modified numerov s method for inverse sturm liouville problems in this paper we propose a new modified numerov s method for recovering from eigenvalues a symmetric potential of a sturm liouville operator with dirichlet boundary conditions we use interpolation to refine the mesh sufficiently for numerov s method to be effective even without the asymptotic correction technique of andrew and paine accuracy and stability of the method are investigated convergence of the method is established our method is extended to deal with natural boundary conditions numerical experiments confirm its competitiveness

asymptotic correction of numerov s eigenvalue estimates with natural boundary conditions abstract asymptotic correction at negligible extra cost greatly improves the accuracy of higher eigenvalues computed by finite difference or finite element methods and generally increases the accuracy of the lower ones as well this paper gives a brief overview of the technique and describes how its previous use with numerov s method may be extended to problems with natural boundary conditions numerical results indicate that it is just as successful as with dirichlet boundary conditions

derivation of a general mixed interpolation formula a general procedure is developed to derive a mixed interpolation formula for approximating any n 1 times differentiable function f x for x in 0 nh by a function f n x of the type f n x au 1 kx bu 2 kx sum n 2 i 0 cix i interpolating points being the ones given by x j jh h 0 j o 1 n where u 1 kx and u 2 kx are the two linearly independent solutions of a suitable second order ordinary differential equation ode and k 0 is an appropriately chosen parameter the results for the particular case when u 1 kx and u 2 kx represent the trigonometric functions follow easily an analysis of the error is also discussed and specific numerical examples are included for the sake of comparison with the known interpolation formulae tables showing the comparison of the maximum errors occuring in the use of the various interpolation formulae has also been presented for some specially chosen functions

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

normal forms and bifurcations of planar vector fields normal forms and bifurcations of planar vector fields by s n chow c li and d wang cambridge university press cambridge and new york 1994 viii 472 pp 69 95 isbn 0 521 37226 7

decentralized adaptive fuzzy output feedback control of stochastic nonlinear large scale systems with dynamic uncertainties in this paper centralized and decentralized adaptive fuzzy output feedback control schemes are investigated for a class of stochastic nonlinear interconnected large scale systems with dynamic uncertainties and unmeasured states fuzzy systems are used to approximate the unknown nonlinear functions decentralized k filters are designed to estimate the unmeasured states an available dynamic signal is introduced to dominate the unmodeled dynamics by combining dynamic surface control dsc technique with backstepping design the condition in which the approximation errors are assumed to be bounded is avoided using the defined compact set in the stability analysis the unknown smooth interconnections and black box functions are effectively dealt with using it o formula and chebyshev s inequality it is shown that all the signals in the closed loop system are bounded in probability and the error signals are semi globally uniformly ultimately bounded in mean square or the sense of four moment simulation results demonstrate the effectiveness of the proposed approach

a smart nonstandard finite difference scheme for second order nonlinear boundary value problems a new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems an artificial parameter is introduced in the scheme symbolic computation is proposed for the construction of the scheme local truncation error of the method is discussed numerical examples are illustrated numerical results show that the method is very effective

adaptive fuzzy tracking control of switched uncertain nonlinear systems with unstable subsystems this paper is concerned with the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems where the solvability of the fuzzy tracking control problem for subsystems is unnecessary fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a new adaptive fuzzy tracking control technique for the problem under study is set up based on the technique above we design a switching law and construct adaptive fuzzy controllers of subsystems explicitly by a recursive design algorithm to guarantee that all the signals in the resulting closed loop system remain bounded and the system output of the closed loop system converges to a small neighborhood of the reference signal multiple lyapunov functions are exploited to get out of the conservativeness caused by adoption of a common lyapunov function for all subsystems which is commonly required when applying the backstepping like recursive design scheme a mass spring damper system is provided to demonstrate the effectiveness of the proposed design method

peaking free output feedback adaptive neural control under a nonseparation principle high gain observers have been extensively applied to construct output feedback adaptive neural control anc for a class of feedback linearizable uncertain nonlinear systems under a nonlinear separation principle yet due to static gain and linear properties high gain observers are usually subject to peaking responses and noise sensitivity existing adaptive neural network nn observers cannot effectively relax the limitations of high gain observers this paper presents an output feedback indirect anc strategy under a nonseparation principle where a hybrid estimation scheme that integrates an adaptive nn observer with state variable filters is proposed to estimate plant states by applying a single lyapunov function candidate to the entire system it is proved that the closed loop system achieves practical asymptotic stability under a relatively low observer gain dominated by controller parameters our approach can completely avoid peaking responses without control saturation while keeping favourable noise rejection ability simulation results have shown effectiveness and superiority of this approach

