the leja method revisited backward error analysis for the matrix exponential the leja method is a polynomial interpolation procedure that can be used to compute matrix functions in particular computing the action of the matrix exponential on a given vector is a typical application this quantity is required e g in exponential integrators the leja method essentially depends on three parameters the scaling parameter the location of the interpolation points and the degree of interpolation we present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy additional aspects that are required for an efficient and reliable implementation are discussed numerical examples illustrating the performance of our matlab code are included

the linear exponential quadratic gaussian control for discrete systems with application to reliable stabilization in this paper we derive the discrete linear exponential quadratic gaussian leqg controller which can take both the system and measurement noise covariances into consideration comparing with the traditional linear quadratic gaussian lqg design the leqg has the wilder design freedom the proposed discrete leqg control scheme is then applied to the study of reliable control which can tolerate abnormal operation within some pre specified set of actuators this is achieved by suitable modification of the algebraic riccati equation for the design of the controller the bounds of gain margins for the feedback control gains of reliable stabilization are also derived the stability of the overall system is preserved despite the abnormal operation of actuators within a pre specified subset in the bounds of gain margins

approaches to t s fuzzy affine model based reliable output feedback control for nonlinear ito stochastic systems this paper deals with the problem of reliable and robust mathscr h infty static output feedback sof controller synthesis for continuous time nonlinear stochastic systems with actuator faults the nonlinear stochastic plant is expressed by an ito type takagi sugeno fuzzy affine model with parametric uncertainties and a markov process is employed to model the occurrence of actuator fault the purpose is to design an admissible piecewise sof controller such that the resulting closed loop system is stochastically stable with a prescribed disturbance attenuation level in an mathscr h infty sense specifically based on a markovian lyapunov function combined with ito differential formula s procedure and some matrix inequality convexification procedures two new approaches to the reliable sof controller analysis and synthesis are proposed for the underlying stochastic fuzzy affine systems it is shown that the existence of desired reliable controllers is fully characterized in terms of strict linear matrix inequalities finally simulation examples are presented to illustrate the effectiveness and advantages of the developed methods

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

accurate computation of divided differences of the exponential function abstract the traditional recurrence for the computation of exponential divided differences along with a new method based on the properties of the exponential function are studied in detail in this paper our results show that it is possible to combine these two methods to compute exponential divided differences accurately a hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference author

a massively parallel exponential integrator for advection diffusion models this work considers the real leja points method relpm m caliari m vianello l bergamaschi interpolating discrete advection diffusion propagators at spectral leja sequences j comput appl math 172 2004 79 99 for the exponential integration of large scale sparse systems of odes generated by finite element or finite difference discretizations of 3 d advection diffusion models we present an efficient parallel implementation of relpm for polynomial interpolation of the matrix exponential propagators exp dta v and f dta v f z exp z 1 z a scalability analysis of the most important computational kernel inside the code the parallel sparse matrix vector product has been performed as well as an experimental study of the communication overhead as a result of this study an optimized parallel sparse matrix vector product routine has been implemented the resulting code shows good scaling behavior even when using more than one thousand processors the numerical results presented on a number of very large test cases gives experimental evidence that relpm is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

overcoming order reduction in diffusion reaction splitting part 2 oblique boundary conditions splitting methods constitute a well established class of numerical schemes for the time integration of partial differential equations their main advantages over more traditional schemes are computational efficiency and superior geometric properties in the presence of nontrivial boundary conditions however splitting methods usually suffer from order reduction and some additional loss of accuracy for diffusion reaction equations with inhomogeneous oblique boundary conditions a modification of the classic second order strang splitting is proposed that successfully solves the problem of order reduction the same correction also improves the accuracy of the classic first order lie splitting the proposed modification only depends on the available boundary data and in the case of non dirichlet boundary conditions on the computed numerical solution consequently this modification can be implemented in an efficient way which makes the modified splitting schemes superior to their classic versions the fr

