uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

learning on dynamic statistical manifolds hyperbolic balance laws with uncertain random parameters and inputs are ubiquitous in science and engineering quantification of uncertainty in predictions derived from such laws and reduction of predictive uncertainty via data assimilation remain an open challenge that is due to nonlinearity of governing equations whose solutions are highly non gaussian and often discontinuous to ameliorate these issues in a computationally efficient way we use the method of distributions which here takes the form of a deterministic equation for spatio temporal evolution of the cumulative distribution function cdf of the random system state as a means of forward uncertainty propagation uncertainty reduction is achieved by recasting the standard loss function i e discrepancy between observations and model predictions in distributional terms this step exploits the equivalence between minimization of the square error discrepancy and the kullback leibler divergence the loss function is regularized by adding a lagrangian constraint enforcing fulfilment of the cdf equation minimization is performed sequentially progressively updating the parameters of the cdf equation as more measurements are assimilated

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions this study proposes a framework for uncertainty analysis by incorporating explicit numerical solutions of governing equations for flood wave propagation with the expectation operator it aims at effectively evaluating the effect of variations in initial and boundary conditions on the estimation of flood waves spatiotemporal semi variogram models are employed to quantify the correlation of the variables in time and space the 1 d nonlinear kinematic wave equation for the overland flow named evo ns kwe is applied in the model development model validation is made by comparison with the monte carlo simulation mcs model in the calculation of statistical properties of model outputs i e flow depths that is the mean standard deviation and coefficient of variation cv the results from the model validation show that the evo ns kwe model can produce excellent approximations of the mean and less satisfactory approximations of the standard deviation and cv compared with those obtained using the mcs model it concludes that the uncertainties of flow depths in the domain are significantly affected by variations in the boundary condition future application of the evo ns kwe model enable the evaluation of uncertainty in model outputs induced by the initial and boundary condition subject to uncertainty and will also provide corresponding probabilistic information for risk quantification method

decay of hankel singular values of analytic control systems we show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a hankel operator that belongs to the schatten class sp for all positive p this implies that the hankel singular values converge to zero faster than any polynomial rate this in turn implies fast convergence of balanced truncations as a corollary decay rates for the eigenvalues of the controllability and observability gramians are also provided applications to the heat equation and a plate equation are given

exact pdf equations and closure approximations for advective reactive transport mathematical models of advection reaction phenomena rely on advective flow velocity and bio chemical reaction rates that are notoriously random by using functional integral methods we derive exact evolution equations for the probability density function pdf of the state variables of the advection reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions these pdf equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly correlated random field our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large eddy diffusivity led closure approximation 1 we find that the solution to the led based pdf equation which is exact for uncorrelated reaction rates is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

differintegral interpolation from a bandlimited signal s samples the whittaker shannon cardinal series dictates that any l 2 bandlimited signal is defined everywhere by its sufficiently closely spaced sample values this paper derives those interpolation functions necessary for direct evaluation of such a signal s derivatives integrals and fractional derivatives directly from the sample values generation and recursion formulas for these interpolation functions are presented

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

uncertainty quantification in kinematic wave models we develop a probabilistic approach to quantify parametric uncertainty in first order hyperbolic conservation laws kinematic wave equations the approach relies on the derivation of a deterministic equation for the cumulative density function cdf of a system state in which probabilistic descriptions probability density functions or pdfs of system parameters and or initial and boundary conditions serve as inputs in contrast to pdf equations which are often used in other contexts cdf equations allow for straightforward and unambiguous determination of boundary conditions with respect to sample variables the accuracy and robustness of solutions of the cdf equation for one such system the saint venant equations of river flows are investigated via comparison with monte carlo simulations

a frequency domain iterative identification algorithm using general orthonormal basis functions in this paper a two step frequency domain identification algorithm is proposed in which the identified model is parameterized in terms of general orthonormal basis functions furthermore an update scheme is presented where the poles of the basis functions are adjusted iteratively and the convergence properties of this scheme with respect to unknown but bounded noise are studied the algorithm is applied to experimental frequency response data obtained from a compact disk player

convergence and convergence rate of the balanced realization truncations for infinite dimensional discrete time systems abstract in this paper we study approximation via balanced realizations for a large class of infinite dimensional discrete time linear systems we give properties of the truncated system and prove that the approximation is l 2 convergent in the case of nuclear systems we prove convergence in nuclear norm and give an estimation of the l convergence rate