a survey of adaptive fuzzy controllers nonlinearities and classifications adaptive fuzzy controllers afcs i e fuzzy controllers using heuristic expert knowledge and equipped with adaption ability are widely used in the industry in the present days as they are capable of dealing with ill formulated nonlinear systems where exact mathematical formulas are not known or partially described nonlinear systems afcs have been widely deployed to control many nonlinear systems with different types of critical nonlinearities over the last two decades many different approaches toward adaptive fuzzy control have been reported in this paper we recapitulate the latest research works related to afcs and provide an up to date survey on the latest development made in adaptive fuzzy control theory and applications based on different types of fuzzy systems considered and nonlinearities associated with the nonlinear system to be controlled we categorize the different approaches based on different types of nonlinearities and consider basic nonlinear systems nonlinear systems with external disturbances nonlinear systems with some special type of nonlinearities nonaffine nonlinear systems and strict feedback nonlinear systems it turns out that many existing methods are subject to various limitations and none of them is capable of controlling a general nonlinear systems hence it is still an open problem to design and develop a universal afc for a general class of nonlinear systems

adaptive observers with arbitrary exponential rate of convergence for nonlinear systems concerns the same class of linearly parameterized single output nonlinear systems that the authors previously identified in 1992 in terms of differential geometric conditions when persistency of excitation conditions are satisfied the adaptive observers presented in this note guarantee arbitrarily fast exponential convergence both of parameter and state estimates to actual parameters and states while previous adaptive observers guarantee only exponential not arbitrarily fast convergence this extends earlier results for linear systems

fuzzy adaptive control with state observer for a class of nonlinear discrete time systems with input constraint in this paper an adaptive fuzzy controller is constructed for a class of nonlinear discrete time systems with unknown functions and bounded disturbances the main characteristics of the systems are that they take into account the effect of discrete time dead zone and the system states are not required to be measurable the stability problem of this class of systems is for the first time to be addressed in this paper due to the unavailability of the states and the presence of the discrete time dead zone the controller design becomes more difficult to stabilize the uncertain nonlinear discrete time systems the fuzzy logic systems are used to approximate the unknown functions a fuzzy state observer is designed to estimate the immeasurable states and the effect caused by discrete time dead zone can be solved via establishing an adaptation auxiliary signal based on the lyapunov approach it is proved that all the signals of the closed loop system are the semiglobal uniformly ultimately bounded and the tracking error is made within a small neighborhood around zero the feasibility of the developed control scheme is verified via two simulation examples

on fitted modifications of runge kutta nystrom pairs abstract modification of only four coefficients of the standard runge kutta nystrom pairs is enough for producing a method of the same orders that integrates exactly the harmonic oscillator as model problem these new coefficients are o 2 perturbations of the initial ones with the frequency of the problem theoretical investigation for the possibilities of order reduction is given numerical results over standard pairs of orders 6 4 and 8 6 justify our efforts

exponential fitting collocation methods for a class of volterra integral equations abstract in this paper we propose a collocation method for a class of volterra integral equations whose solutions contain periodic functions since the exponential fitting interpolation has an advantage in approximating periodic functions we consider employing it with collocation method to construct our scheme the global convergence analysis of the scheme is also presented based on the interpolation error the theoretical results as well as the superiority of the method are verified in the numerical part

on a class of modified newton cotes quadrature formulae based upon mixed type interpolation abstract we present quadrature rules which integrate exactly not only polynomials up to a certain degree but also the functions sin kx and cos kx where k is a free parameter the formulae we obtain are modified newton cotes formulae they are derived by replacing the integrand by an interpolation function of the form a cos kx b sin kx n 2j 0cjxj based on equally spaced nodes the total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified simpson s 3 8 rule numerical examples show the efficiency of the modified rules and the importance of the error term

a three point formula for numerical quadrature of oscillatory integrals with variable frequency abstract a simple three point formula is constructed for the evaluation of general oscillatory integrals a rigorous derivation of the local error term is presented and the implications to high frequency oscillations are discussed simple examples given include integrals with variable frequency for which the usual filon formula would be inappropriate for cases where filon s formula is appropriate the new formula appears to be computationally more efficient the main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach comparison with exact results is possible from the work of stoker 16 for a case which whilst special in the physical sense does not simplify the integral involved in all cases the implementation of the formula is as straightforward as the implementation of the ordinary simpson rule

on a new type of mixed interpolation abstract we approximate every function f by a function f n x of the form a cos kx b sin kx n 2 i 0 c i x i so that f jh f n jh for the n 1 equidistant points jh j 0 n that interpolation function f n x is proved to be unique and can be written as the sum of the n th degree interpolation polynomial based on the same points and two correction terms the error term is also discussed the results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0 this new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations

trigonometric polynomial or exponential fitting approach the application of a trigonometric polynomial and an exponential fitting approach is compared for a three point formula for second order derivatives for simpson s quadrature rule and for numerov s scheme for second order differential equations the expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied the errors for specific problems obtained in both the approaches as a function of the frequency are compared

