closed loop system under the dynamic feedback controller to that under a given static feedback linearizing controller arbitrarily fast the proposed anti windup method allows thus to attenuate the effect of windup arbitrarily fast and guarantee that the region of closed loop asymptotic stability under the dynamic controller is limited only by the corresponding region under the static feedback linearizing controller the application and advantages of the method are illustrated through a simulation study on a methyl methacrylate polymerization reactor

adaptive wavelet schemes for nonlinear variational problems we develop and analyze wavelet based adaptive schemes for nonlinear variational problems we derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal our approach is based on a new paradigm that has been put forward recently for a class of linear problems the original problem is transformed first into an equivalent one which is well posed in the euclidean metric ell 2 then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step this iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances the main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions we prove asymptotically optimal complexity for adaptive realizations of first order iterations and of newton s method

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

harmonic wavelets towards the solution of nonlinear pde in this paper harmonic wavelets which are analytically defined and band limited are studied together with their differentiable properties harmonic wavelets were recently applied to the solution of evolution problems and more generally to describe evolution operators in order to consider the evolution of a solitary profile and to focus on the localization property of wavelets it seems to be more expedient to make use of functions with limited compact support either in space or in frequency the connection coefficients of harmonic wavelets are explicitly computed in the following at any order and characterized by some recursive formulas in particular they are functionally and finitely defined by a simple formula for any order of the basis derivatives

time response shaping using output to input saturation transformation for linear systems the control law design is often performed so that the resulting closed loop meets specific frequency domain requirements however in many cases it may be observed that the obtained controller does not enforce time domain requirements amongst which the objective of keeping a scalar output variable in a given interval in this article a transformation is proposed to convert prescribed bounds on an output variable into time varying saturations on the synthesised linear scalar control law this transformation uses some well chosen time varying coefficients so that the resulting time varying saturation bounds do not overlap in the presence of disturbances using an anti windup approach it is obtained that the origin of the resulting closed loop is globally asymptotically stable and that the constrained output variable satisfies the time domain constraints in the presence of an unknown finite energy bounded disturbance an application to a linear ball and beam model is presented

adaptive wavelet methods for saddle point problems optimal convergence rates in this paper an adaptive wavelet scheme for saddle point problems is developed and analyzed under the assumption that the underlying continuous problem satisfies the inf sup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity this means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the error of the best wavelet n term approximation of the solution with respect to the relevant norms moreover the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom it is remarkable that compatibility constraints on the trial spaces such as the ladyzhenskaya babuska brezzi lbb condition do not arise in the second part the general results are applied to the stokes problem aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon the regularity of the solutions in certain besov scales is analyzed these results reveal under which circumstances the work accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements this in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the stokes problem

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

refinable functions functionals and iterated function systems invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage that although usually considered in different contexts are deeply linked we outline these links in particular with respect to existence and regularity of these objects

a rankin selberg integral for the adjoint representation of gl3 in this paper we construct steps 1 4 and par t of 2 as formulated at g sh p 19 for the funct ion l s r here is a cusp form on g l a and r is the eight dimensional adjoint representa t ion of g l a the l group of g l 3 to do so we construct a rankin selberg integral using the embedding of sl 3 in gz let us ment ion tha t the langland s conjucture for this l function can not be obta ined by the langlands shahid i method the paper is organized as follows in the first section we collect some informat ion on the s t ructure of gz and compute the space of double cosets p gz sl3 where p is a maximal parabol ic subgroup of g2 in sect two the global integral is const ructed and is shown to be euler ian with whi t t ake r model finally in sect three we carry out the unramif ied computa t ion using the local computa t ion of the integral for gl3 x gl3 as const ructed in i j s for simplicity we shall assume tha t rc has a trivial central character

adaptive frame methods for nonlinear variational problems in this paper we develop adaptive numerical solvers for certain nonlinear variational problems the discretization of the variational problems is done by a suitable frame decomposition of the solution i e a complete stable and redundant expansion the discretization yields an equivalent nonlinear problem on the space of frame coefficients the discrete problem is then adaptively solved using approximated nested fixed point and richardson type iterations we investigate the convergence stability and optimal complexity of the scheme a theoretical advantage for example with respect to adaptive finite element schemes is that convergence and complexity results for the latter are usually hard to prove the use of frames is further motivated by their redundancy which at least numerically has been shown to improve the conditioning of the discretization matrices also frames are usually easier to construct than riesz bases we present a construction of divergence free wavelet frames suitable for applications in fluid dynamics and magnetohydrodynamics

fast evaluation of nonlinear functionals of tensor product wavelet expansions for a nonlinear functional f and a function u from the span of a set of tensor product interpolets it is shown how to compute the interpolant of f u from the span of this set of tensor product interpolets in linear complexity assuming that the index set has a certain multiple tree structure applications are found in the field of adaptive tensor product solution methods for semilinear operator equations by collocation methods or after transformations between the interpolet and bi orthogonal wavelet bases by galerkin methods

