modification of adaptive control laws using adaptive gain feedback plant bypass has been suggested as a solution to the problems that arise when adaptive systems are used to control non minimum phase plants or plants with unmodelled dynamics it is shown that plant bypass is normally not feasible it is demonstrated that realizable adaptive systems using plant bypass are combined fixed parameter and adaptive feedback systems the fixed parameter feedback provides adaptive gain feedback agf an adaptive control law is modified using agf and the characteristics of this modified adaptive scheme are investigated

recovery type superconvergence and a posteriori error estimates for control problems governed by stokes equations in this paper we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by stokes equations we obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods which are the patch recovery technique and the least squares surface fitting method our results are based on some regularity assumption for the stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions

pointwise error estimates for differences in piecewise linear finite element approximations we consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem new estimates for the differences e x1 e x2 where e x u x u h x is the error and x1 and x2 are any two points in the domain are obtained in terms of weighted l infty norms as a consequence so called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a generalized reaching law with different convergence rates traditionally the convergence rate of the sliding variable given by a discrete reaching law has been fixed in nature this paper presents a generalized algorithm for discrete time sliding mode control systems which offers a flexible convergence rate for the sliding variable for analysis we use the recently developed band approach method which gives us the values of the parameters in the algorithm for the desired convergence to take place once the ultimate band is chosen appropriately

variable structure model reference adaptive control using only input and output measurements abstract a variable structure model reference adaptive controller vs mrac using only the input and output measurements of the plant is proposed stability and transient properties are analysed and discussed remarkable transient behaviour is revealed in particular model following can be perfectly achieved in a finite time the effect of plant input disturbance is investigated and the performance of the vs controller is also shown to be qualitatively superior to that obtained with conventional integral adaptation algorithms a link between the vs mrac and the conventional mrac is suggested simulations are presented to highlight the theoretical results

adaptive sliding mode control in discrete time systems abstract discrete time sliding mode control is considered the case of known parameters is first analyzed leading to a new definition of the so called equivalent control as the piecewise constant control that reduces to zero in finite time the distance of the system state from the sliding manifold in the presence of bounded parametric uncertainties an adaptive control scheme that guarantees the asymptotic satisfaction of the same control objective is presented the main feature of this approach is the reduction of the order of the relevant error equation and the possibility of dealing with the nonmatched uncertainties introduced by the sampling process

robust discrete time chattering free sliding mode control a new control algorithm based on discrete time variable structure systems theory is proposed the basic feature of this algorithm is that trajectories reach the sliding manifold in finite time without chattering apart from stability the robustness of the algorithm with respect to parameter uncertainties as well as external disturbances is considered it is demonstrated that the robustness is improved by decreasing the sampling period the theory is illustrated on a dc servo position system

maximum norm estimates for an immunology model using reaction diffusion equations with stochastic source terms this paper describes numerical methods and the corresponding maximum norm error estimates for a chemotaxis model proposed by kepler upon detecting pathogens immune cells secrete soluble factors that attract other immune cells to the site of the infection the motion of the model cells is stochastic but biased toward the gradient of one or more of the soluble factors the soluble factors are modeled by a system of reaction diffusion equations with sources that are centered on the cells previously i presented a first order splitting in time for solving the reaction diffusion stochastic system numerically the diffusion reaction and stochastic differential equations can be approximated separately to first order in the supremum norm the three dimensional domain is discretized using finite elements and the diffusion is solved using a backward euler scheme combined with multigrid the reaction is solved using a simple semi implicit first order scheme the stochastic differential equations are given by a langevin process which can be simulated exactly the paper concludes by demonstrating first order convergence of the entire simulation and providing a sample simulation of the reaction diffusion stochastic system

pointwise localized error estimates for parabolic finite element equations we derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations the results extend those obtained by a h schatz for stationary elliptic problems in particular they show that the error is more localized for higher order elements

an improved adaptive control for robust adaptation an adaptive control law consisting of an integral and proportional adaptation is suggested it is shown that the addition of the proportional adaptation term to the control law improves the performance of the adaptive system although this is done in the context of a first order system it is believed that this may also be the case for higher order systems the scalar case is only discussed simulation results are presented to complement the theoretical developments

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

on the stability of discrete time sliding mode control systems the stability of discrete time sliding mode control systems is investigated and a new sliding mode condition is suggested it is shown that the control must have upper and lower bounds a numerical example is discussed as an illustration

