an iterative adaptive finite element method for elliptic eigenvalue problems we consider the task of resolving accurately the nth eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation pde using an adaptive finite element method fem conventional adaptive fem algorithms call a generalized eigensolver after each mesh refinement step this is not practical in our situation since the generalized eigensolver needs to calculate n eigenpairs after each mesh refinement step it can switch the order of eigenpairs and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace in order to circumvent these problems we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation and then employs an iterative method to pursue a selected eigenvalue eigenfunction pair on a sequence of locally refined meshes both picard s and newton s variants of the iterative method are presented the underlying partial differential equation pde is discretized with higher order finite elements hp fem but the algorithm also works for standard low order fem the method is described and accompanied with theoretical analysis and numerical examples instructions on how to reproduce the results are provided

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

a posteriori and a priori error analysis for finite element approximations of self adjoint elliptic eigenvalue problems we present a new error analysis for finite element approximations of self adjoint elliptic eigenvalue problems the analysis consists of three steps first we prove a posteriori estimates for the error in the approximate eigenvectors and eigenvalues the error in the eigenvectors is measured both in the l2 and energy norms in these estimates the error is bounded in terms of the mesh size a stability factor and the residual obtained by inserting the approximate eigenpair into the differential equation the stability factor describes relevant stability properties of the continuous problem and we give a precise estimate of its size in terms of the spectrum of the continuous problem the mesh size and the choice of norm next we prove an a priori estimate of the residual in terms of derivatives of the exact eigenvectors and the mesh size finally we obtain precise a priori error estimates by combination of the a posteriori error estimates with the a priori residual estimate the analysis shows that the a posteriori estimates are optimal and may be used for quantitative error estimation and design of adaptive algorithms

a numerically robust state space approach to stable predictive control strategies recent work with predictive control strategies has shown that the stable gpc sgpc approach has significant computational and numerical advantages however sgpc is cast in the transfer function framework which limits its application here we develop a means of extending the results to state space models and show that improved numerical conditioning can be obtained for many stable predictive control strategies

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

a posteriori error estimators for convection diffusion eigenvalue problems a posteriori error estimators for convection diffusion eigenvalue model problems are discussed in heuveline and rannacher 2001 17 in the context of the dual weighted residual method dwr this paper directly addresses the variational formulation rather than the non linear ansatz of becker and rannacher for some convection diffusion model problem and presents a posteriori error estimators for the eigenvalue error based on averaging techniques two different postprocessing techniques attached to the dwr paradigm plus two new dual weighted a posteriori error estimators are also presented the first new estimator utilises an auxiliary raviart thomas mixed finite element method and the second exploits an averaging technique in combination with ideas of dwr the six a posteriori error estimators are compared in three numerical examples and illustrate reliability and efficiency and the dependence of generic constants on the size of the eigenvalue or the convection coefficient

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

properties of generalized predictive control control design using long range prediction based on a dynamic model of the plant has become an important contender for high performance applications the model can be derived analytically but more typically it is provided by simple experiments or in the adaptive context by recursive estimation many methods have been proposed in the literature depending on the assumed model structure and the choice of cost function here the approach called generalized predictive control or gpc is reviewed of special interest is the derivation of prediction equations using an observer polynomial and a demonstration that the selection of particular horizons the costing horizons and the control horizon leads to well understood basic techniques such as dead beat pole placement and lq simulated examples show that gpc is suitable for controlling a complex plant such as unstable inverse unstable systems and how the observer polynomial independently tailors the response to the disturbances preprogrammed set points as with robot trajectory control and actuator nonlinearities can also be catered for a discussion of adaptive applications of gpc to several industrial process concludes that the method is easy to use and effective

brief a receding horizon approach to the nonlinear h control problem the receding horizon rh methodology is extended to the design of a robust controller of h type for nonlinear systems using the nonlinear analogue of the fake h algebraic riccati equation we derive an inverse optimality result for the rh schemes for which increasing the horizon causes a decrease of the optimal cost function this inverse optimality result shows that the input output map of the closed loop system obtained with the rh control law has a bounded l 2 gain robustness properties of the nonlinear h control law in face of dynamic input uncertainty are considered

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

a convergent adaptive method for elliptic eigenvalue problems we prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second order symmetric elliptic partial differential operators the weak form is assumed to yield a bilinear form which is bounded and coercive in h 1 each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation the error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem because of this nonlinearity the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine

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

adaptive discontinuous galerkin methods for eigenvalue problems arising in incompressible fluid flows in this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible navier stokes equations particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries here computable a posteriori error bounds are derived based on employing the generalization of the standard dual weighted residual approach originally developed for the estimation of target functionals of the solution to eigenvalue stability problems the underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution in this way errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented

explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures an a posteriori error estimator is developed for the eigenvalue analysis of heterogeneous elastic structures it constitutes an extension of a well known explicit estimator to heterogeneous structures we prove that our estimates are independent of the variations in material properties and independent of the polynomial degree of finite elements finally we study numerically the effectivity of this estimator on several model problems

a remark on abstract multiplier conditions for robustness problems abstract this paper presents a generalization of recent results of acikmese and corless b acikmese m corless stability analysis with quadratic lyapunov functions some necessary and sufficient multiplier conditions systems control letters 57 2008 78 94 concerning multiplier conditions of quadratic stability of uncertain nonlinear systems abstract full block s procedure results are formulated extending the results of scherer c w scherer lpv control and full block multipliers automatica 37 2001 361 375 for such cases when the sets representing the uncertainties do not necessarily have subspace structure the main contribution of the present work is the investigation of the conditions under which the results of scherer and acikmese and corless can be formulated in a unified framework

robust dynamic programming for min max model predictive control of constrained uncertain systems we address min max model predictive control mpc for uncertain discrete time systems by a robust dynamic programming approach and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the method uses polyhedral representations of the cost to go functions and feasible sets and performs multiparametric programming by a duality based approach in each recursion step we show how to apply the method to robust mpc and give conditions guaranteeing closed loop stability finally we apply the method to a tutorial example a parking car with uncertain mass

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

adaptive finite element methods for computing band gaps in photonic crystals in this paper we propose and analyse adaptive finite element methods for computing the band structure of 2d periodic photonic crystals the problem can be reduced to the computation of the discrete spectra of each member of a family of periodic hermitian eigenvalue problems on a unit cell parametrised by a two dimensional parameter the quasimomentum these eigenvalue problems involve non coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements we propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family in particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate we also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero then so must the error estimator the results hold for eigenvalues of any multiplicity we illustrate the benefits of the resulting adaptive method in practice both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect using the supercell method

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

an improved moving horizon control scheme through lagrange duality this paper presents a computationally tractable moving horizon control scheme where the to be on line solved minimax problem is relaxed to a minimization problem by lagrange duality this helps the avoidance of infeasibility and reduction of conservation caused by the ellipsoid evaluation of time domain constraints disturbance attenuation with performance adaptation is achieved by on line monitoring of the dissipation level and accumulating the energy reserve from performance relaxations that are necessary for the satisfaction of time domain constraints simulation and comparison results confirm the advantages of the proposed algorithm

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

application of a multiplier method to uncertain lur e like systems abstract this paper investigates the conditions under which an abstract matrix multiplier method can be applied to determine guaranteeing cost controls for systems containing nonlinear uncertain elements via linear matrix inequalities lmis quadratically constrained uncertainties and nonlinearities are considered which comprehend the cases of norm bounded positive real and sector bounded uncertainties nonlinearities both the discrete time and the continuous time cases are discussed necessary and sufficient conditions are formulated in the case of unstructured uncertainty the conditions are sufficient in the structured case the cost guaranteeing controls can be determined by solving lmis numerical examples illustrate the effectiveness of the proposed method

coefficients of the singularities for elliptic boundary value problems on domains with conical points iii finite element methods on polygonal domains in the two first parts of this work rairo model math anal numer 24 1990 pp 27e52 rairo model math anal numer 24 1990 pp 343 367 formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated now for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains the solution of such problems by the finite element method is considered in order to approximate the solution or the coefficients different methods are used based on the expressions of the coefficients that were obtained in the first two parts the dual singular function method is also generalized

hp finite elements for elliptic eigenvalue problems error estimates which are explicit with respect to lambda h and p convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form if the mesh width h is fine enough then the eigenvalues resp eigenfunctions converge at some well defined rate in this paper we will determine the maximal mesh width h 0 more precisely the minimal dimension of a finite element space so that the asymptotic convergence estimates hold for h leq h 0 this mesh width will depend on the size and spacing of the exact eigenvalues the spatial dimension and the local polynomial degree of the finite element space for example in the one dimensional case the condition lambda 3 4 h 0 lesssim1 is sufficient for piecewise linear finite elements to compute an eigenvalue lambda with optimal convergence rates as h 0 geq h rightarrow0 it will turn out that the condition for eigenfunctions is slightly more restrictive furthermore we will analyze the dependence of the ratio of the errors of the galerkin approximation and of the best approximation of an eigenfunction on lambda and h in this paper the error estimates for the eigenvalue function are limited to the selfadjoint case however the regularity theory and approximation property cover also the nonselfadjoint case and hence pave the way towards the error analysis of nonselfadjoint eigenvalue function problems

an adaptive homotopy approach for non selfadjoint eigenvalue problems this paper presents adaptive algorithms for eigenvalue problems associated with non selfadjoint partial differential operators the basis for the developed algorithms is a homotopy method which departs from a well understood selfadjoint problem apart from the adaptive grid refinement the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources the first algorithm balances the homotopy discretization and approximation errors with respect to a fixed stepsize in the homotopy the second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance e the outcome of the analysis leads to the third algorithm which allows the complete adaptivity in space homotopy stepsize as well as the iterative algebraic eigenvalue solver all three algorithms are compared in numerical examples