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

applications of fractional calculus in solving abel type integral equations surface volume reaction problem abstract in this paper we consider a class of partial integro differential equations of fractional order motivated by an equation which arises as a result of modeling surface volume reactions in optical biosensors we solve these equations by employing techniques from fractional calculus several examples are discussed furthermore for the first time we encounter an order of the fractional derivative other than 1 2 in an applied problem hence in this paper we explore the applicability of fractional calculus in real world applications further strengthening the true nature of fractional calculus

properties of the katugampola fractional derivative with potential application in quantum mechanics katugampola e print arxiv 1410 6535 recently introduced a limit based fractional derivative d referred to in this work as the katugampola fractional derivative that maintains many of the familiar properties of standard derivatives such as the product quotient and chain rules typically fractional derivatives are handled using an integral representation and as such are non local in character the current work starts with a key property of the katugampola fractional derivative d y t1 dydt and the associated differential operator d t1 d1 these operators their inverses commutators anti commutators and several important differential equations are studied the anti commutator serves as a basis for the development of a self adjoint operator which could potentially be useful in quantum mechanics a hamiltonian is constructed from this operator and applied to the particle in a box model

on galerkin methods for abel type integral equations galerkin methods are studied for the numerical solution of abel type integral equations in order to maintain the causality triangularity of the resulting system of linear equations spline functions of maximal defect are used i e c 1 piecewise polynomials asymptotically optimal error estimates in the supremum norm are proved both robustness and accuracy of the galerkin methods are compared with those of collocation methods some numerical experiments using some cheap quadrature schemes that work quite well under certain circumstances only are shown and some generally applicable quadrature methods are reported

solving fractional integral equations by the haar wavelet method haar wavelets for the solution of fractional integral equations are applied fractional volterra and fredholm integral equations are considered the proposed method also is used for analysing fractional harmonic vibrations the efficiency of the method is demonstrated by three numerical examples

diffusion process modeling by using fractional order models this paper deals with a concept and description of a rc network as an electro analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions it is based on well known analogy between heat and electrical conduction in the paper are compared analytical solution together with numerical solution and experimentally measured data for the first time a fractional order model of diffusion process and its modeling via lumped rc network has been used simple examples of simulations measurements and their comparison are shown

an l convergent method to reduce frequency weighted infinite dimensional discrete time systems abstract in this paper we present a method based on balanced realizations truncations to reduce frequency weighted infinite dimensional discrete time systems the method is shown to be l convergent for a subclass of the nuclear systems class as its numerical computation is not easy we propose a practical method to reduce frequency weighted infinite dimensional systems and prove that it is l convergent for the class of systems whose transfer function has its derivative in the wiener algebra

decay of singular values for infinite dimensional systems with gevrey regularity abstract we consider the decay rate of the singular values of the input map the output map and the hankel operator for a class of infinite dimensional systems this class is characterized by the control operator or the observation operator having a smoothing effect we capture this in the definition of a gevrey operator which generalizes the known concept of a gevrey vector in applications to pdes this abstract assumption on the control operator is typically satisfied when the input is multiplied by a function which is a compactly supported gevrey function in the spatial variable using the theory of polynomial approximation in particular truncated chebyshev expansions we obtain that the singular values decay exponentially in a root of the approximation dimension the power of the root depends on the order of the gevrey operator and on whether the underlying semigroup is nilpotent exponentially stable or polynomially stable

mellin transforms of generalized fractional integrals and derivatives we obtain the mellin transforms of the generalized fractional integrals and derivatives that generalize the riemann liouville and the hadamard fractional integrals and derivatives we also obtain interesting results which combine generalized r m operators with generalized stirling numbers and lah numbers for example we show that 1 1 corresponds to the stirling numbers of the 2nd kind and 2 1 corresponds to the unsigned lah numbers further we show that the two operators r m and m r r m n generate the same sequence given by the recurrence relation s n k i 0 r m m r n 2 k i 1 r i r i s n 1 k i 0 k n with s 0 0 1 and s n 0 s n k 0 for n 0 and 1 min r m n 1 k or k 0 finally we define a new class of sequences for r 1 3 1 4 1 5 1 6 and in turn show that 1 2 1 corresponds to the generalized laguerre polynomials

nuclearity of hankel operators for ultradifferentiable control systems nuclearity of the hankel operator is a known sufficient condition for convergence of lyapunov balanced truncations we show how a previous result on nuclearity of hankel operators of systems with an analytic semigroup can be extended to systems with a semigroup of class dp with p 1 the case p 1 being the analytic case for semigroups that are generated by a dunford schwartz spectral operator we prove that being of class dp is equivalent to being gevrey ultradifferentiable of order p we illustrate how for certain partial differential equations our results lead to an easy way of showing nuclearity of the hankel operator for a wide range of control and observation operators by considering several examples of damped beams

on the eigenvalue decay of solutions to operator lyapunov equations this paper is concerned with the eigenvalue decay of the solution to operator lyapunov equations with right hand sides of finite rank we show that the kth generalized eigenvalue decays exponentially in root k provided that the involved operator a generates an exponentially stable analytic semigroup and a is either self adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis numerical experiments with discretizations of 1d and 2d pde control problems confirm this decay c 2014 elsevier b v all rights reserved