nonlinear tracking control in the presence of state and control constraints a generalized reference governor this paper proposes a new approach to reference governor design as in prior literature the governor accepts input commands and modifies their evolution so that specified pointwise in time constraints on state and control variables are satisfied the new approach applies to general discrete time and continuous time nonlinear systems with uncertainties it relies on safety properties provided by sublevel sets of equilibria parameterized functions these functions need not be lyapunov functions and the corresponding sublevel sets need not be positively invariant technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established the new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors this advantage is illustrated in a nonlinear control problem where off line computer based simulation is the basis for constructing the nonlinear function

nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation summary this paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process in particular evaluation covers here the computation of inner products of such expressions with wavelets which arise for instance in the context of galerkin or petrov galerkin schemes for the solution of differential equations the central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion i e the number of nonzero wavelet coefficients as little as possible the following issues are addressed first motivated by previous treatments of the subject we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of besov norms the principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets the analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal we conclude with a simple numerical example

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

an anti windup design for linear systems with input saturation abstract an anti windup design problem is posed and it is shown that all observer based anti windup modifications solve this problem at least locally sufficient conditions for an observer based anti windup modification to solve the global version of this problem are presented a novel observer based anti windup design is then proposed such that the anti windup problem can be interpreted in terms of a reduced order system in particular the proposed anti windup design induces an invariant subspace for the dynamic behavior of the mismatch between the constrained and unconstrained closed loops the dynamics in this invariant subspace are identical to the behavior of the plant with input saturation starting at the origin stabilized by linear state feedback and driven by the mismatch between the unconstrained input and this input passed through a saturation function the second part of the paper shows how the original dynamic compensator can be modified while retaining those dynamic features that produce a desirable closed loop steady state response to ensure that the requisite invariant subspace exists and it is reasonably tuned for input saturation two case studies are carried out on systems that have been investigated in the literature

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

on a rankin selberg integral of the l function for widetilde mathrm sl _2 times mathrm gl _2 sl 2 gl2 we present a rankin selberg integral on the exceptional group g 2 which represents the l function for generic cuspidal representations of widetilde mathrm sl 2 times mathrm gl 2 as an application we show that certain fourier jacobi type periods on g 2 are non vanishing

adaptive transient solution of nonuniform multiconductor transmission lines using wavelets this paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations the discretization of the governing equations is obtained through a weak formulation using biorthogonal wavelet bases as trial and test functions it is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes a simple yet effective time space adaptive algorithm capable of selecting the minimal number of unknowns at each time iteration is described numerical results show the high degree of adaptivity of the proposed scheme

antiwindup for stable linear systems with input saturation an lmi based synthesis this paper considers closed loop quadratic stability and l sub 2 performance properties of linear control systems subject to input saturation more specifically these properties are examined within the context of the popular linear antiwindup augmentation paradigm linear antiwindup augmentation refers to designing a linear filter to augment a linear control system subject to a local specification called the unconstrained closed loop behavior building on known results on h sub spl infin and lpv synthesis the fixed order linear antiwindup synthesis feasibility problem is cast as a nonconvex matrix optimization problem which has an attractive system theoretic interpretation the lower bound on the achievable l sub 2 performance is the maximum of the open and unconstrained closed loop l sub 2 gains in the special cases of zero order static and plant order antiwindup compensation the feasibility conditions become convex linear matrix inequalities it is shown that if and only if the plant is asymptotically stable plant order linear antiwindup compensation is always feasible for large enough l sub 2 gain and that static antiwindup compensation is feasible provided a quasi common lyapunov function between the open loop and unconstrained closed loop exists using the solutions to the matrix feasibility problems the synthesis of the antiwindup augmentation achieving the desired level of l sub 2 performance is then accomplished by solving an additional lmi

fast computation of adaptive wavelet expansions in this paper we describe and analyze an algorithm for the fast computation of sparse wavelet coefficient arrays typically arising in adaptive wavelet solvers the scheme improves on an earlier version from dahmen et al numer math 86 49 101 2000 in several respects motivated by recent developments of adaptive wavelet schemes the new structure of the scheme is shown to enhance its performance while a completely different approach to the error analysis accommodates the needs put forward by the above mentioned context of adaptive solvers the results are illustrated by numerical experiments for one and two dimensional examples

robust control of constrained linear systems with bounded disturbances this technical note develops a novel robust control algorithm for linear systems subject to additive and bounded disturbances the approach is based on the constraint tightening method while this problem can be tackled using existing robust model predictive control techniques the proposed method has an advantage in that it is computationally efficient and avoids the need to solve repeatedly an online optimization problem while the optimization problem solved at initialization is a simple linear programming problem the algorithm elaborated in this technical note guarantees convergence to a minimal disturbance invariant set and the terminal predicted state constraint set is allowed to be larger than the minimal disturbance invariant set as an illustration the developed algorithm is applied to constrained roll control of a ship operating in a wave field simulation results show that the proposed approach reduces the ship roll motion while the input and dynamic stall constraints are satisfied

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization