a power series method for computing singular solutions to nonlinear analytic systems given a system of analytic equations having a singular solution we show how to develop a power series representation for the solution this series is computable and when the multiplicity of the solution is small highly accurate estimates of the solution can be generated for a moderate computational cost in this paper a theorem is proven using results from several complex variables which establishes the basis for the approach then a specific numerical method is developed and data from numerical experiments are given

brief state space representation for fractional order controllers this article proposes an infinite dimensional state space realization for linear filters with transfer function c d s c 0 1 s w b 1 s w h d where 0 w b w h and 0

robust stability and stabilization of fractional order interval systems with the fractional order alpha the 0 alpha 1 case this technical note firstly presents a sufficient and necessary condition for the robust asymptotical stability of fractional order interval systems with the fractional order satisfying 0 1 and then a sufficient condition for the robust asymptotical stabilization of such fractional order interval systems is derived all the results are obtained in terms of linear matrix inequalities finally two illustrative examples are given to show that our results are effective for checking the robust stability and designing the robust stabilizing controller for fractional order interval systems

robust stability and stabilization of fractional order interval systems an lmi approach this technical note presents necessary and sufficient conditions for the stability and stabilization of fractional order interval systems the results are obtained in terms of linear matrix inequalities two illustrative examples are given to show that our results are effective and less conservative for checking the robust stability and designing the stabilizing controller for fractional order interval systems

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

robust stability test of a class of linear time invariant interval fractional order system using lyapunov inequality this paper provides a new analytical robust stability checking method of fractional order linear time invariant interval uncertain system this paper continues the authors previous work yangquan chen hyo sung ahn i podlubny robust stability check of fractional order linear time invariant systems with interval uncertainties in proceedings of the ieee conference on mechatronics and automation niagara falls canada july 2005 pp 210 215 where matrix perturbation theory was used for the new robust stability checking lyapunov inequality is utilized for finding the maximum eigenvalue of a hermitian matrix through numerical examples the usefulness and the effectiveness of the newly proposed method are verified

second and fourth indices of plethysms the direct product of several copies of a representation decomposes into a direct sum of components each with a definite permutation symmetry the decomposition of any of the components into a direct sum of irreducible representations is the computation of a plethysm the decomposition is often simply effected when the dimension and its analogs the second and fourth indices of the plethysm are known the paper contains formulas for second and fourth indices of many specific plethysms as well as a prescription for the general plethysm the same formula is valid for the plethysm based on any finite representation of any semisimple lie algebra applications are illustrated by decomposition of all plethysms of degree 3 based on the e8 representation of dimension 3875 all fourth degree e8 scalars are enumerated

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

frequency domain synthesis of a filter using viete root functions this paper proposes a noniterative method to fit a rational transfer function to a specified frequency response a method using a smoothing of width modulated pulses of the phase asymptotic diagram is modified to provide an exact algebraic method this is then used to synthesize a robust controller and a nonrational transfer function with a fractional differentiation order

boundary fractional derivative control of the wave equation the wave equation with fractional derivative feedback at the boundary is studied the existence and uniqueness as well as the asymptotic decay of the solution towards zero is proved the method used is motivated by the fact that the input output relationship as generalized diffusion equations defined on the infinite spatial domain r with collocated sensor and actuator control can be expressed in terms of fractional integrals compared to other methods the payoff is as follows 1 the proofs are simpler and 2 the method used can easily be adapted to a wide class of problems involving fractional derivative or integral operators of the time variable

computing singular solutions to nonlinear analytic systems a method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented since manyreal systems extend naturally tocomplex analytic systems this porvides a method for generating approximations to singular solutions to real systems examples include systems of polynomials and systems made up of trigonometric exponential and polynomial terms the theorem on which the method is based is proven using results from several complex variables no special conditions on the derivatives of the system such as restrictions on the rank of the jacobian matrix at the solution are required the numerical method itself is developed from techniques of homotopy continuation and 1 dimensional quadrature a specific implementation is given and the results of numerical experiments in solving five test problems are presented

robust asymptotical stability of fractional order linear systems with structured perturbations this paper considers the robust asymptotical stability problem of fractional order linear systems with structured perturbations which are more general than the investigated fractional order interval systems based on the kronecker product and m analysis necessary and sufficient conditions for the robust asymptotical stability are established by transforming such a problem into checking the nonsingularity of a class of uncertain matrices furthermore the robustness bounds with respect to parametric perturbations to preserve the asymptotical stability are given in terms of the structured singular values finally illustrative examples are given to show the effectiveness of the proposed approach

su m n weight systems and superprojection matrices representations of the su m n superalgebra are studied in terms of the kac dynkin weight systems superprojection matrices are introduced for the possible branching patterns

a note on the use of the lambert w function in the stability analysis of time delay systems the lambert w function is defined to be the multivalued inverse of the function w we w z this function has been used in an extremely wide variety of applications including the stability analysis of fractional order as well as integer order time delay systems the latter application is based on taking the mth power and or nth root of the transcendental characteristic equation tce and representing the roots of the derived tce s in terms of w functions in this note we re examine such an application of using the lambert w function through actually computing the root distributions of the derived tces of some chosen orders it is found that the rightmost root of the original tce is not necessarily a principal branch lambert w function solution and that a derived tce obtained by taking the mth power of the original tce introduces superfluous roots to the system with these observations some deficiencies displayed in the literature chen y q moore k l 2002a analytical stability bound for delayed second order systems with repeating poles using lambert w function automatica 38 5 891 895 chen y q moore k l 2002b analytical stability bound for a class of delayed fractional order dynamic systems nonlinear dynamics 29 1 4 191 200 are pointed out moreover we clarify the correct use of lambert w function to stability analysis of a class of time delay systems this will actually enlarge the application scope of the lambert w function which is becoming a standard library function for various commercial symbolic software packages to time delay systems

a numerical algorithm for stability testing of fractional delay systems this paper presents an effective numerical algorithm for testing the bibo stability of fractional delay systems described by fractional order delay differential equations it is based on using cauchy s integral theorem and solving an initial value problem the algorithm has a reliable result which is illustrated by several examples and hence is practically useful in the analysis and design of feedback control for both integer and fractional order systems having time delays

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

congruence classes of finite representations of simple lie superalgebra the concept of congruence of representations of lie algebras is generalized and applied to the finite dimensional representations of lie superalgebras

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

a note on time domain simulation of feedback fractional order systems the study of feedback fractional order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional order models and that fractional order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer order controllers owing to the lack of effective analytic methods for the time domain analysis and simulation of linear feedback fractional order systems we suggest in this paper two reliable and accurate numerical methods for inverting fractional order laplace transforms one is based on computing bromwich s integral with a numerical integration scheme capable of accuracy control and the other is based on expanding the time response function in a b spline series in order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the grunwald letniknov approximation method and podlubny s analytic formulas which are in a form of double infinite series the time domain simulations of the feedback control of a fractional order process with a pd sup spl mu controller and a fractional order band limited lead compensator are worked out the simulation results indicate that a convergence problem indeed occurs in using podlubny s infinite series expressions and that the problem could not be overcome by a series acceleration scheme

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

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

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

robust stability for fractional order systems with structured and unstructured uncertainties the issues of robust stability for uncertain fractional order systems of two types of order a 0 1 are dealt with in this paper for the polytope type uncertainty case a less conservative sufficient condition for robust stability is given for the norm bounded uncertainty case a sufficient and necessary condition for robust stability is presented both of these conditions can be checked by solving sets of linear matrix inequalities two numerical examples are presented to confirm the proposed conditions

a methodology for solving chemical equilibrium systems this paper describes a methodology for solving chemical equilibrium systems the methodology is especially appropriate when many systems with the same structure must be solved quickly as in finite difference models of fluid flow and combustion the approach features identifying a canonical form for such systems and exploiting this canonical form to effect a preliminary algebraic reduction then numerical scaling and iterative solution techniques are evoked both homotopy continuation and a variant of newton s method are effective solution methods this methodology is a significant advance over previous approaches to solving chemical equilibrium systems the problem of solving such systems has heretofore been regarded as challenging solution techniques have tended to emphasize case by case analyses and hit or miss numerical methods we show that the problem can be solved by a unified approach quickly and reliably