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

a construction of rational wavelets and frames in hardy sobolev spaces with applications to system modeling using the daubechies wavelet theory we establish rational wavelet decompositions of the hardy sobolev classes on the half plane the decay of wavelet coefficients is analyzed and error bounds for approximation are given we give applications to the modeling of linear systems and to the model reduction of infinite dimensional systems

neutral fuzzy fractional functional differential equations abstract in this paper a class of fuzzy fractional functional differential equations of neutral type is investigated the existence uniqueness and iterative formula in the case of linear equations are obtained meanwhile the continuous dependence of solutions on initial data parameters and functions involved in right hand side of equation is also discussed several examples are given to illustrate the applications of our results

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

subspace based identification of infinite dimensional multivariable systems from frequency response data a new identification algorithm which identifies low complexity models of infinite dimensional systems from equidistant frequency response data is presented the new algorithm is a combination of the fourier transform technique with the recent subspace techniques given noisefree data finite dimensional systems are exactly retrieved by the algorithm when noise is present it is shown that identified models strongly converge to the balanced truncation of the identified system if the measurement errors are covariance bounded several conditions are derived on consistency illustrating the trade offs in the selection of certain parameters of the algorithm two examples are presented which clearly illustrate the good performance of the algorithm

lqg balancing for continuous time infinite dimensional systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for continuous time infinite dimensional systems lqg balanced realizations are those for which the optimal cost operator for the system and its dual system are equal and diagonal the class of systems we consider is that of distributional resolvent linear systems which includes well posed linear systems as a subclass we prove the existence of lqg balanced realizations under a finite cost condition for both the system and its dual system we also show that an lqg balanced realization of a well posed transfer function is well posed we further show that approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the optimal cost operator of a system and its dual system is independent of the particular realization our method of proof shows the connections with coprime factorizations lyapunov balanced realizations and discrete time systems the main reason for studying lqg balanced realizations is that truncated lqg balanced realizations provide a good approximation of the original system we show that under certain conditions this is also true in the infinite dimensional case by proving an error bound in the gap metric

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

continuity of the spectral factorization on a vertical strip the continuity of the mapping which associates a spectral factor to a spectral density is investigated this mapping can be defined on several classes of spectral densities and spectral factors for the usual largest class of spectral densities i e essential bounded functions on the imaginary axis that are bounded away from zero it is known that this mapping is not continuous it is shown here that for slightly smaller but still generic class the mapping becomes continuous

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

the hilbert schmidt property of feedback operators in this article we prove new sufficient conditions under which the feedback operator associated with the linear quadratic regulator control design for distributed parameter systems is nuclear or hilbert schmidt examples illustrating the main results are also given

hermite hadamard and hermite hadamard fejer type inequalities for generalized fractional integrals abstract in this paper we obtain the hermite hadamard and hermite hadamard fejer type inequalities for fractional integrals which generalize the two familiar fractional integrals namely the riemann liouville and the hadamard fractional integrals into a single form we prove that in most cases we obtain the riemann liouville and the hadamard equivalence just by taking limits when a parameter 1 and 0 respectively

approximation and identification of continuous time systems an approximation approach to the identification of stable continuous time possibly infinite dimensional systems is studied for this purpose concrete exponential leg endre series methods are proposed and analysed here for the l1 and l2 approximations of continuous time systems rate of approximation results are given for the proposed methods for an important class of delay systems the proposed exponential legendre series method for the l2 approximation has a certain optimal rate of approximation property for this class of delay systems these results show that exponential legendre series techniques have promising properties in connection with the l1 and l2 identifications of stable continuous time systems the proposed l1 identification method has a direct application to the robust design of control systems

