high order methods for state dependent delay differential equations with nonsmooth solutions this work presents a theoretical basis for high order numerical methods to solve state dependent delay differential equations of the form begin gathered dot x t f t x t x alpha t x t quad text for t in a b hfill alpha t x t leqq t hfill x t phi t quad text for t in bar a a hfill end gathered where bar a min alpha t x t for t in a b the solutions to such equations typically have derivative jump discontinuities jump points which propagate from the initial jump point t a thus high order methods require an accurate determination of the location of jump discontinuities in lower order derivatives of the solution x t the locations of these jump points are characterized as the zeros of certain nonlinear equations which themselves depend on x t because of this interdependence the following nontrivial question arises can these unknown jump points be determined accurately enough to develop high order methods this question

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

numerical solution of neutral functional differential equations by adams methods in divided difference form a variable step and variable order algorithm for the numerical solution of neutral functional differential equations is described this general purpose algorithm is based on the variable step formulation of the adams methods represented in divided difference form the adams bashforth and adams moulton methods are implemented in predictor corrector mode the detection of derivative discontinuities relies on the estimates of the local discretization errors the restarting of the integration at each discontinuity point relies on the step size and order changing strategy based on the estimates of the local discretization errors this algorithm reduces the overhead cost by implementing the adams methods in divided difference form this algorithm increases the reliability and efficiency by obtaining asymptotically correct estimates of the local discretization errors for the lower adjacent orders without any extra function evaluation by using local extrapolation the results from three test examples are presented

a highly stable deferred correction scheme with interpolant for systems of nonlinear two point boundary value problems iterated deferred correction is a very popular approach to the numerical solution of general first order systems of nonlinear two point boundary value problems typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived this marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates in the present paper we show that richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation guided by this analysis we are able to derive for the first time a lobatto deferred correction code which is very efficient for the solution of stiff problems particularly in cases where the mirk code twpbvp f is unstable furthermore we are able to derive efficient interpolants for this lobatto code which are also applicable to mirk formulae and we consider the problem of estimating the error in these interpolants this completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems this new lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software

stability analysis for linear discretisations of the advection equation with runge kutta time integration for the 1 dim linear advection problem stability limits of runge kutta rk methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented the analysis can be carried out in a rather general way by introduction of a broad class of runge kutta methods here called linear case runge kutta lc rk methods which behave completely similar for linear time independent and homogeneous ode systems and contain the classical order stage rk methods the set of conditions for the coefficients of these lc rk schemes could be derived explicitly for arbitrary order n from an efficiency viewpoint the lc rk 3rd order methods in combination with upwind 3rd or 5th order or the lc rk 4th order scheme with 4th order centered difference advection are a good choice the analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented

stepsize control and continuity consistency for state dependent delay differential equations abstract derivative discontinuities occur frequently in the solutions of delay differential equations even when the functions defining them are all c unless correctly treated such discontinuities can undermine the continuity assumptions made by ode software in this paper we discuss some specific difficulties arising from this aspect in the treatment of state dependent problems a necessary property continuity consistency is introduced along with a class of methods for which it is satisfied

computer solutions of state dependent delay differential equations abstract an adaptation of the runge kutta verner 5 6 formula pair is used to construct a numerical method for the solution of state dependent delay differential equations with nonvanishing lag a fifth degree divided difference newton backward interpolation polynomial is used to find the location of the derivative jump discontinuities of the solution in order to maintain the sixth order accuracy of the rkv pair the value of the solution at the delay is approximated by a three point hermite polynomial this new method is tested on some real life problems a fortran program called sysdel is available from the authors

a sufficient condition for gpn stability for delay differential equations in this paper the author considers a linear test delay differential equation with non constant coefficients related to the definitions of pn and gpn stability for numerical methods he defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for gpn stability

time point relaxation runge kutta methods for ordinary differential equations bellen a z jackiewicz and m zennaro time point relaxation runge kutta methods for ordinary differential equations journal of computational and applied mathematics 45 1993 121 137 we investigate convergence order and stability properties of time point relaxation runge kutta methods for systems of ordinary differential equations these methods can be implemented in gauss jacobi or gaussseidel modes denoted by trgjrk k or trgsrk k respectively where k stands for the number of picard lindelof iterations as k f 0 these modes tend to the same one step method for ordinary differential equations called diagonal split runge kutta dsrk method it is proved that these methods are convergent with order min k p where p is the order of the underlying runge kutta method recurrence relations resulting from application of trgjrk k trgsrk k and dsrk methods to the two dimensional test system u au pu u pu au t 3 0 where a and p are real parameters are derived and stability regions in the h p plane are plotted for some methods using a variant of the boundary locus method in most cases stability regions increase as the number of picard lindeldf iterations k becomes larger

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

chattering avoidance by second order sliding mode control relying on the possibility of generating a second order sliding motion by using as control the first derivative of the control signal instead of the actual control a new solution to the problem of chattering elimination in variable structure control vsc is presented such a solution inspired by the classical bang bang optimal control strategy is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second order uncertain system with unavailable velocity then the applicability of the algorithm is extended via suitable modifications to the case of nonlinear systems with uncertainties of more general types the proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold

chattering reduction and error convergence in the sliding mode control of a class of nonlinear systems to reduce chattering in sliding mode control a boundary layer around the switching surface is used and a continuous control is applied within the boundary the effects of various control laws within the boundary layer on chattering and error convergence in different systems are studied new functions for chattering reduction and error convergence inside the boundary layer are proposed which are discontinuous in magnitude but not in sign the internal model principle has been used to generalize the design for the class of nonlinear systems being considered

dynamic output feedback h control synthesis for discrete time t s fuzzy systems via switching fuzzy controllers this paper studies the problem of dynamic output feedback h control of discrete time takagi sugeno t s fuzzy systems via a switched dynamic parallel distributed compensation dpdc scheme the considered switched dpdc control structure is determined based on the values of membership functions some new properties of membership functions are obtained and exploited to derive sufficient conditions for designing switching dpdc controllers that guarantee the stability and h performances of closed loop nonlinear systems the design conditions are given in terms of solvability of a set of linear matrix inequalities lmis it is shown that the new approach provides results at least as good as the corresponding dpdc methods a numerical example is given to illustrate the effectiveness of the proposed approach

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

high accuracy large step explicit runge kutta hale rk schemes for computational aeroacoustics in many realistic calculations the computational grid spacing required to resolve the mean flow gradients is much smaller than the grid spacing required to resolve the unsteady propagating waves of interest because of this the high temporal resolution provided by existing optimized time marching schemes can be excessive due to the small time step required for stability in regions of clustered grid in this work explicit fourth order accurate runge kutta time marching schemes are optimized to increase the inviscid stability limit rather than the accuracy at large time steps single and multiple step optimized schemes are developed and analyzed the resulting schemes are validated on several realistic benchmark problems

output feedback model reference sliding mode control of uncertain multivariable systems this note considers the robust output tracking problem using a model reference sliding mode controller for linear multivariable systems of relative degree one it is shown that the closed loop system is globally exponentially stable and the performance is insensitive to bounded input disturbances and parameter uncertainties the strategy is based on output feedback unit vector control to generate sliding mode the only required a priori information about the plant high frequency gain matrix k sub p is the knowledge of a matrix s sub p such that k sub p s sub p is hurwitz which relaxes the positive definiteness requirement usually needed by other methods

comparison of monolithic and splitting solution schemes for dynamic porous media problems 5 summary proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents we compare the monolithic and splitting solution of the different multi field formulations feasible in porous media dynamics because of the inherent solid fluid momentum 9 interactions one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint here the 11 resulting set of differential algebraic equations dae is solved by the finite element method fem following two different strategies 1 an implicit monolithic approach where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators 2 a semi explicit implicit splitting scheme in the sense of a fractional step method 15 where the dae are first discretized in time split using intermediate variables and then discretized in space using linear equal order approximations for all primary unknowns finally a one and a two dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies therefore several test cases differing in the used multi field formulation the 19 monolithic time stepping method and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter in the end we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem copyright q 2009 john wiley sons ltd 23

fuzzy adaptive output feedback control for mimo nonlinear systems in this paper two observer based adaptive fuzzy output feedback control schemes are presented for a class of uncertain continuous time multi input multi output mimo nonlinear dynamics systems whose states are not available within these schemes fuzzy logic systems are employed to approximate the plant s unknown nonlinear functions and then the state observer is designed for estimating the states of the plant upon which a fuzzy adaptive output feedback controller is firstly investigated in order to overcome the controller singularity problem and relax the requirement of bounding parameter values a second modified fuzzy adaptive output feedback controller is proposed by using a regularized inverse and a robustifying control term all parameter adaptive laws and robustifying control terms are derived based on lyapunov stability analysis so that convergence to zero of tracking errors and boundedness of all signals in the closed loop system can be guaranteed simulations performed on a two link robot manipulator illustrate the approach and exhibit its performance