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

sliding mode control of a discrete system abstract conventional sliding mode control designed on the basis of a continuous system is known to be robust to the plant uncertainties a realized digital system however not only yields chattering but also may become unstable by a long sampling interval this paper presents a stable discrete sliding mode control insensitive to the choice of sampling interval and not yielding chattering the control system is designed on the basis of a discrete lyapunov function and a sufficient condition of the control gain to make the system stable is given contrary to the continuous case the derived switching plane of the control law is different from the sliding mode and in its neighborhood the control law is given by the linear state feedback simulations show the effectiveness of the proposed method

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

generalization of gao s reaching law for higher relative degree sliding variables in this paper a generalized discrete time reaching law for sliding variables with arbitrary relative degree is proposed the reaching law is based on gao s strategy and ensures higher order switching type sliding motion which is a novel concept in the field of discrete time plants the controller synthesis process is simplified by performing a specific system state transformation it is demonstrated that the width of the layer to which the representative point of the system is driven decreases as the relative degree of the sliding variable gets bigger it has further been proven that achieving a narrower quasi sliding mode band width is reflected in reduced state error finally the reaching phase elimination for variables with arbitrary relative degree is considered this is obtained by implementing a time varying sliding surface in a finite number of initial time instants uniform ultimate boundedness of all state variables is guaranteed

robust redesign of adaptive control effects of unmodeled high frequency dynamics on stability and performance of adaptive control schemes are analyzed in the regulation problem global stability properties are no longer guaranteed but a region of attraction exists for exact adaptive regulation the dependence of the region of attraction on unmodeled parasitics is examined first then the general case of model reference adaptive control is considered in which parasitics can destroy stability and boundedness properties a modified adaptive law is proposed guaranteeing the existence of a region of attraction from which all signals converge to a residual set which contains the equilibrium for exact tracking the size of this set depends on design parameters the frequency range of parasitics and the reference input signal characteristics

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

a generalized reaching law for discrete time sliding mode control the paper presents a generalized algorithm for the reaching and boundedness of the sliding variable in case of discrete time sliding mode control systems the algorithm is general as it not only comprises functions in the sliding variable but also functions in other variables which are known with the help of the proposed generalized reaching law an important problem of a reaching law with uncertainty bounded by known functions is studied the convergence of the sliding variable to an ultimate bounding function in presence of such an uncertainty is shown by the band approach method this is achieved by appropriately selecting the functions in the proposed generalized reaching law

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

enhancing finite element approximation for eigenvalue problems by projection method abstract this paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems the projection method is a postprocessing procedure that constructs a new approximation by using the least squares method the results are based on some regularity assumption for the elliptic problem and are applicable to the finite element approximations of self adjoint elliptic eigenvalue problems with general quasi regular partitions therefore the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes

strict lyapunov functions for the super twisting algorithm a method to construct a family of strict lyapunov functions i e with negative definite derivative for the super twisting algorithm without or with perturbations is provided this second order sliding modes algorithm is widely used to design controllers observers and exact differentiators the proposed lyapunov functions ascertain finite time convergence provide an estimate of the convergence time and ensure the robustness of the finite time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm since the lyapunov functions and their derivatives are quadratic forms the operation with them is as simple as for linear time invariant systems

implicit euler numerical scheme and chattering free implementation of sliding mode systems in this paper it is shown that the implicit euler time discretization of some classes of switching systems with sliding modes yields a very good stabilization of the trajectory and of its derivative on the sliding surface therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used are avoided moreover the method an event capturing or time stepping algorithm allows for multiple switching surfaces i e a sliding surface of codimension 2 the details of the implementation are given and numerical examples illustrate the developments this method may be an alternative method for chattering suppression keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces links with discrete time sliding mode controllers are studied

