on a lyapunov equation for polynomial continuous time systems the primary goal of this paper is to describe the classes of all polynomial vector fields such that the associated dynamical system admits some polynomial lyapunov function v x by fixing or constraining the time derivative of v x in different ways several different cases are studied hints are given about how to use the results for stability analysis and for non polynomial systems

pid controller design for fractional order systems with time delays abstract classical proper pid controllers are designed for linear time invariant plants whose transfer functions are rational functions of s where 0 1 and s is the laplace transform variable effect of input output time delay on the range of allowable controller parameters is investigated the allowable pid controller parameters are determined from a small gain type of argument used earlier for finite dimensional plants

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

multivariable fractional system approximation with initial conditions using integral state space representation in this paper we consider the approximation of general multivariable non commensurate fractional systems by integer order state space models this work contains two main contributions first a new state space representation using the fractional integral operator is introduced second the approximate model carries explicitly the initial conditions of the system two examples are given to illustrate the accuracy of the approximation

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

p _1 nonconforming finite elements on triangulations into triangles and quadrilaterals the p 1 nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected lipschitz domains an explicit a priori analysis for the combination of the park sheen and the crouzeix raviart nonconforming finite element methods is given for second order elliptic pdes with inhomogeneous dirichlet boundary conditions

system initial conditions vs derivative initial conditions the alternative system initial conditions versus the derivative initial conditions is focused in this paper it is shown that riemann liouville and caputo initial conditions result from the corresponding derivative and not necessarily from the system at hand to setup the correct system initialization a formulation generalizing the integer order approach is presented this is based on a generalization to the fractional environment of the well known jump formula the obtained scheme is very general and does not depend on any transform besides it can also be used in the time variant case the riemann liouville and caputo initial conditions are interpreted in terms of this general framework and deduced equations where they are correct

a newton like algorithm to compute the inverse of a nonlinear map that converges in finite time abstract this paper deals with the problem of inverting a nonlinear map the proposed solution consists in a nonlinear state observer which mimics a newton like algorithm that allows to determine the inverse of a given diffeomorphism in finite time the results are illustrated by application to the inverse kinematics of a three dof planar manipulator

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

an optimally convergent adaptive mixed finite element method we prove convergence and optimal complexity of an adaptive mixed finite element algorithm based on the lowest order raviart thomas finite element space in each step of the algorithm the local refinement is either performed using simple edge residuals or a data oscillation term depending on an adaptive marking strategy the inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator

on the uniform algebraic observability of multi switching linear systems the main objective of this paper is the characterisation of the notion of uniform algebraic observability for multi switching linear systems in particular it is shown that a multi switching linea

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

stabilization of polynomial nonlinear systems by algebraic geometry techniques this technical note applies methods from algebraic geometry namely polynomial ideals and sub modules to find parameterized sets of state feedback stabilizing control laws for polynomial continuous time affine nonlinear systems the proposed method is illustrated by simple examples

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

observability of the discrete state for dynamical piecewise hybrid systems abstract in this paper we deal with the observability of piecewise affine hybrid systems our aim is to give sufficient conditions to observe the discrete and continuous states in terms of algebraic and geometrical conditions firstly we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems secondly we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems throughout this paper we illustrate our propositions with examples and simulations

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

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

dead beat controllability of polynomial systems symbolic computation approaches state and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition qepcad symbolic computation program the method is unified for a very large class of systems and can handle one or two sided control constraints families of minimum time state output dead beat controllers are obtained the computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system we may beat the curse of complexity a computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented necessary and sufficient conditions are given they are still very difficult to check therefore a number of easier to check sufficient conditions are also provided the latter are based on the grobner basis method and qepcad it is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system

observability for switched linear systems characterization and observer design this paper presents a characterization of observability and an observer design method for switched linear systems with state jumps a necessary and sufficient condition is presented for observability globally in time when the system evolves under predetermined mode transitions because this characterization depends upon the switching signal under consideration the existence of singular switching signals is studied alongside developing a sufficient condition that guarantees uniform observability with respect to switching times furthermore while taking state jumps into account a relatively weaker characterization is given for determinability the property that concerns with recovery of the original state at some time rather than at all times assuming determinability of the system a hybrid observer is designed for the most general case to estimate the state of the system and it is shown that the estimation error decays exponentially since the individual modes of the switched system may not be observable the proposed strategy for designing the observer is based upon a novel idea of accumulating the information from individual subsystems contrary to the usual approach dwell time between switchings is not necessary but the proposed design does require persistent switching for practical purposes the calculations also take into account the time consumed in performing computations

stable finite element pair for stokes problem and discrete stokes complex on quadrilateral grids in this paper we construct a stable finite element pair for incompressible stokes problem and a discrete stokes complex associated with that pair on general quadrilateral grids the finite element spaces involved consist of piecewise polynomials only and the divergence free condition is imposed in a primal formulation these constructions can be generated onto grids that consist of both triangular and quadrilateral cells

numerical calculation of the left and right fractional derivatives the calculation of fractional derivatives is an important topic in scientific research while formal definitions are clear from the mathematical point of view they pose limitations in applied sciences that have not been yet tackled this paper addresses the problem of obtaining left and right side derivatives when adopting numerical approximations the results reveal the relationship between the resulting distinct values for different fractional orders and types of signals

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

switching signal estimator design for a class of elementary systems the goal of this technical note is to design a switching signal estimator for a class of elementary continuous time switching or switched systems first the elementary system is recast into a polynomial form and secondly some tools borrowed from algebraic geometry are used to express the switching signal as a function of the time derivatives of the output and of the input

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

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

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

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

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

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem

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