the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

order statistics and probabilistic robust control abstract order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution also the concept of distribution free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution to overcome the limitations imposed by the continuity assumption in the existing order statistics theory we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied by applying this inequality we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound or the lower bound and the range of a quantity we also give conditions which are much weaker than the absolute continuity assumption for the existence of such minimum sample size furthermore the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established this guideline can be applied in general without continuity assumption

factorizing the factorization a spectral element solver for elliptic equations with linear operation count abstract the paper proposes a novel factorization technique for static condensation of a spectral element discretization matrix that yields a linear operation count of just 13n multiplications for the residual evaluation where n is the total number of unknowns in comparison to previous work it saves a factor larger than 3 and outpaces unfactored variants for all polynomial degrees using the new technique as a building block for a preconditioned conjugate gradient method yields linear scaling of the runtime with n which is demonstrated for polynomial degrees from 2 to 32 this makes the spectral element method cost effective even for low polynomial degrees moreover the dependence of the iterative solution on the element aspect ratio is addressed showing only a slight increase in the number of iterations for aspect ratios up to 128 hence the solver is very robust for practical applications

robustness analysis with full structured uncertainties this paper deals with problems involving full structured rather than block diagonal uncertainty i e uncertainty blocks where each sub block or element may be an independent uncertainty rearranging this problem into standard block diagonal form results in an explosion in the required computation and so a number of researchers have proposed more efficient bounds in particular those based on non similarity scaling for such problems here we show how to map such problems into reduced size standard problems the standard bounds applied to the reduced size problem are then shown to be at least as accurate and require the same computational effort as earlier techniques with the added bonus that the standard upper bound is convex moreover this new approach is applicable in a much more general setting allowing one to efficiently compute both robust stability and robust performance for problems involving multiple real and complex uncertainty blocks any number of which may be full structured

symmetric linear systems an application of algebraic systems theory dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations to the stability augmentation of pairs of helicopters lifting a large mass linear models for such systems display certain obvious symmetries in this paper we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems such systems are inherently related to the representation theory of algebras over fields we will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra

distributed control for identical dynamically coupled systems a decomposition approach we consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems if the state space matrices of these systems satisfy a certain structural property then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant this procedure is basically a multiobjective optimization under linear matrix inequality constraints with system norms as performance indices the explicit expressions for computing these controllers are given for both h infin or h 2 performance and both for static state feedback and dynamic output feedback in discrete time at the end of the paper two application examples illustrate the effectiveness of the approach

a posteriori error estimator for obstacle problems in this paper we consider a posteriori error estimators for obstacle problems the variational inequality is reformulated as a mixed problem in terms of a discrete nodewise defined but variationally consistent lagrange multiplier locally defined equilibrated fluxes and an h mathrm div conforming lifting define our estimator to obtain a better local upper bound for the estimator we introduce a different elementwise defined lagrange multiplier although the upper and lower bounds are established for affine obstacles we present generalizations to nonsmooth obstacles and to nonmatching meshes different numerical examples show the efficiency and reliability of our estimator due to its flexible construction principle and abstract framework it can be also applied as an error indicator to more complex obstacle problems such as e g american option pricing in financial mathematics

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

randomized algorithms for robust controller synthesis using statistical learning theory by now it is known that several problems in the robustness analysis and synthesis of control systems are np complete or np hard these negative results force us to modify our notion of solving a given problem an approach that is recently gaining popularity is that of using randomized algorithms which can be used to solve a problem approximately most of the time we begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters it is argued that in order to assess the performance of a controller as the plant varies over a prespecified family it is better to use the average performance of the controller as the objective function to be minimized rather than its worst case performance as the worst case objective function usually leads to rather conservative designs then it is shown that a property from statistical learning theory known as uniform convergence of empirical means ucem plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems in particular whenever the ucem property holds there exists an efficient i e polynomial time randomized algorithm using very recent results in statistical learning theory it is shown that the ucem property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities in particular several problems such as robust stabilization and weighted h 2 h norm minimization are amenable to the randomized approach

