sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

an observability form for linear systems with unknown inputs in this paper is given a constructive algorithm that transforms a linear system with unknown inputs into a novel observability form this form is useful to derive finite time observers even if the system does not satisfy some matching conditions usually required for the design of unknown input observers the practical example of an aircraft subject to actuator faults is provided to show the efficiency of the approach

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

robust fault reconstruction in uncertain linear systems using multiple sliding mode observers in cascade in observer based fault reconstruction one of the necessary conditions is that the first markov parameter from the fault to the output must be full rank this paper seeks to relax that requirement by using multiple sliding mode observers in cascade signals from an observer are used as the output of a fictitious system whose input is the fault another observer is then designed and implemented for the fictitious system this process is repeated until the first markov parameter of the fictitious system with respect to the fault is full rank the result is that robust fault reconstruction can be carried out for a wider class of systems compared to other works that also seek to relax the requirement of a full rank first markov parameter in addition this paper has also investigated and presented the necessary and sufficient conditions as easily testable conditions and also the precise number of observers required a simulation example verifies the effectiveness of the scheme

hybrid dead beat observers for a class of nonlinear systems this paper studies both the full order and the reduced order dead beat observer problem for a class of nonlinear systems linear in the unmeasured states a novel hybrid observer design strategy is proposed with the help of the notion of strong observability in finite time the proposed methodology is applied for the estimation of the frequency of a sinusoidal signal the results show that accurate estimation can be provided even if the signal is corrupted by high frequency noise a brief discussion of the robustness properties of the proposed observer with respect to measurement errors is also provided

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

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

high gain observers in the state and estimation of robots having elastic joints the paper deals with the state and parameter estimation of robots having elastic joints the estimation of the unknown parameters of the robot is reduced to the estimation of its state variables by a state space immersion the luenberger observer is used in the state estimation of the extended robot model the use of high gains is studied in the cancellation of nonlinearities in order to simplify the observer design the high gain induces a time scale separation between the robot model and the observer and therefore the singular perturbation theory can be used in the stability analysis of the estimate error dynamics in particular it is shown that the error dynamics reaches the stable equilibrium in a very fast transient ensuring that the slow dynamics of the observer is just that of the given robot

time scaling for observer design with linearizable error dynamics in this paper we consider the problem of observer design for dynamical systems with scalar output by linearization of the error dynamics via coordinate change output injection and time scaling we present necessary and sufficient conditions which guarantee the existence of a coordinate change and output dependent time scaling such that in the new coordinates and with respect to the new time the system has linear error dynamics

a continuous time observer which converges in finite time it is shown that a continuous time observer which comprises two standard nth order observers and a delay d can observe the state of an nth order linear system in finite time d exactly in particular almost any convergence time d can be assigned independent of the observer eigenvalues

global exponential observers for two classes of nonlinear systems abstract this paper develops sufficient conditions for the existence of global exponential observers for two classes of nonlinear systems i the class of systems with a globally asymptotically stable compact set and ii the class of systems that evolve on an open set in the first class the derived continuous time observer also leads to the construction of a robust global sampled data exponential observer under additional conditions two illustrative examples of applications of the general results are presented one is a system with monotone nonlinearities and the other is a chemostat system

on sliding mode observers via equivalent control approach in this paper sliding mode observer design principles based on the equivalent control approach are discussed for a linear time invariant system both in continuous and discrete time for the continuous case the observer is designed using a recursive procedure however the observer is eventually expressed as a replica of the original system with an additional auxiliary input with a certain nested structure a direct discrete time counterpart of the sliding mode realization of a reduced order asymptotic observer using the discrete time equivalent control is also developed simulation of a linearized truck trailer during a manoeuvre illustrates the approach results show the effectiveness and the finite time convergence characteristics of the proposed discrete and continuous time sliding mode observers

on the robust design of sliding observers for linear systems abstract the restrictive character of well known structural constraints related to the matching conditions for the sliding mode feedforward state reconstruction problem in linear time invariant perturbed systems is critically reexamined from a new perspective it is shown that in generalized state space coordinates a matched canonical state space realization exists which always allows discontinuous asymptotic stabilization of the observation error dynamics the well known structural conditions thus become largely irrelevant and robust asymptotic state estimation is shown to be feasible for any perturbed observable system by means of sliding mode observers

an immersion based observer design for rank observable nonlinear systems in this note an observer design for a wide class of nonlinear systems is proposed it is based on some high gain design for the particular case of state affine systems up to triangular nonlinearity on the one hand and immersion of nonlinear systems into such a form on the other hand conditions for the observer convergence are given in terms of appropriate excitation and a constructive procedure for the immersion is presented the methodology is illustrated by examples of application