on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

robust controller design for stochastic nonlinear systems via convex optimization this paper presents a new nonlinear control design framework called convex optimization based stochastic steady state tracking error minimization cvstem for a class of ito stochastic nonlinear systems including stochastic lagrangian systems this state feedback controller design involves the use of multiple state dependent coefficient sdc models of a nonlinear system equation and computes its controller parameters by solving a convex optimization problem in order to minimize an upper bound of the steady state tracking error this problem is shown to be equivalent to the original nonlinear optimization problem of minimizing the upper bound the exponential stability and robustness properties due to the state dependent riccati inequality are proven using stochastic contraction analysis discrete time stochastic incremental contraction analysis with respect to a state and time dependent metric is also presented along with its explicit connection to continuous time contraction analysis a result of simulation is used to validate the superiority of the proposed controller compared to a known nonlinear controller a pid controller and an h infinity controller

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

a necessary condition for nonlinear h control via measurement feedback abstract in a series of recent papers it was shown that the solution of the problem of disturbance attenuation for nonlinear system is related to the existence of solutions of a pair of hamilton jacobi inequalities in n independent variables associated with state feedback and output injection design the purpose of the present paper is to discuss the necessity of the existence of solutions of the hamilton jacobi inequality which determines the output injection gain

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

low rank second order splitting of large scale differential riccati equations we apply first and second order splitting schemes to the differential riccati equation such equations are very important in e g linear quadratic regulator lqr problems where they provide a link between the state of the system and the optimal input the methods can also be extended to generalized riccati equations e g arising from lqr problems given in implicit form in contrast to previously proposed schemes such as bdf or rosenbrock methods the splitting schemes exploit the fact that the nonlinear and affine parts of the problem when considered in isolation have closed form solutions we show that if the solution possesses low rank structure which is frequently the case then this is preserved by the method this feature is used to implement the methods efficiently for large scale problems the proposed methods are expected to be competitive as they at most require the solution of a small number of linear equation systems per time step finally we apply our low rank implementations to the riccati equations arising from two lqr problems the results show that the rank of the solutions stay low and the expected orders of convergence are observed

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

comparative performance of exponential implicit and explicit integrators for stiff systems of odes exponential integrators have enjoyed a resurgence of interest in recent years but there is still limited understanding of how their performance compares with that of state of the art integrators most notably the commonly used newton krylov implicit methods in this paper we present comparative performance analysis of krylov based exponential implicit and explicit integrators on a suite of stiff test problems and demonstrate that exponential integrators have computational advantages compared to the other methods particularly as problems become larger and more stiff we argue that the faster convergence of the krylov iteration within exponential integrators accounts for the main proportion of the computational savings that they provide and illustrate how the structure of these methods ensures such efficiency in addition we demonstrate the computational advantages of the newly introduced tokman and loffeld 2010 17 exponential propagation runge kutta epirk fifth order methods the detailed analysis of the performance of the methods that is presented provides guidelines for the construction and implementation of efficient exponential methods and the quantitative comparisons inform the selection of appropriate schemes for other problems

interpolating discrete advection diffusion propagators at leja sequences we propose and analyze the relpm real leja points method for evaluating the propagator tb v via matrix interpolation polynomials at spectral leja sequences here b is the large sparse nonsymmetric matrix arising from stable 2d or 3d finite difference discretization of linear advection diffusion equations and z is the entire function z ez 1 z the corresponding stiff differential system y t by t g y 0 y 0 is solved by the exact time marching scheme y i 1 y i t i t i b by i g i 0 1 where the time step is controlled simply via the variation percentage of the solution and can be large numerical tests show substantial speed ups up to one order of magnitude with respect to a classical variable step size crank nicolson solver

reliable control of nonlinear systems via variable structure scheme this study proposes a class of variable structure stabilizing laws which make the closed loop system be capable of tolerating the abnormal operation of actuators within a pre specified subset of actuators the ranges of acceptable change in control gain magnitude that preserves system s stability are estimated for the whole set of actuators these ranges are shown to be able to be made larger than those obtained by linear quadratic regulator lqr reliable design veillette 1995 and liang 2000 by the choice of control parameters besides this approach does not need the solution of hamilton jacobi hj equation or inequality which is essential for optimal approaches such as lqr and h infin reliable designs as a matter of fact this approach can also relax the computational burden for solving the hj equation or inequality

