experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

an unfitted finite element method for the numerical approximation of void electromigration microelectronic circuits usually contain small voids or cracks and if those defects are large enough to sever the line they cause an open circuit we present a numerical method for investigating the migration of voids in the presence of both surface diffusion and electric loading our mathematical model involves a bulk interface coupled system with a moving interface governed by a fourth order geometric evolution equation and a bulk where the electric potential is computed thanks to a novel approximation of the interface equidistribution of its vertices is guaranteed and no remeshing is necessary in addition the used curvature approximation means that our method is unconditionally stable for the evolution by surface diffusion only various examples are performed to demonstrate the accuracy of the method

stability and incremental improvement of suboptimal mpc without terminal constraints the stability of suboptimal model predictive control mpc without terminal constraints is investigated for continuous time nonlinear systems under input constraints exponential stability and decay of the optimization error are guaranteed if the number of optimization steps in each sampling instant satisfies a lower bound that depends on the convergence ratio of the underlying optimization algorithm the decay of the optimization error shows the incremental improvement of the suboptimal mpc scheme

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

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

partitioned versus global krylov subspace iterative methods for fe solution of 3 d biot s problem finite element analysis of 3 d biot s consolidation problem needs fast solution of discretized large 2 2 block symmetric indefinite linear systems in this paper partitioned iterative methods and global krylov subspace iterative methods are investigated and compared the partitioned iterative methods considered include stationary partitioned iteration and non stationary prevost s pcg procedure the global krylov subspace methods considered include minres and symmetric qmr sqmr two efficient preconditioners are proposed for global methods numerical experiments based on a pile group problem and simple footing problems with varied soil profiles are carried out numerical results show that when used in conjunction with suitable preconditioners global krylov subspace iterative methods are more promising for large scale computations and further improvement could be possible if significant differences in the solid material properties are addressed in these preconditioned iterative methods 2007 elsevier b v all rights reserved

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

model predictive control of continuous time nonlinear systems with piecewise constant control a new model predictive control mpc algorithm for nonlinear systems is presented the plant under control the state and control constraints and the performance index to be minimized are described in continuous time while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times in so doing the optimization is performed with respect to sequences as in discrete time nonlinear mpc but the continuous time evolution of the system is considered as in continuous time nonlinear mpc

a real time iteration scheme for nonlinear optimization in optimal feedback control an efficient newton type scheme for the approximate on line solution of optimization problems as they occur in optimal feedback control is presented the scheme allows a fast reaction to disturbances by delivering approximations of the exact optimal feedback control which are iteratively refined during the runtime of the controlled process the contractivity of this real time iteration scheme is proven and a bound on the loss of optimality compared with the theoretical optimal solution is given the robustness and excellent real time performance of the method is demonstrated in a numerical experiment the control of an unstable system namely an airborne kite that shall fly loops

unconstrained receding horizon control of nonlinear systems it is well known that unconstrained infinite horizon optimal control may be used to construct a stabilizing controller for a nonlinear system we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control the key idea is to approximate the tail of the infinite horizon cost to go using as terminal cost an appropriate control lyapunov function roughly speaking the terminal control lyapunov function clf should provide an incremental upper bound on the cost in this fashion important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers the absence of constraints allows a significant speedup in computation furthermore it is shown that in order to guarantee stability it suffices to satisfy an improvement property thereby relaxing the requirement that truly optimal trajectories be found we provide a complete analysis of the stability and region of attraction operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class it is shown that the guaranteed region of operation contains that of the clf controller and may be made as large as desired by increasing the optimization horizon restricted of course to the infinite horizon domain moreover it is easily seen that both clf and infinite horizon optimal control approaches are limiting cases of our receding horizon strategy the key results are illustrated using a familiar example the inverted pendulum where significant improvements in guaranteed region of operation and cost are noted

further results on robustness of possibly discontinuous sample and hold feedback we demonstrate that sample and hold state feedback control possibly discontinuous with respect to the state is robust when the closed loop system possesses an appropriate lyapunov function we first show that if a lyapunov decrease over sampling periods exists for the nominal system this decrease can be maintained with some degradation relative to a sufficiently small additive perturbation we then proceed to catalog several applications of this robustness e g robustness to measurement noise computational delays or fast actuator dynamics