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

a three stage vsvo hermite birkhoff taylor ode solver the new variable step variable order ode solver hbt p of order p presented in this paper combines a three stage runge kutta method of order 3 with a taylor series method of order p 2 to solve initial value problems y f t y y t 0 y 0 where y r r d and f rxr d r d the order conditions satisfied by hbt p are formulated and they lead to vandermonde type linear algebraic systems whose solutions are the coefficients in the formulae for hbt p a detailed formulation of variable step hbt p in both fixed order and variable order modes is presented the new method and the taylor series method have similar regions of absolute stability to obtain high accuracy results at high order this method has been implemented in multiple precision

on multi input chattering free second order sliding mode control a solution to the problem of eliminating the chattering effect which is always associated with practical implementations of variable structure control is presented with reference to a class of uncertain multi input nonlinear systems the solution procedure relies on the application of an original control approach capable of enforcing a second order sliding mode i e a sliding regime on a surface s x t 0 in the system state space with s spl dot x t identically equal to zero a regime enforced by a control signal depending on s x t but directly acting only on s spl uml x t such an approach in its original formulation only applies to single input nonlinear systems with particular types of uncertainties in the present paper its validity is extended to multi input nonlinear systems characterized by uncertainties of more general nature covering a wide class of real processes

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

static output feedback controller design for fuzzy systems an ilmi approach this paper examines the problem of static output feedback control of a takagi sugeno ts fuzzy system the existence of a static output feedback control law is given in terms of the solvability of bilinear matrix inequalities an iterative algorithm based on the linear matrix inequality is proposed to compute the static output feedback gain to reduce the conservatism of the design the structural information of the membership function of the fuzzy rules is incorporated numerical examples are used to illustrate the validity of the methods

parallel iteration across the steps of high order runge kutta methods for nonstiff initial value problems abstract for the parallel integration of nonstiff initial value problems ivps three main approaches can be distinguished approaches based on parallelism across the problem on parallelism across the method and on parallelism across the steps the first type of parallelism does not require special integration methods and can be exploited within any available ivp solver the method parallelism approach received much attention particularly within the class of explicit runge kutta methods originating from fixed point iteration of implicit runge kutta methods of gaussian type the construction and implementation on a parallel machine of such methods is extremely simple since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors this type of parallelism is most suitable for shared memory systems the required number of processors is roughly half the order of the generating runge kutta method and the speed up with respect to a good sequential ivp solver is about a factor 2 the third type of parallelism step parallelism can be achieved in any ivp solver based on predictor corrector iteration and requires the processors to communicate after each full iteration if the iterations have sufficient computational volume then the step parallel approach may be suitable for implementation on distributed memory systems most step parallel methods proposed so far employ a large number of processors but lack the property of robustness due to a poor convergence behaviour in the iteration process hence the effective speed up is rather poor the dynamic step parallel iteration process proposed in the present paper is less massively parallel but turns out to be sufficiently robust to achieve speed up factors up to 15

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

reoptimization of steiner trees given an instance of the steiner tree problem together with an optimal solution we consider the scenario where this instance is modified locally by adding one of the vertices to the terminal set or removing one vertex from it in this paper we investigate the problem of whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance our results are as follows i we prove that these reoptimization variants of the steiner tree problem are np hard even if edge costs are restricted to values from 1 2 ii we design 1 5 approximation algorithms for both variants of local modifications this is an improvement over the currently best known approximation algorithm for the classical steiner tree problem which achieves an approximation ratio of 1 ln 3 2 1 55 iii we present a ptas for the subproblem in which the edge costs are natural numbers 1 k for some constant k

