the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

adaptive systems and time varying plants abstract the behaviour of four adaptive algorithms lms equation error output error identification and the goodwin ramadge caines adaptive control algorithm are examined in the face of plant parameter variations the key point to note is that persistency of excitation conditions in the time invariant parameter case guarantees exponential convergence then standard and newly developed lyapunov stability results allow us to guarantee tracking error and parameter error boundedness when the plant parameters are actually time varying

performance limitations of adaptive parameter estimation and system identification algorithms in the absence of excitation we address the problem of fundamental performance limitations in adaptive parameter estimation and system identification occurring in environments where perturbations are present but there is lack of sufficient excitation we construct a simple yet general bursting scenario to derive an analytical lower bound on the worst case peak steady state error for a wide class of parameter estimation and system identification algorithms our results show that in the absence of any input constraints arbitrarily small perturbations impose a serious performance limitation in the sense that the worst case performance deteriorates proportionally with the size of the parametric uncertainty set

collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers a numerical scheme is proposed to solve singularly perturbed two point boundary value problems with a turning point exhibiting twin boundary layers the scheme comprises a b spline collocation method on a uniform mesh which leads to a tridiagonal linear system asymptotic bounds are established for the derivative of the analytical solution of a turning point problem the analysis is done on a uniform mesh which permits its extension to the case of adaptive meshes which may be used to improve the solution the design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem some relevant numerical examples are also illustrated to verify computationally the theoretical aspects

robustness of unmodified stochastic adaptive control algorithms originally adaptive control theory was developed for the ideal system models i e linear system models under the assumption that relative degree and upper bounds on the order of the systems are known in the early 1980s adaptive control algorithms designed for such ideal system models were strongly attacked by many researchers due to lack of robustness in the presence of unmodeled dynamics and external disturbances the purpose of the present paper is to relax existing constant pressure on the adaptive control algorithms originally designed for the ideal system models it is shown that such adaptive control algorithms are globally stable and robust with respect to unmodeled dynamics and external disturbances without any modifications such as i spl sigma modification spl epsiv sub 1 modification relative dead zone projection of the parameter estimates etc global stability of the unmodified algorithms is established by requiring the reference signal to be persistently exciting

robustness of adaptive control for stochastic systems robustness of the stochastic adaptive systems with unmodelled dynamics and external disturbances is discussed in this paper an adaptive controller is designed using the els algorithm except for the finite switches it is shown that the closed loop system is globally stable both the estimation error and the tracking error approach their minima when the unmodelled dynamics diminish that is the asymptotically adaptive system operating with the els algorithm has a good robustness with respect to unmodelled dynamics and external disturbances

parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers this article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers classical finite difference schemes do not yield parameter uniform convergent results on a uniform mesh in general robust computational techniques for boundary layers chapman hall london crc press boca raton fl 2000 in order to overcome this difficulty we propose an appropriate piecewise uniform shishkin mesh and apply the classical finite difference schemes on this mesh error estimates are derived by decomposing the solution into smooth and singular components the present method is layer resolving as well as parameter uniform convergent numerical examples are presented to show the applicability and efficiency of the method

variations on a theme of boole and stein weiss abstract we give an alternative proof of a theorem of stein and weiss the distribution function of the hilbert transform of a characteristic function of a set e only depends on the lebesgue measure e of such a set we exploit a rational change of variable of the type used by george boole in his paper on the comparison of transcendents with certain applications to the theory of definite integrals together with the observation that if two functions f and g have the same l p norm in a range of exponents p 1 p p 2 then their distribution functions coincide

robustness of an adaptive pole placement algorithm in the presence of bounded disturbances and slow time variation of parameters the robustness properties of a specific adaptive pole placement scheme in the presence of bounded disturbances and slow time variation of parameters are studied it is shown that the least squares algorithm used in this scheme retains the global stability of the overall system in the presence of bounded disturbances of small magnitude and slow time variation of system parameters within a compact set

a parameter uniform grid equidistribution method for singularly perturbed degenerate parabolic convection diffusion problems abstract we consider a class of singularly perturbed degenerate parabolic convection diffusion problems on a rectangular domain a numerical method is constructed using the implicit euler scheme on a uniform mesh in the time direction and the upwind finite difference scheme on a layer adaptive non uniform mesh in the spatial direction the layer adaptive non uniform mesh in the spatial direction is generated through the equidistribution of a suitably chosen monitor function we perform error analysis through the truncation error and barrier function approach and prove that the method is uniformly convergent with first order in both time and space numerical results are given in support of theoretical findings

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

transient performance and robustness of direct adaptive control it is shown that a discrete adaptive controller with a fixed gain gradient estimator can be applied to linear systems that are undermodeled provided that the mismatch is small and the estimated parameters are projected into a compact convex set which contains stabilizing control parameters and excludes the possibility of dividing with small numbers the latter implies that the sign and a lower bound for the high frequency gain of the modeled part of the system are known the performance of the adaptive system is related to the size of the external perturbations and the ideal result as these tend to zero is obtained the main conclusion is that while convergence stability and robustness may be achieved the transient performance may not be acceptable and the gradient estimator should probably not be used unless modifications are added design guidelines are established the results extend readily to indirect adaptive laws like pole assignment linear quadratic optimal and predictive controllers

the p version of the finite element method for a singularly perturbed boundary value problem the p version of the finite element method is applied to solve the singularly perturbed two point boundary value problem with or without turning point with the special choice of mesh points global error estimates are derived in some cases the exponential rate of convergence is obtained some numerical results are given to show the performance of the proposed method

robust adaptive control of plants with unknown order and high frequency gain abstract an adaptive control scheme is presented which has a simple structure and guarantees global stability of the closed loop plant and asymptotic zero tracking error for a class of reference signals the plants to be controlled are of minimum phase and relative degree one no a priori knowledge for the order the parameters of the plant or the sign of the high frequency gain is required the number of adjustable control parameters is proportional to the number of frequencies in the reference signal and is independent of the order of the plant the proposed algorithm is then modified by using a variable structure for generating the control signal to guarantee robust stability of the closed loop plant and asymptotic zero tracking error in the presence of bounded disturbances and unmodeled dynamics

robust adaptive lq control schemes the certainty equivalence principle is used to combine a robust adaptive law with a control structure derived from the linear quadratic lq control problem the resulting adaptive control scheme is applicable to minimum and nonminimum phase continuous time plants and is robust with respect to unmodeled dynamics and bounded disturbances the computational complexity of the continuous time adaptive lq control scheme is improved by using a hybrid adaptive law which requires the solution of an algebraic riccati equation at each interval of time rather than at each time t

robustness of model reference adaptive controllers an input output approach a general input output approach is employed to study the direct model reference adaptive control problem in the presence of parametric and unstructured uncertainty a sufficient condition for the boundedness of the closed loop signals is developed relating an appropriately defined size of the unstructured uncertainty to the size of the parametric uncertainty this condition involves the intuitively appealing gains of various sensitivity operators and allows for the study of the effect of the design parameters on the boundedness properties of the closed loop system the results of this work indicate a direction towards a unified and systematic theory in adaptive control

paper an input output view of robustness in adaptive control the stability and robustness properties of adaptive control systems are examined using input output stability theory i e passivity and small gain theory a generic adaptive error system is developed based on the concept of a tuned system an ideal converged nonadaptive closed loop system using this error system with passivity theory gives conditions for global stability where only boundedness in norm is required on the external inputs e g disturbance reference and initial conditions small gain theory is used to develop local stability results where the magnitudes of the external inputs are restricted in the global case a particular system operator not the plant is required to be strictly passive a condition which is unlikely to hold in actual use due to unmodeled dynamics the local results however are not so restricted and allow for unmodeled dynamics in this latter case an estimate of the stability margin is given under a persistent excitation condition