on the algorithm by al mohy and higham for computing the action of the matrix exponential a posteriori roundoff error estimation abstract the algorithm by al mohy and higham 2011 2 computes an approximation to e a b for given a and b where a is an n by n matrix and b is for example a vector of dimension n it uses a scaling together with a truncated taylor series approximation to the exponential of the scaled matrix in this paper a method is developed for estimating the roundoff error of the computed solution an asymptotic expansion of this error for small values of the unit roundoff is the basis of the method the roundoff error is further expressed in terms of sums of rounding errors which occur during the computation a second approximation to e a b which is computed with a lower precision than the first one is used to evaluate these rounding errors in practice the result is an upper bound on the normwise relative roundoff error further an algorithm is proposed for computing the error bound the cost for performing this algorithm depends on the type of problem and the accuracy which is required for the error estimate in case that all computations are performed with standard precisions this cost can be expressed in terms of the number of computed matrix vector products and is bounded from above by two times the cost for computing e a b

a new class of exponential propagation iterative methods of runge kutta type epirk we propose a new class of the exponential propagation iterative methods of runge kutta type epirk the epirk schemes are exponential integrators that can be competitive with explicit and implicit methods for integration of large stiff systems of odes introducing the new more general than previously proposed ansatz for epirk schemes allows for more flexibility in deriving computationally efficient high order integrators recent extension of the theory of b series to exponential integrators 1 is used to derive classical order conditions for schemes up to order five an algorithm to systematically solve these conditions is presented and several new fifth order schemes are constructed several numerical examples are used to verify the order of the methods and to illustrate the performance of the new schemes

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

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

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

efficient orthogonal matrix polynomial based method for computing matrix exponential the matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields this paper presents an efficient method for computing matrix exponentials based on hermite matrix polynomial expansions hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods a backward error analysis of the approximation in exact arithmetic is given this analysis is used to provide a theoretical estimate for the optimal scaling of matrices two algorithms based on this method have been implemented as matlab functions they have been compared with matlab functions funm and expm obtaining greater accuracy in the majority of tests a careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs reaching in numerical tests relative higher average costs than expm of only 4 43 for the final hermite selected order and obtaining better accuracy results in the 77 36 of the test matrices the matlab implementation of the best hermite matrix polynomial based algorithm has been made available online

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated

new scaling squaring taylor algorithms for computing the matrix exponential the matrix exponential plays a fundamental role in linear differential equations arising in engineering mechanics and control theory the most widely used and the most generally efficient technique for calculating the matrix exponential is a combination of scaling and squaring with a pade approximation for alternative scaling and squaring methods based on taylor series we present two modifications that provably reduce the number of matrix multiplications needed to satisfy the required accuracy bounds and a detailed comparison of the several algorithmic variants is provided

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

matrix oriented discretization methods for reaction diffusion pdes comparisons and applications abstract systems of reaction diffusion partial differential equations rd pdes are widely applied for modeling life science and physico chemical phenomena in particular the coupling between diffusion and nonlinear kinetics can lead to the so called turing instability giving rise to a variety of spatial patterns like labyrinths spots stripes etc attained as steady state solutions for large time intervals to capture the morphological peculiarities of the pattern itself a very fine space discretization may be required limiting the use of standard vector based ode solvers in time because of excessive computational costs by exploiting the structure of the diffusion matrix we show that matrix based versions of time integrators such as implicit explicit imex and exponential schemes allow for much finer problem discretizations we illustrate our findings by numerically solving the schnakenberg model prototype of rd pde systems with turing pattern solutions and the dib morphochemical model describing metal growth during battery charging processes

nonparametric estimation of kullback leibler divergence in this letter we introduce an estimator of kullback leibler divergence based on two independent samples we show that on any finite alphabet this estimator has an exponentially decaying bias and that it is consistent and asymptotically normal to explain the importance of this estimator we provide a thorough analysis of the more standard plug in estimator we show that it is consistent and asymptotically normal but with an infinite bias moreover if we modify the plug in estimator to remove the rare events that cause the bias to become infinite the bias still decays at a rate no faster than further we extend our results to estimating the symmetrized kullback leibler divergence we conclude by providing simulation results which show that the asymptotic properties of these estimators hold even for relatively small sample sizes