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

adaptive control based barrier lyapunov functions for a class of stochastic nonlinear systems with full state constraints abstract an adaptive control scheme is developed in the paper for nonlinear stochastic systems with unknown parameters all the states in the systems are required to be constrained in a bounded compact set i e the full state constraints are considered in the systems it is for the first time to control nonlinear stochastic systems with the full state constraints in contrast to deterministic systems the stochastic systems with the full state constraints are more difficult to be stabilized and the design procedures are more complicated by constructing barrier lyapunov functions blf in symmetric and asymmetric forms it can be ensured that all the states of the stochastic systems are not to transgress their constraint bounds thus the proposed scheme not only solves the stability problem of stochastic systems but also overcomes the effect of the full state constraints on the control performance finally it is proved that all the signals in the closed loop system are semi global uniformly ultimately bounded sguub in probability the system output is driven to follow the reference signals and all the states are ensured to remain in the predefined compact sets the validity of the proposed scheme is verified by a simulation example

exponentially fitted singly diagonally implicit runge kutta methods it is the purpose of this paper to derive diagonally implicit exponentially fitted ef runge kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations whose solutions are supposed to exhibit an exponential behaviour in addition to the standard approach for the derivation of ef methods we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages the derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages as classically done in the standard derivation of multistage ef based methods standard and revised ef methods are then compared in terms of linear stability and numerical performances

switched observer based adaptive neural control of mimo switched nonlinear systems with unknown control gains in this paper the problem of adaptive neural output feedback control is addressed for a class of multi input multioutput mimo switched uncertain nonlinear systems with unknown control gains neural networks nns are used to approximate unknown nonlinear functions in order to avoid the conservativeness caused by adoption of a common observer for all subsystems an mimo nn switched observer is designed to estimate unmeasurable states a new switched observer based adaptive neural control technique for the problem studied is then provided by exploiting the classical average dwell time adt method and the backstepping method and the nussbaum gain technique it effectively handles the obstacle about the coexistence of multiple nussbaum type function terms and improves the classical adt method since the exponential decline property of lyapunov functions for individual subsystems is no longer satisfied it is shown that the technique proposed is able to guarantee semiglobal uniformly ultimately boundedness of all the signals in the closed loop system under a class of switching signals with adt and the tracking errors converge to a small neighborhood of the origin the effectiveness of the approach proposed is illustrated by its application to a two inverted pendulum system

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

design of a unified adaptive fuzzy observer for uncertain nonlinear systems this paper presents an adaptive fuzzy observer for a class of uncertain nonlinear systems more precisely we propose a unified approach for designing such an observer with some design flexibility so that it can be easily adaptable and employed either as a high gain or a sliding mode observer by selecting its gain appropriately additionally we derive a suitable parameter adaptation law so that the proposed observer is robust with respect to ubiquitous fuzzy approximation errors and external disturbances we also show that the observation error is ultimately bounded using a lyapunov approach without having recourse to the usual strictly positive real spr condition or a suitable observation error filtering the effectiveness of the proposed observers is illustrated through two simulation case studies taken from the adaptive fuzzy control literature

fuzzy tracking adaptive control of discrete time switched nonlinear systems this paper is concerned with the problem of fuzzy tracking adaptive control for a class of discrete time switched uncertain nonlinear systems with arbitrary switching based on the common lyapunov function method and by utilizing the fuzzy logic systems to approximate the unknown nonlinear functions an adaptive fuzzy controller is firstly constructed for a class of uncertain nonlinear discrete time switched systems meanwhile the proposed adaptive control algorithm reduces the amount of online adjustable parameters based on the above techniques the constructed fuzzy controllers and the adaptive law can guarantee that all the signals are bounded and the system output can converge to a small neighborhood of the reference signal in the closed loop system an illustrative example is provided to demonstrate the effectiveness of the proposed approaches

extended exponentially fitted interpolation formulas for oscillatory functions we consider exponentially fitted interpolation formulas that use not only the pointwise values of an w dependent function f but also of its derivatives up to the nth order at the two boundary nodes of a closed interval a b the function f is of the form f x f 1 x cos wx f 2 x sin wx email protected a b where the functions f 1 and f 2 are smooth on the interval as n increases numerical results about the errors for the formulas are illustrated and compared some properties of the coefficients of the formulas at the two nodes are presented and extended to the case with three nodes we investigate if the properties of the coefficients can be generalised to the cases with more than three nodes finally some other numerical results regarding the errors are given

exponentially fitted numerov methods the error in the estimate of the kth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh length h is o k 6h 4 it is shown that the error can be reduced to o k 3h 4 by using one of the three versions of the exponentially fitted numerov method numerical examples demonstrate the usefulness of this approach even for low values of k

finite dimensional guaranteed cost sampled data fuzzy control for a class of nonlinear distributed parameter systems in this paper a finite dimensional guaranteed cost sampled data fuzzy control gcsdfc problem is addressed for a class of nonlinear parabolic partial differential equation pde systems initially applying the galerkin s method to the pde system a nonlinear ordinary differential equation ode system is derived the resulting nonlinear ode system can accurately describe the dominant dynamics of the pde system which is subsequently expressed by the takagi sugeno t s fuzzy model then a guaranteed cost sampled data fuzzy controller is developed to stabilize exponentially the closed loop slow fuzzy system while providing an upper bound for the quadratic cost function a novel time dependent functional is constructed to derive the condition for the existence of the proposed controller which is presented by bilinear matrix inequalities bmis moreover a suboptimal gcsdfc problem to minimize the cost bound can be formulated as a bmi optimization problem a local optimization algorithm that views the bmi as a double linear matrix inequality lmi is given to solve this bmi optimization problem in which a latin hypercube sampling lhs method is proposed to find an initially feasible solution for starting the algorithm furthermore it is shown that the proposed controller can ensure the exponential stability of the closed loop pde system finally simulation results on the fisher equation and the temperature profile of a catalytic rod show that the proposed design strategy is effective