on the parameterized complexity of dynamic problems in a dynamic version of a base problem x it is assumed that some solution to an instance of x is no longer feasible due to changes made to the original instance and it is required that a new feasible solution be obtained from what remained from the original solution at a minimal cost in the parameterized version of such a problem the changes made to an instance are bounded by an edit parameter while the cost of reconstructing a solution is bounded by some increment parameter capitalizing on the recent initial work of downey et al on the dynamic dominating set problem we launch a study of the dynamic versions of a number of problems including vertex cover maximum clique connected vertex cover and connected dominating set in particular we show that dynamic vertex cover is w 1 hard and the connected versions of both dynamic vertex cover and dynamic dominating set become fixed parameter tractable with respect to the edit parameter while they remain w 2 hard with respect to the increment parameter moreover we show that dynamic independent dominating set is w 2 hard with respect to the edit parameter we introduce the reoptimization parameter which bounds the difference between the cardinalities of initial and target solutions we prove that while dynamic maximum clique is fixed parameter tractable with respect to the edit parameter it becomes w 1 hard if the increment parameter is replaced with the reoptimization parameter finally we establish that dynamic dominating set becomes w 2 hard when the target solution is required not to be larger than the initial one even if the edit parameter is exactly one

numerical treatment of delay differential equations by hermite interpolation a class of numerical methods for the treatment of delay differential equations is developed these methods are based on the wellknown runge kutta fehlberg methods the retarded argument is approximated by an appropriate multipoint hermite interpolation the inherent jump discontinuities in the various derivatives of the solution are considered automatically problems with piecewise continuous right hand side and initial function are treated too real life problems are used for the numerical test and a comparison with other methods published in literature

output feedback control of uncertain nonlinear systems using a self structuring adaptive fuzzy observer this paper describes the design of an output feedback controller based on an adaptive fuzzy observer for uncertain single input single output nonlinear dynamical systems we have focused on the realization of a minimal dynamic order for the adaptive fuzzy observer for this purpose we adopt an adaptive fuzzy observer in which no strictly positive real spr condition is needed and we combine a self structuring scheme with an on line estimation of fuzzy parameters by using this proposed scheme we can reduce the dynamic order of the adaptive output feedback fuzzy control system the lyapunov synthesis approach is used to guarantee a global uniform ultimate boundedness property of the state estimation error and tracking error as well as all other signals in the closed loop system no a priori knowledge of an upper limits on the uncertainties including optimal parameters and modeling errors is required the theoretical results are illustrated through simulation examples

linear multistep methods for functional differential equations a new way to define linear multistep methods for functional differential equations is presented and some of their properties are analyzed the asymptotic behavior of the global discretization error is investigated finally milne s device is generalized to functional differen tial equations the effect of the nonsmoothness of the exact solution is taken into account 1 introduction consider the functional differential equation fde 1 x t f t xt to t t xto o where x t e ri 0 e c t 0 rn fe c to t x c t 0 rn ri and xt 9 x t 9 for 0 e t 0 t e to t it is well known that the solution x of 1 is usually not smooth that is even if f and 4 are c functions x may have jump discontinuities in its derivatives the occurrence of these jump discontinuities may lead to order breakdown for numerical methods if no special provisions are made it seems that all available techniques require information about the location of the jumps if the delay is not state dependent then these locations may be known a priori if not then they may be calculated numerically see feldstein and neves 6 when the location of a jump discontinuity is known two different techniques are available i make the location of the jump a grid point and restart in practice this

reoptimization of maximum weight induced hereditary subgraph problems the reoptimization issue studied in this paper can be described as follows given an instance i of some problem p an optimal solution opt for p in i and an instance i resulting from a local modification of i that consists of insertions or removals of a small number of data we wish to use opt in order to solve p in i the aim is to achieve either a better approximation ratio or a better running time or both than guaranteed by ex nihilo computation we use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems the main problems studied are max independent set maxk colorable subgraph and max split subgraph under vertex insertions and deletions

stability of generalized runge kutta methods for stiff kinetics coupled differential equations a stability and efficiency improved class of generalized runge kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and or nonlinear coupled differential equations the determination of the coefficients required by the method is precisely obtained from the so called equations of condition which in turn are derived by an approach based on butcher series since the equations of condition are fewer in number free parameters can be chosen for optimizing any desired feature of the process a further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced in addition the a stability properties of the method are investigated implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