robust model reference adaptive control we propose a new model reference adaptive control algorithm and show that it provides the robust stability of the resulting closed loop adaptive control system with respect to unmodeled plant uncertainties the robustness is achieved by using a relative error signal in combination with a dead zone and a projection in the adaptive law the extra a priori information needed to design the adaptive law are bounds on the plant parameters and an exponential bound on the impulse response of the inverse plant transfer function

richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers we consider an interior turning point problem with two exponential boundary layers its discretisation on a piecewise uniform shishkin mesh yields a scheme which is uniformly convergent measured in the discrete maximum norm of almost order one richardson extrapolation improves the accuracy to o n 2 ln 2 n both can be proved under the assumption e c n 1

robust continuous time adaptive control by parameter projection the problem of adaptive control of a continuous time plant of arbitrary relative degree in the presence of bounded disturbances as well as unmodeled dynamics is addressed the adaptation considered is the usual gradient update law with parameter projection the latter being the only robustness enhancement modification employed it is shown that if the unmodeled dynamics which consists of multiplicative as well as additive system uncertainty is small enough then all the signals in the closed loop system are bounded this shows that extra modifications are not necessary for robustness with respect to bounded disturbances and small unmodeled dynamics in the nominal case where unmodeled dynamics and disturbances are absent the asymptotic error in tracking a given reference signal is zero moreover the performance of the adaptive controller is also robust

towards a link between the factorization approach and mimo robust adaptive control based on a doubly coprime factorization theorem nett et al 1984 an adaptive controller for discrete time mimo systems consisting of a not necessarily minimum phase modelled part and fraction type unmodelled dynamics which are not necessarily stable non linear and time varying is developed then by virtue of a stability result which is itself of interest it is shown that the adaptive closed loop system is bibo stable even in the presence of small but not necessarily stable unmodelled dynamics and arbitrarily bounded initial conditions consequently an initial but important step for applying the factorization approach to mimo robust adaptive control is completely established

some multidimensional integrals related to many body systems with the 1 r2 potential an n dimensional integral evaluated by k aomoto 1988 is shown to represent the density matrix for an impurity particle in the 1 r2 quantum many body problem on a line the value of the n dimensional integral representing the same density matrix in periodic boundary conditions is conjectured as too is the value of an n dimensional integral which represents a two point correlation function in the system also the partition function of a related classical hamiltonian is evaluated by formulating a conjecture which asserts that the sum of jacobians of a certain change of variables in n dimensions is a constant

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

continuous and numerical analysis of a multiple boundary turning point problem a singularly perturbed boundary value problem with a multiple turning point at a boundary is considered a representation of the solution is given and it is used in the construction of a uniform finite difference scheme the scheme is a first order exponentially fitted one an improved modification on a special discretization mesh is given

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

a globally stable apcs in the presence of bounded noises and disturbances this paper presents a globally stable adaptive predictive control system apcs for discrete linear processes with pure time delays in the presence of bounded measurement and process noise plus unmeasured disturbances the main contribution is the proof of stability in the context of discrete adaptive control schemes this proof is based on the minimization of the a posteriori estimation error and the convergence of the estimated parameters

alternating direction numerical scheme for singularly perturbed 2d degenerate parabolic convection diffusion problems abstract in this article we study the numerical solution of singularly perturbed 2d degenerate parabolic convection diffusion problems on a rectangular domain the solution of this problem exhibits parabolic boundary layers along x 0 y 0 and a corner layer in the neighborhood of 0 0 first we use an alternating direction implicit finite difference scheme to discretize the time derivative of the continuous problem on a uniform mesh in the temporal direction then to discretize the spatial derivatives of the resulting time semidiscrete problems we apply the upwind finite difference scheme on a piecewise uniform shishkin mesh we derive error estimate for the proposed numerical scheme which shows that the scheme is e uniformly convergent of almost first order up to a logarithmic factor in space and first order in time some numerical results have been carried out to validate the theoretical results