decentralized adaptive output feedback control for a class of nonlinear large scale systems with unknown time varying delayed interactions in this paper the authors investigate a decentralized adaptive output feedback controller design for large scale nonlinear systems with input saturations and time delayed interconnections unmatched in control inputs the interaction terms with unknown time varying delays are bounded by unknown nonlinear bounding functions including all states of subsystems this point is a main contribution of this paper compared with previous output feedback control approaches which assume that the time delayed bounding functions only depend on measurable output variables the bounding functions are compensated by using appropriate lyapunov krasovskii functionals and the function approximation technique based on neural networks the observer dynamic surface design technique is employed to design the proposed memoryless local controller for each subsystem in addition we prove that all signals in the closed loop system are semiglobally uniformly bounded and control errors converge to an adjustable neighborhood of the origin finally an example is provided to illustrate the effectiveness of the proposed control system

on the error and its control in a two parameter generalised newton cotes rule a quadrature formula containing two free phase parameters k k and recently written by van daele et al 13 is rederived using an extension of lagrange s identity by using this method a closed form expression is determined for the local error term e f and the relevant peano kernel is given explicitly sufficient conditions are established under which this kernel remains definite thus allowing a particularly simple expression to be deduced for e f which reduces to the classical result for boole s rule in the limit k k 0 recent work on optimisation of global error is extended to include the new rule it is shown that this error can be reduced by a factor o h 2 on a certain curve in the phase space a further reduction by a factor o h 2 is sometimes possible by choosing the phase parameters on the intersection of this curve and another such when these curves intersect only at complex values the reduction is still achieved a closed form expression for the global error is also derived under these circumstances and this is seen to be asymptotically o h 1 0 as h 0 a second limiting form of the five point formula is found to reduce to the generalised boole s rule written by vanden berghe et al 14 and a third limiting form is also written this third rule which is a special case of a one parameter family of generalised boole s rules is seen to perform better than the other two in the two examples studied several numerical examples are given with extensive diagrams to illustrate all uses of the techniques proposed

barrier lyapunov functions for nussbaum gain adaptive control of full state constrained nonlinear systems abstract in this paper an adaptive controller design is studied for single input single output siso nonlinear systems with parameter uncertainties and the systems are enforced to subject to the full state constraints a remarkable feature of the constrained systems is that the so called control direction is unknown or in other words the signs of control gains are unknown in the existing results we discover that all the state constraint control results are required to determine a priori knowledge of control direction it will be inevitable to bring about a different design procedure and a difficult task when no a priori knowledge of control direction is known to stabilize this class of systems two parameter adaptive controllers with nussbaum gain technique are constructively framed to overcome the unknown control direction problem and the novel symmetric and asymmetric barrier lyapunov functions blfs are adopted to guarantee that the states are not to violate their constraints then the proposed blf strategy can be to conquer the conservatism of the traditional blf based controls for the full state constraints finally two theorems are provided to show that all the signals in the closed loop system are bounded the outputs are driven to follow the reference signals and all the states are ensured to remain in the predefined compact sets the effectiveness of the proposed scheme is performed via a simulation example

analytic normalization of analytic integrable systems and the embedding flows abstract this paper provides the normal forms of analytic integrable differential systems and diffeomorphisms via analytic normalizations furthermore we consider the existence of embedding flows of an analytic integrable diffeomorphism

stable adaptive observers for nonlinear time varying systems an adaptive observer identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form is described sufficient conditions for stability of this observer are provided these conditions are in terms of the structure of the system and canonical form the boundedness of the parameter variations and the sufficient richness of some signals the scope of the canonical form and the use of the observer identifier is motivated by the presentation of applications to time invariant bilinear systems nonlinear systems in phase variable form a biotechnological process and a robot manipulator in each case the specific stability conditions are presented

adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems an adaptive fuzzy control approach is proposed for a class of multiple input multiple output mimo nonlinear systems with completely unknown nonaffine functions the mimo systems are composed of n subsystems and each of subsystems is in the nested lower triangular form it is difficult and complicated to control this class of systems due to the existence of unknown nonaffine functions and the couplings among the nested subsystems this difficulty is overcome by introducing some special type lyapunov functions and taking advantage of the mean value theorem the backstepping design method and the approximation property of the fuzzy systems the proposed control approach can guarantee that all the signals in the closed loop system are bounded a simulation experiment is utilized to verify the feasibility of the proposed approach

