finite time stabilization of a perturbed double integrator part i continuous sliding mode based output feedback synthesis the twisting and supertwisting algorithms generating important classes of second order sliding modes sosms are well recognized for their finite time stability and robustness properties in the present paper a continuous modification of the twisting algorithm and an inhomogeneous perturbation of the supertwisting algorithm are introduced to extend the class of sosm s that present the afore mentioned attractive features thus modified the twisting and supertwisting algorithms are utilized in the state feedback synthesis and respectively velocity observer design made for the finite time stabilization of a double integrator if only output measurements are available performance and robustness issues of the resulting output feedback synthesis are illustrated by means of numerical simulations

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

improved conditioning of infinite elements for exterior acoustics an optimized version of the so called mapped wave envelope elements also known as astley leis elements is introduced these elements extend to infinity in one dimension and therefore provide an approach to the simulation of exterior acoustical problems in both frequency and time domains their formulation is improved significantly through the proper choice of polynomial bases in the direction of radiation in particular certain jacobi polynomials are identified which behave well with respect to conditioning of the system matrices as a consequence the size of the finite element discretization may reduce considerably without any loss of accuracy in addition the new polynomial bases lead to superior performance of the infinite elements in conjunction with iterative solvers

convergence rate for a curse of dimensionality free method for a class of hjb pdes in previous work of the first author and others max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems although max plus basis expansion and max plus finite element methods can provide substantial computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes where the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we consider a previously obtained numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution although previous work indicated that the method was not subject to the curse of dimensionality it did not indicate any error bounds or convergence rate here we obtain specific error bounds

differential games with ergodic payoff we address a zero sum differential game with ergodic payoff we study this problem via the viscosity solutions of an associated hamilton jacobi isaacs equation under certain condition we establish the existence of a value and prove certain representation formulae

calculation of sound radiation in infinite domain using a meshless method a meshless method coupling with a variable order infinite acoustic wave envelope element for sound radiation calculation in infinite domain is presented with the aim of accurately calculating the acoustic radiation and improving computational efficiency it is based on using the element free galerkin method in the inner region enclosing the radiator and a variable order infinite acoustic wave envelope element in the outer region for the proper modeling of the pressure amplitude decay the details are provided for the derivation and implementation of this method the factors of influencing the performance of the method which include the shape function constructing the number of integration points the weight functions and the support domain are discussed a hybrid adaptive gauss legendre quadrature is devised to obtain good integration accuracy the suitable radius of the support domain for the acoustic field calculation in free space is also determined by use of numerical experiments a complex structure is designed for simulation to validate the method the results illustrate the accuracy applicability and effectiveness of this method

numerical approximations for stochastic differential games the markov chain approximation method is a widely used robust relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high versions of these methods have been used in applications to various two player differential and stochastic dynamic games for a long time and proofs of convergence are available for some cases mainly using pde type techniques in this paper purely probabilistic proofs of convergence are given for a broad class of such problems where the controls for the two players are separated in the dynamics and cost function and which cover a substantial class not dealt with in previous works discounted and stopping time cost functions are considered finite horizon problems and problems where the process is stopped on first hitting an a priori given boundary can be dealt with by adapting the methods of h j kushner and p dupuis numerical methods for stochastic control problems in continuous time 2nd ed springer verlag berlin new york 2001 as done in this paper for the treated problems the essential conditions are the weak sense existence and uniqueness of solutions an almost everywhere continuity condition and that a weak local consistency condition holds almost everywhere for the numerical approximations just as for the control problem there are extensions to problems with controlled variance and jumps

continuity of the upper and lower value of slow growth differential games abstract some continuity properties are proved for a zero sum differential game with an unbounded control and no coercivity assumptions

on performance criteria two theorems generalize previous work lack of performance m p is defined as the least upper bound of the absolute error as time t ranges over the real numbers r and the input ranges over a space fm this space contains all r r such that t 0 for t o and such that sup d m p if and only if the transfer function has a zero at the origin and all its poles have negative real parts some of the implications of the theorems in control theory and design are discussed

computation of peak output for inputs satisfying many bounding conditions on magnitude and slope the evaluation of peak outputs is an essential component of control systems design in which the outputs are required to remain within their prescribed bounds in the presence of all possible inputs characterising a possible set with many bounding conditions can reduce conservatism thereby yielding a better design however this gives rise to difficulty in computing the peak outputs using analytical techniques this article develops a practical method for computing the peak outputs of linear time invariant systems for a class of possible sets characterised with many bounding conditions on the two and or the infinity norms of the inputs and their slopes the original infinite dimensional convex optimisation problem is approximated as a large scale convex programme defined in a euclidean space with sparse matrices which can be solved efficiently in practice a case study of control design for a building subject to seismic disturbances is reinvestigated where three bounding conditions are used the numeri

a three dimensional infinite element for wave propagation conditions aux limites ouvertes dans les problemes acoustiques utilisation de la methode des elements finis

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

cal h _ infty control of nonlinear systems differential games and viscosity solutions dealing with disturbances is one of the most important questions for controlled systems cal h infty optimal control theory is a deterministic way to tackle the problem in the presence of unfavorable disturbances the theory of differential games and the study of the associated hamilton jacobi isaacs equation appear to be basic tools of the theory we consider a general nonlinear system and prove that the existence of a continuous local viscosity supersolution of the isaacs equation corresponding to the cal h infty control problem is sufficient for its solvability we also show that the existence of a lower semicontinuous viscosity supersolution is necessary

robust optimal switching control for nonlinear systems we formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls we formulate the problem both in an l2 gain dissipative system framework and in a game theoretic framework we show that under appropriate assumptions a continuous switching storage function is characterized as a viscosity supersolution of the appropriate system of quasi variational inequalities the appropriate generalization of the hamilton jacobi bellman isaacs equation for this context and that the minimal such switching storage function is equal to the continuous switching lower value function for the game finally we show how a prototypical example with one dimensional state space can be solved by a direct geometric construction

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

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

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

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

the principle of matching practical conditions for systems with inputs restricted in magnitude and rate of change the main objective of matching a principle of design is to ensure that the set of possible inputs to a system is a subset of the set of tolerable inputs those which are judged to produce an acceptable response from the system this note presents practical conditions some necessary and sufficient and others only sufficient for a matched design when the set of possible inputs is characterized by restrictions on the magnitude and the rate of change of the input and the set of tolerable inputs is characterized by magnitude bounds on the response of the system

discrete differentiators based on sliding modes abstract sliding mode based differentiation of the input f t yields exact estimations of the derivatives f f n provided an upper bound l t of f n 1 t is available in real time in practice it involves discrete sampling and numerical integration of the internal variables between the measurements accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling whereas sampling and integration steps are independently variable or constant proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous time counterparts event triggered sampling is considered extensive numeric experiments are presented and analyzed

a survey of finite element methods for time harmonic acoustics many of the current issues and methodologies related to finite element methods for time harmonic acoustics are reviewed the effective treatment of unbounded domains is a major challenge most prominent among the approaches that have been developed for this purpose are absorbing boundary conditions infinite elements and absorbing layers standard computational methods are unable to cope with wave phenomena at short wave lengths due to resolutions required to control dispersion and pollution errors leading to prohibitive computational demands since computation naturally separates the scales of a problem according to the mesh size multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them other issues addressed are related to the efficient solution of systems of specialized algebraic equations and inverse problems of acoustics the tremendous progress that has been made in all of the above areas in recent years will surely continue leading to many more exciting developments

comparison theorems for viscosity solutions of a system of quasivariational inequalities with application to optimal control with switching costs in this paper we prove a comparison principle between a viscosity sub and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching cost control problem is characterized as the minimal nonnegative continuous viscosity supersolution of the sqvi