reliable mixed l2 h fuzzy static output feedback control for nonlinear systems with sensor faults this paper is concerned with the design of reliable mixed l 2 h static output feedback sof fuzzy controllers for nonlinear continuous time systems with complete sensor faults the takagi and sugeno t s fuzzy model is employed to represent a nonlinear system a sufficient condition for the existence of reliable mixed l 2 h sof fuzzy controllers is presented in terms of a set of quadratic matrix inequalities qmis which not only guarantees that the closed loop fuzzy system satisfies a desired h disturbance attenuation constraint for all admissible operating regimes including the normal and sensor fault cases but also provides different upper bounds on the l 2 performance criterion for different operating regimes a suboptimal reliable fuzzy controller is obtained by the proposed iterative linear matrix inequality ilmi algorithm for minimizing the normal l 2 performance bound while maintaining acceptable lower levels of the bounds in the sensor fault cases finally a numerical example is given to illustrate the effectiveness of the proposed design method

guaranteed performance in semi global trajectory tracking for matched disturbance nonlinear systems in this article we present a semi global trajectory tracking approach that guarantees a priori computable l 2 and l performance bounds for matched disturbance control affine systems the proposed controller is derived by combining a standard inverse control technique with an extended nonlinear robust state feedback the latter is based on a control lyapunov function used for stabilising one operating point inside the considered state space a difference gradient formulation of this lyapunov function is then applied to prove stabilisation along any trajectory in the considered state space results for l 2 and l bounded disturbances will be presented and further extended to the case of actuator uncertainties and disturbance offsets the theoretical contributions are verified applying them to a numerical example

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

remarks regarding the gap between continuous lipschitz and differentiable storage functions for dissipation inequalities appearing in h control abstract this paper deals with the regularity of solutions of the hamilton jacobi inequality which arises in h control it shows by explicit counterexamples that there are gaps between existence of continuous and locally lipschitz positive definite and proper solutions and between lipschitz and continuously differentiable ones on the other hand it is shown that it is always possible to smooth out solutions provided that an infinitesimal increase in gain is allowed

guaranteed robustness bounds for matched disturbance nonlinear systems this paper considers input affine nonlinear systems with matched disturbances and shows how to compute an a priori upper bound of the h attenuation level achieved by the optimal l 2 controller and the suboptimal h central controller the case where the disturbance contains a constant term is also discussed these bounds are shown to depend only on the function mapping the control input to the performance variable this result is used to derive a robust control design for a special but practically important class of non input affine nonlinear systems consisting of the series connection of a nonlinear state and input dependent map and of a nonlinear input affine dynamical system approximate inversion of the nonlinear static map leads to a robust control problem which fits into the framework the effectiveness of the theoretical results is shown by its use for the robust control design of a diesel engine test bench

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

spl hscr sub spl infin control of nonlinear systems via output feedback controller parameterization the standard state space solutions to the spl hscr sub spl infin control problem for linear time invariant systems are generalized to nonlinear time invariant systems a class of local nonlinear output feedback spl hscr sub spl infin controllers are parameterized as nonlinear fractional transformations on contractive stable nonlinear parameters as in the linear case the spl hscr sub spl infin control problem is solved by its reduction to state feedback and output estimation problems together with a separation argument sufficient conditions for spl hscr sub spl infin control problem to be locally solved are also derived with this machinery

rosenbrock methods for solving riccati differential equations the riccati differential equation rde arises in several fields like optimal control optimal filtering h control of linear time varying systems differential games etc in the literature there is a large variety of approaches to compute its solution particularly for stiff rdes matrix valued versions of the standard multi step methods for solving ordinary differential equations have given good results in this technical note we discuss a particular class of one step methods these are the linear implicit runge kutta methods or rosenbrock methods we show that they offer a practical alternative for solving stiff rdes they can be implemented with good stability properties and allow for a cheap step size control the matrix valued version of the rosenbrock methods for rdes requires the solution of one sylvester equation in each stage of the method for the case in which the coefficient matrices of the sylvester equation are dense the bartels stewart method can be efficiently applied for solving the equations the computational cost computing time and memory requirements is smaller than for multi step methods

