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

brief paper pseudo state feedback stabilization of commensurate fractional order systems this paper addresses the problem of pseudo state feedback stabilization of commensurate fractional order systems fos in the proposed approach linear matrix inequalities lmi formalism is used to check if the pseudo state matrix eigenvalues belong to the fos stability region of the complex plane a review of lmi stability conditions is first proposed for fractional order 0 n 1 and 1 n 2 the paper then focuses particularly on the case 0 n 1 as the stability region is non convex and associated lmi condition is not as straightforward to obtain as in the case 1 n 2 a new lmi stability condition is thus proposed based on this condition a necessary and sufficient lmi method for the design of stabilizing controllers is given this method paves the way for extension to fos of various lmi based results among these possible extensions a first result on robust control of polytopic fractional order systems is given in this paper

intersections of valuation rings in k x y we associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base field as an application we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field is in a suitable sense the branch of an algebraic curve

normalization and stabilization for rectangular singular fractional order t s fuzzy systems abstract this paper considers the normalization and stabilization of rectangular singular fractional order takagi sugeno t s fuzzy systems with order 0 1 by using the proportional and derivative type dynamic compensator the rectangular singular fractional order t s fuzzy systems are transformed into an augmented square singular fractional order t s fuzzy systems and the admissibility of the augmented square singular fractional order t s fuzzy systems can guarantee the normalization and stabilization of the rectangular singular fractional order t s fuzzy systems two sufficient conditions are expressed in terms of a set of bilinear matrix inequalities which can be efficiently solved by an iterative lmi algorithm a numerical example is given to verify the effectiveness of the results proposed

technical communique fractional order proportional derivative controller for a class of fractional order systems recently fractional order systems fos have attracted more and more attention in various fields but the control design techniques available for the fos suffer from the lack of direct systematic approaches in this paper we focus on a given type of simple model of fos a fractional order proportional derivative fo pd controller is proposed for this class of fos and a practical and systematic tuning procedure has been developed for the proposed fo pd controller synthesis the fairness issue in comparing with other controllers such as the traditional integer order pid io pid controller and the fractional order proportional derivative fo pd controller has been addressed under the same number of design parameters and the same specifications fair comparisons of the three controllers i e io pid fo pd and fo pd via the simulation tests illustrate that the io pid controller designed may not always be stabilizing to achieve flat phase specification while both fo pd and fo pd controllers designed are always stabilizing furthermore the proposed fo pd controller outperforms fo pd controller for the class of fractional order systems

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

dynamical compactifications of c 2 we find good dynamical compactifications for arbitrary polynomial mappings of c 2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula for maps of low topological degree we prove that the green function is well behaved for maps of maximum topological degree we give normal forms

temperature dependent fluctuations in the two dimensional xy model we present a detailed investigation of the probability density function pdf of order parameter fluctuations in the finite two dimensional xy 2dxy model in the low temperature critical phase of this model the pdf approaches a universal non gaussian limit distribution in the limit t 0 our analysis resolves the question of temperature dependence of the pdf in this regime for which conflicting results have been reported we show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously derived graphical expansion this is confirmed by numerical simulations on two controlled approximations to the 2dxy model the harmonic and harmonic xy models the harmonic model has no kosterlitz thouless berezinskii ktb transition and the pdf becomes progressively less skewed with increasing temperature until it closely approximates a gaussian function above t approximate to 4 pi near to that temperature we find some evidence of a phase transition although our observations appear to exclude a thermodynamic singularity

stabilisability analysis of high order unstable processes by fractional order controllers in this article the sufficient stabilisability criteria for unstable time delay processes by fractional order controllers are investigated the process is a high order system with a single unstable pole which also has a stable zero the adopted approach to derive the sufficient conditions for stability is based on the well known nyquist stability criterion stabilisability is studied by applying fractional order proportional integral and proportional derivative controllers and the results are given in terms of the maximum allowable value of time delay additionally limitations on the parameters of the controllers which must be taken into account in the controller design are proposed