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

non linear discrete variable structure systems in quasi sliding mode abstract the problem of inducing convergent quasi sliding regimes on smooth state space surfaces of non linear single input single output discrete time controlled systems is addressed in full generality a suitable extension of the notion of relative degree is used in establishing the most salient features of quasi sliding motions in non linearly controlled systems several examples are given

output tracking control of uncertain nonlinear second order systems abstract the solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum time optimal control problem two different types of uncertainty are considered in the paper in the first case a constant bound on the uncertain dynamics is assumed to be known in the second case the bound is a function of both the measurable and the unmeasurable state variable of the system in both cases the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features the resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one or to a vicinity of this latter is proved also in the uncertain case

an o t sup 2 boundary layer in sliding mode for sampled data systems the use of a discontinuous control law typically sign functions in a sampled data system will bring about chattering phenomenon in the vicinity of the sliding manifold leading to a boundary layer with thickness o t where t is the sampling period however by proper consideration of the sampling phenomenon in the discrete time sliding mode control design the thickness of the boundary layer can be reduced to o t sup 2 in contrast to discontinuous control for continuous time vss the discrete time sliding mode control need not be of switching type

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

analysis of chattering in systems with second order sliding modes a systematic approach to the chattering analysis in systems with second order sliding modes is developed the neglected actuator dynamics is considered to be the main cause of chattering in real systems the magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second order sliding mode control generalized suboptimal 2 smc g so algorithms is evaluated sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding poincare maps for linear systems driven by 2 smc g so algorithms analysis tools based on the frequency domain methods are developed the first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions the second technique represents a modified tsypkin s method and provides for a relatively simple theoretically exact approach to evaluate the periodic motion parameters examples of analysis and simulation results are given throughout this paper

perturbations of forms and error estimates for the finite element method at a point with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point we first derive a variety of local error estimates for u uh at a point x0 where uh belongs to a finite element space shr and is an approximation to u satisfying the local equations a u u h varphi f varphi for all varphi in shr with compact support in a neighborhood of x0 here the a cdot cdot are bilinear forms associated with second order elliptic equations and the f are linear functionals in the case that f equiv 0 our results coincide with those of schatz siam j numer anal 38 2000 pp 1269 1293 but are improvements when f neq 0 we apply these results to improve the superconvergence error estimates obtained by schatz sloan and wahlbin siam j numer anal 33 1996 pp 505 521 at points x0 where the subspaces are symmetric with respect to x0

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

lessons from an example in on the stability of discrete time sliding mode control systems initial conditions are presented for which the indicated example in the above paper by sarpturk et al ibid vol ac 32 1987 is shown to fail to meet the paper s theoretical requirements the importance of this error is that the example now represents a counterexample to the suggested form of the sliding mode control demonstrates a necessary restriction on any such quasi sliding mode control and illustrates that restrictions presented in another paper are not global

chattering free digital sliding mode control with state observer and disturbance rejection in this paper a novel discrete time implementation of sliding mode control systems is proposed which fully exploits the multivaluedness of the dynamics on the sliding surface it is shown to guarantee a smooth stabilization on the discrete sliding surface in the disturbance free case hence avoiding the chattering effects due to the time discretization in addition when a disturbance acts on the system the controller attenuates the disturbance effects on the sliding surface by a factor h where h is the sampling period most importantly this holds even for large h the controller is based on an implicit euler method and is very easy to implement with projections on the interval 1 1 or as the solution of a quadratic program the zero order hold zoh method is also investigated first and second order perturbed systems with a disturbance satisfying the matching condition without and with dynamical disturbance compensation are analyzed with classical and twisting sliding mode controllers

local a posteriori estimates on a nonconvex polygonal domain a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems on a nonconvex polygonal domain the results are extensions of previous results for smooth domains w hoffmann et al math comp 70 2001 pp 897 909 a h schatz and l b wahlbin math comp 73 2004 pp 517 523

reaching law for dsmc systems with relative degree 2 switching variable abstractthis paper presents the first attempt to design a reaching law based discrete time sliding mode controller with a relative degree 2 switching variable the current value of this variable is only affected by the control signal and disturbance generated two time instants ago it is demonstrated that the new reaching law based strategy offers a smaller quasi sliding mode band width when compared to a similar control scheme with relative degree 1 switching variable this in turn leads to reduced system output error in the control system proposed in the paper

a new adaptive law for robust adaptation without persistent excitation a new adaptive law motivated by the work of loannou and kokotovic 1983 is proposed for the robust adaptive control of plants with unknown parameters in this adaptive law the output error plays a dual role in the adjustment of the control parameter vector the advantages of using the adaptive law over others proposed in the literature are discussed in the ideal case the adaptive system has bounded solutions in addition the origin of the error equations is exponentially stable when the reference input is persistently exciting and has a sufficiently large amplitude the adaptive system is also shown to be robust under bounded external disturbances finally it is shown that by suitably modifying the adaptive law the overall system can be made robust in the presence of a class of unmodeled dynamics of the plant simulation results are presented throughout the paper to complement the theoretical developments

piecewise linear finite element methods are not localized recent results of schatz show that standard galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error these results do not indicate that such localization occurs when piecewise linear elements are used we show via simple one dimensional examples that schatz s estimates are sharp in that localization indeed does not occur when piecewise linear elements are used

asymptotically exact functional error estimators based on superconvergent gradient recovery the use of dual adjoint problems for approximating functionals of solutions of pdes with great accuracy or to merely drive a goal oriented adaptive refinement scheme has become well accepted and it continues to be an active area of research the traditional approach involves dual residual weighting drw in this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact the first is of drw type and is derived for meshes in which most triangles satisfy an inlinemediaobject not available see fulltext approximate parallelogram property the second functional estimator involves dual error estimate weighting dew using any superconvergent gradient recovery technique for the primal and dual solutions several experiments are done which demonstrate the asymptotic exactness of a dew estimator which uses a gradient recovery scheme proposed by bank and xu and the effectiveness of refinement done with respect to the corresponding local error indicators

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second order quasilinear elliptic problems two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second order quasilinear elliptic dirichlet boundary value problems over convex polyhedral domains omega in space dimension n ge 2 we first give a residual estimator which is equivalent to nabla u u h l infty omega up to higher order terms the second type of residual estimator is designed to control nabla u u h locally over any subdomain of omega it is a novel a posteriori counterpart to the localized or weighted a priori estimates of sch98 this estimator is shown to be equivalent up to higher order terms to the error measured in a weighted global norm which depends on the subdomain of interest all estimates are proved for general shape regular meshes which may be highly graded and unstructured the constants in the estimates depend on the unknown solution u in the nonlinear case but in a fashion which places minimal restrictions on the regularity of u

output feedback sliding mode control for sampled data systems we address the output feedback sliding mode control problem for a sampled data linear system with external disturbances of the matching type by taking into account the disturbance compensation a deadbeat high gain output feedback control strategy with additional dynamics is able to attenuate the disturbances in the framework of singular perturbation analysis it is shown that the closed loop system exhibits good robustness against exogenous disturbances

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part ii the piecewise linear case we extend results from part i about estimating gradient errors elementwise a posteriori given there for quadratic and higher elements to the piecewise linear case the key to our new result is to consider certain technical estimates for differences in the error e x 1 e x 2 rather than for e x itself we also give a posteriori estimators for second derivatives on each element

recovery based error estimator for interface problems conforming linear elements this paper studies a new recovery based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems instead of recovering the gradient in the continuous finite element space the flux is recovered through a weighted l 2 projection onto h mathrm div conforming finite element spaces the resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results this paper also proposes an adaptive finite element method based on either the recovery based estimators or the edge estimator through local mesh refinement and establishes its convergence in particular it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps

ultraconvergence of high order fems for elliptic problems with variable coefficients in this paper we investigate local ultraconvergence properties of the high order finite element method fem for second order elliptic problems with variable coefficients under suitable regularity and mesh conditions we show that at an interior vertex which is away from the boundary with a fixed distance the gradient of the post precessed kth k ge 2 k 2 order finite element solution converges to the gradient of the exact solution with order mathcal o h k 2 mathrm ln h 3 o hk 2 lnh 3 the proof of this ultraconvergence property depends on a new interpolating operator some new estimates for the discrete green s function a symmetry theory derived in 26 and the richardson extrapolation technique in 20 numerical experiments are performed to demonstrate our theoretical findings

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

a posteriori error estimator based on gradient recovery by averaging for discontinuous galerkin methods we consider some anisotropic and piecewise constant diffusion problems in domains of r 2 approximated by a discontinuous galerkin method with polynomials of any fixed degree we propose an a posteriori error estimator based on gradient recovery by averaging it is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability the lower bound is also established moreover these additional terms are negligible when the recovered gradient is superconvergent the reliability and efficiency of the proposed estimator is confirmed by some numerical tests