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

robust adaptive control a modified scheme abstract a normalizing signal that does not require the knowledge of the stability margin of the unmodelled dynamics is used in robust model reference adaptive control mrac for plants with an enlarged class of unmodelled dynamics the modified robust mrac scheme guarantees that all signals in the closed loop plant are uniformly bounded and the tracking error is of the order of the modelling error in the mean

discrete time saturation constrained adaptive pole assignment control the performance of a discrete time adaptive pole assignment controller subject to a saturation control constraint is analyzed closed loop system stability is established for a class of type i plants in the presence of plant modeling uncertainties and disturbances it is shown that the stability is maintained by implementing a simple fixed deadzone in the parameter estimator if the plant is free from the modeling uncertainties and disturbances the deadzone threshold can be set to zero then the closed loop system can be characterized by the desired poles when the control input is not saturated

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

numerical approximation of 2d time dependent singularly perturbed convection diffusion problems with attractive or repulsive turning points abstract in this work we are interested in approximating the solution of 2d parabolic singularly perturbed problems of convection diffusion type the convective term of the differential equation associated to the initial and boundary value problem is such that each one of its components has an interior simple turning point which can be of attractive or repulsive type we describe a numerical method to discretize the continuous problem which combines the fractional implicit euler method defined on a uniform mesh to discretize in time and the classical upwind finite difference scheme defined on a nonuniform mesh of shishkin type to discretize in space the fully discrete algorithm has a low computational cost from a numerical point of view we see that the method is efficient and uniformly convergent with respect to the diffusion parameter in both cases when the source term is a continuous function or it has a first kind discontinuity at the turning points some numerical results for different test problems are showed from them we deduce the good properties of the numerical method

robust adaptive systems and self stabilization a methodology for the global stability analysis and consequently for the design of robust deterministic and stochastic adaptive control filtering and prediction is introduced the methodology represents a mathematical formalization of the self stabilization mechanism which is a natural characteristic of every properly designed adaptive system the underlying idea is the construction of a suitable lyapunov function for different periods of adaptation the effectiveness of the proposed approach is demonstrated by solving the robust deterministic and stochastic adaptive control problems it is shown that very small algorithm gains may produce very large signals in the adaptive loop which are unacceptable for practical applications the intensity of the admissible unmodeled dynamics does not depend on the algorithm gain and it is specified in terms of the corresponding h sup infinity norm in order to establish goal stability persistency exciting conditions are not required in the present results

a new class of nonlinear robust adaptive controllers recently a new class of adaptive control schemes motivated from nonlinear systems theory has been proposed for minimum phase linear time invariant plants under certain assumptions on the plant transfer function these schemes guarantee uniform signal boundedness and good transient and steady state response for the regulation or tracking error despite the good performance properties of the proposed schemes their practicality relies very much on their robustness properties with respect to plant model uncertainties these robustness properties have to be established and understood for application purposes the purpose of this paper is to analyse the stability properties of this new class of adaptive controllers in the presence of reasonable classes of plant model uncertainties we first indicate that the same non robust behaviour that appears in the case of the traditional adaptive controllers without modifications may be exhibited by this new class of adaptive controller we then propose design mod

stability of discrete model reference adaptive control revisited abstract we demonstrate that the model reference adaptive controller is robust with respect to small model plant mismatch the sign and a lower bound on the high frequency gain are known and the control parameters are constrained to belong to a compact set which contains the parameters of a stabilizing controller data normalization and deadzones do not play a role in the analysis a constructive proof is used to develop bounds for the allowable gain of the model mismatch when the algorithm state is far from equilibrium the signals are measured by a decaying exponential the rate of convergence is estimated an expression which can be used to calculate the size of an overbounding set is developed its size scales uniformly with respect to the external disturbances the inverse of the adaptive gain and the reference the ideal case is obtained as these tend to zero this result holds true even when there is a model mismatch

an asymptotic error analysis of identifiers and adaptive observers in the presence of parasitics the behavior of identifiers and adaptive observers in the presence of parasitics is analyzed bounds for the state parameter error are established the adaptive schemes considered are shown to be robust provided the input signal is rich for the dominant modes but does not contain high frequencies in the parasitic range the bounds indicate possibilities for reducing the error by a proper choice of the input signal

brief paper stability analysis of adaptively controlled systems subject to bounded disturbances an approach for studying stability of adaptively controlled systems subject to bounded disturbances with unknown statistics is given a shorter proof of some results established for minimum phase systems is also given new results are established for nonminimum phase systems in each case the uniform boundedness of all signals is proven under a condition that is discussed examples of adaptive schemes that can be treated with this approach are given