uniformly continuous differintegral sliding mode control of nonlinear systems subject to holder disturbances an integral sliding mode controller based on fractional order differintegral operators is proposed this controller generalizes the classical discontinuous integer order integral sliding mode scheme by using differintegral operators their topological properties lead to a uniformly continuous controller that enforces an integral sliding mode for any initial condition in addition it is demonstrated that the proposed scheme is robust against matched holder continuous but not necessarily differentiable disturbances and uncertainties also asymptotic convergence of tracking errors is assured for any initial condition by means of an ideal controller even in presence of anomalous but holder continuous disturbances it is worth to mention that the salient properties of our proposal i e invariance at any initial condition uniform continuity and robustness to non differentiable disturbances are not provided by any existing integer order sliding mode based controller of the literature the viability of the proposed scheme is shown in a representative simulation study

modeling and numerical analysis of fractional order bloch equations this paper deals with the bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time these equations were introduced by felix bloch in 1946 and they are used for a description of the nuclear magnetic resonance nmr this physical phenomenon is used in medicine chemistry physics and engineering to study complex material fractional order generalization of the bloch equations was presented by richard magin et al in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous porous or composite materials in this paper we describe numerical and simulation models created for matlab simulink of the classical and the fractional order bloch equations the behaviour and stability analysis of the bloch equations are presented as well

quadratic lyapunov functions for stability analysis in fractional order systems with not necessarily differentiable solutions abstract solutions of fractional order differintegral equations are generally not necessarily integer order differentiable neither in the strong nor in the weak sense thus limiting the stability analysis in systems based on the most conventional fractional order operators in this paper a consistent and well posed definition for fractional order systems is performed based on the study of alternative fractional order operators that preserve the most interesting and useful properties of differintegrals even in the case of not necessarily integer order weakly differentiable functions in addition it is shown that these operators comply to a recently verified well known inequality which allows us to demonstrate mittag leffler stability in a more general class of fractional order systems considering quadratic lyapunov functions by demonstrating a generalization of the lyapunov direct method for a class of fractional order nonlinear systems illustrative examples are given to highlight the feasibility of the proposed method and a multivariable fractional integral sliding mode control application is presented

stabilizing and robust fractional order pi controller synthesis for first order plus time delay systems for all the stable first order plus time delay foptd systems a fractional order proportional integral fopi or a traditional integer order proportional integral derivative iopid controller can be designed to fulfill a flat phase constraint and two design specifications simultaneously gain crossover frequency and phase margin in this paper a guideline for choosing two feasible or achievable specifications and a new fopi iopid controller synthesis are proposed for all the stable foptd systems using this synthesis scheme the complete feasible region of two specifications can be obtained and visualized in the plane with this region as the prior knowledge all combinations of two specifications can be verified before the controller design especially it is interesting to compare the areas of these two feasible regions for the iopid and fopi controllers this area comparison reveals for the first time the potential advantages of one controller over the other in terms of achievable performances a simulation illustration is presented to show the effectiveness and the performance of the designed fopi controller compared with the optimized integer order pi controller and the iopid controller designed following the same synthesis for the fopi in this paper

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

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

robustness analysis and design of fractional order iλ dμ controllers using the small gain theorem in this paper a simple method is proposed to tune the parameters of fractional integral fractional derivative fife i d controllers based on the bode diagram the proposed technique provides a

on contractible curves on normal surfaces we give characterizations of contractible curves on proper normal algebraic surfaces in terms of complementary weil divisors from this we obtain some generalizations of the classical criteria for contractibility of castelnuovo and artin furthermore we will derive a finiteness result on homogeneous spectra defined by weil divisors on proper normal algebraic surfaces

optimal periodic gain fractional delayed state feedback control for linear fractional periodic time delayed systems this paper develops the fundamentals of optimal tuning periodic gain fractional delayed state feedback control for a class of linear fractional order periodic time delayed systems although there exist techniques for the state feedback control of linear periodic time delayed systems by discretization of the monodromy operator there is no systematic method to design state feedback control for linear fractional periodic time delayed fptd systems this paper is devoted to defining and approximating the monodromy operator for a steady state solution of fptd systems it is shown that the monodromy operator cannot be achieved in a closed form for fptd systems and hence the short memory principle along with the fractional chebyshev collocation method is used to approximate the monodromy operator the proposed method guarantees a near optimal solution for fptd systems with fractional orders close to unity the proposed technique is illustrated in examples specifically in finding optimal linear periodic gain fractional delayed state feedback control laws for the fractional damped mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers in which it is demonstrated that the use of time periodic control gains in the fractional feedback control generally leads to a faster response

