Error bounds for Newton refinement of solutions to algebraic riccati equations

Kenney, Charles; Laub, Alan J.; Wette, M.

doi:10.1007/bf02551369

Cited by 31 publications

(14 citation statements)

References 16 publications

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“…A standard reference for numerical algorithms for solving the algebraic Riccati equation is the work of Laub [108]. We also mention more recent work by Kenney, Laub and Wette [98], and Laub and Gahinet [109].…”

Section: Notes and Referencesmentioning

confidence: 99%

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Control Theory for Linear Systems

Trentelman

Stoorvogel

Hautus

2001

Communications and Control Engineering

487

462

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PrefaceThis book originates from several editions of lecture notes that were used as teaching material for the course 'Control Theory for Linear Systems', given within the framework of the national Dutch graduate school of systems and control, in the period from 1987 to 1999. The aim of this course is to provide an extensive treatment of the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs.One of the important themes of control is the design of controllers that, while achieving an internally stable closed system, make the influence of certain exogenous disturbance inputs on given to-be-controlled output variables as small as possible. Indeed, in the appropriate sense this theme is covered by the classical linear quadratic regulator problem and the linear quadratic Gaussian problem, as well as, more recently, by the H 2 and H ∞ control problems. Most of the research efforts on the linear quadratic regulator problem and the linear quadratic Gaussian problem took place in the period up to 1975, whereas in particular H ∞ control has been the important issue in the most recent period, starting around 1985.In, roughly, the intermediate period, from 1970 to 1985, much attention was attracted by control design problems that require to make the influence of the exogenous disturbances on the to-be-controlled outputs equal to zero. The static state feedback versions of these control design problems, often called disturbance decoupling, or disturbance localization, problems were treated in the classical textbook 'Linear Multivariable Control: A Geometric Approach', by W.M. Wonham. Around 1980, a complete theory on the disturbance decoupling problem by dynamic measurement feedback became available. A central role in this theory is played by the geometric (i.e., linear algebraic) properties of the coefficient matrices appearing in the system equations. In particular, the notions of (A, B)-invariant subspace and (C, A)-invariant subspace play an important role. These notions, and their generalizations, also turned out to be central in understanding and classifying the 'fine structure' of the system under consideration. For example, important dynamic properties such as system invertibility, strong observability, strong detectability, the minimum phase property, output stabilizability, etc., can be characterized in terms of these geometric concepts. The notions of (A, B)-invariance and (C, A)-invariance also turned out to be instrumental in other synthesis problems, like observer design, problems of tracking and regulation, etc. In this book, we will treat both the 'pre-1975' approach represented by the linear quadratic regulator problem and the H 2 control problem, as well as the 'post-1985' approach represented by the H ∞ control problem and its applications to robust control. However, we feel that a textbook dedicated to control theory for linear state space systems should also contain the central issues of the 'geometric approach', namely a treatment...

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Section: Notes and Referencesmentioning

confidence: 99%

“…We refer to Roberts [154] and Byers [26]. For solving Riccati equations we also mention more recent work by Kenney, Laub and Wette [98] and Laub and Gahinet [109].…”

Section: Notes and Referencesmentioning

confidence: 99%

Control Theory for Linear Systems

Trentelman

Stoorvogel

Hautus

2001

Communications and Control Engineering

487

462

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“…With respect to rounding and truncation errors, the sign function has been found to often work just as well as invariant subspace techniques, but it can be shown that occasionally ill-conditioned sign iterations will cause numerical instability ( [52], [53]), and with intermediate low order approximations, such numerical difficulties are even more prominent. Hence we cannot 'solve' the Lyapunov and Riccati equations; there will always be non-zero residuals, , , the norms of which are not necessarily a good measure of the backwards error [54], [55], and the suggested criteria are not easily calculable using SSS methods. Furthermore, due to the potential fragility of optimal controllers [56] it is not acceptable to blindly use an approximated controller and assume that it will have the expected closed loop stability and performance; some kind of closed loop test is needed.…”

Section: B Numerical Issues Relating To the Approximationsmentioning

confidence: 99%

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

Rice

Verhaegen

2009

IEEE Trans. Automat. Contr.

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Abstract-We consider the problem of designing controllers for spatially-varying interconnected systems distributed in one spatial dimension. The matrix structure of such systems can be exploited to allow fast analysis and design of centralized controllers with simple distributed implementations. Iterative algorithms are provided for stability analysis, analysis and sub-optimal controller synthesis. For practical implementation of the algorithms, approximations can be used, and the computational efficiency and accuracy are demonstrated on an example.

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“…Chen [3] has derived global perturbation bounds for the solution, and Xu [18] has improved and sharped Chen's results. Kenney, Laub and Wette [7] have derived residual error bounds associated with Newton refinement of approximate solutions. Ghavimi and Laub [4] have presented a new backward error criterion, together with a sensitivity measure, for assessing solution accuracy of the ARE (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…An important problem is: How near isX to the exact solution X ? Kenney, Laub and Wette [7] have studied the real ARE (1.1) (i.e., the matrices of (1.1) are in R n×n , the set of real n × n matrices), and proved the following result: …”

Section: Introductionmentioning

confidence: 99%

Residual bounds of approximate solutions of the algebraic Riccati equation

Sun

1997

Numerische Mathematik

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LetX ≥ 0 approximate the unique Hermitian positive semi-definite solution X to the algebraic Riccati equation (ARE)be the residual of the ARE with respect toX , and define the linear operator L byBy applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solutionX , we obtained the following result: If A − FX is stable, and if 4 L −1 L −1R F < 1 for any unitarily invariant norm , then

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Error bounds for Newton refinement of solutions to algebraic riccati equations

Cited by 31 publications

References 16 publications

Control Theory for Linear Systems

Control Theory for Linear Systems

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

Residual bounds of approximate solutions of the algebraic Riccati equation

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