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

robustness of discrete time adaptive controllers an input output approach a general input output approach that can be used for the design and analysis of robust discrete time adaptive control schemes is presented in particular a model reference adaptive controller is considered the design approach involves the use of the certainty equivalence principle to combine a robust control structure with a robust adaptive law using exponentially weighted i sub 2 norms and exploiting the properties of the adaptive law and the control structure robust stability is established for the closed loop adaptive control scheme

a higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers abstract in this article a parameter uniform numerical method is presented to solve one dimensional singularly perturbed parabolic convection diffusion multiple turning point problems exhibiting two exponential boundary layers we study the asymptotic behaviour of the solution and its partial derivatives the problem is discretized using the implicit euler method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized shishkin mesh the scheme is shown to be e uniformly convergent of order one in time direction and order two in spatial direction upto a logarithmic factor numerical experiments are conducted to validate the theoretical results comparison is done with the upwind scheme on a uniform mesh as well as on the standard shishkin mesh to demonstrate the higher order accuracy of the proposed scheme on a generalized shishkin mesh

theory and design of robust direct and indirect adaptive control schemes a general approach for designing and the theory for analysing robust direct and indirect adaptive control schemes for continuous time plants is presented the design approach involves the development of a general robust adaptive law and the use of the certainty equivalence principle to combine it with robust model reference and pole placement control structures the global stability properties and robustness of the developed adaptive control schemes are established by using a general theory which relates the properties of signals in the mean sense over intervals of time the developed theory and design approach are used to analyse and compare the robustness properties and performance of a wide class of robust adaptive laws which employ a dead zone fixed e1 and a switching modification as well as their variations

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

comments on the robust stability analysis of adaptive controllers using normalizations some results on robustness to unmodeled dynamics of adaptive controllers using normalization factors are analyzed it is shown that the proofs given by various authors fail to establish that the proposed adaptive controllers provide a nonvanishing robustness margin for the class of unmodeled dynamics considered

robust adaptive regulation with minimal prior knowledge a robust adaptive regulator is designed under minimal prior knowledge namely the order of a nominal plant model in a well defined sense the involved robustness means that the closed loop stability is guaranteed in the presence of some class of time varying parameters and unmodeled dynamics the main design feature of the proposed adaptive regulator consists in its self excitation capability together with an appropriate identification stabilization time splitting unlike in the available literature concerning robust adaptive control the involved self excitation is established independently of the parameter estimator properties the use of the robustness oriented modifications as deadzone and parameter projection or contraction which are aimed at achieving the parameter estimates boundedness is no longer required beforehand thereby reducing the underlying prior knowledge

pseudo burst phenomenon in ideal adaptive systems abstract a phenomenon is described which is similar to the well known burst phenomenon with examples it is shown that the responses of adaptive systems may exhibit large transients of the error signal separated by quiet periods in contrast to the burst phenomenon the responses observed in this paper correspond to ideal systems that is adaptive systems without noise or unmodeled dynamics also in accordance with the theory the bursts do not occur indefinitely two examples are discussed the first is shown to be very similar to the burst phenomenon the second is found to be quite different with its origin being in the internal excitation generated through the interaction of adaptation and control an averaging analysis is provided which highlights the uses and limitations of averaging for the analysis of adaptive systems

robust parameter adjustment with nonparametric weighted ball in h sup infinity uncertainty using a parameter adjustment law robust monotonic parameter error reduction is proved for a linear parameter estimation problem in which the signal is corrupted by the presence of nonparametric dynamical uncertainty the nonparameterized dynamics are characterized by a known frequency domain magnitude bound a nonconservative bound on the finite time energy of the nonparametric dynamics output is constructed in real time when the error single energy is small enough to be due to nonparametric dynamics alone the parameter adjustment is shut off to avoid misadjustment the approach is an extension of existing adaptive control error dead zone ideas to the case of frequency domain bounded uncertainty