mixed interpolation methods with arbitrary nodes abstract previous work on interpolation by linear combinations of the form ac x bs x i 0 n 2 i x i where c and s are given functions and the coefficients a b and j are determined by the interpolation conditions was restricted to uniformly spaced interpolation nodes here we derive both newtonian and lagrangian formulae for the interpolant for arbitrarily chosen distinct nodes in the newtonian form the interpolating function is expressed as the sum of the interpolating polynomial based on the given nodes and two correction terms involving an auxiliary function for which a recurrence relation is obtained each canonical function for the lagrangian form may be expressed as a product of the corresponding lagrange polynomial and a function which depends on divided differences of c x and s x

observer design for a class of mimo nonlinear systems a simple observer is proposed for a large class of mimo nonlinear systems which includes many physical models the main characteristic of the proposed observer lies in the easiness of its implementation and calibration indeed the gain of this observer does not necessitate the resolution of any dynamical system and its expression is given moreover its calibration is achieved through the choice of a single parameter a simulation example is given in order to illustrate the performance of the proposed observer

run time efficient observer based fuzzy neural controller for nonaffine multivariable systems with dynamical uncertainties abstract in this paper a novel hierarchical structure with run time efficiency is developed to solve the rule explosion problem of fuzzy neural network control for a class of uncertain nonaffine multivariable systems the parameters of the hybrid adaptive controller are on line tuned by the derived update laws under the constraint that only system outputs are available for measurement compared with the previous approaches the proposed design process is more flexible and requires less computation time according to the stability analysis the overall control scheme guarantees that the closed loop systems can obtain successful system control effective state observer and desired tracking performance finally illustrative examples are provided to show the effectiveness of the proposed approach

output feedback control of uncertain nonlinear systems using a self structuring adaptive fuzzy observer this paper describes the design of an output feedback controller based on an adaptive fuzzy observer for uncertain single input single output nonlinear dynamical systems we have focused on the realization of a minimal dynamic order for the adaptive fuzzy observer for this purpose we adopt an adaptive fuzzy observer in which no strictly positive real spr condition is needed and we combine a self structuring scheme with an on line estimation of fuzzy parameters by using this proposed scheme we can reduce the dynamic order of the adaptive output feedback fuzzy control system the lyapunov synthesis approach is used to guarantee a global uniform ultimate boundedness property of the state estimation error and tracking error as well as all other signals in the closed loop system no a priori knowledge of an upper limits on the uncertainties including optimal parameters and modeling errors is required the theoretical results are illustrated through simulation examples

fuzzy adaptive robust backstepping stabilization for siso nonlinear systems with unknown virtual control direction in this paper a direct fuzzy adaptive robust control approach is proposed for a class of siso nonlinear systems with completely unknown virtual control directions unknown nonlinearities unmodeled dynamics and dynamic disturbances in the backstepping recursive design fuzzy logic systems are employed to approximate the combined nonlinear uncertainties a dynamic signal and nussbaum gain technique are introduced into the control scheme to dominate the dynamic uncertainties and solve the unknown signs of virtual control directions respectively it is proved that the proposed robust fuzzy adaptive scheme can guarantee the all signals in the closed loop system are semi globally uniformly ultimately bounded the effectiveness of the proposed approach is illustrated via three examples

global adaptive observers for nonlinear systems via filtered transformations the problem of determining global adaptive observers for a class of single output nonlinear systems which are linear with respect to an unknown constant parameter vector is addressed sufficient conditions are given to observe asymptotically an equivalent state without persistency of excitation under additional geometric conditions the original state can be observed as well the results obtained are based on a nonlinear change of coordinates driven by auxiliary filters filtered transformations

on the error estimation for a mixed type of interpolation we analyse the error induced by the approximation of a function by an interpolation function which is a combination of algebraic and first order trigonometric polynomials we prove that under certain conditions this error can be expressed in a similar form as in the purely polynomial case as an application we establish in closed form the local truncation error for a class of extended linear multistep methods of the adams type

state observer based adaptive fuzzy output feedback control for a class of uncertain nonlinear systems this paper aims to develop state observer based adaptive fuzzy control techniques for controlling a class of uncertain nonlinear systems with bounded external disturbances an adaptive fuzzy observer is proposed to estimate the system state variables it is shown that the observation errors obtained from the observer are uniformly ultimately bounded applying the estimated system state for design of an output feedback controller the uniformly ultimate boundedness of the tracking errors for the resulting closed loop system can be guaranteed a typical robot arm system is employed in our simulation studies and the results demonstrate the usefulness and effectiveness of the proposed techniques for controlling nonlinear systems with bounded external disturbances