new adaptive exponential propagation iterative methods of runge kutta type exponential integrators have emerged as an efficient alternative to commonly used time integrators recently a new class of exponential propagation iterative methods of runge kutta type epirk has been introduced m tokman j comput phys 230 2011 pp 8762 8778 these schemes possess a structure that makes them computationally advantageous compared to other exponential methods in addition the general epirk formulation offers flexibility that allows derivation of new efficient techniques in this paper we use this feature to derive new epirk methods which are particularly designed to take advantage of the adaptive krylov algorithm j niesen and w wright http arxiv org abs 0907 4631 2010 the adaptive krylov method significantly reduces the computational complexity of evaluating products of matrix varphi functions and vectors necessary for implementing an exponential integrator we present the derivation of the new adaptive epirk methods construct new schemes and illustrate the co

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

an efficient spde approach for el nino abstract we consider the numerical approximation of stochastic partial differential equations spdes based models for a quasi periodic climate pattern in the tropical pacific ocean known as el nino phenomenon we show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation respectively in this context we provide a numerical framework to approximate these parameters directly we compare this method to stochastic differential equations and spdes based models from the literature solved by taylor methods and stochastic galerkin methods respectively numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 last nino event validate the efficiency of our approach

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

brief solutions of nonlinear optimal and robust control problems via a mixed collocation dae s based algorithm a new algorithm for computing the solutions of nonlinear optimal and robust h control problems is proposed the algorithm is based on the use of the collocation method to transform the pde s into ode s some convergence results are given and several examples are presented

global stabilization of nonlinear cascade systems abstract we present sufficient conditions for the global stabilizability of two cascade connected nonlinear systems these are based on general results concerning global asymptotic stability of triangular systems which are proved in the last section for polynomial systems in particular the stabilizing feedback is given explicitly

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

fuzzy model based reliable static output feedback mathscr h _ infty control of nonlinear hyperbolic pde systems this paper investigates the problem of output feedback robust mathscr h infty control for a class of nonlinear spatially distributed systems described by first order hyperbolic partial differential equations pdes with markovian jumping actuator faults the nonlinear hyperbolic pde systems are first expressed by takagi sugeno fuzzy models with parameter uncertainties and then the objective is to design a reliable distributed fuzzy static output feedback controller guaranteeing the stochastic exponential stability of the resulting closed loop system with certain mathscr h infty disturbance attenuation performance based on a markovian lyapunov functional combined with some matrix inequality convexification techniques two approaches are developed for reliable fuzzy static output feedback controller design of the underlying fuzzy pde systems it is shown that the controller gains can be obtained by solving a set of finite linear matrix inequalities based on the finite difference method in space finally two examples are presented to demonstrate the effectiveness of the proposed methods

on the fly backward error estimate for matrix exponential approximation by taylor algorithm abstract in this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on the fly without the need to precompute in high precision quantities related to specific accuracies in this way the scaling parameter s and the degree m of the truncated taylor series the underlying method are adapted to the input matrix and not to a class of matrices sharing some spectral properties such as with the classical backward error analysis the result is a very flexible method which can approximate the matrix exponential at any desired accuracy moreover the risk of overscaling that is the possibility to select a value s larger than necessary is mitigated by a new choice as the sum of two powers of two finally several numerical experiments in matlab with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives

a new approach to constructing efficient stiffly accurate epirk methods the structural flexibility of the exponential propagation iterative methods of runge kutta type epirk enables construction of particularly efficient exponential time integrators while the epirk methods have been shown to perform well on stiff problems all of the schemes proposed up to now have been derived using classical order conditions in this paper we extend the stiff order conditions and the convergence theory developed for the exponential rosenbrock methods to the epirk integrators we derive stiff order conditions for the epirk methods and develop algorithms to solve them to obtain specific schemes moreover we propose a new approach to constructing particularly efficient epirk integrators that are optimized to work with an adaptive krylov algorithm we use a set of numerical examples to illustrate the computational advantages that the newly constructed epirk methods offer compared to previously proposed exponential integrators

reliable h control of multiple time scales singularly perturbed systems with sensor failure this paper studies the reliable h control for linear time invariant multiparameter singularly perturbed systems against sensor failures by time scale decomposition the full order system is decomposed into slow and fast subsystems after designing a reliable h controller for the global system three reduced reliable h sub controllers based on the slow and fast problems are obtained through the manipulation of the algebraic riccati equations the resulting control systems are reliable in that they provide guaranteed asymptotic stability and h performance when all control components are operational as well as when sensor failures occur

global l 2 gain design for a class of nonlinear systems abstract it is known that the so called h control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback in this paper we prove that this condition suffices to solve also a global h control problem for a fairly large class of nonlinear systems if one is free to choose a state dependent weight of the control input using a two way backward and forward recursive induction argument we simultaneously construct starting from a solution of the riccati algebraic equation a global solution of the hamilton jacobi isaacs partial differential equation arising in the nonlinear h control as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems

on the stability of some algorithms for computing the action of the matrix exponential abstract this paper deals with the stability of some one step methods for computing the action of a matrix exponential on a vector i e e a b where a is an n by n matrix and b is a vector of dimension n the methods are based on certain assumptions made for the case that a is a matrix having sufficiently small norm if the norm of a is larger a fixed step size is used to perform a scaling of a an analysis of the roundoff error of the computed e a b shows that these methods are backward stable if a is hermitian and forward stable if for example a is skew hermitian or if a is real and essentially nonnegative and b is real and nonnegative in addition an upper bound on the forward error of the computed e a b is obtained for all a and b this bound which is expressed in terms of the condition number is evaluated for several examples of the data the results of this paper partly apply to the algorithm proposed by al mohy and higham 2011 for computing e a b

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

a novel approach to reliable output feedback control of fuzzy affine systems with time delays and sensor faults this paper proposes a novel system augmentation approach to the delay dependent reliable piecewise affine mathscr h infty static output feedback control for nonlinear systems with time varying delay and sensor faults in the piecewise markovian lyapunov functional based framework the nonlinear plant is described by a continuous time takagi sugeno fuzzy affine model with parametric uncertainties and the sensor faults are characterized by a markov process specifically by applying a state input augmentation technique the original closed loop system is first reformulated into a descriptor fuzzy affine system based on a new piecewise markovian lyapunov krasovskii functional combined with a wirtinger based integral inequality improved reciprocally convex inequality and s procedure a novel bounded real lemma is then derived for the underlying closed loop system furthermore by taking advantage of the redundancy of descriptor system formulation together with a linearization procedure the piecewise affine controller synthesis is carried out it is shown that the desired piecewise affine controller gains can be attained by solving a linear matrix inequality based optimization problem finally simulation examples are performed to confirm the effectiveness and less conservatism of the presented approach

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

high performance computing of the matrix exponential this work presents a new algorithm for matrix exponential computation that significantly simplifies a taylor scaling and squaring algorithm presented previously by the authors preserving accuracy a matlab version of the new simplified algorithm has been compared with the original algorithm providing similar results in terms of accuracy but reducing processing time it has also been compared with two state of the art implementations based on pade approximations one commercial and the other implemented in matlab getting better accuracy and processing time results in the majority of cases

using nonorthogonal lanczos vectors in the computation of matrix functions the lanczos algorithm uses a three term recurrence to construct an orthonormal basis for the krylov space corresponding to a symmetric matrix a and a nonzero starting vector varphi the vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes including solving linear systems au varphi and computing the matrix exponential e ta varphi although the vectors produced in finite precision arithmetic are not orthogonal we show why they can still be used effectively for these purposes