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

computational framework for coupling compressible and low mach number codes a hybrid multicode computational method is developed that allows combining compressible and low mach number codes into a single computational solver the proposed methodology can be used for integrated simulations of multicomponent engineering problems the unsteady coupling between the two codes is performed via exchanging time dependent state information through the interfaces using overlapping meshes the proposed coupling procedure is validated on laminar steady and unsteady test problems both constant density and variable density regimes of the low mach number code are investigated results of the test cases performed in frameworks of one way coupling and two way coupling are documented several methods of implementation of interface conditions for the compressible code are compared by looking at the levels of dilatation as well as numerical errors the injection method when all the variables are simply transformed from one code to another through an interpolation procedure is chosen for its superior stability accuracy and simplicity of implementation numerical results for the coupled calculations obtained with the injection method show good agreement with both standalone calculations and analytical solutions

a spectral multidomain method for the numerical simulation of turbulent flows the primitive variable formulation of the unsteady incompressible navier stokes equations in three space dimensions is discretized with a combined fourier legendre spectral method a semi implicit pressure correction scheme is applied to decouple the velocity from the pressure the arising elliptic scalar problems are first diagonalized in the periodic fourier direction and then solved by a multidomain legendre collocation method in the two remaining space coordinates in particular both an iterative and a direct version of the so called projection decomposition method pdm are introduced to separate the equations for the internal nodes from the ones governing the interface unknowns the pdm method first introduced by v agoshkov and e ovtchinnikov and later applied to spectral methods by p gervasio e ovtchinnikov and a quarteroni is a domain decomposition technique for elliptic boundary value problems which is based on a galerkin approximation of the steklov poincare equation for the unknown variables associated to the grid points lying on the interface between subdomains after having shown the exponential convergence of the proposed discretization technique some issues on the efficient implementation of the method are given finally as an illustration of the potentialities of the algorithm for the numerical simulation of turbulent flows the results of a direct numerical simulation dns of a fully turbulent plane channel flow are presented

deflation accelerated preconditioning of the poisson neumann schur problem on long domains with a high order discontinuous element based collocation method a combination of block jacobi and deflation preconditioning is used to solve a high order discontinuous element based collocation discretization of the schur complement of the poisson neumann system as arises in the operator splitting of the incompressible navier stokes equations the preconditioners and deflation vectors are chosen to mitigate the effects of ill conditioning due to highly elongated domains typical of simulations of strongly non hydrostatic environmental flows and to achieve generalized minimum residual method gmres convergence independent of the size of the number of elements in the long direction the ill posedness of the poisson neumann system manifests as an inconsistency of the schur complement problem but it is shown that this can be accounted for with appropriate projections out of the null space of the schur complement matrix without affecting the accuracy of the solution the block jacobi preconditioner is shown to yield gmres convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition the combined deflation and block jacobi preconditioning is compared with two level non overlapping block jacobi preconditioning of the schur problem and while both methods achieve convergence independent of the grid size deflation is shown to require half as many gmres iterations and 25 less wall clock time for a variety of grid sizes and domain aspect ratios the deflation methods shown to be effective for the two dimensional poisson neumann problem are extensible to the three dimensional problem assuming a fourier discretization in the third dimension a fourier discretization results in a two dimensional helmholtz problem for each fourier component that is solved using deflated block jacobi preconditioning on its schur complement here again deflation is shown to be superior to two level non overlapping block jacobi preconditioning requiring about half as many gmres iterations and 15 less time

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

probabilistic robustness analysis explicit bounds for the minimum number of samples in this paper we study robustness analysis of control systems affected by bounded uncertainty motivated by the difficulty to perform this analysis when the uncertainty enters into the plant coefficients in a nonlinear fashion we study a probabilistic approach in this setting the uncertain parameters q are random variables bounded in a set q and described by a multivariate density function f q we then pose the following question given a performance level what is the probability that this level is attained the main content of this paper is to derive explicit bounds for the number of samples required to estimate this probability with a certain accuracy and confidence a priori specified it is shown that the number obtained is inversely proportional to these thresholds and it is much smaller than that of classical results finally we remark that the same approach can be followed to study several problems in a control system context for example we can evaluate the worst case h norm of the sensitivity function of multi input multi output systems or compute when the robustness margin is of interest