analytic integrable systems analytic normalization and embedding flows abstract in this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems more details we will prove that any complete analytic integrable diffeomorphism f x b x f x in c n 0 with b having eigenvalues not modulus 1 and f x o x 2 is locally analytically conjugate to its normal form meanwhile we also prove that any complete analytic integrable differential system x a x f x in c n 0 with a having nonzero eigenvalues and f x o x 2 is locally analytically conjugate to its normal form furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow we note that parts of our results are the improvement of moser s one in j moser the analytic invariants of an area preserving mapping near a hyperbolic fixed point comm pure appl math 9 1956 673 692 and of poincare s one in h poincare sur l integration des equations differentielles du premier order et du premier degre ii rend circ mat palermo 11 1897 193 239 these results also improve the ones in xiang zhang analytic normalization of analytic integrable systems and the embedding flows j differential equations 244 2008 1080 1092 in the sense that the linear part of the systems can be nonhyperbolic and the one in n t zung convergence versus integrability in poincare dulac normal form math res lett 9 2002 217 228 in the way that our paper presents the concrete expression of the normal form in a restricted case

nonlinear system fault diagnosis based on adaptive estimation an approach to fault diagnosis for a class of nonlinear systems is proposed in this paper it is based on a new adaptive estimation algorithm for recursive estimation of the parameters related to faults this algorithm is designed in a constructive manner through a nontrivial combination of a high gain observer and a recently developed linear adaptive observer without resort to any linearization its global exponential convergence is ensured by an easy to check persistent excitation condition a numerical example is presented for illustration

error analysis for frequency dependent interpolation formulas using first derivatives abstract we call attention to the entire errors which result when frequency dependent interpolation formulas are utilized to approximate oscillatory functions f x at some x on the domain of interest with a frequency of the form f x f 1 x cos x f 2 x sin x where the functions f 1 and f 2 are smooth enough to be approximated by polynomials the interpolation formulas to be considered utilize not only the pointwise values of the function f but also of its derivative f at two or three nodes on a closed and bounded interval in particular investigations about the interpolation formulas i or i using three equally spaced nodes or three unequally spaced nodes enable us to construct i or i related composite formulas which are obtained from applying the formulas i or i onto subintervals where the union of all the subintervals is the domain of interest numerical results show that newly constructed composite formulas are superior in their accuracy to other approximations to interpolate the oscillatory functions finally the entire errors with respect to the interpolation formulas using the derivative information at two or three nodes are obtained

fuzzy basis functions comparisons with other basis functions fuzzy basis functions fbf s which have the capability of combining both numerical data and linguistic information are compared with other basis functions because a fbf network is different from other networks in that it is the only one that can combine numerical and linguistic information comparisons are made when only numerical data is available in particular a fbf network is compared with a radial basis function rbf network from the viewpoint of function approximation their architectural interrelationships are discussed additionally a rbf network which is implemented using a regularization technique is compared with a fbf network from the viewpoint of overcoming ill posed problems a fbf network is also compared with specht s probabilistic neural network and his general regression neural network grnn from an architectural point of view a fbf network is also compared with a gaussian sum approximation in which gaussian functions play a central role finally we summarize the architectural relationships between all the networks discussed in this paper

the optimal exponentially fitted numerov method for solving two point boundary value problems second order boundary value problems are solved by means of exponentially fitted numerov methods these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe l gr ixaru g vanden berghe exponential fitting kluwer academic publishers dordrecht 2004 special attention is paid to the expression of the error term of such methods an algorithm concerning the choice of the best suited method and its parameter is discussed several numerical examples are given to sustain the theory

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

observer based adaptive control for uncertain time delay systems in this paper we will focus on investigating the observer based controlling problem of time delay systems first we investigate a class of simple time delay systems the corresponding adaptive observer and controller are designed which are both independent of the time delays based on lyapunov stability theory we prove that the closed loop system is asymptotically stable next we further consider the interconnected time delay system case the corresponding adaptive observer and controller are designed we prove that the resulting closed loop system is also asymptotically stable simulations on controlling time delay systems and interconnected systems are investigated and the results show that the designed controllers are feasible and efficient

gevrey asymptotics of series in mourtada type compensators used for linearization of an analytic 1 1 resonant saddle abstract given a 1 1 resonant saddle singularity of a planar analytic vector field we provide a linearization procedure using a series expansion in compensators of mourtada type and show that this series has gevrey 1 asymptotics in case of an analytic poincare dulac normal form we show that this transformation is analytic as a function of the compensators

orbital linearization around equilibria of vector fields in rn and lie symmetries abstract in this paper we study some relationships between analytic vector fields x in r n which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set n x of its locally analytic normalizers analytic infinitesimal generators of lie point orbitally symmetries in that neighborhood characterization and necessary and sufficient conditions for the orbital linearization of x via the existence of special elements in n x are given

robust adaptive critic control of nonlinear systems using fuzzy basis function networks an lmi approach abstract this paper proposes an adaptive critic tracking control design for a class of nonlinear systems using fuzzy basis function networks fbfns the key component of the adaptive critic controller is the fbfn which implements an associative learning network aln to approximate unknown nonlinear system functions and an adaptive critic network acn to generate the internal reinforcement learning signal to tune the aln another important component the reinforcement learning signal generator requires the solution of a linear matrix inequality lmi which should also be satisfied to ensure stability furthermore the robust control technique can easily reject the effects of the approximation errors of the fbfn and external disturbances unlike traditional adaptive critic controllers that learn from trial and error interactions the proposed on line tuning algorithm for aln and acn is derived from lyapunov theory thereby significantly shortening the learning time simulation results of a cart pole system demonstrate the effectiveness of the proposed fbfn based adaptive critic controller

an efficient method to approximate eigenfunctions and high index eigenvalues of regular sturm liouville problems the computation of the eigenvalues of a sturm liouville problem is a difficult task when high index eigenvalues are computed in most previous methods it can be seen that the uncertainty of the results increases as the estimated eigenvalues grow larger this paper is to present some new methods in which not only the error of calculating the higher eigenvalues does not grow but it also vanishes as eigenvalues tend to infinity moreover the proposed method gives good estimates of eigenfunctions corresponding to high eigenvalues