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

adaptive continuous higher order sliding mode control this paper is concerned with the development of an adaptation structure which can be applied to conventional super twisting and higher order sliding mode schemes the objective is to alter the modulation gains associated with these schemes in such a way that they are as small as possible to mitigate chattering effects but large enough to ensure that sliding can be maintained in the presence of bounded and derivative bounded uncertainties in all the proposed schemes the equivalent control is used to drive the adaptive mechanism the approach is based on a novel dual layer nested adaptive methodology which is quite different to the existing schemes proposed in the sliding mode literature the new adaptive schemes do not require knowledge of the minimum and maximum allowed values of the adaptive gain and in their most general form do not require information about the bound on the disturbances and their derivatives

supervisory acceleration of convergence for homogeneous systems the rate of convergence to the origin for a chain of integrators stabilized by homogeneous feedback is accelerated by a supervisory switching of control parameters the proposed acceleration algorithm ensures a fixed time convergence for otherwise exponentially or finite time stable homogeneous closed loop systems bounded disturbances are taken into account the results are especially useful in the case of exponentially stable systems widespread in the practice the proposed switching strategy is illustrated by computer simulation

fully discrete spectral methods for boundary integral equations on slender spheroids we develop fully discrete spectral boundary integral algorithms to solve interior and exterior potential boundary value problems on three dimensional slender spheroids our approach is based on identifying and using analytical solutions of associated generalised eigenvalue problems of boundary integral operators we generalise approximation theory results on the sphere using spheroidally appropriate eigenfunctions and solve an open problem of stability of spectral methods for first kind boundary integral equations on spheroids

inverse optimal design of input to state stabilizing nonlinear controllers we show that input to state stabilizability as defined by sontag 1989 1995 is both necessary and sufficient for the solvability of a hamilton jacobi isaacs equation associated with a meaningful differential game problem similar to but more general than the nonlinear h sub spl infin problem the significance of the result stems from the fact that constructive solutions to the input to state stabilization problem are available presented in the paper and that as shown here inverse optimal controllers possess margins on input to state stability against a certain class of input unmodeled dynamics rather than completion of squares the main tools in our analysis are legendre fenchel transformations and the general form of young s inequality

homogeneous time varying systems robustness analysis the problem of stability robustness with respect to time varying perturbations of a given frequency spectrum is studied applying homogeneity framework the notion of finite time stability over time intervals of finite length i e short finite time stability is introduced and used for that purpose the results are applied to analyze the super twisting algorithm sta behavior under time varying perturbations some simulation examples illustrate these robustness properties

computation of the h norm for nonlinear systems a convergence result in this paper we show that h norm for a class of nonlinear continuous time systems is the limit of the h norm for approximating discrete time systems the convergence result is proved by means of a characterization of the h norm in terms of the value for an ergodic control problem

mapped infinite elements for exterior wave problems recently a new type of infinite elements which uses r 1 decay was proposed they were applied to exterior wave problems and good results were obtained in two dimensional problems however it was necessary to move the origin of the r 1 decay in order to model the outgoing wave more accurately because it decays roughly as r 1 2 in this paper the mapped infinite elements with r 1 2 decay and the necessary numerical integration procedure are presented these elements do not require any artificial movement of the origin several example problems are solved the results show that the infinite elements with r 1 2 decay here give much more accurate values than the infinite elements with exponential decay and any damper elements

homogeneity approach to high order sliding mode design it is shown that a general uncertain single input single output regulation problem is solvable only by means of discontinuous control laws giving rise to the so called high order sliding modes the homogeneity properties of the corresponding controllers yield a number of practically important features in particular finite time convergence is proved and asymptotic accuracy is calculated in a very general way in the presence of input noises discrete measurements and switching delays a robust homogeneous differentiator is included in the control structure thus yielding robust output feedback controllers with finite time convergence it is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high order sliding mode controllers finally simulation results are presented