some properties of an upper bound for spl mu a convex upper bound of the mixed structured singular value spl mu is analyzed the upper bound is based on a multiplier method it is simple it can exploit low rank properties and it is shown to be less conservative than the well known d g scaling a direct relationship with d g scaling is given the upper bound can be modified to one that is continuous with an explicit lipschitz constant

frequency sweeping tests for stability independent of delay in this paper we study the stability properties of linear time invariant delay systems given in a state space form we consider specifically the notion of asymptotic stability independent of delay systems with both commensurate and noncommensurate delays are investigated we present for each class of systems a necessary and sufficient condition in terms of structured singular values and further we demonstrate how these conditions may be extended to study stability independent of delay for uncertain systems our results consist of several frequency sweeping tests that can be systematically implemented and that should complement the previous work

unstructured spectral element methods for simulation of turbulent flows in this paper we present a spectral element fourier algorithm for simulating incompressible turbulent flows in complex geometries using unstructured quadrilateral meshes to this end we compare two different interface formulations for extending the conforming spectral element method in order to allow for surgical mesh refinement and still retain spectral accuracy the zanolli iterative procedure and variational patching based on auxiliary mortar functions we present an interpretation of the original mortar element method as a patching scheme and develop direct and iterative solution techniques that make the method efficient for simulations of turbulent flows the properties of the new method are analyzed in detail by studying the eigenspectra of the advection and diffusion operators we then present numerical results that illustrate the flexibility as well as the exponential convergence of the new algorithm for nonconforming discretizations we conclude with simulation studies of the turbulent cylinder wake at re 1000 external flow and turbulent flow over riblets at re 3280 internal flow

superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle abstract superconvergence analysis of nonconforming finite element methods fems are discussed for solving the second order variational inequality problem with displacement obstacle the elements employed have a common typical character i e the consistency error can reach order o h 3 2 nearly 1 2 order higher than their interpolation error when the exact solution of the considered problem belongs to h 5 2 for any e 0 by making full use of special properties of the element s interpolations and bramble hilbert lemma the superconvergence error estimates of order o h 3 2 in the broken h1 norm are derived finally some numerical results are provided to confirm the theoretical results

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

continuity properties of the real complex structured singular value the structured singular value function mu is defined with respect to a given uncertainty set this function is continuous if the uncertainties are allowed to be complex however if some uncertainties are required to be real then it can be discontinuous it is shown that mu is always upper semi continuous and conditions are derived under which it is also lower semi continuous with these results the real parameter robustness problem is reexamined a related although not equivalent problem is formulated which is always continuous and the relationship between the new problem and the original real mu m problem is made explicit a numerical example and results obtained via this related problem are presented

computational complexity of real structured singular value in l sub p setting this paper studies a generalized real structured singular value spl mu problem where uncertain parameters are bounded by an l sub p norm two results are presented the first one shows that this generalized spl mu problem is np hard for any given rational number p spl isin 1 spl infin the np hardness holds as long as le the size of the largest repeated block exceeds one this result generalizes the known np hardness result for the conventional spl mu problem with p spl infin the second result which strengthens the first one considers the approximability problem of the generalized spl mu we show that the problem of obtaining an estimate for the generalized spl mu with some guaranteed bounds on the relative error remains to be np hard regardless how large this bound is

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

sequential design of decentralized controllers abstract a procedure for sequential design of decentralized controllers for linear systems is presented it is shown how to include a simple estimate of the effect of closing subsequent loops into the design problem for the loop which is to be closed in the examples the robust performance in terms of the structured singular value is used as the measure of control performance but the procedure may be applied also for other performance measures

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

the spectral projection decomposition method for elliptic equations in two dimensions the projection decomposition method pdm is invoked to extend the application area of the spectral collocation method to elliptic problems in domains compounded of rectangles theoretical and numerical results are presented demonstrating the high accuracy of the resulting method as well as its computational efficiency

first order least squares method for the obstacle problem we define and analyse a least squares finite element method for a first order reformulation of the obstacle problem moreover we derive variational inequalities that are based on similar but non symmetric bilinear forms a priori error estimates including the case of non conforming convex sets are given and optimal convergence rates are shown for the lowest order case we provide a posteriori bounds that can be used as error indicators in an adaptive algorithm numerical studies are presented

quasi optimal schwarz methods for the conforming spectral element discretization the spectral element method is used to discretize self adjoint elliptic equations in three dimensional domains the domain is decomposed into hexahedral elements and in each of the elements the discretization space is the set of polynomials of degree n in each variable a conforming galerkin formulation is used the corresponding integrals are computed approximately with gauss lobatto legendre gll quadrature rules of order n and a lagrange interpolation basis associated with the gll nodes is used fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method the conforming finite element space on the gll mesh consisting of piecewise q1 or p1 functions produces a stiffness matrix kh that is known to be spectrally equivalent to the spectral element stiffness matrix kn kh is replaced by a preconditioner tilde k h which is well adapted to parallel computer architectures the preconditioned operator is then tilde k h 1 k n techniques for nonregular meshes are developed which make it possible to estimate the condition number of tilde k h 1 k n where tilde k h is a standard finite element preconditioner of kh based on the gll mesh two finite element based preconditioners the wirebasket method of smith and the overlapping schwarz algorithm for the spectral element method are given as examples of the use of these tools numerical experiments performed by pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions

svd controllers for h 2 h and m optimal control0 plant structure is utilized for the simplification of system analysis and controller synthesis for plants where the directionality is independent of frequency the singular value decompositioin svd is used to decouple the system into nominally independent subsystems of lower dimension in h2 and h optimal control the controller synthesis can then be performed for each of these subsystems independently and the resulting overall svd controller will be optimal the same will hold for any norm that is invariant under unitary transformations in h optimal control the resulting controller is also super optimal since a controller of dimension n n will minimize the norm in n directions for robust control in terms of the structured singular value the svd controller is optimal for a practically relevant class of block diagonal structures and uncertainty and performance weights

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

distributed control in multiple dimensions a structure preserving computational technique we consider the problem of analysis and control of discretely distributed systems in multidimensional arrays for spatially invariant systems in one spatial dimension we build an efficient arithmetic that preserves the rational laurent operator structure leading to fast iterative methods of solving lyapunov and riccati equations and block diagonalizations and thus arbitrarily non conservative stability analysis and h2 and h distributed controller synthesis these one dimensional results are then used to build an efficient arithmetic and controller synthesis procedure in n dimensions by induction the extension of these techniques from laurent operators with rational symbols to sequentially semi separable matrices yields a procedure for linear computational complexity analysis and controller synthesis for finite extent heterogeneous multidimensional systems with boundary conditions the procedures are demonstrated on two computational examples

efficient variants of the vertex space domain decomposition algorithm several variants of the vertex space algorithm of smith for two dimensional elliptic problems are described the vertex space algorithm is a domain decomposition method based on nonoverlapping subregions in which the reduced schur complement system on the interface is solved using a generalized block jacobi type preconditioner with the blocks corresponding to the vertex space edges and a coarse grid two kinds of approximations are considered for the edge and vertex space subblocks one based on fourier approximation and another based on an algebraic probing technique in which sparse approximations to these subblocks are computed our motivation is to improve the efficiency of the algorithm without sacrificing the optimal convergence rate numerical and theoretical results on the performance of these algorithms including variants of an algorithm of bramble pasciak and schatz are presented

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

brief an efficient off line formulation of robust model predictive control using linear matrix inequalities the practicality of model predictive control mpc is partially limited by its ability to solve optimization problems in real time moreover on line computational demand for synthesizing a robust mpc algorithm will likely grow significantly with the problem size in this paper we use the concept of an asymptotically stable invariant ellipsoid to develop a robust constrained mpc algorithm which gives a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off line one within another in state space this off line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed loop system and substantial reduction of the on line mpc computation the controller design is illustrated with two examples

analysis and design for ill conditioned plants part 1 lower bounds on the structured singular value abstract a study is made of a special case of the robust performance problem given by freudenberg 1989 when the weightings used to describe the uncertainty and performance specifications vary only with frequency then it is possible to strengthen the results in the above mentioned work by deriving both upper and lower bounds upon the structured singular value both sets of bounds are stated in terms of the coupling coefficients introduced by freudenberg 1989 and essentially yield necessary and sufficient conditions for the structured singular value to be small this information is used to suggest a strategy for compensator design to achieve robust performance despite plant ill conditioning applying this strategy to an example it can be seen how the design trade offs quantified by the bode gain phase relation manifest themselves in the robust performance problem finally the design is compared with one obtained using the synthesis approach

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

multivariable system identification for integral controllability integral controllability is necessary and sufficient for a multivariable model to be usable in a decoupling controller with integral action that can be arbitrarily detuned without jeopardizing closed loop robust stability the design of experiments for identification of integral controllable models is challenging because it must satisfy cumbersome eigenvalue inequalities involving a coupling between the real system and its model to address this challenge an optimization based mathematical framework is developed that characterizes efficient identification experiments ensuring integral controllability the proposed framework recovers well known experiment designs but also produces new ones of both theoretical and practical interest such designs are expressed either analytically or as a result of numerical optimization and are demonstrated in a number of examples these designs can be easily implemented in industrial practice by combining additional objectives or constraints of interest the proposed framework can further serve as a basis for new experiment designs in future work

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

controller design with real parametric uncertainty a number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty in particular the complex synthesis procedure has been successfully applied to a number of engineering problems however the presence of real parametric uncertainty in the problem description substantially complicates matters so that standard complex synthesis techniques are no longer adequate in this paper we develop a procedure to tackle the mixed real and complex synthesis problem this procedure involves a d g k iteration between computing the mixed upper bound and solving an h optimal control problem and has guaranteed convergence to a local minimum of the nonconvex problem the procedure has been implemented in software and several controller designs are compared with the corresponding complex synthesis designs

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

algebraic two level preconditioners for the schur complement method the solution of elliptic problems is challenging on parallel distributed memory computers since their green s functions are global to address this issue we present a set of preconditioners for the schur complement domain decomposition method they implement a global coupling mechanism through coarse space components similar to the one proposed in bramble pasciak and shatz math comp 47 1986 pp 103 134 the definition of the coarse space components is algebraic they are defined using the mesh partitioning information and simple interpolation operators these preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration the numerical and parallel scalability of those preconditioners are illustrated on two dimensional model examples that have anisotropy and or discontinuity phenomena

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

timely communication diagonal edge preconditioners in p version and spectral element methods domain decomposition preconditioners for high order galerkin methods in two dimensions are often built from modules associated with the decomposition of the discrete space into subspaces of functions related to the interior of elements individual edges and vertices the restriction of the original bilinear form to a particular subspace gives rise to a diagonal block of the preconditioner and the action of its inverse on a vector has to be evaluated in each iteration each block can be replaced by a preconditioner in order to decrease the cost knowledge of the quality of this local preconditioner can be used directly in a study of the convergence rate of the overall iterative process the schur complement of an edge with respect to the variables interior to two adjacent elements is considered the assembly and factorization of this block matrix are potentially expensive especially for polynomials of high degree it is demonstrated that the diagonal preconditioner of one such block has a condition number that increases approximately linearly with the degree of the polynomials numerical results demonstrate that the actual condition numbers are relatively small

computation of the structured singular value via moment lmi relaxations the structured singular value ssv provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties unfortunately computing the ssv is an np hard problem and the polynomial time algorithms available in the literature are only able to provide except for some special cases upper and lower bounds on the exact value of the ssv in this work we present a new algorithm to compute an upper bound on the ssv in case of mixed real complex uncertainties the underlying idea of the developed approach is to formulate the ssv computation as a nonconvex polynomial optimization problem which is relaxed into a sequence of convex optimization problems through moment based relaxation techniques two heuristics to compute a lower bound on the ssv are also discussed the analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the robust control toolbox in matlab and it provides in most of the cases coincident lower and upper bounds on the structured singular value

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

on problems involving eigenvalues for uncertain matrices by structured singular values this technical note considers problems that involve eigenvalues of uncertain matrices expressed in linear fractional form using the structured singular value mu and the skewed structured singular value nu in particular positive definiteness conditions maximum and minimum eigenvalues and generalized eigenvalues of uncertain matrices are expressed using mu and nu the obtained results allow us to acquire information of an uncertain matrix efficiently by the existing computational tools that provide practically useful approximations to the values of mu and nu

a gain based lower bound algorithm for real and mixed µ problems a new lower bound algorithm for real and mixed m problems is presented the basic idea of this algorithm is to use a related worst case gain problem to compute the real blocks and if the block structure is mixed the standard power iteration to compute the complex blocks numerical tests indicate that the algorithm is fast and provides good lower bounds for both real and mixed m problems of small to moderate size

stability analysis of interface temporal discretization in grid overlapping methods we investigate the stability of a temporal discretization of interface terms in grid overlapping methods a matrix stability analysis is performed on a model problem of the one dimensional diffusion equation on overlapping grids the scheme stability is first analyzed theoretically and a proof of the unconditional stability of the first order interface extrapolation scheme with the first and second order time integration for any overlap size is presented for the higher order schemes we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap the influence of iterations on stability properties is also investigated numerical experiments are then presented relating the obtained stability bounds to the observed numerical values semidiscrete analysis confirms the derived scaling for the stability bounds

constrained robustness analysis by randomized algorithms this paper shows that many robust control problems can be formulated as constrained optimization problems and can be tackled by using randomized algorithms two different approaches in searching reliable solutions to robustness analysis problems under constraints are proposed and the minimum computational efforts for achieving certain reliability and accuracy are investigated and bounds for sample size are derived moreover the existing order statistics distribution theory is extended to the general case in which the distribution of population is not assumed to be continuous and the order statistics is associated with certain constraints

linear mechanical systems and dyadic transfer function matrices for some linear mechanical systems which can be written as a dyadic transfer function matrix dtm mimo controller design can be reduced to a number of siso designs internal stability of the siso designs guarantees stability of the resulting mimo controller by relating the insights from modal analysis to system theory necessary and sufficient conditions are derived for linear mechanical systems to have the dtm property the main theorems are illustrated on an active suspension design of an agricultural spray boom

an efficient and reliable residual type a posteriori error estimator for the signorini problem we derive a new a posteriori error estimator for the signorini problem it generalizes the standard residual type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non linearity remarkably these additional contact related terms vanish in the case of so called full contact we prove reliability and efficiency for two and three dimensional simplicial meshes moreover we address the case of non discrete gap functions numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two and three dimensional case for simplicial as well as for unstructured mixed meshes

stability of non linear delay systems delay independent small gain theorem and frequency domain interpretation of the lyapunov krasovskii method the purpose of this note is to study the relationship between a certain stability criterion for non linear delay systems obtained via the lyapunov krasovskii method and a delay independent version of the small gain theorem we show that contrary to the delay free case in which the kalman yakubovich popov lemma ensures the equivalence of the two approaches the first method assumes stronger hypothesis than the second one however numerical verification of the former is in general np hard whereas the latter leads to linear matrix inequalities the difference between the two approaches is precisely stated and among other benefits this permits to exhibit classes of problems for which the lyapunov krasovskii method is not conservative

a necessary and sufficient minimality condition for uncertain systems a necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations lfts on structured operator sets this condition is based on the existence of a rank deficient solution to either of a pair of linear matrix inequalities which generalize lyapunov equations the notion of gramians is thus also generalized to uncertain systems as well as kalman like decomposition structures a related minimality condition the converse of the reducibility condition may then be inferred from these results and the equivalence class of all minimal lft realizations defined these results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems subsequent results such as the definition of and rank tests on structured controllability and observability matrices are also given the minimality results described are applicable to multidimensional system realizations as well as to uncertain systems connections to formal powers series representations also exist

theorems of perron frobenius type for matrices without sign restrictions abstract the paper attempts to solve a problem which was called a challenge for the future in linear algebra appl we define and investigate a new quantity for real matrices the sign real spectral radius and derive various characterizations bounds and properties of it in certain aspects our quantity shows similar behavior to the perron root of a nonnegative matrix it is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix relations to the perron root of the entrywise absolute value of the matrix and to the number are given as well

global optimization for h sub spl infin control with constant diagonal scaling this paper considers the h sub spl infin control problem with constant diagonal scaling related to the robust control synthesis for systems with structured time varying uncertainties the problem is not convex in general and hence it is difficult to find a global solution the purpose of this paper is to provide an algorithm to find a suboptimal solution with any specified small tolerance from the globally optimal solution for the optimization problem the algorithm based on the triangle covering method is proposed the computational complexity analysis shows that its worst case order is polynomial in the inverse of the tolerance and the size of an a priori given interval of scaling with a fixed number of uncertainty blocks