decentralized optimal control of a class of interconnected nonlinear discrete time systems by using online hamilton jacobi bellman formulation in this paper the direct neural dynamic programming technique is utilized to solve the hamilton jacobi bellman equation forward in time for the decentralized near optimal regulation of a class of nonlinear interconnected discrete time systems with unknown internal subsystem and interconnection dynamics while the input gain matrix is considered known even though the unknown interconnection terms are considered weak and functions of the entire state vector the decentralized control is attempted under the assumption that only the local state vector is measurable the decentralized nearly optimal controller design for each subsystem consists of two neural networks nns an action nn that is aimed to provide a nearly optimal control signal and a critic nn which evaluates the performance of the overall system all nn parameters are tuned online for both the nns by using lyapunov techniques it is shown that all subsystems signals are uniformly ultimately bounded and that the synthesized subsystems inputs approach their corresponding nearly optimal control inputs with bounded error simulation results are included to show the effectiveness of the approach

positive definiteness in the numerical solution of riccati differential equations in this work we address the issue of integrating symmetric riccati and lyapunov matrix differential equations in many cases typical in applications the solutions are positive definite matrices our goal is to study when and how this property is maintained for a numerically computed solution there are two classes of solution methods direct and indirect algorithms the first class consists of the schemes resulting from direct discretization of the equations the second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy we show first that using a direct algorithm a one step scheme or a strictly stable multistep scheme explicit or implicit limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite then we show two ways to obtain positive definite higher order approximations by using indirect algorithms the first is to apply a symplectic integrator to an associated hamiltonian system the other uses stepwise linearization

reliable control of nonlinear systems in this paper we extend veillette s results 1995 to the study of reliable linear quadratic regulator problem for nonlinear systems this is achieved by employing the hamilton jacobi inequality in the nonlinear case instead of algebraic riccati equation in the linear one the proposed state feedback controllers are shown to be able to tolerate the outage of actuators within a prespecified subset of actuators both the gain margins of guaranteeing system stability and retaining a performance bound are estimated

h sub spl infin tuning for task space feedback control of robot with uncertain jacobian matrix the h sub spl infin tuning problem for task space feedback control of a robot with uncertain kinematics and jacobian matrix from joint space to task space is formulated and solved h sub spl infin tuning methods are presented for two task space feedback controllers when the systems are subject to external disturbance it is shown that the conditions for the establishment of h sub spl infin tuning can be obtained in simple forms without solving the hamilton jacobi inequalities even in the presence of kinematics uncertainty

operator based preconditioning of stiff hyperbolic systems we introduce an operator based scheme for preconditioning stiff components encountered in implicit methods for hyperbolic systems of pdes posed on regular grids the method is based on a directional splitting of the implicit operator followed by a characteristic decomposition of the resulting directional parts this approach allows for the solution of any number of characteristic components from the entire system to only the fastest stiffness inducing waves we apply the preconditioning method to stiff hyperbolic systems arising in magnetohydrodynamics and gas dynamics we then present numerical results showing that this preconditioning scheme works well on problems where the underlying stiffness results from the interaction of fast transient waves with slowly evolving dynamics scales well to large problem sizes and numbers of processors and allows for additional customization based on the specific problems under study

a normal law for the plug in estimator of entropy this paper establishes a sufficient condition for the asymptotic normality of the plug in estimator of shannon s entropy defined on a countable alphabet the sufficient condition covers a range of cases with countably infinite alphabets for which no normality results were previously known

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

discretized dynamical low rank approximation in the presence of small singular values low rank approximations to large time dependent matrices and tensors are the subject of this paper these matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations based on splitting the orthogonal projection onto the tangent space of the low rank manifold novel time integrators for obtaining approximations by low rank matrices and low rank tensor trains were recently proposed by standard theory the lie trotter and strang projector splitting methods are first and second order accurate respectively but the usual error bounds break down when the low rank approximation has small singular values this happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated on the other hand the integrators are exact when given time dependent matrices or tensors are already of the prescribed rank we provide an error analysis which unifies these properties we show that in c