adaptivity and error estimation for the finite element method applied to convection diffusion problems a detailed analysis is performed for a finite element method applied to the general one dimensional convection diffusion problem piecewise polynomials are used for the trial space the test space is formed by locally projecting l spline basis functions onto upwinded polynomials the error is measured in the l p mesh dependent norm the method is shown to be quasi optimal provided that the input data is piecewise smooth a reasonable assumption in practice a posteriors error estimates are derived having the property that the effectivity index theta error estimate true error converges to one as the maximum mesh size goes to zero these error estimates are composed of locally computable error indicators providing for an adaptive mesh refinement strategy numerical results show that theta is nearly one even on coarse meshes and optimal rates of convergence are attained by the adaptive procedure the robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th

a modified model reference adaptive control scheme for improved transient performance the author proposes a modified model reference adaptive control mrac scheme aimed at improving the transient performance of adaptive systems while maintaining the ideal asymptotic properties possessed by standard mrac in the modified scheme the estimation error which is generated by the identification scheme is used directly as a control signal to counteract errors resulting from the certainty equivalence design it is shown that the modified scheme provides essentially the same stability and robustness properties as the standard adaptive mrac approach but with the transient performance substantially improved simulation results are presented to illustrate the effectiveness of the proposed scheme

stable model reference adaptive control in the presence of bounded disturbances the adaptive control of a linear time invariant plant in the presence of bounded disturbances is considered in addition to the usual assumptions made regarding the plant transfer function it is also assumed that the high frequency gain k p of the plant and an upper bound on the magnitude of the controller parameters are known under these conditions the adaptive controller suggested assures the boundedness of all signals in the overall system

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

on the existence and uniqueness of solutions in adaptive control systems the issue of existence and uniqueness of solutions which arises in the analysis of adaptive control systems is investigated sufficient conditions that guarantee existence and uniqueness in positive time of solutions to differential equations describing a wide class of adaptive schemes are given the concept of filippov s solution is introduced as a method of dealing with discontinuous adaptive systems

variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer s abstract a numerical method based on cubic spline with nonuniform grid is given for the singularly perturbed two point boundary value problems having turning point the scheme derived in this paper is second order accurate numerical examples are given to support the predicted theory

a remarkable property of definite integrals we show that the integral of f u is the same as the integral of f x both with respect to x for a variety of functions u in this note f will denote any integrable function and all integral signs denote the cauchy principal value the formula oo 00 f u dx f x dx 00 oo where 2 u x 1 x goes back at least as far as cauchy and can be proved as follows from a graph of the relation 2 one sees that the solution for x has two branches 1 x y u iju2 4 oo t oo thus fo or00 r f u dx f u dx f u xi x du oo j0 oo but x x i 1 1 1 ju2 4 iu2 4 j thus for example exp x2 1 x2 dx exp x l x dx o 2e oo 1 r00 r2 w e x dx 2e2 oo 2e2 equations 1 2 clearly represent a useful property of the definite integral that deserves to be widely known what is more remarkable is that 1 remains true when 2 is replaced by v aj 3 u x2 y r j x s received june 21 1982 1980 mathematics subject classification primary 26a06 26a42 1983 american mathematical society 0025 5718 82 0000 0898 01 50 561 license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

robust adaptive pole placement for linear time varying systems this paper studies the robustness properties of a basic adaptive control algorithm with respect to plant parameter variation as well as modeling errors and bounded disturbances the algorithm consists of a projected gradient estimator and a pole assignment controller a unified method of proof is presented for robust stability of both discrete and continuous time adaptively controlled time varying systems the robust performance of an adaptive pole assignment controller is also discussed