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

optimal under actuated kinematic motion planning on the ϵ group abstract a global motion planning method is described based on the solution of minimum energy type curves on the frame bundle of connected surfaces of arbitrary constant cross sectional curvature applying the geometric framing of pontryagin s principle gives rise to necessary conditions for optimality in the form of a boundary value problem this arbitrary dimensional boundary value problem is solved using a numerical shooting method derived from a general lax pair solution the paper then specializes to the 3 dimensional case where the lax pair equations are integrable a semi analytic method for matching the boundary conditions is proposed by using the analytic form of the extremal solutions and a closed form solution for the exponential map this semi analytical approach has the advantage that an analytic description of the control accelerations can be derived and enables actuator constraints to be incorporated via time reparametrization the method is applied to two examples in space mechanics the attitude control of a spacecraft with two reaction wheels and the spacecraft docking problem

a feedback min max mpc algorithm for lpv systems subject to bounded rates of change of parameters a novel closed loop model based predictive control mpc strategy for input saturated polytopic linear parameter varying lpv discrete time systems is proposed it is postulated that the plant belongs to a polytopic family of linear systems each member of which being parameterized by the value that a parameter vector assumes in the unit simplex such a parameter can be measured online and used for feedback while a bound on its rate of change is known and exploited for predictions the paper extends the mpc scheme presented by lu et al 2000 for the restricting case of 1 step long control horizons to the general case of control horizons of arbitrary length n this is done by suitably modifying the robust mpc scheme presented by casavola et al 2000 for uncertain polytopic systems the feasibility and closed loop stability of this strategy are proved and a numerical example is also presented in order to show how the freedom of extending the control horizon and knowledge of the parameter is significant in order to improve the performance of the control strategy

nonconvex optimization problem the infinite horizon linear quadratic control problem with quadratic constraints abstract we present a general method for the solution of the problem of minimization of a quadratic functional in a hilbert space over the intersection of some subspace and a set defined by special quadratic constraints many optimal control problems can be formulated in this abstract form as an example a stationary infinite horizon linear quadratic optimization problem with quadratic constraints is posed and solved

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

quasi min max mpc algorithms for lpv systems in this paper a new model predictive controller mpc is developed for polytopic linear parameter varying lpv systems we adopt the paradigm used in gain scheduling and assume that the time varying parameters are measured on line but their future behavior is uncertain and contained in a given polytope at each sampling time optimal control action is computed by minimizing the upper bound on the quasi worst case value of an infinite horizon quadratic objective function subject to constraints on inputs and outputs the mpc algorithm is called quasi because the first stage cost can be computed without any uncertainty this allows the inclusion of the first move u k k separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints proposed optimization problems are solved by semi definite programming involving linear matrix inequalities it is shown that closed loop stability is guaranteed by the feasibility of the linear matrix inequalities a numerical example demonstrates the unique features of the mpc design

path planning on compact lie groups using a homotopy method abstract in this paper we address the issue of solving the motion planning problems mpp for short for a class of left invariant control systems whose state spaces are semisimple compact lie groups the sub riemannian metrics induced by the dynamics of admit nontrivial abnormal extremals this fact a priori represents an obstruction for the procedure we use to tackle the mpp which consists of a homotopy or continuation method we are however able to provide complete answers for the mpp

robust model predictive control using tubes a form of feedback model predictive control mpc that overcomes disadvantages of conventional mpc but which has manageable computational complexity is presented the optimal control problem solved on line yields a tube and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty computational complexity is linear rather than exponential in horizon length asymptotic stability of the controlled system is established

min max feedback model predictive control for constrained linear systems min max feedback formulations of model predictive control are discussed both in the fixed and variable horizon contexts the control schemes the authors discuss introduce in the control optimization the notion that feedback is present in the receding horizon implementation of the control this leads to improved performance compared to standard model predictive control and resolves the feasibility difficulties that arise with the min max techniques that are documented in the literature the stabilizing properties of the methods are discussed as well as some practical implementation details

