nonasymptotic pseudo state estimation for a class of fractional order linear systems this paper aims at designing a nonasymptotic and robust pseudo state estimator for a class of fractional order linear systems which can be transformed into the brunovsky s observable canonical form of pseudo state space representation with unknown initial conditions first this form is expressed by a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output based on which the modulating functions method is applied then the former initial values and the fractional derivatives of the output are exactly given by algebraic integral formulae using a recursive way which are used to nonasymptotically estimate the pseudo state of the system in noisy environment second the pseudo state estimator is studied in discrete noisy case which contains the numerical error due to a used numerical integration method and the noise error contribution due to a class of stochastic processes then the noise error contribution is analyzed where an error bound useful for the selection of design parameter is provided finally numerical examples illustrate the efficiency of the proposed pseudo state estimator where some comparisons with the fractional order luenberger like observer and a new fractional order mathcal h infty like observer are given

work by robert kalaba on automated sensitivity analysis this paper reviews work by robert kalaba on the development of a program nasa for the automated sensitivity analysis of parametrized nonlinear systems over parameter intervals nasa incorporates a fast and efficient algorithm feed for higher order partial derivative evaluation as well as an adaptive homotopy continuation procedure for obtaining all required initial conditions applications are envisioned for fields such as economics and the biological sciences where models tend to be complex and closed form solutions are difficult to obtain

irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise in this paper we derive several results related to the long time behavior of a class of stochastic semilinear evolution equations in a separable hilbert space h d mathfrak u t rm a mathfrak u t rm b mathfrak u t mathfrak u t dt dl t quad mathfrak u 0 x in rm h here a is a positive self adjoint operator and b is a bilinear map and the driving noise l is basically a d rm a 1 2 valued levy process satisfying several technical assumptions by using a density transformation theorem type for levy measure we first prove a support theorem and an irreducibility property of the ornstein uhlenbeck processes associated to the nonlinear stochastic problem second by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain gamma in 0 1 4 b is continuous on d rm a gamma times d rm a gamma with values in d rm a 1 2 using a coupling argument the exponential ergodicity is also proved under the stronger assumption that b is continuous on rm h times rm h while the latter condition is only satisfied by the nonlinearities of goy and sabra shell models the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2d navier stokes magnetohydrodynamics equations the 3d leray varvec alpha model the goy and sabra shell models

variable order fractional numerical differentiation for noisy signals by wavelet denoising in this paper a numerical method is proposed to estimate the variable order fractional derivatives of an unknown signal in noisy environment firstly the wavelet denoising process is adopted to reduce the noise effect for the signal secondly polynomials are constructed to fit the denoised signal in a set of overlapped subintervals of a considered interval thirdly the variable order fractional derivatives of these fitting polynomials are used as the estimations of the unknown ones where the values obtained near the boundaries of each subinterval are ignored in the overlapped parts finally numerical examples are presented to demonstrate the efficiency and robustness of the proposed method an efficient method is proposed to numerically estimate the variable order fractional derivatives of a noisy signal wavelet denoising is adopted to reduce the noise effect in a signal polynomials are constructed to fit the denoised signal in a set of overlapped subintervals of the considered interval the variable order fractional derivatives of a noisy signal are numerically obtained based on fitting polynomials numerical examples of four cases are considered to demonstrate the efficiency and robustness of the proposed method

frequency domain analysis and tuning of the algebraic differentiators abstractthe algebraic differentiators introduced by mboup join and fliessare studied in the frequency domain the study complements the time domain developments of the previous reference where a least squares interpretation has allowed to explain the main properties of the method and also to devise efficient differentiators the frequency domain analysis gives answers to the central question of the choice of the algebraic differentiator parameters also an interpretation of the algebraic differentiators in terms of wavelet decomposition is given

numerical experiments using sukhanov s initial value method for nonlinear two point boundary value problems iv on considere un probleme aux 2 points limites pour un systeme d ordre 4 d equations differentielles non lineaires

automatic differentiation of functions of derivatives in a recent paper an algorithm feed was introduced for the systematic exact evaluation of higher order partial derivatives of functions of many variables the present paper demonstrates that feed can be applied to a much broader class of functions than envisioned previously specifically feed can be used to evaluate the higher order partial derivatives of functions which are defined in terms of the derivatives of other functions a task required in many applications

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

differentiation by integration with jacobi polynomials in this paper the numerical differentiation by integration method based on jacobi polynomials originally introduced by mboup et al 19 20 is revisited in the central case where the used integration window is centered such a method based on jacobi polynomials was introduced through an algebraic approach 19 20 and extends the numerical differentiation by integration method introduced by lanczos 1956 21 the method proposed here rooted in 19 20 is used to estimate the nth email protected n order derivative from noisy data of a smooth function belonging to at least c n 1 q email protected n in 19 20 where the causal and anti causal cases were investigated the mismodelling due to the truncation of the taylor expansion was investigated and improved allowing a small time delay in the derivative estimation here for the central case we show that the bias error is o h q 2 where h is the integration window length for email protected c n q 2 in the noise free case and the corresponding convergence rate is o d q 1 n 1 q where d is the noise level for a well chosen integration window length numerical examples show that this proposed method is stable and effective

fractal system as represented by singularity function a fractional slope on the log log bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole in order to represent and study its dynamical behavior a singularity function method is presented which consists of cascaded branches of a number of pole zero negative real pairs or simple rc section the distribution spectrum of the system can also be easily calculated and its accuracy depends on a prescribed error specified in the beginning the method is then extended to a multiple fractal system which consists of a number of fractional power poles the result can be simulated by a combination of singularity functions each representing a single fractal system

solving nonlinear equations by adaptive homotopy continuation standard homotopy continuation methods for solving systems of nonlinear equations require the continuation parameter to move from 0 to 1 along the real line difficulties can occur however if a point of singularity is encountered during the course of the integration to ameliorate these difficulties this paper proposes extending the continuation parameter to complex values and adaptively computing a continuation path in the complex plane that avoids points giving rise to singularities specifically it is proposed that the continuation parameter move from 0 0i to 1 0i along a spider web grid centered at 1 0i in the complex plane the actual path through the grid is determined step by step in accordance with two objectives short path length and avoidance of singular points a two phase homotopy continuation is used to study the implementation of this idea numerical examples are presented which indicate the effectiveness of the approach

stabilization of quasi integrable hamiltonian systems with fractional derivative damping by using fractional optimal control an innovative procedure for designing fractional optimal control to asymptotically stabilize with probability one quasi integrable hamiltonian systems with fractional derivative dampings is proposed it is proved that the systems indeed can be stabilized by using the proposed procedure an example is given to illustrate the procedure and its effectiveness

numerical experiments using sukhanov s initial value method for nonlinear two point boundary value problems in a recent paper sukhanov derived a new method for transforming a nonlinear two point boundary value problem into an initial value problem sukhanov s equations involve only the solution of ordinary differential equations and not partial differential equations an earlier paper by the authors presented their interpretation of sukhanov s method an alternative method is presented in this paper numerical results are given

a computer program to minimize a function with many variables using computer evaluated exact higher order derivatives this paper describes a computer program for an optimization algorithm using first up to the rth order derivatives to find the optimum of r continuously differentiable functions of many variables the algorithm developed by kalaba and tishler 6 7 obtains the exact values of the derivatives required for the optimization from the table algorithm presented in 4 and 5 the optimization algorithm described here collapses to the well known newton raphson algorithm when only first and second order derivatives are used

the small time chung wichura law for levy processes with non vanishing brownian component we give a small time functional version of chung s other law of the iterated logarithm for levy processes with non vanishing brownian component this is an analogue of the other law of the iterated logarithm at large times for levy processes and random walks with finite variance as extended to a functional version by wichura as one of many possible applications we mention a functional law for a two sided passage time process