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

sliding mode estimators for a class of non linear systems affected by bounded disturbances the problem of state estimation for a class of non linear systems with lipschitz non linearities is addressed using sliding mode estimators stability conditions have been found to guarantee asymptotic convergence to zero of the estimation error in the absence of noise and non divergence if the state perturbations and measurement noise are bounded a method is proposed to find a suitable solution to the algebraic riccati equation on which the design of the estimator is based the performance of the resulting sliding mode filter minimizes an upper bound on the asymptotic estimation error based on such an approach a sliding mode estimator may be designed so as to outperform the extended kalman filter in practical applications with models affected by uncertainty and strong possibly unknown non linearities as shown by means of simulations

a globally convergent frequency estimator online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications in this paper the authors provide a solution to the problem of ensuring a globally convergent estimation more specifically they propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features 1 all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values 2 the authors obtain a simple tuning procedure for the estimator design parameters which trades off the adaptation tracking capabilities with noise sensitivity ensuring exponential stability of the desired orbit and 3 transient performance is considerably enhanced even for small or large frequencies as witnessed by extensive simulations to reveal some of the stability instability mechanisms of the existing algorithms and motivate our modifications the authors make appeal to a novel nonlinear state dependent time scaling the main advantage of working in the new time scale is that they remove the coupling between the parameter update law and the filter itself decomposing the system into a feedback form where the required modifications to ensure stability become apparent even though they limit their attention here to the simplest case of a single constant frequency without noise the algorithm is able to track time varying frequencies preserve local stability in the presence of multiple sinusoids and is robust with respect to noise

total p differentials on schemes over z p2 abstract for a scheme x defined over the length 2 p typical witt vectors w 2 k of a characteristic p field we introduce total p differentials which interpolate between frobenius twisted differentials and buium s p differentials they form a sheaf over the reduction x 0 and behave as if they were the sheaf of differentials of x over a deeper base below w 2 k this allows us to construct the analogues of gauss manin connections and kodaira spencer classes as in the katz oda formalism we make connections to frobenius lifts borger weiland s biring formalism and deligne illusie classes

output feedback sliding mode control in the presence of unknown disturbances this paper examines the use of a sliding mode controller and an equivalent output injection sliding mode observer to control a nonlinear plant in the presence of an unknown disturbance with only the measured output available for feedback the observer reconstructs the full state in finite time and provides an estimate of the unknown disturbance the controller ensures asymptotic reference trajectory tracking or alternatively asymptotic convergence to the origin for the plant asymptotic stability of the closed loop system is proved under a set of nonrestrictive assumptions on the plant and the disturbance these results are verified numerically through simulation

a global high gain finite time observer a global finite time observer is designed for nonlinear systems which are uniformly observable and globally lipschitz this result is based on a high gain approach combined with recent advances on finite time stability using lyapunov function and homogeneity concepts

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

observer design for nonlinear triangular systems with unobservable linearisation the article deals with the observer design problem for a wide class of triangular time varying nonlinear systems with unobservable linearisation sufficient conditions are derived for the existence of a luenberger type observer when it is a priori known that the initial state of the system belongs to a given nonempty bounded subset of the state space for the general case the state estimation is exhibited by means of a switching sequence of time varying dynamics

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

a novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions abstract in this paper a novel sliding mode observer is proposed for systems that do not satisfy common conditions required for fault estimation in particular the so called matching condition and minimum phase condition other works attempt to circumvent the matching condition by using multiple observers in cascade or higher order sliding modes whilst the minimum phase condition can be relaxed at the cost of some faults and the states being not estimated or the fault estimation being corrupted by the fault time derivative or other dynamics for the proposed scheme in this paper only one observer is required the minimum phase condition is relaxed to detectability and the matching condition is greatly simplified to a condition related to the system dimensions which makes the scheme also applicable to the case where there are less outputs than faults finally a simulation example is conducted to verify the effectiveness of the proposed observer

locating the peaks of solutions via the maximum principle ii a local version of the method of moving planes let be a bounded smooth domain in r 2n n 2 the well known moser trudinger inequality ensures the nonlinear functional jp u is bounded from below if and only if 2n 2 2n n n 1 w 2n where j u 1 2 f n 2 u 2 log f e u dx in x h n u j u h j 1 n 1 2 and 2n is the area of the unit sphere s 2n 1 in r 2n in this paper we prove the inf u x jp u is always attained for p 2n the existence of minimizers of jp at the critical value p 2n is a delicate problem the proof depends on the blowup analysis for a sequence of bubbling solutions here we develop a local version of the method of moving planes to exclude the boundary bubbling the existence of minimizers for jp at the critical value p 2n is in contrast to the case of two dimensions

