a convenient technique for evaluating angular frequency in some nonlinear oscillations in this paper a convenient technique for evaluating angular frequency in some nonlinear oscillations is proposed it is well known that once the restoring force function is given beforehand the period of motion can be determined by an integral the angular frequency has a relation with the period of motion as well as the integral one makes a little modification for the integrand in the integral and a change of variable it is found that if the three divisions are chosen on the integration interval and the trapezoid quadrature rule is used a higher accurate result for the angular frequency can be achieved for the restoring force being an odd function three numerical examples are presented the eardrum type oscillation is studied as well higher accurate results for the angular frequency are obtained in all those examples

filtering on sampled data systems with parametric uncertainty the paper is concerned with the problem of robust h sub spl infin filtering for a class of systems with parametric uncertainties and unknown time delays under sampled measurements the parameter uncertainties considered are real time varying and norm bounded appearing in the state equation an approach has been proposed for the designing of h sub spl infin filters using sampled measurements which would guarantee a prescribed h sub spl infin performance in the continuous time context irrespective of the parameter uncertainties and unknown time delays both cases of finite and infinite horizon filtering are studied it has been shown that the above robust h sub spl infin filtering problem can be solved in terms of differential riccati inequalities with finite discrete jumps

adaptive fuzzy control for uncertain interconnected time delay systems in this paper the problem of control for a class of uncertain interconnected time delay systems is studied the interconnections of the system possess time varying delay characteristics and the bounds of uncertainties are not precisely known a decentralized adaptive feedback controller is designed which is independent of the delays lyapunov krasovskii approach is employed to show that the corresponding closed loop system is asymptotically stable a simulation example is given to demonstrate the potential of the proposed techniques

solution of a duffing harmonic oscillator by the method of harmonic balance abstract the first order harmonic balance method via first fourier coefficient is used to construct an approximate frequency amplitude relation for a duffing harmonic oscillator this relation is in agreement with the result obtained by the ritz procedure

adaptive fuzzy backstepping output feedback tracking control of mimo stochastic pure feedback nonlinear systems with input saturation abstract in this paper the adaptive fuzzy backstepping output feedback tracking control problem is considered for a class of uncertain stochastic multi input and multi output mimo nonlinear systems in pure feedback form the stochastic mimo nonlinear systems under study have unknown nonlinear functions input saturation and immeasurable states by using fuzzy logic systems to identify the uncertain nonlinear system and a smooth function to approximate the input saturation a fuzzy state observer is designed and the estimations of the immeasurable states are obtained based on the backstepping recursive design technique an adaptive fuzzy output feedback tracking control approach is developed it is shown that the proposed control approach guarantees that all the signals of the resulting closed loop system are semi globally uniformly ultimately bounded sguub in mean square in the sense of probability and the observer errors and tracking errors can be regulated to a small neighborhood of the origin by choosing design parameters appropriately a simulation example is provided to show the effectiveness of the proposed approach

assessing prediction error in autoregressive models assessing prediction error is a problem which arises in time series analysis the distinction between the conditional prediction error e and the unconditional prediction error e e has not received much attention in the literature although one can argue that the conditional version is more practical we show in this article that assessing e is nearly impossible in particular we use the correlation coefficient corr e e where e is an estimate of e as a measure of performance and show that limt r v7corr e e c where t is the sample size and c 0 is some constant furthermore the value of c is large only when the process is extremely non gaussian or nearly nonstationary

variable structure adaptive fuzzy control for a class of nonlinear time delay systems abstract in this paper the problem of variable structure adaptive fuzzy control for a class of nonlinear time delay systems is investigated sliding surface dependent of the time delay is constructed based on the lyapunov krasovskii method two adaptive fuzzy logic systems are proposed to approximate two unknown continuous nonlinear functions containing the current state and the delayed state respectively the corresponding reaching law is designed which can drive the state trajectory of the closed system onto the sliding surface within limited time simulation investigation verifies the effect of the main results of the paper

decentralized stabilization of large scale systems with multiple delays in the interconnections in this paper the problem of robust stabilization of a class of time varying large scale systems subject to external disturbances uncertain parameters and multiple time varying delays in the interconnections and non linear inputs is considered linear memory less controllers are developed to guarantee that all solutions of the class of systems considered are uniformly ultimately bounded also non linear memoryless controllers are synthesized to guarantee that the solutions are uniformly ultimately bounded with respect to a bound with any prespecified convergence rate towards it finally an example is included to illustrate the results developed in this paper

properties of predictors for autoregressive time series abstract the prediction of the n s th observation of the pth order autoregressive process is investigated the mean square of the predictor error through terms of order n 1 conditional on yn y n 1 y n p 1 is obtained for the stationary normal process the mean squared error expression is similar to the usual regression formula for the variance of the predictor error the usual regression formula for the estimated variance of a predictor error and its generalization to s period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non stationary processes