an analysis of a uniformly accurate difference method for a singular perturbation problem it will be proven that an exponential tridiagonal difference scheme when applied with a uniform mesh of size h to euxx b x ux f x for o o b and f smooth e in 0 1 and u o and u i given is uniformly second order accurate i e the maximum of the errors at the grid points is bounded by ch2 with the constant c independent of h and e this scheme was derived by el mistikawy and werle by a c patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point the behavior of the approximate solution in between the grid points will be analyzed and some numerical results will also be given

global boundedness of discrete time adaptive control just using estimator projection in this paper we study a discrete time indirect adaptive control algorithm which contains a constrained gradient parameter estimator and a pole assignment control law synthesis module this adaptive control algorithm does not involve modifications like data normalization use of deadzones or injection of persistently excited signals also it requires no a priori knowledge of system modelling errors it is shown that global robustness properties still hold when this simple algorithm is applied to systems with bounded disturbances and arbitrarily small fast parasitic dynamics the problem of indirect decentralized adaptive control of interconnected systems is also considered we use the above adaptive algorithm to design completely decentralized local adaptive controllers for each isolated subsystem by ignoring interactions between subsystems we show that the local controllers designed in this way are robust in the sense that all signals in the closed loop adaptive system are bounded for bounded initial conditions reference inputs disturbances and an arbitrarily small amount of interaction between subsystems and unmodelled dynamics of each subsystem

towards a unified theory of parameter adaptive control ii certainty equivalence and implicit tuning for pt i see ibid vol 35 p 1002 12 1990 it is shown that for a large class of linear multivariable process models a properly designed certainty equivalence controller results in a tunable closed loop parameterized system the importance of this result to both the analysis and synthesis of parameter adaptive control systems is discussed it is demonstrated by means of examples how the connection between certainty equivalence and tunability together with the concept of tunability itself can be used to markedly simplify the stability analysis of adaptive control systems a new family of indirect parameterized controllers is introduced which has capabilities comparable to those of the well known direct parameterized controllers which have served as basic building blocks for adaptive control systems of all types the concept of implicit tuning is formalized and its potential importance to adaptive control is briefly discussed it is explained how the idea of tunability can be used to analyze an implicitly tuned system

canonical partition functions of hamiltonian systems and the stationary phase formula we show that there is a symplectic action of a torus associated to harmonic flows on the cotangent bundle of a semi simple lie algebra this allows to obtain a completely classical proof of the gallavotti marchioro formula by the method of the stationary phase

a new model reference robust adaptive controller in the presence of unmodeled dynamics and bounded disturbances abstract this paper presents a new model reference robust adaptive controller which does not require a priori knowledge of the norm desired controller parameter vector the algorithm is designed according to a reduced order plant which is assumed to be of minimum phase and of an arbitrary relative degree in the adaptive law the output error plays a dual role in the adjustment of the control parameter vector in the absence of unmodeled dynamics and disturbances the tracking error is shown to be zero when reference input is persistently exciting

differential and integral relations involving fractional derivatives of airy functions and applications abstract various differential and integral relations are deduced that involve fractional derivatives of the airy function ai x and the scorer function gi x several new wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the airy functions new fractional derivative conservation laws are derived for equations of the korteweg de vries type

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem a number of results exist in the literature for singularly perturbed differential equations without turning points in particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence this guarantees that the schemes model the boundary layers well we wish to examine whether these schemes will also be uniformly convergent if the equation has turning points to this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes we show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one we confirm these theoretical results by numerical calculations

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

dominant richness and improvement of performance of robust adaptive control abstract several recent robust adaptive control schemes guarantee stability and residual tracking errors which are small in the mean for any bounded initial conditions independent of any persistence of excitation condition since smallness in the mean does not always guarantee small bounds for the tracking error at steady state the tracking performance of these robust schemes may not be acceptable in this paper we show that the convergence and the tracking performance of these globally stable adaptive control schemes can be considerably improved if the reference input signal is chosen to be dominantly rich we show that dominantly rich signals maintain a high level of persistence of excitation relative to the level of the modeling error which guarantees exponential convergence and small bounds for the tracking and parameter error at steady state