observer design using a generalized time scaled block triangular observer form this paper presents an observer design for uncontrolled nonlinear multi output continuous time systems based on a time scaled block triangular observer form tbtof the tbtof generalizes an established block triangular observer form btof by including generalized time scaling functions tsfs which have both state and output dependence introducing these stability preserving tsfs broadens the class of systems which admits a btof based observer design necessary conditions for the tsfs are provided when a tbtof exists an observer can be constructed subsystem at a time sufficient conditions are given to prove estimate error stability a physical example illustrates the tbtof construction and observer design

zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

modular design of nonlinear observers for state and disturbance estimation this work studies the problem of design of nonlinear observers in the presence of exogenous disturbances in particular the present work proposes a systematic design method for nonlinear state and disturbance observers in modular form consisting of an observer for the disturbance free part of the system along with a disturbance observer and a state estimate corrector the modular observer is first defined and characterized in a general setting and then a systematic design method is developed on the basis of exact linearization with eigenvalue assignment necessary and sufficient conditions for feasibility of the proposed nonlinear modular observer design are derived the proposed modular nonlinear observer is tested through simulation studies in an illustrative example involving a biochemical reactor

stability preserving maps for finite time convergence super twisting sliding mode algorithm the super twisting algorithm sta has become the prototype of second order sliding mode algorithm it achieves finite time convergence by means of a continuous action without using information about derivatives of the sliding constraint thus chattering associated to traditional sliding mode observers and controllers is reduced the stability and finite time convergence analysis have been jointly addressed from different points of view most of them based on the use of scaling symmetries homogeneity or non smooth lyapunov functions departing from these approaches in this contribution we decouple the stability analysis problem from that of finite time convergence a nonlinear change of coordinates and a time scaling are used in the new coordinates and time space the transformed system is stabilized using any appropriate standard design method conditions under which the combination of the nonlinear coordinates transformation and the time scaling is a stability preserving map are given provided convergence in the transformed space is faster than o 1 t where t is the transformed time convergence of the original system takes place in finite time the method is illustrated by designing a generalized super twisting observer able to cope with a broad class of perturbations

new algorithm for observer error linearization with a diffeomorphism on the outputs in this paper we give the necessary and sufficient conditions that guarantee the existence of a diffeomorphism which allows one to transform a multi output nonlinear dynamical system into a normal observable canonical form in particular we propose an algorithm that permits one to derive such diffeomorphism the main feature of the canonical form is that it is obtained by allowing a diffeomorphism on the outputs and it also allows one to design an observer with linear error dynamics we first consider multi output nonlinear dynamical system without inputs we then extend the results obtained to multi output nonlinear dynamical system with inputs

multiple time scalings of a multi output observer form time scalings in the multi output observer form for uncontrolled nonlinear continuous time systems are considered in this technical note it is the multi output version of an existing single input result time scaling broadens the class of systems which admits an exact error linearization observer design by including time scaling transformations the existence conditions of the time scaling transformation and the change of state coordinates to time scaled observer form are provided

dynamic observer error linearization in this paper a new framework of observer error linearization problem is proposed the main idea of our approach is twofold the one is to introduce an auxiliary dynamics whose input is the system output and the other is to transform the augmented system into an observable linear system with an injection term which contains the system output as well as the state of the auxiliary dynamics it is a natural extension of the recently developed dynamic observer error linearization where the injection term contains only newly defined output it is also shown that whenever an n dimensional system is immersible into an n d dimensional linear system up to an output injection then it can be also dynamically observer error linearizable in our sense with a d dimensional auxiliary dynamics moreover we show that the converse is not true by providing a counterexample which implies that our approach is applicable to a strictly wider class of systems than that of the system immersion method

reduced order dynamic observer error linearisation a natural extension of observer error linearisation this paper introduces the concept of reduced order dynamic observer error linearisation rdoel for multi output systems which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics the proposed rdoel framework is not only a modified version of dynamic observer error linearisation doel but also a natural extension of observer error linearisation oel we provide three necessary conditions for the rdoel problem and then derive a necessary and sufficient condition described in terms of lie algebras of vector fields furthermore from the result we also give a geometric necessary and sufficient condition for the oel problem which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered

constructive algorithm for system immersion into non linear observer form there exists a class of non linear systems which cannot be transformed into a non linear observer form an observable linear system up to output injection via diffeomorphism but can be immersed into a higher dimensional non linear observer form this class of systems can be characterized by a differential equation called characteristic equation if the system is an n dimensional system and it is immersible into n m dimensional observer form the characteristic equation involves n m 1 unknowns where n m unknowns are for the state immersion and one for the output transformation in general one should solve these unknowns simultaneously which makes the characterization difficult after investigating the algebraic structure of the characteristic equation we present an algorithm to check the immersibility under a constant rank assumption using the algorithm one can check the immersibility iteratively since only one unknown is involved at each step of the algorithm

