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Some linear algebra issues concerning the implementation of blended implicit methods

numerical linear algebra WITH applications 2005年第2-3期12卷 305-314页

作者： Brugnano, L Magherini, C Univ Florence Dipartimento Matemat I-50134 Florence Italy

In this paper we discuss some linear algebra issues concerning the implementation of blended implicit methods (J. Comput. Appl. Math. 2000;116:41-62, Appl. Numer. Math. 2002;42:29-45, J. Comput. Appl. Math. 2004;164-165:145-158, In Recent Trends in numerical Analysis, Trigiante D (ed.), Nova Science Publication Inc.: New York, 2001;81-105) for the numerical solution of ODEs. In particular, we describe the strategies, used in the numerical code BiM (J. Comput. Appl. Math. 2004;164-165:145-158), for deciding whether re-evaluating the Jacobian and/or the factorization involved in the non-linear splitting for solving the discrete problem. Copyright (c) 2004 John Wiley I Sons, Ltd.

关键词： numerical methods for ODEs stiff problems iterative solution of linear systems nonlinear splittings computational codes

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Eigenvalue assignment with optimal spectral conditioning by static output feedback

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IEE PROCEEDINGS-CONTROL theory AND applications 2005年第6期152卷 621-627页

作者： Franzè, G Carotenuto, L Muraca, P Univ Calabria DEIS I-87036 Arcavacata Di Rende CS Italy

This paper presents a method for the robust assignment of the eigenvalues of a linear system by static output feedback, based on the minimisation of the spectral condition number of the closed-loop dynamic matrix. The procedure relies on an assignment method recently proposed by the authors, that enables the explicit parameterisation of the set of the feedback gain matrices which assign the prescribed closed-loop spectrum, and on the column-scaling properties of eigenvector matrices. The efficiency of the approach is demonstrated using several extensive numerical experiments.

关键词： algebra linear system feedback gain matrices robust assignment static output feedback optimal spectral conditioning eigenvector matrices explicit parameterisation column-scaling properties eigenvalue assignment eigenstructure assignment closed-loop dynamic matrix closed loop systems matrix algebra Control system analysis and synthesis methods linear systems feedback

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Two-grid methods for banded linear systems from DCT III algebra

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numerical linear algebra WITH applications 2005年第2-3期12卷 241-249页

作者： Chan, RH Serra-Capizzano, S Tablino-Possio, C Univ Insubria Sede Como Dipartimento Fis & Matemat I-22100 Como Italy Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Univ Milan Dipartimento Matemat & Applicaz I-20126 Milan Italy

We describe a two-grid and a multigrid method for linear systems whose coefficient matrices are point or block matrices from the cosine algebra generated by a polynomial. We show that the convergence rate of the two-grid method is constant independent of the size of the given matrix. numerical examples from differential and integral equations are given to illustrate the convergence of both the two-grid and the multigrid method. Copyright (c) 2004 John Wiley W Sons, Ltd.

关键词： DCT-III matrix algebra band matrices two-grid method multigrid method

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Two-step waveform relaxation methods for implicit linear initial value problems

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numerical linear algebra WITH applications 2005年第2-3期12卷 293-304页

作者： Pan, JY Bai, ZZ Ng, MK Chinese Acad Sci Acad Math & Syst Sci Inst Comp Math & Sci Engn State Key Lab Sci Engn Comp Beijing 100080 Peoples R China Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

A class of two-step waveforrn relaxation methods is established and studied for solving implicit linear initial value problems, which, in particular, includes the alternating direction implicit waveform relaxation (ADIWR) methods. The convergence property of the ADIWR methods is studied in depth under suitable conditions, and their computing behaviour is illustrated by numerical examples. Moreover, we discuss about the choice of the optimal parameters involved in the ADIWR methods. Copyright (c) 2004 John Wiley & Sons, Ltd.

关键词： initial value problems waveform relaxation alternating direction implicit iteration convergence analysis

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Symmetric schemes and Hamiltonian perturbations of linear Hamiltonian problems

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numerical linear algebra WITH applications 2005年第2-3期12卷 171-179页

作者： Amodio, P Iavernaro, F Trigiante, D Univ Bari Dipartimento Matemat I-70125 Bari Italy Univ Florence Dipartimento Energet I-50134 Florence Italy

We analyse the behaviour of symmetric boundary value methods applied to Hamiltonian evolutionary problems. We derive a forward-backward error analysis which allows its to state that the obtained numerical solution may be essentially regarded as the exact solution of a perturbed Hamiltonian problem of the same type as the original one and vice versa. A numerical test emphasizes the theoretical results. Copyright (c) 2004 John Wiley & Sons, Ltd.

关键词： Hamiltonian evolutionary problems symmetric schemes forward-backward error

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H ∞ model reduction of linear systems with distributed delay

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IEE PROCEEDINGS-CONTROL theory AND applications 2005年第6期152卷 662-674页

作者： Lam, J Gao, H Wang, C Univ Hong Kong Dept Mech Engn Hong Kong Hong Kong Peoples R China Harbin Inst Technol Space Control & Inertial Technol Res Ctr Harbin 150001 Peoples R China

This paper is concerned with the problem of H-infinity model reduction for linear distributed-delay systems. For a given stable system containing both discrete and distributed delays in the state, our attention is focused on the construction of reduced-order model of the same structure, which guarantees the corresponding error system to be asymptotically stable and has a prescribed R. norm error performance. This problem is solved through a delay-dependent approach. By defining a new Lyapunov functional and making use of the Newton-Leibniz formula, a delay-dependent bounded real lemma is first established in terms of linear matrix inequalities (LMIs). Based on this performance criterion and by employing the projection lemma, sufficient conditions are formulated for the existence of admissible reduced-order models. Owing to the presence of matrix inverse constraints, these conditions are intrinsically nonconvex. An algorithm based on the sequential linear programming matrix method is proposed for verifying these conditions, which can be readily implemented by using standard numerical software. Desired reduced-order models can be constructed if these conditions are satisfied. The effectiveness of the proposed model reduction method is illustrated via a numerical example.