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

homotopy perturbation method for a conservative x1 3 force nonlinear oscillator the homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x 1 3 we find this method works very well for the whole range of initial amplitudes excellent agreement of the approximate frequencies with the exact ones has been demonstrated and discussed only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0 60 for small and large values of oscillation amplitude while this relative error is as low as 0 050 for the second iteration comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators

harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable abstract the second order harmonic balance method is used to construct three approximate frequency amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable two procedures are used to solve the nonlinear differential equation approximately in the first the differential equation is rewritten in a form that does not contain the y 1 expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first one and the discrepancy between the approximate frequency and the exact one is lower than 1 28

linearized galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators abstract linearized harmonic balance galerkin methods for the determination of the periodic solutions of nonlinear oscillators are presented and the results are compared with those of an artificial parameter method based on the introduction of a linear stiffness term and the linstedt poincare technique the linearized harmonic balance methods presented in this paper are not iterative and are based on taylor series expansions the neglect of nonlinear terms in the resulting series a fourier approximation to the solution and orthogonality conditions it is shown that the first approximation provided by the linearized harmonic balance technique is identical to that of the artificial parameter method whereas the second one depends on how the equation is linearized and written and the discrepancies have been found to be caused by the neglect of the nonlinear terms in the taylor series expansion because a fourier series expansion of these terms introduces harmonics which are not accounted for in the linearization procedure it is also shown that the first approximation of the artificial parameter method and the linearized harmonic balance procedure are asymptotically equivalent to that derived from a modified linstedt poincare method when the nonlinearities are small and that these techniques may not be applicable to some oscillators

technical communique stabilization of a class of uncertain large scale stochastic systems with time delays this paper focuses on a class of large scale interconnected bilinear stochastic systems with time delays and time varying parameter uncertainties the problem we address is to design a robust decentralized controller such that the closed loop interconnected stochastic system is globally asymptotically stable in probability for all admissible uncertainties and time delays first robust stability analysis is given in terms of a set of linear matrix inequalities lmis then it is shown that the robust decentralized stabilization can be solved by an lmi based design method

harmonic balance approach to periodic solutions of non linear jerk equations the method of harmonic balance is applied to non linear jerk equations which involve the third order time derivative for many types of cubic non linearities the method yields good estimates of the period and displacement amplitude of oscillations for a range of values of initial velocity amplitude when compared with numerical solutions some limitations notably the restriction to zero initial acceleration as well as implications and possible extensions are discussed

sequential stability analysis and observer design for distributed ts fuzzy systems many complex physical systems are the interconnection of lower dimensional subsystems for such systems distributed stability analysis and observer design presents several advantages with respect to centralized approaches such as modularity easier analysis and design and reduced computational complexity applications include distributed process control traffic and communication networks and economic systems in this paper we propose sequential stability analysis and observer design for distributed systems where the subsystems are represented by takagi sugeno ts fuzzy models the analysis and design are done sequentially for the subsystems allowing for the online addition of new subsystems the conditions are formulated as lmis and are therefore easy to solve the approach is illustrated on simulation examples

decentralized adaptive robust control for a class of large scale systems including delayed state perturbations in the interconnections the problem of decentralized stabilization is considered for a class of large scale time varying systems with delayed state perturbations in the interconnections in this note the upper bounds of the uncertainties in the interconnections are assumed to be unknown the adaptation laws are proposed to estimate such unknown bounds and by making use of their updated values a class of decentralized memoryless state feedback controllers is constructed based on lyapunov stability theory and lyapunov krasovskii functional it is shown that the solutions to the resulting adaptive closed loop large scale time delay system can be guaranteed to be uniformly ultimately bounded finally a numerical example is given to demonstrate the validity of the results

control of markovian jump discrete time systems with norm bounded uncertainty and unknown delay this paper studies the problem of control for discrete time delay linear systems with markovian jump parameters the system under consideration is subjected to both time varying norm bounded parameter uncertainty and unknown time delay in the state and markovian jump parameters in all system matrices we address the problem of robust state feedback control in which both robust stochastic stability and a prescribed h sub spl infin performance are required to be achieved irrespective of the uncertainty and time delay it is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution

decentralized adaptive tracking control of nonaffine nonlinear large scale systems with time delays this paper addresses the problem of decentralized tracking control of large scale systems with uncertain nonaffine nonlinear isolated subsystems and nonlinear interconnections with time varying delays based on lyapunov krasovskii functional approach and implicit function theorem a delay independent decentralized tracking controller is proposed due to functional approximation capability of fuzzy logic systems fls neither strict structure restrictions on the isolated subsystems nor a priori knowledge of the strong interconnections with time varying delays is required in our control design furthermore transient performance of the resulting closed loop system is also addressed under an analytical framework finally two numerical examples are provided to show the effectiveness of the proposed controller

higher accuracy analytical approximations to the duffing harmonic oscillator a new method has been presented for analytically solving the duffing harmonic oscillator the method is obtained by combining newton s method with the harmonic balance method by using the method one obtains linear algebraic equations instead of nonlinear algebraic equations the complexity of the hb method is greatly simplified iteration of procedure yields rapid convergence with respect to exact solution the results are valid for the complete range of oscillation amplitude including the limiting cases of amplitude approaching zero and infinity

a finite extensibility nonlinear oscillator abstract the dynamics of a finite extensibility nonlinear oscillator feno is studied analytically by means of two different approaches a generalized decomposition method gdm and a linearized harmonic balance procedure lhb from both approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation within the generalized decomposition method two different versions are presented which provide different kinds of approximate analytical solutions in the first version it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes the second version which expands the nonlinear term in taylor s series around the equilibrium point exhibits a little lower degree of accuracy but it supplies an infinite series as the approximate solution on the other hand a linearized harmonic balance method is also employed and the comparison between the approximate period and the exact one numerically calculated is slightly better than that obtained by both versions of the gdm in general the agreement between the results obtained by the three methods and the exact solution numerically integrated for amplitudes a between 0 a 0 9 is very good both for the period and the amplitude of oscillation for the rest of the amplitude range 0 9 a l 2 error demonstrates that all three approximations do not represent a good description for the feno and higher order perturbation solutions are needed instead as a complement very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation

stable adaptive fuzzy sliding mode control of interconnected systems the problem of stable adaptive fuzzy control for a class of interconnected systems with unknown control gains is studied in this paper by using fuzzy modeling method a design scheme of an adaptive fuzzy controller is proposed the design is capable of incorporating linguistic and numerical information into controllers by theoretical analysis the closed loop adaptive fuzzy control system is proved to be globally stable in the sense that all signals involved are bounded with tracking errors converging to a neighborhood of zero

what is an analysis of variance the analysis of variance is usually regarded as being concerned with sums of squares of numbers and independent quadratic forms of random variables in this paper an alternative interpretation is discussed for certain classes of dispersion models for finite or infinite arrays of random variables a form of generalized spectral analysis is described and its intuitive meaning explained the analysis gives a spectral decomposition of each dispersion in the class incorporating an analysis of the common variance and an associated orthogonal decomposition of each of the random variables one by product of this approach is a clear understanding of the similarity between the spectral decomposition for second order stationary processes and the familiar linear models with random effects

a successive integration technique for the solution of the duffing oscillator equation with damping and excitation abstract in this paper a successive integration technique is suggested for solving the duffing oscillator equation with damping and excitation in the technique one performs integrations from some initial values over a wider range some of them may even be divergent however some of them will give a convergent result such that the periodic condition of motion is satisfied in fact the convergent result represents a stable periodic solution for the motion if a convergent result or a stable periodic solution is obtained the stability test is passed a harmonic balance method in conjunction with the successive integration technique abbreviated as hbmsit is also suggested in the method the initial values are obtained from the harmonic balance method therefore the hbmsit belongs to the successive integration techniques many examples with calculated results are presented

several numerical solution techniques for nonlinear eardrum type oscillations three numerical solution techniques for nonlinear eardrum type oscillations with an even function in the restoring force are studied the first suggested technique is named the target function technique in the technique the second zeros of the target function is the circular frequency of motion the second suggested technique is named the multiple parameter technique and the involved parameters are evaluated from the governing equation the initial conditions and properties of motion the third suggested technique is named the direct integration technique all techniques depend on the computer computation significantly and provide very accurate numerical results finally numerical examples are given

adaptive fuzzy decentralized output feedback control for nonlinear large scale systems with unknown dead zone inputs in this paper the problem of adaptive fuzzy decentralized backstepping control is considered for a class of nonlinear large scale strict feedback systems with unknown dead zones and immeasurable states fuzzy logic systems are used to approximate the unknown nonlinear functions and a state filter is designed to estimate the immeasurable states applying an adaptive backstepping design technique and combining it with the dead zone inverse method an adaptive fuzzy decentralized output feedback backstepping control is developed it is proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and tracking errors converge to a small neighborhood of the origin by appropriate choice of design parameters simulation results are given to demonstrate that the proposed adaptive decentralized control approach has a satisfactory control performance

an artificial parameter decomposition method for nonlinear oscillators applications to oscillators with odd nonlinearities abstract a method for obtaining series solutions of nonlinear second order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well known adomian s decomposition technique the method is formulated in both integral and differential forms for the determination of the limit cycle of oscillators with odd nonlinearities two differential forms and one integral form of the artificial parameter method are presented these versions are based on introducing a linear stiffness term with an unknown frequency and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator the three formulations provide identical results and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified linstedt poincare methods however the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation it is also shown that two and three level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined

harmonic balance and iteration calculations of periodic solutions to y y 1 0 we use two methods harmonic balance and iteration to calculate analytical approximations to the periodic solutions of a nonlinear singular oscillator

the bias of autoregressive coefficient estimators abstract this article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary but known finite order the results include models both with and without a constant term the effects of overspecification of the model order on the bias are described the emphasis is on least squares and yule walker estimators but the methods extend to other estimators of similar design although only the order t 1 component of the bias is captured where t is the series length this asymptotic approximation is shown to be very accurate for least squares estimators through some numerical simulations the simulations examine fourth order autoregressions chosen to resemble some data series from the literature the order t 1 bias approximations for yule walker estimators need not be accurate especially if the zeros of the associated polynomial have moduli near 1 examples are given where the approximation is accurate and where it is useless the bias expressions are

asymptotic representations of the period for the nonlinear oscillator this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp006 007

prescribed performance adaptive fuzzy output feedback dynamic surface control for nonlinear large scale systems with time delays the adaptive fuzzy decentralized output feedback control method is proposed for a class of nonlinear large scale systems the considered nonlinear systems contain unstructured uncertainties unknown time varying delays and immeasurable states fuzzy logic systems are used to approximate the unstructured uncertainties and a fuzzy state observer is designed to estimate the unmeasured states based on the backstepping recursive design technique and the prescribed performance technique a new adaptive fuzzy output feedback control method is developed in order to overcome the problem of explosion of complexity inherent in the backstepping control design the dynamic surface control dsc technique is introduced into the control scheme based on lyapunov krasovskii functional it is proved that all the signals in the closed loop system are bounded and the tracking errors remain an adjustable neighborhood of the origin with the prescribed performance bounds the simulation examples and comparison with the previous control methods are provided to show the effectiveness of the proposed control approach

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

solution of the relativistic an harmonic oscillator using the harmonic balance method this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp 2007 020

on the uniqueness of angular frequency using harmonic balance from the equation of motion and the energy relation two factors motivated the present exercise one is whether the lowest order harmonic solution is the cause of the disagreement in the values of w as obtained from the equation of motion and the energy relation or whether the reason lies elsewhere

robust stabilization of a class of nonlinear time delay systems the problem of robust stabilization for a class of uncertain dynamic systems with multiple delayed state perturbations is considered it is assumed that perturbations of the time delayed sections are not bounded by first order linear functions but bounded by high order functions with unknown gains and the time delay considered is time varying two classes of controllers are proposed when the time derivative of each time varying time delay is less than 1 a class of adaptive state feedback controllers are proposed based on lyapunov krasovskii method which can render the closed loop systems uniformly ultimately bounded stable novel nonlinear feedback controllers are developed by employing razumikhin lemma and the controller also can render the closed loop systems stable in the sense of uniform ultimate boundedness finally several examples are given to show the potential of the proposed techniques

exact solutions of the saturable discrete nonlinear schrodinger equation exact solutions to a nonlinear schrodinger lattice with a saturable nonlinearity are reported for finite lattices we find two different standing wave like solutions and for an infinite lattice we find a localized soliton like solution the existence requirements and stability of these solutions are discussed and we find that our solutions are linearly stable in most cases we also show that the effective peierls nabarro barrier potential is nonzero thereby indicating that this discrete model is quite likely nonintegrable