technical communique on type number concept in fractional order systems in this note the type number concept is defined for fractional order systems based on this definition it is shown that fractional order systems having type numbers more than 1 can not track some classes of reference inputs without any overshoot

integral sliding mode control for nonlinear systems with matched and unmatched perturbations we consider the problem of designing an integral sliding mode controller to reduce the disturbance terms that act on nonlinear systems with state dependent drift and input matrix the general case of both matched and unmatched disturbances affecting the system is addressed it is proved that the definition of a suitable sliding manifold and the generation of sliding modes upon it guarantees the minimization of the effect of the disturbance terms which takes place when the matched disturbances are completely rejected and the unmatched ones are not amplified a simulation of the proposed technique applied to a dynamically feedback linearized unicycle illustrates its effectiveness even in presence of nonholonomic constraints

measuring nonextensitivity parameters in a turbulent couette taylor flow we investigate probability density functions of velocity differences at different distances r measured in a couette taylor flow for a range of reynolds numbers re there is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics where the entropies are nonadditive for independent subsystems we extract the scale dependent nonextensitivity parameter q r mathrm re from the laboratory data

representation of the absorption of nonlinear waves by fractional derivatives fractional derivatives of arbitrary order are incorporated into a model nonlinear equation in order to simulate the absorption mechanism in a finite frequency domain that makes it possible to take into account a power dependence of the absorption coefficient on the frequency at any real degree the model gives a mathematical description of a large set of well known absorption mechanisms the method is applied to calculate weakly nonlinear oscillations in a nonrigid tube as a result the value of the absorption coefficient and its dependence on the frequency are found based on experimental data

beyond fractional derivatives local approximation of other convolution integrals dynamic systems involving convolution integrals with decaying kernels of which fractionally damped systems form a special case are non local in time and hence infinite dimensional straightforward numerical solution of such systems up to time t needs o t 2 computations owing to the repeated evaluation of integrals over intervals that grow like t finite dimensional and local approximations are thus desirable we present here an approximation method which first rewrites the evolution equation as a coupled in finite dimensional system with no convolution and then uses galerkin approximation with finite elements to obtain linear finite dimensional constant coefficient approximations for the convolution this paper is a broad generalization based on a new insight of our prior work with fractional order derivatives singh chatterjee 2006 nonlinear dyn 45 183 206 in particular the decaying kernels we can address are now generalized to the laplace transforms of known functions of these the power law kernel of fractional order differentiation is a special case the approximation can be refined easily the local nature of the approximation allows numerical solution up to time t with o t computations examples with several different kernels show excellent performance a key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system as distinct from numerically approximating just the solution for the given initial values this allows non standard uses of the approximation e g in stability analyses

on convergence time and disturbance rejection of super twisting control super twisting algorithm is one of the versions of high order sliding mode control the interest to this algorithm is explained by its attractive properties continuous control input finite convergence time disturbance rejection in this paper the upper bound of admissible unknown disturbances and low bound of the convergence time are found and shown that both the values can be achieved with any desired accuracy

on the computation of darboux first integrals of a class of planar polynomial vector fields abstract we study the class of planar polynomial vector fields admitting darboux first integrals of the type i 1 r f i i where the i s are positive real numbers and the f i s are polynomials defining curves with only one place at infinity we show that these vector fields have an extended reduction procedure and give an algorithm which from a part of the extended reduction of the vector field computes a darboux first integral for generic exponents

limitations of frequency domain approximation for detecting chaos in fractional order systems abstract in this paper we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems the number and location of equilibria and also the stability of these points are compared between the original system and its frequency based approximated counterpart it is shown that the original system and its approximation are not necessarily equivalent according to the number location and stability of the fixed points this problem can cause erroneous results in special cases for instance to prove the existence of chaos in fractional order systems numerical simulations have been largely based on frequency domain approximations but in this paper we show that this method is not always reliable for detecting chaos this approximation can numerically demonstrate chaos in the non chaotic fractional order systems or eliminate chaotic behavior from a chaotic fractional order system