complete comparative static differential equations equations having form 1 arise as first order conditions in microeconomic constrained optimization models as defining characterizations for general equilibrium and macroeconomic models and as steady state solution characterizations for descriptive and optimal growth models see silberberg 12 hansen 7 cass and shell 3 and brock l if y r r is continuously differentiable over a neighborhood of a point x0 a in r n l for which y xo lx 0 and iy x0 or 0 then the implicit function theorem guarantees the existence of a continuously differentiable function x n cr r over some open neighborhood n cro of cx in r such that dx z ix yx x x a lyjx a x g e n cp 2

small deviations of general levy processes we study the small deviation problem log p sup t 0 1 x t e as e 0 for general levy processes x the techniques enable us to determine the asymptotic rate for general real valued levy processes which we demonstrate with many examples as a particular consequence we show that a levy process with nonvanishing gaussian component has the same strong asymptotic small deviation rate as the corresponding brownian motion

non asymptotic fractional order differentiator for a class of fractional order linear systems this paper aims at designing a non asymptotic fractional order differentiator for a class of fractional order linear systems to estimate the riemannliouville fractional derivatives of the output in discrete noisy environment the adopted method is a recent algebraic method originally introduced by fliess and sira ramirez firstly the fractional derivative of the output of an arbitrary order is exactly given by a new algebraic formula in continuous noise free case without knowing the initial conditions of the considered system secondly a digital fractional order differentiator is introduced in discrete noisy cases which can provide robust estimations in finite time then some error analysis is given where an error bound useful for the selection of the design parameter is provided finally numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator

a spline based method of solution of nonlinear two point boundary value problems abstract a method is presented that is meant to solve nonlinear two point boundary value problems the method is based on a spline interpolation of the function of interest with undetermined supporting points these are determined from the differential equation written at the supporting points which leads together with the boundary conditions to a nonlinear algebraic system this is next solved resorting to a newton type method with continuation to ensure convergence the procedure is illustrated with examples arising from computer aided design and robotics applications

an ada library for automatic evaluation of derivatives requirements for the evaluation of high order partial derivatives arise in many applications including optimal control system identification and the numerical solution of nonlinear algebraic equations two point boundary value problems and integral equations these evaluations have long been shunned by numerical analysts because of the difficulty in forming symbolic expressions for the derivatives this paper describes in detail an implementation in ada of a set of simple procedures for evaluating derivatives exactly without the requirement to form their symbolic expressions and without resorting to finite difference methods an expression whose derivatives are desired is written in the usual way using standard mathematical operators and intrinsic functions and values of all desired derivatives are returned automatically two examples of parameter identification are shown one from economics and one from guidance and control a complete listing of all ada code is included

parameter and differentiation order estimation in fractional models this paper deals with continuous time system identification using fractional differentiation models an adapted version of the simplified refined instrumental variable method is first proposed to estimate the parameters of the fractional model when all the differentiation orders are assumed known then an optimization approach based on the use of the developed instrumental variable estimator is presented two variants of the algorithm are proposed either all differentiation orders are set as integral multiples of a commensurate order which is estimated or all differentiation orders are estimated the former variant allows to reduce the number of parameters and can be used as a good initial hit for the latter variant the performances of the proposed approaches are evaluated by monte carlo simulation analysis finally the proposed identification algorithms are used to identify thermal diffusion in an experimental setup

a finite algorithm for the exact evaluation of higher order partial derivatives of functions of many variables abstract a formal algorithm is given for the systematic exact evaluation of higher order partial derivatives of functions of many variables the algorithm improves upon wengert s method in two key respects applications are envisioned wherever gradients jacobians hessians and power series expansions could be employed

nonlocal automated sensitivity analysis previously a complete ordinary differential equation ode system was developed for tracking the solution x of a parameterized system of nonlinear equations 0 x over an interval 0 this paper develops a two phase complex homotopy continuation method for obtaining the required initial conditions at t an initial short artificial continuation is followed by a continuation which essentially proceeds through the physically meaningful function which can ameliorate the problem of artificially induced singularities also the path in the complex plane followed by the continuation parameter in each phase evolves sequentially in an attempt to keep the path both short minimal number of integration steps and numerically stable avoidance of singular points a fortran program nasa has been developed for solving the complete ode system starting with a two phase initialization the program incorporates a fast and efficient procedure for the automatic evaluation of partial derivatives numerical experiments are reported which illustrate the program s effectiveness in ref 1 an analytically complete system of ordinary differential equations odes was developed for finding a solution vector x for a parameterized system of nonlinear equations 0 p x the well known ode dx d x was supplemented with odes for the adjoint a and the determinant 6 of the jacobian matrix vx x this differential system in principle permits nonlocal sensitivity analysis i e the tracking of the solution vector x and the sensitivity vector dx d together with the adjoint a and the determinant i over any closed interval 0 where the determinant i remains nonzero a fortran program was used successfully in ref 1 to solve this differential system for several illustrative examples subsequently the program was modified to incorporate a fast and efficient algorithm feed for the automatic evaluation of higher order partial derivatives 2 nevertheless to make the program a truly practical tool a simple and reliable procedure was still needed for obtaining the initial conditions x a and c required by the differential system at the initial parameter point ct this paper develops a simple but effective homotopy continuation method for obtaining these initial conditions which differs in two principal respects from standard homotopy continuation methods first the proposed continuation essentially proceeds through the function rather than through an artificial construct letting f x d x t a standard continuation method for finding a solu tion vector x x to the system of equations 0 f x would generally involve solving a system of equations of the form

innovative fractional derivative estimation of the pseudo state for a class of fractional order linear systems in this paper a non asymptotic and robust method is proposed to estimate the fractional integral and derivative of the pseudo state for a class of fractional order linear systems in noisy environment with unknown initial conditions to the best of our knowledge no method has been developed for such estimation firstly the estimation problem is transformed into estimating the fractional integral and derivative of the output and a set of fractional derivative initial values then algebraic integral formulas are exactly derived for the sought estimators by applying different modulating functions with specified properties in particular a design parameter is introduced in the formulas of fractional derivative initial values which can improve the robustness by choosing appropriate values secondly it is shown how to construct the required modulating functions in an efficient way where another design parameter is involved moreover some error analysis is given to choose the design parameters finally numerical simulations are provided to demonstrate the efficiency and the robustness to noises of the proposed method

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

on tuning fractional order proportional derivative controllers for a class of fractional order systems abstract this paper deals with a method recently proposed for tuning fractional order proportional derivative fo pd controllers using this tuning method the tuned fo pd controller can ensure the desired phase margin the desired gain crossover frequency and the flatness of the phase bode plot at such a frequency in the present paper the achievable region of this tuning method in the gain crossover frequency phase margin plane is obtained analytically also the continuity of this region and uniqueness of the tuned parameters are investigated moreover the achievable region of the aforementioned tuning method in the presence of time delay in the feedback loop is found

fractional order differentiation by integration and error analysis in noisy environment the integer order differentiation by integration method based on the jacobi orthogonal polynomials for noisy signals was originally introduced by mboup join and fliess we propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals firstly two fractional order differentiators are deduced from the jacobi orthogonal polynomial filter using the riemann liouville and the caputo fractional order derivative definitions respectively exact and simple formulae for these differentiators are given by integral expressions hence they can be used for both continuous time and discrete time models in on line or off line applications secondly some error bounds are provided for the corresponding estimation errors these bounds allow to study the design parameters influence the noise error contribution due to a large class of stochastic processes is studied in discrete case the latter shows that the differentiator based on the caputo fractional order derivative can cope with a class of noises whose mean value and variance functions are polynomial time varying thanks to the design parameters analysis the proposed fractional order differentiators are significantly improved by admitting a time delay thirdly in order to reduce the calculation time for on line applications a recursive algorithm is proposed finally the proposed differentiator based on the riemann liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises

parameters and fractional differentiation orders estimation for linear continuous time non commensurate fractional order systems abstract this paper proposes a two steps algorithm for the joint estimation of parameters and fractional differentiation orders of a linear continuous time fractional system with non commensurate orders the proposed algorithm combines the modulating functions and the first order newton methods sufficient conditions ensuring the convergence of the method are provided moreover the method is extended to the joint estimation of smooth unknown input and fractional differentiation orders a potential application of the proposed algorithm consists in estimating the fractional differentiation orders of a fractional neurovascular model along with the neural activity considered as an input for this model to assess the performance of the proposed method different numerical tests are conducted

on the laguerre rational approximation to fractional discrete derivative and integral operators this note ties the laguerre continued fraction expansion of the tustin fractional discrete time operator to irreducible jacobi tri diagonal matrices the aim is to prove that the laguerre approximation to the tustin fractional operator s or s is stable and minimum phase for any value 0 1 of the fractional order it is also shown that zeros and poles of the approximation are interlaced and lie in the unit circle of the complex z plane keeping a special symmetry on the real axis the quality of the approximation is analyzed both in the frequency and time domain truncation error bounds of the approximants are given

finite time stabilization of fractional order uncertain chain of integrator an integral sliding mode approach in this technical note a novel methodology for robust finite time stabilization of a chain of uncertain fractional order integrator is proposed this is achieved by first designing a nominal controller which stabilizes the system in finite time an integral sliding mode like surface and a switching controller is proposed such that when the system is on the surface the equivalent value of the integral sliding mode control is the negative of the disturbance and hence the disturbance is cancelled an improved strategy with more general kind of uncertainty is also proposed numerical examples are presented to illustrate the proposed methods

local and nonlocal comparative static analysis of economic systems a complete differential equation system has been developed for the local and nonlocal comparative static analysis of general parametrized economic systems j x a 0 the differential equations allow both qualitative and quantitative sensitivity studies at parameter points or over parameter intervals a fortran program comstat is now available for the numerical integration of the differential equations the only input requirements are j and initial conditions initialization is rendered routine by a two stage application of comstat derivatives are automatically evaluated via a generalized wengert method freeing the user from ever having to form analytical derivatives comstat is applied to a problem in optimal economic growth to show the sensitivity of the steady state capital labor ratio to changes in the time preference parameter

fractional order sliding mode based extremum seeking control of a class of nonlinear systems this paper develops a fractional order sliding mode based extremum seeking controller fo sm esc for the optimization of nonlinear systems the proposed fo sm esc involving an fo derivative function 0 d t q sgn e 0 q 1 is shown to achieve faster tracking and higher control accuracy than the integer order io sm esc the tradeoff between control performance and parameters selection is analyzed and visualized the comparison between the fo io sm esc is given to reveal the potential advantages of the fo controller over the io controller simulation and experimental results show that the fo sm esc can have a faster convergence speed and a smaller neighborhood around the optimal operational point

nonlinear least squares via automatic derivative evaluation this paper further develops the table algorithm kalaba et al 1981 for exact evaluation of higher order partial derivatives of functions of many variables we present and apply the table algorithm in the least squares estimation of the parameters of a ces production function the actual use of the algorithm proved to be easy and accurate

an improved pseudo state estimator for a class of commensurate fractional order linear systems based on fractional order modulating functions abstract in this paper a non asymptotic pseudo state estimator for a class of commensurate fractional order linear systems is designed in noisy environment different from existing modulating functions methods the proposed method is based on the system model with fractional sequential derivatives by introducing fractional order modulating functions by applying the fractional order integration by parts formula and thanks to the properties of the fractional order modulating functions a set of fractional derivatives and fractional order initial values of the output are analogously obtained by algebraic integral formulas then an explicit formula of the pseudo state is accomplished by using the fractional sequential derivatives of the output computed based on the previous results this formula does not contain any source of errors in continuous noise free case and can be used to non asymptotically estimate the pseudo state in discrete noisy case the construction of the fractional order modulating functions is also shown which is independent of the time finally simulations and comparison results demonstrate the efficiency and robustness of the proposed method