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions abstract in this paper the main objective is to describe an approximate scheme for solving nonlinear quadratic integral equations nqies of fractional order the method is based on piecewise linear functions that are called hat functions hfs by applying the hfs approximation their properties and operational matrices nonlinear equation becomes to a nonlinear system of equations which can be solved by using simple arithmetic methods also the error analysis of hfs method and convergence analysis of proposed model are given at the end two nonlinear examples are presented to illustrate the validity and applicability of the explained approach

a note on a system theoretic approach to a conjecture by peller khrushchev based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented given a monotonically decreasing sequence of positive numbers n n 1 does there exist a hankel operator whose sequence of singular values is n n 1

riccati equations and normalized coprime factorizations for strongly stabilizable infinite dimensional systems the first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic riccati equation for a class of infinite dimensional systems of the form a b s 1 2b d where a is dissipative and all the other operators are bounded these systems are not exponentially stabilizable and so the standard theory is not applicable the second part uses the riccati equation results to give formulas for normalized coprime factorizations over h for positive real transfer functions of the form d s 1 2b author a 1 b

numerical solution based on two dimensional orthonormal bernstein polynomials for solving some classes of two dimensional nonlinear integral equations of fractional order abstract in this paper we develop a numerical scheme based on two dimensional orthonormal bernstein polynomials 2d obps to solve two dimensional nonlinear integral equations of fractional order the fractional integral considered here is in the riemann liouville sense by using definition of riemann liouville fractional integral two dimensional nonlinear fractional integral equations is transformed into two dimensional nonlinear ordinary integral equations operational matrices method based on 2d obps are applied to obtain an approximate solution with high accuracy for these equations in addition error analysis of the proposed method is discussed and an upper error bound is provided under weak assumptions some linear and nonlinear examples are given to demonstrate the accuracy efficiency and speed of the suggested method

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

fractional differential equations of caputo katugampola type and numerical solutions abstract this paper is concerned with a numerical method for solving generalized fractional differential equation of caputo katugampola derivative a corresponding discretization technique is proposed numerical solutions are obtained and convergence of numerical formulae is discussed the convergence speed arrives at o t 1 numerical examples are given to test the accuracy

l approximation and nuclearity of delay systems abstract the l convergence of partial fraction decompositions of single input single output delay systems is studied conditions for the nuclearity of such systems are derived and examples given

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

what is a fractional derivative this paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals two criteria for required by a fractional operator are formulated the grunwald letnikov riemann liouville and caputo fractional derivatives and the riesz potential are accessed in the light of the proposed criteria a leibniz rule is also obtained for the riesz potential

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

correction to what is a fractional derivative by ortigueira and machado journal of computational physics volume 293 15 july 2015 pages 4 13 special issue on fractional pdes there is a debate among contemporary mathematicians about what it really means by a fractional derivative the question arose as a consequence of introducing a new definition of a fractional derivative in 1 in a reply ortigueira and machado 2 came up with several very important criteria to determine whether a given derivative is a fractional derivative according to their criterion the new fractional derivative called conformable fractional derivative introduced by khalil et al 1 turns out not to be a fractional derivative but rather a controlled or conformable derivative in proving the claim the authors in 2 use an example 2 p 6 it turns out that the explanation given there needs some corrections and it is the sole purpose of this note

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

fractional integral associated to generalized cookie cutter set and its physical interpretation this paper is based on nigmatullin s study when the residual memory set is a generalized cookie cutter set on using various hypotheses it is proved that the fractional exponent of a fractional integral is not uniquely determined by the fractal dimension of the generalized cookie cutter set it is determined by of self similar measure or infinite self similar measure on this generalized cookie cutter set and can run over all positive real numbers

hankel operators that commute with second order differential operators abstract suppose that is a continuous and self adjoint hankel operator on l 2 0 with kernel x y and that l f d d x a x d f d x b x f x with a 0 0 if a and b are both quadratic hyperbolic or trigonometric functions and satisfies a suitable form of gauss s hypergeometric differential equation or the confluent hypergeometric equation then l l the paper catalogues the commuting pairs and l including important cases in random matrix theory there are also results proving rapid decay of the singular numbers of hankel integral operators with kernels that are analytic and of exponential decay in the right half plane

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

new approach to a generalized fractional integral abstract the paper presents a new fractional integration which generalizes the riemann liouville and hadamard fractional integrals into a single form conditions are given for such a fractional integration operator to be bounded in an extended lebesgue measurable space semigroup property for the above operator is also proved we give a general definition of the fractional derivatives and give some examples

compactness and nuclearity of the hankel operator and internal stability of infinite dimensional state linear systems we examine the relationships between the exponential or strong stability of certain classes of regular linear systems and the compactness and nuclearity properties of the hankel operator new sufficient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup

the suboptimal nehari problem for well posed linear systems we solve the suboptimal nehari problem for a transfer function that has a state space realization as a system stable input output and input output stable well posed linear system we obtain an explicit solution in terms of the state space parameters