estimating the error of the classic runge kutta formula abstract a high quality error estimator is presented for the classic four stage fourth order runge kutta formula that is free when using this estimator the solution can be approximated accurately throughout the step at no additional cost

brief variable structure output feedback control design for a class of uncertain dynamic systems in this paper we consider the problem of designing a variable structure output feedback control vsofc law for a class of uncertain systems with mismatched uncertainty in the state matrix we derive existence conditions of linear sliding surfaces in terms of constrained linear matrix inequalities lmis and we give vsofc strategies we also give a simple design procedure and parameterizations of the linear switching surfaces together with solvability conditions for the given constrained lmis finally we give a design example in order to show the effectiveness of our method

robust h fuzzy static output feedback control of t s fuzzy systems with parametric uncertainties this paper addresses robust h fuzzy static output feedback control problem for t s fuzzy systems with time varying norm bounded uncertainties sufficient conditions for synthesis of a fuzzy static output feedback controller for t s fuzzy systems are derived in terms of a set of linear matrix inequalities lmis in comparison with the existing literatures the proposed approach not only simplifies the design procedure but also achieves a better h performance three drawbacks existing in the previous papers such as coordinate transformation same output matrices and bmi problem have been eliminated the effectiveness of the proposed design method is demonstrated by an example for the control of a truck trailer system

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

cluster expansion for abstract polymer models new bounds from an old approach we revisit the classical approach to cluster expansions based on tree graphs and establish a new convergence condition that improves those by koteck preiss and dobrushin as we show in some examples the two ingredients of our approach are i a careful consideration of the penrose identity for truncated functions and ii the use of iterated transformations to bound tree graph expansions

output feedback stabilization of uncertain fuzzy systems using variable structure system approach based on the variable structure system theory an output feedback fuzzy control design method is developed for a class of uncertain nonlinear systems that can be represented by a t s takagi sugeno fuzzy model the uncertain fuzzy systems under consideration may have mismatched uncertainties in the state matrix as the local controller a compensator based sliding mode output feedback controller with a nonlinear discontinuous term is used in terms of linear matrix inequalities lmis a sufficient condition is derived for the existence of the fuzzy controller guaranteeing a stable sliding motion it is shown that the compensator gain and the switching surface parameter matrix can be characterized in terms of the solution of the lmi existence condition finally an lmi based design algorithm and numerical design examples are given

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

a one step 7 stage hermite birkhoff taylor ode solver of order 11 a one step 7 stage hermite birkhoff taylor method of order 11 denoted by hbt 11 7 is constructed for solving nonstiff first order initial value problems y f t y y t 0 y 0 the method adds the derivatives y to y 6 used in taylor methods to a 7 stage runge kutta method of order 6 forcing an expansion of the numerical solution to agree with a taylor expansion of the true solution to order 11 leads to taylor and runge kutta type order conditions these conditions are reorganized into vandermonde type linear systems whose solutions are the coefficients of the method the new method has a larger scaled interval of absolute stability than the dormand prince dp87 and a larger unscaled interval of absolute stability than the taylor method t11 of order 11 hbt 11 7 is superior to dp87 and t11 in solving several problems often used to test higher order ode solvers on the basis of the number of steps cpu time and maximum global error numerical results show the benefit of adding high order derivatives to runge kutta methods

designing efficient software for solving delay differential equations in this paper the efficient implementation of numerical software for solving delay differential equations is addressed several strategies that have been developed over the past 25 years for improving the efficiency of delay differential equation solvers are described of particular interest is a new method of automatically constructing the network dependency graph used in tracking derivative discontinuities

observer based fuzzy adaptive control for strict feedback nonlinear systems in this paper a new fuzzy adaptive control approach is developed for a class of siso strict feedback nonlinear systems with unmeasured states using fuzzy logic systems to approximate the unknown nonlinear functions a fuzzy adaptive observer is introduced for state estimation as well as system identification under the framework of the backstepping design fuzzy adaptive output feedback control is constructed recursively it is proven that the proposed fuzzy adaptive control approach guarantees the semi global boundedness property for all the signals and the tracking error to a small neighborhood of the origin simulation studies are included to illustrate the effectiveness of the proposed approach

state observation of nonlinear uncertain dynamical systems this note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties the design of these observers utilizes techniques related to variable structure systems theory a measure for the rate at which the estimates converge to the actual states is derived