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

on linear programming and robust modelpredictive control using impulse responses abstract campo and morari have derived a linear programming problem with a potentially large number of constraints which is equivalent to a min max formulation for robust model predictive control of linear systems that formulation involves minimization with respect to the controls of the maximum with respect to the system s impulse response from a set of possible impulse responses of the infinity norm of the error between the predicted and required system output sequences here an alternative linear programming problem is derived which has a smaller number of constraints and is therefore potentially more convenient for on line control

convergence and quasi optimality of adaptive finite element methods for harmonic forms numerical computation of harmonic forms typically called harmonic fields in three space dimensions arises in various areas including computer graphics and computational electromagnetics the finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms in this work we study the convergence properties of adaptive finite element methods afem for computing harmonic forms we show that a properly defined afem is contractive and achieves optimal convergence rate beginning from any initial conforming mesh this result is contrasted with related afem convergence results for elliptic eigenvalue problems where the initial mesh must be sufficiently fine in order for afem to achieve any provable convergence rate

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

nonconforming finite element analysis for poisson eigenvalue problem the main aim of this paper is to study a nonconforming quadrilateral finite element named modified quasi wilson element approximation to poisson eigenvalue problem firstly by employing a special property of this element when u h 3 the consistency error is of order o h 2 which is one order higher than its interpolation error o h and the interpolation postprocessing technique the superclose and superconvergence results of order o h 2 for the exact solution of eigenvector u in broken h 1 norm are deduced on generalized rectangular meshes and rectangular meshes respectively secondly it is proved that the consistency error even can reach o h 4 order for arbitrary quadrilateral meshes when u h 5 this is a new astonishing feature which has never been discovered subsequently based on the above characteristic and some asymptotic expansions of the conforming bilinear finite element the extrapolation solution of order o h 4 for eigenvalue is derived finally some numerical results are provided to verify the theoretical analysis

a feasible moving horizon cal h _ infty control scheme for constrained uncertain linear systems this note presents a robust moving horizon hinfin control scheme that achieves disturbance attenuation with performance adaptation or for a prescribed level the on line solved optimization problem is a lagrange relaxation of the constrained minimax problem with the benefit of achieving feasibility in the moving horizon implementation uncertainties are incorporated by the use of the full block multiplier technique simulation and comparison results are given

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

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a multilevel correction type of adaptive finite element method for eigenvalue problems an adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme different from the standard adaptive finite element method which requires solving ei

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

a global steering method for nonholonomic systems abstract in this paper we present an iterative steering algorithm for nonholonomic systems also called driftless control affine systems and we prove its global convergence under the sole assumption that the lie algebraic rank condition larc holds true everywhere that algorithm is an extension of the one introduced in jean et al 2005 21 for regular systems the first novelty here consists in the explicit algebraic construction starting from the original control system of a lifted control system which is regular the second contribution of the paper is an exact motion planning method for nilpotent systems which makes use of sinusoidal control laws and which is a generalization of the algorithm described in murray and sastry 1993 29 for chained form systems

an optimal adaptive fem for eigenvalue clusters the analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods a first analysis for multiple eigenvalues of the recent work dai et al arxiv preprint 1210 1846v2 2013 introduces an adaptive method whose marking strategy is based on the element wise sum of local error estimator contributions for multiple eigenvalues this paper proves the optimality of a practical adaptive algorithm based on a lowest order conforming finite element method for eigenvalue clusters for the eigenvalues of the laplace operator in terms of nonlinear approximation classes all estimates are explicit in the initial mesh size the eigenvalues and the cluster width to clarify the dependence of the involved constants

a new nonconforming mixed finite element scheme for second order eigenvalue problem a new nonconforming mixed finite element method mfem for short is established for the laplace eigenvalue problem firstly the optimal order error estimates for both the eigenvalue and eigenpair the original variable u and the auxiliary variable p u are deduced the lower bound of eigenvalue is estimated simultaneously then by use of the special property of the nonconforming e q 1 r o t element the consistency error is of order o h2 in broken h1 norm which is one order higher than its interpolation error the techniques of integral identity and interpolation postprocessing we derive the superclose and superconvergence results of order o h2 for u in broken h1 norm and p in l2 norm furthermore with the help of asymptotic expansions the extrapolation solution of order o h3 for eigenvalue is obtained finally some numerical results are presented to validate our theoretical analysis

brief an improved approach for constrained robust model predictive control in this paper we present a new technique to address constrained robust model predictive control the main advantage of this new approach with respect to other well known techniques is the reduced conservativeness specifically the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several lyapunov functions each one corresponding to a different vertex of the uncertainty s polytope

an oscillation free adaptive fem for symmetric eigenvalue problems a refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first order adaptive finite element method afem is presented the h 1 stability of the l 2 projection provides reliability and efficiency of the edge contribution of standard residual based error estimators for p 1 finite element methods in fact the volume contributions and even oscillations can be omitted for courant finite element methods this allows for a refined averaging scheme and so improves mao et al in adv comput math 25 1 3 135 160 2006 the proposed afem monitors the edge contributions in a bulk criterion and so enables a contraction property up to higher order terms and global convergence numerical experiments exploit the remaining l 2 error contributions and confirm our theoretical findings the averaging schemes show a high accuracy and the afem leads to optimal empirical convergence rates

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored