Error bounds for Newton refinement of solutions to algebraic riccati equations

Recent error bounds derived from the Schur method of solving algebraic Riccati equations (ARE) complement residual error bounds associated with Newton refinement of approximate solutions. These approaches to the problem of error estimation not only work well together but also represent the first computable error bounds for the solution of Riccati equations. In this paper the closed-loop Lyapunov operator is seen to be central to the question of whether Newton refinement will improve an approximate solution (region of convergence), as well as providing a means of bounding the actual error in terms of the residual error. In turn, both of these issues are related to the condition of the ARE and the damping of the associated closed-loop dynamical system. Numerical results are given for seven problems taken from the literature. This research was supported by the National Science Foundation (and AFOSR) under Grant No. ECS87-18897 and the National Science Foundation under Grant No. DMS88-00817.     

采用幂次趋近律的滑模控制稳态误差界

对一类不确定非线性系统采用幂次趋近律的滑模跟踪控制,分别推导了采用Slotine形式的传统滑模面和积分滑模面时的稳态跟踪误差的界.首先,基于Lyapunov方法求出了滑模误差的最终界和系统不确定性、幂次趋近律参数之间定量的数学关系.其次,利用有界输入有界输出稳定的方法,分别求出了采用Slotine形式的传统滑模面和积分...     

Computation of Sharp Rigorous Componentwise Error Bounds for the Approximate Solutions of Systems of Linear Equations

This paper is concerned with the problem of verifying the accuracy of approximate solutions of systems of linear equations. Recently, fast algorithms for calculating guaranteed error bounds of computed solutions of systems of linear equations have been proposed using the rounding mode controlled verification method and the residual iterative verification method. In this paper, a new verification method for systems of linear equations is proposed. Using this verification method, componentwise verified error bounds of approximate solutions of systems of linear equations can be calculated. Numerical results are presented to illustrate that it is possible to get very sharp error bounds of computed solutions of systems of linear equations whose coefficient matrices are symmetric and positive definite.     

Una variazione del metodo di Halley

Sommario In questa nota si studia una classe di metodi iterativi, per la ricerca delle radici di una equazione, che include il metodo di Halley. Si prova che ogni metodo della classe è del terzo ordine e si calcola la costante di errore per ognuno di essi. Si individuano poi quei metodi della classe che sono globalmente convergenti per funzioni genus zero o uno. Infine is trova un metodo che ha una costante in valore assoluto più piccola di quella di Halley. Tale metodo è di tipo razionale e involge le stesse derivate di quello di Halley.

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We study a class of iterative methods devoted to the search of the roots of an equation. The class includes Halley's method. We prove each method belonging to the class to be of third order and for it evaluate the error constant. Among such methods we identify the ones which are globally convergent for zero or one genus functions. In particulary we find a method which has an error constant whose absolute value is less than the Halley's one. This is a rational one-point method which involves the same derivatives as Halley's method.

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Lavoro svolto nell'ambito del gruppo G.N.I.M.     

Recurrence relations for rational cubic methods II: The Chebyshev method

We continue the analysis of rational cubic methods, initiated in [7]. In this paper, we obtain a system of a priori error bounds for the Chebyshev method in Banach spaces through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of Chebyshev iterates. The error estimates are exact for second degree polynomials. We also discuss some applications.     

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator

In this paper we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Maxwell operator. In particular, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm, are derived. Numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented.     

Accurate recovery-based upper error bounds for the extended finite element framework

This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests.     

Robust adaptive fault‐tolerant tracking control for uncertain linear systems with time‐varying performance bounds

In this paper, the problem of robust adaptive fault‐tolerant tracking control with time‐varying performance bounds is investigated for a class of linear systems subject to parameter uncertainties, external disturbances and actuator failures. In order to ensure the norm of the tracking error less than the user‐defined time‐varying performance bounds, we propose a new control strategy which is predicated on the generalized restricted potential function. Compared with the existing result, a novel method which provides two design freedoms is developed to reduce the tracking error. According to the online estimation information provided by adaptive mechanism, a fault‐tolerant tracking control method guaranteeing time‐varying performance bounds is developed for robust tracking of reference model. It is shown that the closed‐loop signals are bounded and the tracking error within an a priori given, time‐varying performance bounds. A simulation result is provided to demonstrate the efficacy of the proposed fault‐tolerant tracking control method.     

Robust weighted fusion Kalman predictors with uncertain noise variances

In this paper, the problem of designing weighted fusion robust time-varying Kalman predictors is considered for multisensor time-varying systems with uncertainties of noise variances. Using the minimax robust estimation principle and the unbiased linear minimum variance (ULMV) rule, based on the worst-case conservative system with the conservative upper bounds of noise variances, the local and five weighted fused robust time-varying Kalman predictors are designed, which include a robust weighted measurement fuser, three robust weighted state fusers, and a robust covariance intersection (CI) fuser. Their actual prediction error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties of noise variances. Their robustness is proved based on the proposed Lyapunov equation approach. The concept of the robust accuracy is presented, and the robust accuracy relations are proved. The corresponding steady-state robust local and fused Kalman predictors are also presented, and the convergence in a realization between the time-varying and steady-state robust Kalman predictors is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. Simulation results show the effectiveness and correctness of the proposed results.     

Robust stabilization and ultimate boundedness of dynamic surface control systems via convex optimization

In this paper, a new method of analysing the controller gains and filter time constants for dynamic surface control (DSC) is presented. First, since DSC provides linear error dynamics with perturbation terms for a class of non-linear systems, the design method can be used to assign the system matrix eigenvalues of the closed loop error dynamics. Then a procedure for testing the stability and performance of the fixed controller in the face of uncertainties is presented. Finally, an ellipsoidal approximation of the tracking error bounds for a tracking problem is obtained via convex optimization.     