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

fuzzy observer based fuzzy control design for nonlinear systems with persistent bounded disturbances to date nonlinear l gain control problems have not been solved by the conventional control methods for the nonlinear dynamic systems with persistent bounded disturbances this study introduces a fuzzy control design to deal with the nonlinear l gain output feedback control problem first the takagi and sugeno t s fuzzy model is employed to approximate the nonlinear dynamic system next based on the fuzzy model a fuzzy observer based fuzzy controller is developed to minimize the upper bound of l gain of the closed loop system under some linear matrix inequality lmi constraints therefore the nonlinear l gain control problem is transformed into a suboptimal control problem i e to minimize the upper bound of the l gain of the closed loop system subject to some lmi constraints in this situation the nonlinear l gain output feedback control problem can be easily solved by a lmi based optimization method the proposed methods which efficiently attenuate the peak of perturbation due to persistent bounded disturbances extend the l gain control problems from linear dynamic systems to nonlinear dynamic systems

partitioned krylov subspace iteration in implicit runge kutta methods abstract application of the method of lines to partial differential equation leads to very large sparse systems of ordinary differential equations which are usually stiff we consider high order implicit runge kutta methods for the time integration of these systems and propose a partitioned krylov method for the solution of the resulting linear systems it is shown that this method allows for parallel computation in the construction of krylov subspaces and in a few examples it converges much faster than gmres

stabilization of uncertain fuzzy time delay systems via variable structure control approach in view of a recent new application of variable structure control vsc to the stabilization problem for takagi sugeno t s fuzzy models this paper aims to study the stabilization of uncertain fuzzy time delay systems in t s fuzzy model via vsc approach there are mainly two features in this paper one lies in the incorporation of time delays both smooth and nonsmooth delays in which case lyapunov functionals and razumikhin theorem are required to solve the stabilization problem the other feature is that not only matched uncertainties but also mismatched uncertainties in the state variables are considered as a sequence the contribution of this paper consists of various control schemes proposed for the vsc design and the present results are in terms of linear matrix inequalities lmis an illustrative example is given to show the effectiveness of our various results

parallel iterated runge kutta methods for stiff ordinary differential equations abstract for the numerical integration of a stiff ordinary differential equation fully implicit runge kutta methods offer nice properties like a high classical order and high stage order as well as an excellent stability behaviour however such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task this paper discusses an iteration scheme to tackle this problem by means of a suitable choice of the iteration parameters the implicit relations for the stage values as they occur in each iteration can be uncoupled so that they can be solved in parallel the resulting scheme can be cast into the class of diagonally implicit runge kutta dirk methods and similar to these methods requires only one lu factorization per step per processor the stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed we discuss several choices to obtain good stability and fast convergence based on these approaches we wrote two codes possessing local error control and stepsize variation we have implemented both codes on an alliant fx 4 machine four parallel vector processors and shared memory and measured their speedup factors for a number of test problems furthermore the performance of these codes is compared with the performance of the best stiff ode codes for sequential computers like simple lsode and radau5

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

reoptimization in machine scheduling this paper studies reoptimization versions of various min sum scheduling problems the reoptimization setting can generally be described as follows given an instance of the problem for which an optimal solution is provided and given some local perturbations on that instance we search for a near optimal solution for the modified instance requiring very little computation we focus on two kinds of modifications job insertions and job deletions for all reoptimization problems considered we show how very simple reoptimization algorithms can ensure constant approximation ratios and also provide some lower bounds for whole classes of reoptimization algorithms

a general strategy for the optimization of runge kutta schemes for wave propagation phenomena we analyze optimized explicit runge kutta schemes rk for computational aeroacoustics and wave propagation phenomena in general exploiting the analysis developed in s pirozzoli performance analysis and optimization of finite difference schemes for wave propagation problems j comput phys 222 2007 809 831 we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics and provide a general strategy to design runge kutta methods tailored for specific applications we present families of optimized second and third order runge kutta schemes with up to seven stages and describe their implementation in the framework of williamson s 2n storage formulation j h williamson low storage runge kutta schemes j comput phys 35 1980 48 56 numerical simulations of the 1d linear advection equation and of the 2d linearized euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes

stepsize control for delay differential equations using continuously imbedded runge kutta methods of sarafyan the use of continuously imbedded runge kutta sarafyan methods for the solution of ordinary differential equations with either time dependent or state dependent delays is discussed it is shown how to get reliable solutions for such problems in a manner that does not require that the effect of the local approximation error be considered separately from the local integration error it is also shown how to reliably handle derivative discontinuities that arise in the solution of differential equations with delays

how to design a fuzzy adaptive controller based on observers for uncertain affine nonlinear systems this paper focuses on the construction of a fuzzy adaptive output feedback control based on any observer high gain hg observer sliding mode like observer etc for a class of single input single output siso uncertain or ill defined affine nonlinear systems indeed the corrective term of the proposed observer involves a well defined design function which is shown to be satisfied by the commonly used hg based observers namely for the usual hg observers and the sliding mode observers together with their implementable versions the design of the underlying update law as well as the robust control term is based on an appropriate filtering of the output tracking error this particularly allows to overcome the output observation error filtering or the necessity of the famous strictly positive real spr condition

output feedback lmi tracking control conditions with h criterion for uncertain and disturbed t s models this work concerns the tracking problem of uncertain takagi sugeno t s continuous fuzzy model with external disturbances the objective is to get a model reference based output feedback tracking control law the control scheme is based on a pdc structure a fuzzy observer and a h performance to attenuate the external disturbances the stability of the whole closed loop model is investigated using the well known quadratic lyapunov function the key point of the proposed approaches is to achieve conditions under a lmi linear matrix inequalities formulation in the case of an uncertain and disturbed t s fuzzy model this formulation facilitates obtaining solutions through interior point optimization methods for some nonlinear output tracking control problems finally a simulation is provided on the well known inverted pendulum testbed to show the efficiency of the proposed approach

on variable structure output feedback controllers in a recent work by zak hui 1993 a sliding mode controller for mimo systems using static output feedback was proposed very nice geometric conditions for how to design sliding surfaces were given however there are two restrictive assumptions in it one is that the uncertainties in the system must be bounded by a known function of outputs which excludes some possible uncertainties in the a matrix if the system is described by the triple a b c the other one requires a matrix equality to be held which may also be very difficult to satisfy in many systems in this paper we propose a modification of the sliding mode controller for a class of single input single output siso systems which can eliminate the above mentioned limitations and under certain conditions guarantee global closed loop stability hence the range of applicability of zak hui s method can be greatly broadened

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

existence results for an impulsive abstract partial differential equation with state dependent delay in this paper we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state dependent delay

conditions of output stabilization for nonlinear models in the takagi sugeno s form this paper deals with the control law design with an observer for models in the takagi sugeno s ts form in the general case where the nonlinearities and the variables involved are not measurable the results presented in this paper give sufficient conditions of stabilization for the global loop i e ts model stabilized with a state feedback and a fuzzy observer conditions are given in the form of linear matrix inequalities lmi problem

superconvergence of discontinuous galerkin methods for nonlinear delay differential equations with vanishing delay abstract in this paper we investigate the local superconvergence of the discontinuous galerkin dg solutions on quasi graded meshes for nonlinear delay differential equations with vanishing delay it is shown that the optimal order of the dg solution at the mesh points is o h 2 m 1 by analyzing the supercloseness between the dg solution and the interpolation h u of the exact solution we get the optimal order o h m 2 of the dg solution at characteristic points we then extend the convergence results of dg solutions to state dependent delay differential equations numerical examples are provided to illustrate the theoretical results

output tracking control of mimo fuzzy nonlinear systems using variable structure control approach the output tracking control problem for nonlinear systems in the presence of both parameter perturbations and external disturbances is studied our approach is based on the takagi sugeno t s fuzzy modeling method and a variable structure control vsc technique therefore the systems considered are not and need not be in the triangular and parametric strict feedback form which are prevalent among adaptive model following control for nonlinear systems or in the normal form which pervades almost all existing results in neuro fuzzy model following control approach we first study the problem of stabilization of t s fuzzy systems by using a vsc technique a method for the design of a switching surface based on linear matrix inequalities is developed and a stabilizing controller based on a reaching law concept in the presence of both parameter perturbations and external disturbances is proposed then the method is extended to design controllers for output tracking of t s fuzzy nonlinear systems in two cases i e systems which possess the so called strong passive subsystems and strong stable zero dynamics respectively finally illustrative examples are presented to demonstrate the whole design procedure from the original nonlinear systems to their fuzzification and finally to the realization of the desired controllers simulation results show that the goal of output tracking can be achieved by the proposed controllers

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

estimation of time and state dependent delays and other parameters in functional differential equations a parameter estimation algorithm is developed that can be used to estimate unknown time or state dependent delays and other parameters e g initial condition appearing within a nonlinear nonautonomous functional differential equation the original infinite dimensional differential equation is approximated using linear splines which are allowed to move with the variable delay the variable delays are approximated using linear splines as well the approximation scheme produces a system of ordinary differential equations with nice computational properties the unknown parameters are estimated within the approximating systems by minimizing a least squares fit to data criterion convergence theorems are proved for time dependent delays and state dependent delays within two classes which say essentially that fitting the data by using approximations will in the limit provide a fit to the data using the original system numerical test examples are presented that illustrate the method for all types of delay

a global convergence theorem for a class of parallel continuous explicit runge kutta methods and vanishing lag delay differential equations iterated continuous extensions ices are continuous explicit runge kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay differential equations ddes ices have a particular role in the explicit solution of ddes with vanishing lags they may be regarded as parallel continuous explicit runge kutta pcerk methods as they allow one to take advantage of parallel architectures ices can also be related to a collocation method the purpose of this paper is to provide a theorem giving the global order of convergence for variable step implementations of ices applied to state dependent ddes with and without vanishing lags implications of the theory for the implementation of this class of methods are discussed and demonstrated the results establish that our approach allows the construction of pcerk methods whose order of convergence is restricted only by the continuity of the solution

discretizations of nonlinear differential equations using explicit finite order methods abstract in this work we study the appearance of spurious solutions when first order differential equations with unimodal right hand sides are discretized using runge kutta schemes these spurious solutions are explained in terms of the iteration functions schemes that produce good approximating solutions for much longer times are given

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

lmi based nonlinear fuzzy observer controller design for uncertain mimo nonlinear systems based on the variable structure system theory we develop a nonlinear fuzzy observer controller design method for a class of uncertain nonlinear systems that can be represented by a takagi sugeno fuzzy model we use a sliding mode observer and a sliding mode controller as the local state observer and the local controller in terms of linear matrix inequalities lmis we derive sufficient conditions for the existence of the fuzzy observer and the fuzzy controller we also derive lmi conditions for the existence of the fuzzy observer and the fuzzy controller guaranteeing the alpha stability or generalized h2 hinfin performance constraints we show that the observer and the controller can be independently designed i e the separation property holds finally we give simple lmi based design algorithms together with an example

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

robust h control analysis and synthesis for takagi sugeno general uncertain fuzzy systems abstract in this paper robust h control for takagi sugeno general uncertain fuzzy systems where uncertainties come into all the system matrices is considered the main result is to establish equivalent relationships between quadratic stability and quadratic stability with h disturbance attenuation of general uncertain fuzzy systems and h control for fuzzy systems without uncertainty these relationships imply that quadratically stabilizing controllers and quadratically stabilizing controllers with h disturbance attenuation for general uncertain fuzzy systems can be obtained by designing h controllers for fuzzy systems without uncertainties we first give sufficient conditions for the h norm being less than a given number we then consider a general h problem with output feedback controllers and give a design method of h controllers and sufficient conditions which guarantee the required h performance of the closed loop system this design method can be applied to quadratic stabilizing controllers and quadratic stabilizing controllers with h disturbance attenuation for general uncertain fuzzy systems next we analyze quadratic stability and quadratic stability with h disturbance attenuation of general uncertain fuzzy systems and establish equivalent relationships to h control of fuzzy systems without uncertainties based on these relationships we design robust controllers for uncertain fuzzy systems finally examples are given to illustrate the theory