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

the convenient setting for non quasianalytic denjoy carleman differentiable mappings abstract for quasianalytic denjoy carleman differentiable function classes c q where the weight sequence q q k is log convex stable under derivations of moderate growth and also an l intersection see 1 6 we prove the following the category of c q mappings is cartesian closed in the sense that c q e c q f g c q e f g for convenient vector spaces applications to manifolds of mappings are given the group of c q diffeomorphisms is a regular c q lie group but not better

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

quadrature rules using first derivatives for oscillatory integrands we consider the integral of a function y x i y x 11 y x dx and its approximation by a quadrature rule of the form qn y x k 1n wky xk k 1n ky xk i e by a rule which uses the values of both y and its derivative at nodes of the quadrature rule we examine the cases when the integrand is either a smooth function or an dependent function of the form y x f1 x sin x f2 x cos x with smoothly varying f1 and f2 in the latter case the weights wk and k are dependent we establish some general properties of the weights and present some numerical illustrations

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

a dynamic recurrent neural network based adaptive observer for a class of nonlinear systems abstract an adaptive observer for a class of single input single output siso nonlinear systems is proposed using a generalized dynamic recurrent neural network drnn the neural network nn weights are tuned on line with no off line learning required no exact knowledge of non linearities in the observed system is required furthermore no linearity with respect to unknown system parameters is assumed the drnn observer does not assume that nonlinearities in the system are restricted to the system output only the overall adaptive observer scheme is shown to be uniformly ultimately bounded simulation results have verified the performance of the drnn observer

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

adaptive observers for single output nonlinear systems an adaptive version of the nonlinear observer obtained by a j krener et al 1983 is presented this version involves the cancellation of nonlinear terms by output injection as an intermediate step necessary and sufficient conditions are given for transforming a nonlinear system by state space change of coordinates into the special adaptive observer form that was used by y bastin et al 1988 to design adaptive observers

command filtered based fuzzy adaptive control design for mimo switched nonstrict feedback nonlinear systems the adaptive fuzzy tracking control design problem for multi input and multi output uncertain switched nonstrict feedback nonlinear systems with arbitrary switchings is investigated in this paper fuzzy logic systems are introduced to identify the unknown nonlinear functions for state measurable case and model the uncertain nonlinear systems for state immeasurable case both state feedback and observer based output feedback control design schemes are developed based on combined command filter and adaptive fuzzy control technique the proposed adaptive fuzzy controllers not only solve the explosion of complexity problem existing in conventional backstepping control schemes but as well as avoid the calculation of partial derivatives furthermore the stability of the fuzzy control systems under arbitrary switchings is proven based on the common lyapunov function method two simulation examples are presented to further demonstrate the effectiveness of the proposed control strategies

exponentially fitted two step runge kutta methods construction and parameter selection abstract we derive exponentially fitted two step runge kutta methods for the numerical solution of y f x y specially tuned to the behaviour of the solution such methods have nonconstant coefficients which depend on a parameter to be suitably estimated the construction of the methods is shown and a strategy of parameter selection is presented some numerical experiments are provided to confirm the theoretical expectations

adaptive fuzzy asymptotic tracking control of uncertain nonaffine nonlinear systems with non symmetric dead zone nonlinearities this paper is concerned with the problem of robust adaptive fuzzy tracking controller design for a class of uncertain nonaffine nonlinear systems with external disturbances and non symmetric dead zone nonlinearities with the help of fuzzy logic systems flss to approximate the desired nonlinear functions a novel direct adaptive tracking control scheme is established first the implicit function theorem and the mean value theorem are used to deal with the nonaffine term with unknown dead zone input then based on the estimation information for the unknown upper bound of the compounded disturbance an adaptive fuzzy tracking controller with the corresponding parameter updating laws is designed to eliminate the effect of the compounded disturbance furthermore it is proved that all the resulting closed loop system signals are uniformly bounded and the tracking error can asymptotically converge to zero via the lyapunov stability analysis finally a numerical example is provided to verify the effectiveness of the proposed adaptive fuzzy asymptotic tracking design method

adaptive fuzzy tracking control of nonlinear time delay systems with unknown virtual control coefficients in this paper a novel adaptive fuzzy control scheme is proposed for a class of uncertain single input and single output siso nonlinear time delay systems with the lower triangular form fuzzy logic systems are used to approximate unknown nonlinear functions then the adaptive fuzzy tracking controller is constructed by combining lyapunov krasovskii functionals and the backstepping approach the proposed controller guarantees uniform ultimate boundedness of all the signals in the closed loop system while the tracking error converges to a small neighborhood of the origin an advantage of the proposed control scheme lies in that the number of adaptive parameters is not more than the order of the systems under consideration finally simulation studies are given to demonstrate the effectiveness of the proposed design scheme

paper instability analysis and improvement of robustness of adaptive control the effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed five possible types of instability mechanisms parameter drift linear instability fast adaptation instability high frequency instability and throughput instability are analyzed using simple examples a procedure is used to construct lyapunov like functions for a modified adaptive controller applied to a dominant plant of relative degree one in the presence of parasitics and disturbances and obtain sufficient conditions under which none of the five types of instability can occur the modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set the size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast the region of attraction becomes the whole space

truncation errors in exponential fitting for oscillatory problems a generalization of peano s kernel theorem due to ghizzetti and ossicini quadrature formulae birkhauser basel switzerland 1970 provides expressions in the form of integrals for the truncation errors in a variety of exponential fitting formulae for oscillatory problems in some circumstances this leads to an expression analogous to the lagrange form of remainder more generally the error can be expressed as a sum of two terms of lagrange type our examples include formulae for quadrature and numerical differentiation and linear multistep methods for ordinary differential equations two families of exponential fitting quadrature formulae are investigated one with evenly spaced abscissas and the other based on the philosophy of gaussian quadrature in particular the integral representation can be used to determine the asymptotic rate of decay of the error with increasing frequency for a class of oscillatory integrands

on a correction of numerov like eigenvalue approximations for sturm liouville problems for computing approximations for the eigenvalues of sturm liouville problems numerov s method and other finite difference methods are often used giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem recently the authors have derived a modified numerov method with step dependent coefficients which integrates exactly the functions 1 x x2 x3 sin kx and cos kx the parameter k is calculated by minimizing the error term associated to the above modified multistep method equation we shall show that this modified numerov method delivers much more accurate eigenvalues than the classical numerov method which is based on a same grid pattern several numerical experiments will be presented

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

the least squares fit of highly oscillatory functions using eta based functions abstract in this paper we examine the possibility of using the eta functions as a new base for high quality approximations of oscillatory functions with slowly varying weights we focus on the least squares and piecewise least squares approximation of such functions and compare the results obtained by using eta based sets of functions with those obtained by means of the legendre polynomials and fourier series we find out that the accuracies from these are more or less equivalent for small frequencies but they exhibit different behaviors when the frequency is increased the accuracy worsens for the legendre polynomials and fourier series base but it remains bounded for the new base in accordance with the known properties of the eta functions such an advantage makes the new base quite attractive for being used in many other mathematical contexts where highly oscillatory functions are involved

an observer based approach to controlling time delay chaotic systems via takagi sugeno fuzzy model this study presents a fuzzy model based control criterion for time delay chaotic systems with uncertainty the takagi sugeno t s fuzzy model is adopted for representing a chaotic system possessing a time delay a unified approach for designing a fuzzy observer which integrates parallel distributed compensation with an adaptive updating method is first discussed an observer based fuzzy control scheme is then developed to deal with stabilization and tracking problems based on lyapunov stability analysis sufficient conditions that ensure robust control performance of time delay chaotic systems are derived these conditions are represented in terms of linear matrix inequalities and adaptive updating laws finally the proposed approach is validated through numerical synchronization stabilization and model tracking control simulations

a systematic approach to adaptive observer synthesis for nonlinear systems geometric techniques of controller design for nonlinear systems have enjoyed great success a serious shortcoming however has been the need for access to full state feedback this paper addresses the issue of state estimation from limited sensor measurements in the presence of parameter uncertainty an adaptive nonlinear observer is suggested for lipschitz nonlinear systems and the stability of this observer is shown to be related to finding solutions to a quadratic inequality involving two variables a coordinate transformation is used to reformulate this inequality as a linear matrix inequality a systematic algorithm is presented which checks for feasibility of a solution to the quadratic inequality and yields an observer whenever the solution is feasible the state estimation errors then are guaranteed to converge to zero asymptotically the convergence of the parameters however is determined by a persistence of excitation type constraint

on the numerical stability of the exponentially fitted methods for first order ivps abstract in the numerical solution of initial value problems ivps for differential systems exponential fitting ef techniques are introduced to improve the accuracy behaviour of classical methods when some information on the solutions is known in advance typically these ef methods are evaluated by computing their accuracy for some test problems and it is usual to assume that the stability behaviour is similar to the underlying classical methods the aim of this paper it to show that for some standard methods the stability behaviour of their exponentially fitted versions may change strongly furthermore this stability depends on the choice of the fitting space that must be carefully selected in order to assess the quality of the integrators for the ivps under consideration in particular we will show that for the usual fitting space exp x exp x with r the size of the stability domain of the ef method can be much smaller than the one for the original method