Graphs and stability of algorithms

In this paper the method of [22] for roundoff error analysis is extended to the relative case. Two statements are proved for the analysis of backward stability and two corresponding classes of backward stable algorithms are characterized in a constructive way. The error bounds obtained with this approach rely on properties of dependence graphs and are of remarkable generality. A first-order error analysis is provided even though the terms of higher order in the error bounds can be easily estimated. Some examples of the application of the new technique are given in which either we obtain new error bounds or we derive known results in a new way. Received: January 1999 / Accepted: January 2000     

Cone-bounded nonlinearities and mean-square bounds—Smoothing and prediction

Mean-square performance bounds are derived for smoothing and prediction problems associated with the broad class of nonlinear dynamic systems which, when modeled by Ito differential equations, contain drift (·dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. Included in this paper are lower bounds on the error covariance attainable by any smoother or any predictor, including the optimum, and upper bounds on the performance of some simple, implementable predictors reminiscent of the designs which are optimal in the linear case. The lower bounds on smoothing and prediction performance are established using measure-transformation techniques to relate a version of the nonlinear problem to its linearization. The upper bound on prediction performance is constructed by a direct analysis of the estimation error. All the bounds hold for correlated system and observation noises. All are rigorously derived and independent of control or control law. In each case, the computational effort is comparable to that for the corresponding optimum linear smoothing or prediction problem. The bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.     

A geometric approach for computing a posteriori error bounds for the solution of a linear system

A geometric approach for calculating guaranteed error bounds for the solution of a linear system is presented. The error bounds are derived by simple geometric properties of the theory of convex polyhedrons. This approach basically differs from other well-known techniques and gives an optimal geometric characterization of the error bounds.     

Automatic Forward Error Analysis for Floating Point Algorithms

We investigate absolute and relative error bounds for floating point calculations determined by means of sequences of instructions (as, for example, given by a computer program). We get rigorous error bounds on the round-off or generated error due to the actual machine floating point operations, as well as the propagated error from one sequence to the next in a very convenient way by the computer itself. The results stated in the theorems can be used to implement software tools for the automatic computation of a priori worst case error bounds for floating point computations. These automatically computed bounds are valid simultaneously for all data vectors varying in the domain specified and their corresponding machine vectors fulfilling a maximum prescribed error bound.With great success we have used our method in the past to implement a fast interval library for elementary functions called FI_LIB [12]. Further numerical examples often show a high quality of the computed a priori bounds.     

Analysis of validated error bounds of surface-to-surface intersection

In this paper, we address the problem of computing the error bounds of surface-to-surface intersection and propose a novel procedure to reduce them. We formulate the surface-to-surface intersection problem as solving a system of ordinary differential equations and using the validated ODE solver we compute the validated a priori enclosures of an intersection curve, in which the existence and the uniqueness of a solution are guaranteed. Then we use straight line enclosures to reduce the size of the a priori enclosures. These reduced enclosures are again enclosed by bounding curves, which can be used as the reduced error bounds of the intersection curve. We demonstrate our method with tangential and transversal intersections.     

On the a-posteriori error bounds for the solution of ordinary nonlinear differential equations

This paper advances a procedure to compute the a-posteriori error bounds for the solution of a wide class of ordinary differential equations with given initial conditions. The method is based on an iterative scheme which yields upper and lower bounds which approach each other. The convergence of these iterations is proved analytically. Application of the proposed method is demonstrated by several examples. The method is independent of the integration scheme used. Also, separate bounds for each component of the solution are specifically available as a function of time compared to the bound on the norm yielded by conventional methods.     

An Analysis of Quasi-Monte Carlo Integration Applied to the Transillumination Radiosity Method

This paper presents an enhanced transillumination radiosity method that can provide accurate solutions at relatively low computational cost. The proposed algorithm breaks down the double integral of the gathered power to an area integral that is computed analytically and to a directional integral that is evaluated by quasi-Monte Carlo techniques. Since the analytical integration results in a continuous function of finite variation, the quasi-Monte Carlo integration that follows the analytical integration will be efficient and its error can be bounded by the Koksma-Hlawka inequality. The paper also analyses the requirements of the convergence, presents theoretical error bounds and proposes error reduction techniques. The theoretical bounds are compared with simulation results.     

TSK fuzzy function approximators: design and accuracy analysis

Fuzzy systems are excellent approximators of known functions or for the dynamic response of a physical system. We propose a new approach to approximate any known function by a Takagi-Sugeno-Kang fuzzy system with a guaranteed upper bound on the approximation error. The new approach is also used to approximately represent the behavior of a dynamic system from its input-output pairs using experimental data with known error bounds. We provide sufficient conditions for this class of fuzzy systems to be universal approximators with specified error bounds. The new conditions require a smaller number of membership functions than all previously published conditions. We illustrate the new results and compare them to published error bounds through numerical examples.     

Robust weighted state fusion Kalman estimators for networked systems with mixed uncertainties

This paper is concerned with robust weighted state fusion estimation problem for a class of time-varying multisensor networked systems with mixed uncertainties including uncertain-variance multiplicative and linearly correlated additive white noises, and packet dropouts. By augmented state method and fictitious noise technique, the original system is converted into one with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst-case system with the conservative upper bounds of uncertain noise variances, four weighted state fusion robust Kalman estimators (filter, predictor and smoother) are presented in a unified form that the robust filter and smoother are designed based on the robust Kalman predictor. Their robustness is proved by the Lyapunov equation approach in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Their accuracy relations are proved. The corresponding robust local and fused steady-state Kalman estimators are also presented, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method. Finally, a simulation example applied to uninterruptible power system (UPS) shows the correctness and effectiveness of the proposed results.