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

element based preconditioners for elasto plastic problems in geotechnical engineering iterative solvers are widely regarded as the most efficient way to solve the very large linear systems arising from finite element models their memory requirements are small compared to those for direct solvers consequently there is a major interest in iterative methods and particularly the preconditioning necessary to achieve rapid convergence in this paper we present new element based preconditioners specifically designed for linear elasticity and elasto plastic problems the study presented here is restricted to simple associated plasticity but should find wide application in other plasticity models used in geotechnics

receding horizon control of nonlinear systems the receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems the method is especially useful for the control of slow nonlinear systems such as chemical batch processes where it is possible to solve sequentially open loop fixed horizon optimal control problems online the method has been shown to yield a stable closed loop system when applied to time invariant or time varying linear systems it is shown that the method also yields a stable closed loop system when applied to nonlinear systems

a continuation gmres method for fast computation of nonlinear receding horizon control in this paper a fast numerical algorithm for nonlinear receding horizon control is proposed the control input is updated by a differential equation to trace the solution of an associated state dependent two point boundary value problem a linear equation involved in the differential equation is solved by the generalized minimum residual method one of the krylov subspace methods with jacobians approximated by forward differences the error in the entire algorithm is analyzed and is shown to be bounded under some conditions the proposed algorithm is applied to a two link arm whose dynamics is highly nonlinear simulation results show that the proposed algorithm is faster than the conventional algorithms

block preconditioners for symmetric indefinite linear systems this paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a krylov subspace method for solving symmetric indefinite linear systems the focus is on large scale real world problems where block approximations are a practical necessity the main illustration is the performance of the block diagonal constrained and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large scale finite element discretization of biot s consolidation equations this system of equations is of fundamental importance to geomechanics numerical studies show that simple diagonal approximations to the 1 1 block k and inexpensive approximations to the schur complement matrix s may not always produce the most spectacular time savings when k is explicitly available but is able to deliver reasonably good results on a consistent basis in addition the block diagonal preconditioner with a negative 2 2 block appears to be reasonably competitive when compared to the more complicated ones these observation are expected to remain valid for coefficient matrices whereby the 1 1 block is sparse diagonally significant a notion weaker than diagonal dominance moderately well conditioned and has a much larger block size than the 2 2 block copyright 2004 john wiley sons ltd

a modified ssor preconditioner for sparse symmetric indefinite linear systems of equations the standard ssor preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the biot s consolidation equations in this paper a modified block ssor preconditioner combined with the eisenstat trick implementation is proposed for actual implementation a pointwise variant of this modified block ssor preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness numerical experiments show that the proposed modified ssor preconditioned symmetric qmr solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime moreover the proposed modified ssor preconditioners can be generalized to non symmetric biot s systems copyright 2005 john wiley sons ltd

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

performance and robustness of block constraint preconditioners in finite element coupled consolidation problems block constraint preconditioners are a most recent development for the iterative solution to large scale often ill conditioned coupled consolidation problems a major limitation to their practical use however is the somewhat difficult selection of a number of user defined parameters at least 4 in a more or less optimal way the present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters a theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters a number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner mcp combined with the symmetric quasi minimal residual sqmr solver and the mcp triangular variant t mcp combined with the bi conjugate gradient stabilized bi cgstab are efficient and robust tools for the solution to difficult finite element discretized coupled consolidation problems moreover the practical selection of the user defined parameters is relatively easy as a stable behavior is observed for a wide range of fill in degree values the theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination copyright 2009 john wiley sons ltd

novel preconditioners for the iterative solution to fe discretized coupled consolidation equations a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by an inexact constraint preconditioner icp preserving the same block structure as the native fe matrix the conditioning number of the preconditioned coupled problem depends on the quality of the approximation of the block corresponding to the structural stiffness matrix an efficient algorithm to implement icp into a krylov subspace method is developed numerical tests performed on realistic 3d problems reveal that icp typically outperforms standard ilut preconditioners and proves much more robust in severely ill conditioned problems

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

relaxed logarithmic barrier function based model predictive control of linear systems in this paper we investigate the use of relaxed logarithmic barrier functions in the context of linear model predictive control we present results that allow to guarantee asymptotic stability of the corresponding closed loop system and discuss further properties like performance and constraint satisfaction in dependence of the underlying relaxation the proposed stabilizing mpc schemes are not necessarily based on an explicit terminal set or state constraint and allow to characterize the stabilizing control input sequence as the minimizer of a globally defined continuously differentiable and strongly convex function the results are illustrated by means of a numerical example

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

mixed constraint preconditioners for the iterative solution of fe coupled consolidation equations the finite element fe integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix because of ill conditioning the repeated solution in time of the fe equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of krylov subspace methods in the present paper a mixed constraint preconditioner mcp is developed combining implicit and explicit approximations of the inverse of the structural sub matrix with the performance investigated in some representative examples an upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner the mcp is efficiently implemented into a krylov subspace method with the performance obtained in 2d and 3d examples compared to that of inexact constraint preconditioners and least square logarithm scaled ilut preconditioners two variants of mcp t mcp and d mcp developed with the aim at reducing the cost of the preconditioner application are also tested the results show that the mcp variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation fe models and especially so in severely ill conditioned problems

solving large scale semidefinite programs via an iterative solver on the augmented systems the search directions in an interior point method for large scale semidefinite programming sdp can be computed by applying a krylov iterative method to either the schur complement equation sce or the augmented equation both methods suffer from slow convergence as interior point iterates approach optimality numerical experiments have shown that a diagonally preconditioned conjugate residual method on the sce typically takes a huge number of steps to converge however it is difficult to incorporate cheap and effective preconditioners into the sce this paper proposes to apply the preconditioned symmetric quasi minimal residual psqmr method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior point iterates numerical experiments on sdp problems arising from maximum clique and selected sdplib sdp library problems show that moderately accurate solutions can be obtained with a modest number of psqmr steps using the proposed preconditioned reduced augmented equation an sdp problem with 127600 constraints is solved in about 6 5 hours to an accuracy of 10 6 in relative duality gap

software and algorithms for sensitivity analysis of large scale differential algebraic systems sensitivity analysis for dae systems is important in many engineering and scientific applications the information contained in the sensitivity trajectories is useful for parameter estimation optimization model reduction and experimental design in this paper we present algorithms and software for sensitivity analysis of large scale dae systems of index up to two the new software provides for consistent initialization of the solutions and the sensitivities interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations and is capable via mpi of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations

a fixed point iteration scheme for real time model predictive control a simple model predictive control mpc concept for nonlinear systems under input constraints is considered the presented algorithm takes advantage of an mpc formulation without terminal constraints in order to solve the optimality conditions by a fixed point iteration scheme that is easy to implement and of algorithmic simplicity sufficient conditions for the contraction of the fixed point iterations are derived to allow for a real time implementation within an mpc scheme a constant number of fixed point iterations is used in each sampling step and sufficient conditions for asymptotic stability and incremental reduction of the suboptimality are presented

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

a qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

a real time framework for model predictive control of continuous time nonlinear systems a new formulation of model predictive control mpc for continuous time nonlinear systems is developed which allows for the use of ldquoreal timerdquo rt optimization techniques in which the solution to the finite horizon optimal control problem opc evolves within the same timescale as the process dynamics the computational savings of the rt solver are enhanced by the unique framework within which the opc is posed enabling significant reduction in the dimensionality of the search for situations where computational speed takes priority over optimality of the solutions this framework and its associated proof of stability encompasses results on sampled data sd nonlinear model predictive control nmpc implementation as a special case

barrier function based model predictive control a new formulation of nonlinear model predictive control mpc is developed by including a weighted barrier function in the control objective while the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set in addition the resulting optimisation problem to be solved at each control step is effectively unconstrained and thus amenable to elegant optimisation techniques the barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier conventional mpc may be seen as a limiting case of the new class as the barrier weighting itself approaches zero we pay particular attention to the novel approach of fixing the weighting parameter to some positive value possibly large and observe that this provides a degree of controller caution near constraint boundaries we construct an ellipsoidal invariant set by exploiting the geometry of self concordant functions and show nominal closed loop stability for this class of controllers under full state feedback

suboptimal model predictive control feasibility implies stability practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex nonlinear optimization problems are possible in limited computational time in the paper we first establish conditions under which suboptimal model predictive control mpc controllers are stabilizing the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost second we present and analyze two suboptimal mpc schemes that are guaranteed to be stabilizing provided an initial feasible solution is available and for which the computational requirements are more reasonable

existence and multiplicity of symmetric positive solutions for three point boundary value problem in this paper we are concerned with the existence and multiplicity of symmetric positive solutions for the following second order three point boundary value problem u t a t f t u t 0 0 t 1 u t u 1 t u 0 u 1 u 1 2 where a 0 1 0 is symmetric on 0 1 and maybe singular at t 0 and t 1 f 0 1 0 0 is continuous and f u is symmetric on 0 1 for all u 0 growth conditions are imposed on f which yield the existence of at least one or at least two positive solutions our proof based on krasnoselskii s fixed point theorem in a cone