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

higher order finite and infinite elements for the solution of helmholtz problems abstract in recent years the focus in the field of acoustic finite element computations has shifted to the mid frequency regime at higher frequencies however the conventional finite elements with linear shape functions fail to provide reliable results due to so called pollution effects this motivates the use of higher order shape functions and related p fem concepts the solution of large systems of equations arising from engineering problems often involves the use of iterative solution procedures mostly by means of krylov subspace methods the performance of these methods strongly depends on the spectrum of the resulting system matrices which is affected by the polynomial approximation within the finite element formulation the current work shows that finite elements based on bernstein polynomials yield a favorable spectrum of the system matrix and particularly good performance in combination with commonly employed krylov solvers this is shown for interior as well as exterior helmholtz problems where an infinite element formulation is employed to account for the sound radiation

disturbance spaces and sup regulators in discrete time abstract sup output is defined as the supremum value of the absolute output for all time and all disturbances which are known only to the extent that they belong to a function space called the disturbance space a regulator which minimizes the sup output called the sup regulator is determined some relations are found between the disturbance space the sup regulator and the minimal sup output the results are extended to the case of complex disturbances tolerable disturbances are identified

the performance of spheroidal infinite elements a number of spheroidal and ellipsoidal infinite elements have been proposed for the solution of unbounded wave problems in the frequency domain i e solutions of the helmholtz equation these elements are widely believed to be more effective than conventional spherical infinite elements in cases where the radiating or scattering object is slender or flat and can therefore be closely enclosed by a spheroidal or an ellipsoidal surface the validity of this statement is investigated in the current article the radial order which is required for an accurate solution is shown to depend strongly not only upon the type of element that is used but also on the aspect ratio of the bounding spheroid and the non dimensional wave number the nature of this dependence can partially be explained by comparing the non oscillatory component of simple source solutions to the terms available in the trial solution of spheroidal elements numerical studies are also presented to demonstrate the rates at which convergence can be achieved in practice by unconjugated burnett and conjugated astley leis type elements it will be shown that neither formulation is entirely satisfactory at high frequencies and high aspect ratios

analytical approximation methods for the stabilizing solution of the hamilton jacobi equation in this paper two methods for approximating the stabilizing solution of the hamilton jacobi equation are proposed using symplectic geometry and a hamiltonian perturbation technique as well as stable manifold theory the first method uses the fact that the hamiltonian lifted system of an integrable system is also integrable and regards the corresponding hamiltonian system of the hamilton jacobi equation as an integrable hamiltonian system with a perturbation caused by control the second method directly approximates the stable flow of the hamiltonian systems using a modification of stable manifold theory both methods provide analytical approximations of the stable lagrangian submanifold from which the stabilizing solution is derived two examples illustrate the effectiveness of the methods

prolate and oblate spheroidal acoustic infinite elements in a previous paper the first author presented a new 3 d time harmonic prolate spheroidal acoustic infinite element for modeling acoustic fields in exterior domains here we present the development of the logical companion to that element viz an oblate spheroidal infinite element both elements are based on multipole expansions for the acoustic field in spheroidal coordinates developed by the second author in a separate paper recently submitted for publication for completeness an outline of the proofs of convergence of these expansions is included here also essential equations for the previously published prolate spheroidal infinite element are reproduced here in order that this paper can stand as an independent explication of the spheroidal family of infinite elements it is shown that a choice of prolate or oblate spheroidal infinite elements provides a capability for very efficiently modeling acoustic fields surrounding structures with a wide variety of practical shapes

effectiveness and robustness of improved infinite elements for exterior acoustics abstract the practical usability of a recently developed improved infinite element is assessed this improved formulation is based on the so called astley leis elements and especially suitable for the use in conjunction with iterative solvers up to now an extensive numerical study of the robustness and performance of iterative solvers employed in exterior acoustics simulations with infinite elements is missing this contribution summarizes the results of numerous simulations of practical relevance performed with the improved formulation and a standard alternative formulation the results show that the improved infinite elements perform better than the conventional formulation in nearly all simulations

a uniqueness result for p monotone viscosity solutions of hamilton jacobi equations in bounded domains we consider a class of hamilton jacobi equations h x du x 0 with no u dependence and with continuity properties consistent with recent applications in queueing theory continuous viscosity solutions are considered in a compact polyhedral domain with oblique derivative neumann type boundary conditions comparison and uniqueness results are presented which use monotonicity of h x p in the p variable for values of p in the appropriate sub and superdifferential sets of the solution u x several examples illustrate the results

analysis of chattering in continuous sliding mode controllers an analysis of two most popular continuous sliding mode algorithms the power fractional sliding mode algorithm and a second order sliding mode algorithm known as the super twisting is carried out in the frequency domain with the use of the describing function method it is shown that in the presence of an actuator the transient process converges to a periodic motion parameters of this periodic motion are analyzed a few examples are considered to illustrate the obtained results

ℒ2 gain tuning of variable structure siso systems of relative degree n abstractthe present paper extends the l2 gain analysis towards variable structure systems vss with sliding mode sm dynamics the developed analysis is applied to feedback linearisable vss driven by standard sm algorithms of the first order constructive parameter tuning of the resulting closed loop system to possess the l2 gain less than a desired attenuation level is additionally deduced the proposed tuning is shown to be capable not only of rejecting matched uniformly bounded disturbances but also of attenuating unbounded ones including mismatched disturbances numerical simulations of an inverted pendulum driven by a first order sm controller are involved to support the theory in the presence of the position measurement inaccuracy being a source of a mismatched disturbance

a time domain solution for complex multilayered soil model with circular inhomogeneity by the sbfem the scaled boundary finite method sbfem is developed for analyzing wave propagation problem in the two dimensional unbounded domain with rigid bedrock it combines the advantages of the finite element method and boundary element boundary method moreover the original scaling center is replaced by a scaling line which is more suitable for analyzing the multilayered soil model therefore the modified sbfem develops the original sbfem a new derivation of the modified sbfem equation is built in the frame of hamilton system a continued fraction solution of the dynamic stiffness of the soil model with bedrock is obtained for the first time then by introducing the continued fraction solution and auxiliary variables it leads to the model resting on bedrock can be solved in time domain the global equation of motion is solved by the efficient precise time integration method this integral method is firstly employed in the modified sbfem the precision of the proposed method can achieve computer precision therefore an extremely efficient and accurate solution of the modified sbfem in time domain is obtained the results of the complex soil model with circular inhomogeneity show that the proposed method yields excellent results and high accuracy is observed

h control stabilization and input output stability of nonlinear systems with homogeneous properties new relations are established between h control and stabilization and between input output stability and lyapunov stability of nonlinear systems with homogeneous properties these results can be applied to h controller design and stability analysis

mapped wave envelope elements for acoustical radiation and scattering wave envelope elements provide an alternative to boundary element methods for the solution of exterior wave problems applications to date have been to axisymmetric geometries and have involved the use of large but finite wave envelope meshes compatibly matched to denser near field meshes of conventional elements in the present paper the wave envelope idea is extended by the inclusion of an element mapping taken directly from infinite element methodology and by modifications to the residual formulation taken in conjunction these two measures permit the outer boundary of the wave envelope region to be moved to infinity while preserving the boundedness of the acoustic mass and stiffness integrals the present formulation is developed initially with reference to a simple one dimensional model and is then applied in two and three dimensions comparisons between computed results and those obtained from analytic solutions where applicable and from boundary element codes demonstrate the effectiveness of the method

geometric existence theory for the control affine h problem abstract we consider the differential game formulation of the nonlinear state feedback h control problem in which the control term enters linearly in the dynamics and quadratically in the cost under well known conditions on the linearisation of this problem around the equilibrium point at the origin there exists a stable lagrangian manifold this manifold has a generating function s quadratic at infinity a lusternick schnirelman minimax construction produces from s a lipschitz function w over state space we show that for problems in general position w is the lower value function for the h problem and prove existence of a weak globally optimal set valued feedback solution in terms of w the generalised gradient of w this feedback generalises to a maximal region over which is simply connected the classical smooth feedback defined on the neighbourhood of the origin over which has a well defined projection onto state space

brief paper robust stabilization of nonlinear systems via normalized coprime factor representations a reasonably complete theory for the synthesis of robust controllers for a broad class of nonlinear systems is now available we use this theory to generalize the linear theory of normalized coprime factor robustness optimization to the case of affine input nonlinear systems in particular we show that the equilibrium controller may be characterized in terms of the stabilizing and destabilizing solutions of the hamilton jacobi equation used to calculate the normalized right coprime factors of the plant we also show that the optimal robustness margin email protected 1 mn 2 hgeneralizes to the nonlinear case in preparation for the nonlinear analysis we review the linear case in a way which motivates our approach to the nonlinear case and highlights the parallels with it

reaching time estimation for super twisting second order sliding mode controller via lyapunov function designing a method of a lyapunov functions design based on resolving of the first order partial differential equation of a special type is presented a successful resolution of this equation guaranties a finite time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right hand side it is also provides an analytical formula for an upper bound of the corresponding reaching time the suggested method is applied to the problem of a stability analysis and a reaching time estimation of the so called ldquosuper twistingrdquo controller the estimation accuracy is confirmed by numerical example

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

a variational inequality for measurement feedback almost dissipative control in this paper an optimal measurement feedback control problem that yields an almost or practically dissipative closed loop system is considered that is the aim is to consider optimal control problems which when solved yield a closed loop system which almost satisfies a dissipation inequality the main idea is that by weakening the required dissipation inequality a broader class of open loop systems and controllers are admissible leading to broader application in obtaining the main results of this paper dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control this variational inequality can in principle be used to derive the optimal controller in the special case of certainty equivalence an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost dissipative control problem

finite time stability and robust control synthesis of uncertain switched systems stability analysis is developed for uncertain nonlinear switched systems while being asymptotically stable and homogeneous of degree q 0 these systems are shown to approach the equilibrium point in finite time restricted to second order systems this feature is additionally demonstrated to persist regardless of inhomogeneous perturbations based on this fundamental property switched control algorithms are then developed to globally stabilize uncertain minimum phase systems of uniform m vector relative degree 2 2 t the controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts the proposed synthesis procedure is illustrated via application to a friction servo motor

max plus eigenvector representations for solution of nonlinear h sub spl infin problems basic concepts the h sub spl infin problem for a nonlinear system is considered the corresponding dynamic programming equation is a fully nonlinear first order steady state partial differential equation pde possessing a term which is quadratic in the gradient the solutions are typically nonsmooth and further there is nonuniqueness among the class of viscosity solutions in the case where one tests a feedback control to see if it yields an h sub spl infin controller the pde is a hamilton jacobi bellman equation in the case where the optimal feedback control is being determined as well the problem takes the form of a differential game and the pde is in general an isaacs equation the computation of the solution of a nonlinear steady state first order pde is typically quite difficult in this paper we develop an entirely new class of methods for obtaining the correct solution of such pdes these methods are based on the linearity of the associated semigroup over the max plus or in some cases min plus algebra in particular solution of the pde is reduced to solution of a max plus or min plus eigenvector problem for known unique eigenvalue 0 the max plus multiplicative identity it is demonstrated that the eigenvector is unique and that the power method converges to it an example is included

controllability method for the helmholtz equation with higher order discretizations we consider a controllability technique for the numerical solution of the helmholtz equation the original time harmonic equation is represented as an exact controllability problem for the time dependent wave equation this problem is then formulated as a least squares optimization problem which is solved by the conjugate gradient method such an approach was first suggested and developed in the 1990s by french researchers and we introduce some improvements to its practical realization we use higher order spectral elements for spatial discretization which leads to high accuracy and lumped mass matrices higher order approximation reduces the pollution effect associated with finite element approximation of time harmonic wave equations and mass lumping makes explicit time stepping schemes for the wave equation very efficient we also derive a new way to compute the gradient of the least squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm numerical results demonstrate the significant improvements in efficiency due to the higher order spectral elements for a given accuracy spectral element method requires fewer computational operations than conventional finite element method in addition by using higher order polynomial basis the influence of the pollution effect is reduced

an ellipsoidal acoustic infinite element abstract in previous papers the authors presented new 3 d time harmonic prolate and oblate spheroidal acoustic infinite elements based on new prolate and oblate spheroidal multipole expansions for modeling acoustic fields in unbounded domains here we present the development of an ellipsoidal infinite element which is the logical generalization of those elements this development is also based on a new multipole expansion as well as a new system of ellipsoidal coordinates the element stiffness radiation damping and mass matrices are developed and presented in sufficient detail to enable their software implementation both the new coordinate system and the new element include the previous spheroidal coordinate systems and spheroidal elements as limiting cases therefore all the previously reported performance data for the spheroidal elements apply to this ellipsoidal element when used in spheroidal form since the three axes of an ellipsoid can be chosen independently an ellipsoid can circumscribe any structural shape at least as closely and generally more closely than a prolate or oblate spheroid the resulting reduction in size of the finite computational domain will result in even greater computational speeds than those already reported for the spheroidal elements the element may be used to model problems in free space 4 steradians half space 2 quarter space or eighth space 2 since this infinite element provides maximum computational efficiency for structures of all shapes software element libraries would need only this one element for all problems in unbounded domains

dynamic switching surfaces for output sliding mode control the robustness properties of sliding mode and h controllers are exploited to produce a dynamic output feedback controller that is insensitive to matched perturbations and attenuates the unmatched ones the assumptions on the plant differ from the standard assumptions of the riccati state space approach to h control the sliding mode controller drives the state into a reduced order manifold for which the equivalent system does satisfy the standard assumptions and hence the standard theory can be applied the resulting riccati equations are of reduced order

second order sliding mode observer for mechanical systems the super twisting second order sliding mode algorithm is modified in order to design a velocity observer for uncertain mechanical systems the finite time convergence of the observer is proved thus the observer can be designed independently of the controller a discrete version of the observer is considered and the corresponding accuracy is estimated

max plus summation of fenchel transformed semigroups for solution of nonlinear bellman equations abstract max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain advantages they still generally suffer from the curse of dimensionality the natural analog to the laplace transform in ordinary spaces is the fenchel transform over max plus spaces the range space being referred to as the dual space one can transform the semigroup operators into operators on the dual space there are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems natural building blocks correspond to transforms of operators for linear quadratic problems in this paper a method for exploiting operations in the fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time consuming portion of the computations can be hugely reduced although the curse of dimensionality is not entirely avoided the computational cost reductions are very high for some classes of problems

an alternative numerical implementation of the burnett family of acoustic infinite elements in a series of pioneering papers burnett and holford presented the formulation and numerical implementation of three dimensional acoustic infinite elements for modeling acoustic fields in exterior unbounded domains surrounding a structure they are all based on a multipole expansion outside a closed coordinate surface that circumscribes the structure the multipole expansion has a radial variable r and two angular variables that describe the surfaces r const in the finite element representation the pressure field is discretized using a product of radial and angular shape functions the angular variables are discretized directly using nodal angular values however in some cases and especially in problems involving symmetry nodes are located in certain positions where one of the angular variables is undefined thus leading to errors in the value of the angular variable in the interior of the elements connected to these nodes and hence in the computed acoustic field to overcome this problem we propose an alternative numerical formulation in this work which is based on the interpolation of the position vector expressed in the global cartesian coordinate system as in conventional finite elements instead of the interpolation of angular variables we present numerical solutions for a few benchmark problems using the proposed method

mapped spheroidal wave envelope elements for unbounded wave problems this paper describes a family of axisymmetric spheroidal wave envelope elements for modelling exterior wave problems they are of variable radial order and can be used to represent steady and transient wave fields the formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries these offer the prospect of reduced dimensionality in comparison to conventional spherically formulated elements when used to represent wave fields in the vicinity of slender or flat objects conjugated weighting functions are used to give frequency independent acoustic mass stiffness and damping matrices this facilitates a simple extension of the method to transient problems the effectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound fields generated by a rigid sphere in steady harmonic oscillation by a rigid sphere excited from rest and by a circular plate vibrating in a plane baffle

sliding mode based differentiation and filtering the proposed k th order filter is based on sliding modes sms it exactly estimates k input derivatives in the absence of noises and is robust to noises of small magnitudes or having small average values estimation accuracy asymptotics are calculated the filter is applied to the real time accurate estimation of the equivalent control in sm control systems