关键词： H∞ norm error performance H∞ model reduction sufficient conditions asymptotic stability Lyapunov functional linear programming Simulation, modelling and identification delays Optimisation techniques Newton-Leibniz formula linear matrix inequalities Distributed parameter control systems reduced order systems matrix inverse constraints H∞ control Optimal control sequential linear programming matrix method Lyapunov methods linear systems algebra discrete delays reduced-order model Control system analysis and synthesis methods distributed delay systems Stability in control theory

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Quadratic L2 gain performance LPV system design by a LTI controller with ILMI algorithm

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IEE PROCEEDINGS-CONTROL theory AND applications 2005年第2期152卷 125-128页

作者： Xie, W Kitami Inst Technol Satellite Venture Business Lab Kitami Hokkaido 0908507 Japan

A quadratic L2 gain performance LPV control system has been designed using a linear time invariant (LTI) output feedback controller with an iterative linear matrix inequality (ILMI) algorithm. A linear parameter varying (LPV) plant is assumed with immeasurable dependent parameter, which varies in a plant parameter space that is a polytope. Different from gain scheduled LPV control, this is a complex control problem with the non-convex constraint that the controller gain matrix is the same for each vertex plant of the LPV system. To solve this control problem, a heuristic iterative algorithm is presented in the form of a linear matrix inequality (LMI) formulation. An effective method of setting an initial value to the ILMI algorithm proposed here promotes convergence to an admissible solution.

关键词： algebra iterative methods linear time invariant output feedback controller Interpolation and function approximation (numerical analysis) linear matrix inequalities control system synthesis linear parameter varying plant quadratic L2 gain performance iterative linear matrix inequality algorithm heuristic iterative algorithm Control system analysis and synthesis methods linear systems feedback

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An explanation for the logarithmic connection between linear and morphological system theory

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INTERNATIONAL JOURNAL OF COMPUTER VISION 2005年第2-3期64卷 157-169页

作者： Burgeth, B Weickert, J Univ Saarland Fac Math & Comp Sci Math Image Anal Grp D-66041 Saarbrucken Germany

Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. Generalising the conjugacy operation from convex analysis it formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory;a connection that has not been fully understood so far. Our article provides an explanation by disclosing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. The logarithmic Laplace transform makes a natural appearance as it corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the so-called Cramer transform. Originating from stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform relies on the logarithm and constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform.

关键词： linear system theory morphology convex analysis max-plus algebra min-plus algebra slope transform Cramer transform

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Parameter-dependent filter design with guaranteed H_∞ performance

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IEE PROCEEDINGS-CONTROL theory AND applications 2005年第5期152卷 531-537页

作者： Gao, H Lam, J Shi, P Wang, C Harbin Inst Technol Space Control & Inertial Technol Res Ctr Harbin 150001 Peoples R China Univ Hong Kong Dept Mech Engn Hong Kong Hong Kong Peoples R China Univ Glamorgan Sch Technol Pontypridd CF37 1DL M Glam Wales

In the paper, the problem of R-infinity filtering for a class of linear uncertain systems is studied. The parameter uncertainties are assumed to reside in a polytope. The paper is focused on the design of a parameter-dependent filter which guarantees the filtering error system to be asymptotically stable and has a prescribed H-infinity performance. By employing a parameter-dependent Lyapunov function approach, sufficient conditions are established for the existence of the desired filters in terms of linear matrix inequalities, which can be handled easily by using the available toolbox. Both continuous- and discrete-time cases are investigated. It is shown, via a numerical example, that the proposed filter design methods are more effective and less conservative than some existing results.

关键词： parameter uncertainties sufficient conditions parameter-dependent filter design linear uncertain systems linear matrix inequalities parameter-dependent Lyapunov function approach guaranteed H∞ performance Lyapunov methods algebra linear systems continuous time systems Discrete control systems discrete time systems filtering error system H∞ filtering filtering theory uncertain systems Stability in control theory Signal processing theory

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Min-max MPC algorithm for LPV systems subject to input saturation

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IEE PROCEEDINGS-CONTROL theory AND applications 2005年第3期152卷 266-272页

作者： Cao, YY Lin, ZL Zhejiang Univ Inst Modern Control Engn Hangzhou 310027 Peoples R China Univ Virginia Dept Elect & Comp Engn Charlottesville VA 22904 USA

In this paper, a new model predictive control (MPC) algorithm is developed for polytopic linear parameter-varying (LPV) systems subject to input saturation. A set invariance condition for discrete-time LPV systems with input saturation is identified and the invariant set is determined by solving an LMI optimisation problem, which directly incorporates input saturation. Based on this set invariance condition, a min-max MPC algorithm is proposed for the LPV systems with input saturation. A gain-scheduling, MPC algorithm is also proposed for the design of a parameter-dependent controller. numerical examples demonstrate the effectiveness of the algorithms.

关键词： invariance optimisation Combinatorial mathematics gain-scheduling algorithm predictive control control nonlinearities set invariance condition Optimisation techniques set theory linear matrix inequalities polytopic linear parameter-varying systems discrete-time systems Optimal control algebra linear systems Nonlinear control systems Discrete control systems control system synthesis discrete time systems Control system analysis and synthesis methods input saturation min-max MPC algorithm LMI optimisation problem LPV systems

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