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

observer linearization by output dependent time scale transformations we study the problem of observer linearization for single output dynamical systems in the presence of output dependent time scaling changes an alternative algorithm for the solution of the observer linearization problem is introduced the algorithm employs an exterior calculus approach that provides a simple procedure for the solution of the observer linearization problem by means of an output dependent time scale transformation

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

the uniform central limit theorem for theta sums sommes gaussiennes sommes de kloosterman theoreme de limite centrale uniforme pour des series theta theoreme de limite centrale uniformes pour des sommes theta moments sur des intervalles arbitraires

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

extended high gain observer design for a class of mimo non uniformly observable systems abstract this paper deals with the design of an extended high gain observer ehgo for a large class of multiple input multiple output mimo non uniformly observable systems indeed the involved system nonlinearities does not assume the usual triangular structure as it is often the case when a high gain approach is adopted this structure constraint relaxation is achieved by an appropriate use of the characteristic indices associated to the state components there are two fundamental features that are worth to be pointed out firstly the gain of the proposed ehgo is provided by an appropriate riccati ordinary differential equation ode secondly the convergence of the observation error is guaranteed under some persistent excitation condition which can be checked on line simulation results are given to illustrate the performance of the proposed ehgo

nonlinear observer design for state and disturbance estimation a new systematic framework for nonlinear observer design that allows the concurrent estimation of the process state variables together with key unknown process or sensor disturbances is proposed the nonlinear observer design problem is addressed within a similar methodological framework as the one introduced in n kazantzis c kravaris nonlinear observer design using lyapunov s auxiliary theorem systems control lett 34 1998 241 a j krener m xiao nonlinear observer design in the siegel domain siam j control optim 41 2002 932 for state estimation purposes only from a mathematical standpoint the problem under consideration is addressed through a system of first order singular pdes for which a rather general set of solvability conditions is derived a nonlinear observer is then designed with a state dependent gain that is computed from the solution of the system of singular pdes under the aforementioned conditions both state and disturbance estimation errors converge to zero with assignable rates the convergence properties of the proposed nonlinear observer are tested through simulation studies in an illustrative example involving a biological reactor

linearization by generalized input output injection abstract the problem under interest is the linearization of nonlinear mimo systems by generalized input output injection in order to design observers with linear error dynamics the method is based on the study of the structure of the input output differential equations thus the problem is solved as a realization problem in this note one considers the linearization under two kinds of inout output injection in the first case the transformation depends on the output and time derivatives of the input whereas in the second case derivatives of both the input and the output are considered necessary and sufficient conditions are obtained which generalize the ones on standard input output injection linearization

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

reduced order state reconstruction for nonlinear dynamical systems in the presence of model uncertainty abstract a new approach to the reduced order state reconstruction problem for nonlinear dynamical systems in the presence of model uncertainty is proposed the problem of interest is conveniently formulated and addressed within the context of singular first order non homogeneous partial differential equations pde theory leading to a reduced order nonlinear state estimator that is constructed through the solution of a system of singular pdes a set of necessary and sufficient conditions is derived that ensure the existence and uniqueness of a locally analytic solution to the above system of pdes and a series solution method is developed that is easily programmable with the aid of a symbolic software package such as maple furthermore the convergence of the estimation error or the mismatch between the actual unmeasurable states and their estimates is analyzed and characterized in the presence of model uncertainty finally the performance of the proposed reduced order state reconstruction method is evaluated in a case study involving a biological reactor that exhibits nonlinear dynamic behavior

extended output depending normal form this paper concerns the design of a nonlinear observer through a transformation of a nonlinear system into an observer form that supports a high gain observer sufficient geometrical condition is deduced to guarantee the existence of change of coordinates allowing the transformation of a nonlinear system into the proposed normal form the result is highlighted by studying the susceptible exposed infected and recovered seir model