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

high order sliding mode observer for fractional commensurate linear systems with unknown input abstract in this paper a high order sliding mode observer hosmo is proposed for the joint estimation of the pseudo state and the unknown input of fractional commensurate linear systems with single unknown input and a single output the convergence of the proposed observer is proved using a lyapunov based approach in addition an enhanced variant of the proposed fractional hosmo is introduced to avoid the peaking phenomenon and thus to improve the estimation results in the transient phase simulation results are provided to illustrate the performance of the proposed fractional observer in both noise free and noisy cases the effect of the observer s gains on the estimated pseudo state and unknown input is also discussed

pdf of velocity fluctuation in turbulence by a statistics based on generalized entropy an analytical formula for the probability density function pdf of the velocity fluctuation in fully developed turbulence is derived non perturbatively by assuming that its underlying statistics is the one based on the generalized measures of entropy the renyi entropy or the tsallis entropy the parameters appearing in the pdf including the index q in the generalized measures are determined self consistently with the help of the observed value of the intermittency exponent the derived pdf explains quite well the experimentally observed density functions

discretization algorithm for fractional order integral by haar wavelet approximation abstract a discretization algorithm is proposed by haar wavelet approximation theory for the fractional order integral in this paper the integration time is divided into two parts one presents the effect of the past sampled data calculated by the iterative method and the other presents the effect of the recent sampled data at a fixed time interval calculated by the haar wavelet this method can reduce the amount of the stored data effectively and be applied to the design of discrete time fractional order pid controllers finally several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm

the statistical distributions of one dimensional turbulence we study a one dimensional discrete analog of the von karman flow widely investigated in turbulence made of a lattice of anharmonic oscillators excited by both ends in the presence of a dissipative term proportional to the second order finite difference of the velocities similar to the viscous term in a fluid the dynamics of the model shows striking similarities with an actual turbulent flow both at local and global scales calculations of the probability distribution function of velocity increments extensively studied in turbulence with a very large number of points in order to determine accurately the statistics of rare events allow us to provide a meaningful comparison of different theoretical expressions of the pdfs

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

power fluctuations in a closed turbulent shear flow we study experimentally the power consumption p of a confined turbulent flow at constant reynolds number re we analyze in details its temporal dynamics and statistical properties in a setup that covers two decades in reynolds numbers we show that nontrivial power fluctuations occur over a wide range of amplitudes and that they involve coherent fluid motions over the entire system size as a result the power fluctuations do not result from averaging of independent subsystems and its probability density function pi p is strongly non gaussian the shape of pi p is reynolds number independant and we show that the relative intensity of fluctuations decreases very slowly as re increases these results are discussed in terms of an analogy with critical phenomena

nyquist based stability analysis of non commensurate fractional order delay systems abstract as a generalization of the first and second order models the elementary fractional order models have been widely used in various engineering fields however most of the previous studies only focus on commensurate fractional order models in this paper a general non commensurate elementary fractional order delay system is investigated first the stability of the studied fractional order delay system is analyzed based on nyquist theorem then a series of sufficient stability conditions are presented for different combinations of parameters including the fractional orders time delay pseudo damping factor and natural frequency 0 finally three examples are given to show the effectiveness of the presented results

identification of parameters of a half order system this correspondence presents the half order system behavior and its parameter identification the identification is based on fitting the measured data using the mittag leffler function the data were collected for a discharge of a half order system the values of parameters obtained by a new identification method are in good agreement with the calculated interval for theoretical values which takes into account the manufacturing tolerances of the used electrical elements

hybrid projective synchronization of chaotic fractional order systems with different dimensions abstract the hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper it is shown that the slave system can be synchronized with the projection of the master system generated through state transformation based on the stability theorem of linear fractional order systems a suitable controller for achieving the synchronization is given the hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order the corresponding numerical results verify the effectiveness of the proposed method