optimization based non linear observers revisited optimization based non linear observers are those where the integrated output prediction error is used to define the dynamic of the observer s state their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold here we present some new theoretical results concerning these observers despite the title we use our observer is not based on the search for a global minimum of a cost function it is based on a descent like approach without the use of the hessian matrix a nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations besides the theoretical results some implementation issues are discussed in particular we propose to use the post stabilization approach well known when dealing with differential systems with invariant submanifolds this enables to increase the sampling period while keeping very good precision two illustrative examples are presente

topological degree for solutions of fourth order mean field equations we consider the following fourth order mean field equation with navier boundary condition delta 2 u rho frac h x e u int omega h e u rm in omega quad u delta u 0 rm on partial omega quad quad quad quad quad quad quad quad quad quad where h is a c2 positive function is a bounded and smooth domain in mathbb r 4 we prove that for rho in 32m sigma 3 32 m 1 sigma 3 the degree counting formula for is given by d rho left begin array ll frac 1 m chi omega 1 cdot cdot cdot chi omega m rm for m 0 1 rm for m 0 end array right where is the euler characteristic of similar result is also proved for the corresponding dirichlet problem delta 2 u rho frac h x e u int omega h e u quad rm in omega quad u nabla u 0 quad rm on partial omega quad quad quad quad quad quad quad quad quad quad

parameter and state estimation of nonlinear systems using a multi observer under the supervisory framework we present a hybrid scheme for the parameter and state estimation of nonlinear continuous time systems which is inspired by the supervisory setup used for control state observers are synthesized for some nominal parameter values and a criterion is designed to select one of these observers at any given time instant which provides state and parameter estimates assuming that a persistency of excitation condition holds the convergence of the parameter and state estimation errors to zero is ensured up to a margin which can be made as small as desired by increasing the number of observers to reduce the potential computational complexity of the scheme we explain how the sampling of the parameter set can be dynamically updated using a zoom in procedure this strategy typically requires a fewer number of observers for a given estimation error margin compared to the static sampling policy the results are shown to be applicable to linear systems and to a class of nonlinear systems we illustrate the applicability of the approach by estimating the synaptic gains and the mean membrane potentials of a neural mass model

nonlinear observer design using lyapunov s auxiliary theorem abstract the present work proposes a new approach to the nonlinear observer design problem based on the early ideas that influenced the development of the linear luenberger observer theory the proposed approach develops a nonlinear analogue the formulation of the observer design problem is realized via a system of singular first order linear pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the above system of pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package

nonlinear observers for autonomous lipschitz continuous systems this paper considers the state observation problem for autonomous nonlinear systems an observation mapping is introduced which is defined by applying a linear integral operator rather than a differential operator to the output of the system it is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems and to construct observers of a remarkably simple structure a linear state variable filter followed by a nonlinearity the observer is established by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist and by giving an explicit expression for its nonlinearity it is demonstrated that the existence conditions are satisfied and hence our results include a new observer which is not high gain for the wide class of smooth systems it is shown that the observer can as well be designed to realize an arbitrary finite accuracy rather than ultimate exactness on a compact region of the state space this requires only observability of the system a corresponding numerical design procedure is described which is easy to implement and computationally feasible for low order systems

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

generalised observer design for dissipative lipschitz nonlinear systems in the presence of measurement noise this paper presents two novel observer concepts first it develops a globally exponentially stable nonlinear observer for noise free dissipative nonlinear systems second for a dissipative nonlinear system with measurement noise the paper develops an observer to guarantee a desired performance namely an upper limit on the ratio of the square of the weighted l2 norm of the error to the square of the weighted l2 norm of the measurement noise the necessary and sufficient conditions for both observers are reformulated as algebraic riccati equations ares so that standard solvers can be utilised in addition the paper presents necessary and sufficient conditions to be satisfied by the nonlinear system in order to ensure that the are and hence the observer design problem has a solution the use of the methodology developed in this paper is demonstrated through illustrative examples in literature there is no previous observer for dissipative system that provides both necessary and sufficient condition

existence and multiplicity results for a fourth order mean field equation in this article we consider the following fourth order mean field equation on smooth domain r4 2u keu keuin u u 0on where r and 0 k c2 through a refined blow up analysis we characterize the critical points at infinity of the associated variational problem and compute their contribution of the difference of topology between the level sets of the associated euler lagrange functional we then use topological and dynamical methods to prove some existence and multiplicity results