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

high gain observer for a class of non triangular systems this paper presents a high gain observer for a class of mimo nonlinear systems involving some uncertainties the latter is particularly composed of cascade subsystems where each subsystem is associated with a subset of the output variables and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems the main contribution consists in extending the available results to allow more interconnections between the subsystems of fundamental interest it is shown that the underlying observation error exponentially converges to zero in the absence of uncertainties moreover the observation error can be made as small as desired by properly specifying the observer design parameter in the case where uncertainties are considered

partial observer normal form for nonlinear system in this paper we investigate the estimation problem for a class of partially observable nonlinear systems for the proposed partial observer normal form ponf necessary and sufficient conditions are deduced to guarantee the existence of a change of coordinates which can transform the studied system into the proposed ponf examples are provided to illustrate the effectiveness of the proposed results

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

asymptotic relations among fourier coefficients of automorphic eigenfunctions a detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane z y 1 2 k ir 2 my e 2 imx z x iy r 2 the eigenvalue s 2 m 1 2 and k ir the macdonald bessel function the phase velocity of on s imz 1 is a double valued vector field the tangent field to the pencil of geodesics g tangent to the horocycle s imz 1 for a sl 2 r a multi term stationary phase expansion is presented in a for az e 2 in rez uniform in parameters an application is made to find an asymptotic relation for the fourier coefficients of a family of automorphic eigenfunctions in particular for automorphic with coefficients an and eigenvalue a it is shown for the special range n 1 2 that a n is o 1 4 e 1 2 2 for a large improving by an order of magnitude for this special range upon the estimate from the general hecke bound o n 1 2 1 4 e 1 2 2 an exposition of the wkb asymptotics of the macdonald bessel functions is presented

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

observer with linear error dynamics for nonlinear multi output systems based upon nonlinear state coordinate transformations and a condensed dual brunovsk form for multi output systems a method for designing nonlinear state observers with linear error dynamics is presented necessary and sufficient conditions are provided for identifying the class of nonlinear systems for which state observers with linear error dynamics can be obtained

asymptotic relations among fourier coefficients of real analytic eisenstein series following wolpert we find a set of asymptotic relations among the fourier coefficients of real analytic eisenstein series the relations are found by evaluating the integral of the product of an eisenstein series pi with an exponential factor along a horocycle we evaluate the integral in two ways by exploiting the automorphicity of pir the first of these evaluations immediately gives us one coefficient while the other evaluation provides us with a sum of fourier coefficients the second evaluation of the integral is done using stationary phase asymptotics in the parameter a a r2 is the eigenvalue of pir for a cubic phase as applications we find sets of asymptotic relations for divisor functions

an algorithm for orbital feedback linearization of single input control affine systems feedback linearization is an effective design and analysis tool used in the study of nonlinear control systems however cases arise where linearizability conditions cannot be met for systems that are not linearizable by classical techniques orbital feedback linearization or feedback linearization by state dependent time scaling has been proposed to relax these conditions unfortunately approaches proposed to date have led to conditions that tend to be more difficult to check then conditions for state feedback linearization in this paper necessary and sufficient conditions for orbital feedback linearizability is presented for a class of single input nonlinear systems the conditions are simple and can be checked directly from the data of the problem using an exterior calculus approach a simple algorithm is developed to compute state dependent time scaling that yield state feedback linearizable systems it is shown that orbital feedback linearizability generalizes the concept state feedback linearizability to deal with locally weakly accessible control systems

nonlinear observer design in the presence of delayed output measurements the present work proposes a new approach to the nonlinear observer design problem in the presence of delayed output measurements the proposed nonlinear observer possesses a state dependent gain which is computed from the solution of a system of first order singular partial differential equations and in particular consists of a chain of state observation algorithms that reconstruct the unmeasurable state vector at different delayed time instants within the time delay window introduced by the available output measurements therefore the proposed nonlinear observer exhibits a chain like structure that explicitly reflects and takes into account the magnitude of the output delay furthermore a set of conditions is derived under which convergence of the estimation error to zero is established finally the performance of the proposed observer and its convergence properties are evaluated in an illustrative biological reactor example

immersion of non linear systems into linear systems up to output injection characteristic equation approach there exists a class of non linear systems which cannot be transformed into non linear observer form via diffeomorphism but can be immersed into higher dimensional non linear observer form this paper investigates when n dimensional non linear systems can be immersed into n m dimensional non linear systems in non linear observer form and provides a necessary and sufficient condition we also present an algorithm to check the immersibility in general at the first step of the algorithm one should solve a differential equation with m 1 unknowns rather than solving one with n m unknowns simultaneously when a solution to the differential equation can be obtained the immersibility can be checked by a recursive algorithm moreover when 2m n a straightforward algorithm is provided to check the immersibility