reliable rm h_infty fuzzy control for continuous time nonlinear systems with actuator failures this paper is concerned with the design of reliable hinfin fuzzy controllers for continuous time nonlinear systems with actuator failures the takagi and sugeno fuzzy model is employed to represent a nonlinear system the objective is to find a stabilizing state feedback fuzzy controller such that the nominal hinfin performance is optimized while satisfying a prescribed hinfin performance constraint in the actuator failure cases based on the linear matrix inequality lmi techniques two efficient methods for the design of a suboptimal reliable hinfin fuzzy controller are proposed different lyapunov functions are used during the design for the nominal and actuator failure cases which lead to a less conservative controller design in the first method a single lyapunov function is used for the actuator failure cases the second method adopts a parameter dependent lyapunov function for the actuator failure cases which further reduces the conservatism of the design finally numerical simulations on the chaotic rossler system are given to illustrate the effectiveness of the proposed design methods

a fully implicit numerical method for single fluid resistive magnetohydrodynamics we present a nonlinearly implicit conservative numerical method for integration of the single fluid resistive mhd equations the method uses a high order spatial discretization that preserves the solenoidal property of the magnetic field the fully coupled pde system is solved implicitly in time providing for increased interaction between physical processes as well as additional stability over explicit time methods a high order adaptive time integration is employed which in many cases enables time steps ranging from one to two orders of magnitude larger than those constrained by the explicit cfl condition we apply the solution method to illustrative examples relevant to stiff magnetic fusion processes which challenge the efficiency of explicit methods we provide computational evidence showing that for such problems the method is comparably accurate with explicit time simulations while providing a significant runtime improvement due to its increased temporal stability

optimality of the paterson stockmeyer method for evaluating matrix polynomials and rational matrix functions abstract many state of the art algorithms reduce the computation of transcendental matrix functions to the evaluation of polynomial or rational approximants at a matrix argument this task can be accomplished efficiently by resorting to the paterson stockmeyer method an evaluation scheme originally developed for polynomials of matrices that extends quite naturally to rational functions an important feature of this technique is that the number of matrix multiplications required to evaluate an approximant of order n grows slower than n itself with the result that different approximants yield the same asymptotic computational cost we analyze the number of matrix multiplications required by the paterson stockmeyer method and by two widely used generalizations one for evaluating diagonal pade approximants of general functions and one specifically tailored to those of the exponential in all the three cases we identify the approximants of maximum order for any given computational cost

an approach to h control of a class of nonlinear systems this paper considers robust h control of a class of nonlinear systems that can be represented by a heterogeneous model a new controller design method using both state feedback is proposed based on quadratic stability theory a number of necessary and sufficient conditions are derived for quadratic stabilizability with disturbance attenuation of the control system concerned an example is also given to demonstrate the performance of the proposed controller design methods

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

nonlinear magnetohydrodynamics simulation using high order finite elements a conforming representation composed of 2d finite elements and finite fourier series is applied to 3d nonlinear non ideal magnetohydrodynamics using a semi implicit time advance the self adjoint semi implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments growth rates for resistive tearing modes with experimentally relevant lundquist number are computed accurately with time steps that are large with respect to the global alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree p two or larger an error diffusion method controls the generation of magnetic divergence error convergence studies show that this approach is effective for continuous basis functions with p 2 where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity is computed accurately with p 3 without mesh alignment a simulation of tearingmode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3d magnetic topologies

common stabilizers for linear control systems in the presence of actuators outage abstract issue concerning the existence of common stabilizers for a linear control system experiences actuators outage are presented the possible outage of actuators examined in this study are not restricted to a pre specified set by finding common quadratic type lyapunov functions we obtain sufficient conditions for the existence of common stabilizers for cases of which all the possible failed actuators belonged to a pre specified set the results presented in this paper agree with those obtained by veillette in 1995 the control gain of common stabilizer for non nested case is explicitly derived to guarantee system stability a simplified checking condition for the existence of common stabilizers is also obtained for the extreme case when only single actuator can normally operate

h disturbance attenuation for state delayed systems in this note the differential game and dissipation inequality are applied to the disturbance attenuation or h control for linear systems with delayed state firstly a simple sufficient condition on the existence of a suboptimal h state feedback controller is given which is independent of delay and an observer based dynamic output feedback solution is presented in terms of riccati inequalities or riccati equations secondly a sufficient condition on the existence of a delay dependent state feedback is presented and the criterion is presented by a matrix inequality which can be solved by numerical methods