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

improved zz a posteriori error estimators for diffusion problems conforming linear elements abstract in cai and zhang 2009 we introduced and analyzed an improved zienkiewicz zhu zz estimator for the conforming linear finite element approximation to elliptic interface problems the estimator is based on the piecewise constant flux recovery in the h div conforming finite element space this paper extends the results of cai and zhang 2009 to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear h div space

comments on on the stability of discrete time sliding mode control systems with reply in the above paper by s z sarpturk and y istefanopulos see ieee trans automat contr vol ac 32 p 930 2 1987 stability conditions for discrete time sliding mode control systems are presented which require the stabilizing control to be upper and lower bounded the present author shows that these bounds depend on the distance of the system state from the sliding manifold and that they converge as the state approaches the manifold sarpturk and istefanopulos agree that the bounds of the discrete time control guaranteeing the stability of the sliding mode system depend also on the distance of the state from the sliding manifold therefore in order to guarantee the stability of the sliding mode system the controls must be chosen so as to satisfy the convergence and sliding bounds derived in the paper

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

a combined direct indirect and variable structure method for robust adaptive control direct methods and indirect methods for the adaptive control of linear plants with unknown parameters are currently well known methods for combining direct and indirect methods have also been suggested in the paper the latter method is combined with the variable structure method new adaptive laws are suggested which overcome the principal drawbacks of the variable structure method while the direct and indirect components of the new method assure stability the variable structure component improves the transient response of the system simulation results are presented to show that the overall system is asymptotically stable in the ideal case and exhibits robustness and improved performance in the presence of perturbations

on discrete time variable structure sliding mode control the purpose of this paper is to show the limitations of discrete time variable structure sliding mode control and that the equivalent control must be used in order to have sliding in a neighborhood of the switching surface conflicting requirements for the sliding mode controller behavior in the continuous and discrete time domains are revealed and analyzed a linear control law for an uncertain discrete time linear plant with bounded uncertainties is analyzed and its superiority over nonlinear controllers is demonstrated the conclusion of the obtained results is that in the discrete time variable structure sliding mode controller design unlike in the continuous time the designer may have limited flexibility in selecting controller architectures

pointwise error estimates for finite element solutions of the stokes problem in this paper pointwise error estimates for general finite element approximations of the stokes problem are established on quasi uniform grids in r n the results obtained in this paper improve and extend the existing error estimates in the maximum norm for the stokes problem the new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by product provide logarithm free bounds for all errors except the error of the velocity approximation of the lowest order

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

brief digital second order sliding mode control for uncertain nonlinear systems in this note we analyze the discrete time implementation of a second order sliding mode control 2 smc scheme the treatment is detailed for a simple class of feedback linearizable nonlinear systems expressed in the brunowsky normal form first it is shown that the direct discretization of a continuous time 2 smc scheme guarantees the finite time attainment of a motion in an o t 2 boundary layer of the sliding manifold t being the sampling period then a suitable iterative learning procedure that leads to the asymptotic reduction of the boundary layer to o t 3 is proposed simulation results are reported at the end of the paper

the local ultraconvergence for high degree galerkin finite element methods abstract in this paper we investigate the local ultraconvergence of k degree k 3 finite element methods for the second order elliptic boundary value problem with constant coefficients over a family of uniform rectangular triangular meshes t h on a bounded rectangular domain d the k degree finite element estimates are developed for the green s function and its derivatives they are employed to explore the relationship among d i s t x d d i s t x m and the ultraconvergence of k degree finite element methods at vertex x where m is the set of corners of d numerical examples are conducted to demonstrate our theoretical results

a new robust mrac using variable structure design for relative degree two plants abstract variable structure design concept has been shown not only useful in dealing with uncertain systems but also useful in establishing controllers in the adaptive context continuing the work this research proposed a new robust model reference adaptive control for single input single output linear plants with relative degree two a class which encompasses a large number of linearized mechanical systems when compared with the conventional mrac schemes the current approach drastically improves the transient behavior and the tracking performance of the closed loop system from several numerical simulations furthermore if the reference input satisfies the persistently exciting condition or if the prior knowledge about the interval in which the true parameter vector lies is available the magnitude of the input force can be considerably reduced into a more realistic level in the presence of mild unmodelled dynamics and bounded output disturbances the proposed scheme possesses similar well behaved properties it can be shown that the tracking error will in fact fall into a residual set whose size can be directly related to the size of unmodelled dynamics and of output disturbances

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires