Sensitivity Reduction and Robustness
01 Jan 1994-pp 297-518
TL;DR: In Chapters Two to Five the authors devote their attentions primarily to issues regarding parameter-sensitivity analysis, and incorporate in control synthesis algorithm any information regarding possible discrepancies in system “output”, in order to generate new controllers that can stand against the consequences of some possible uncertainties.
Abstract: In Chapters Two to Five we devote our attentions primarily to issues regarding parameter-sensitivity analysis. In spite of different uncertainties which may exist in a dynamical system, we expect that our controller will ultimately maintain qualitatively and/or quantitatively the overall system operation as desired. Or it will maintain this operation within a prespecified class of such operations which this is really the key issue in any robustness methodology. Therefore we must incorporate in control synthesis algorithm any information regarding possible discrepancies in system “output”, in order to generate new controllers that can stand against the consequences of some possible uncertainties. This incorporation is by no means a simple task, and before we let our expectations exceed our means, we examine some of our earlier assumptions and review the class of problems which currently can be analyzed.
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01 Jan 1991
TL;DR: In this article, the standard problem of finding an internally stabilizing controller for finite-dimensional linear time-varying continuous-time plants is considered, and it is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions.
Abstract: In this paper the standard problem of $H^\infty $ control theory for finite-dimensional linear time-varying continuous-time plants is considered. The problem is: given a real number $\gamma > 0$, find (if one exists) an internally stabilizing controller such that the closed-loop operator norm is less than $\gamma $. Under rather weak assumptions on the plant model, it is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space formulae for one solution to the problem are also given.
180 citations
Posted Content•
TL;DR: In this paper, the Hilbert-Schmidt-Hankel norm (HSH-norm) of a transfer function of a stable system is shown to be equal to the square root of the area enclosed by the oriented Nyquist diagram of the transfer function.
Abstract: It is shown that the Hilbert-Schmidt-Hankel norm (HSH-norm) of a transfer function of a stable system is equal, up to a constant factor, to the square root of the area enclosed by the oriented Nyquist diagram of the transfer function (multiplicities included). A generalization is presented for the case of systems which have no poles on the stability boundary, but otherwise have no restrictions on the pole locations. >
31 citations
01 Jan 1964
TL;DR: Axiomatic approach to control system theory as generalization of dynamic systems, noting weak and strong stability and Liapunov function.
Abstract: Axiomatic approach to control system theory as generalization of dynamic systems, noting weak and strong stability and Liapunov function
8 citations
DOI•
01 Jul 1986
TL;DR: In this paper, the problem of controller design for systems described by linear discrete-time models, with a known bound for the approximation error introduced by state equations, has been solved using necessary conditions for saddle-point equilibrium of a two-person zero-sum linear quadratic game.
Abstract: A problem of controller design for systems described by linear discrete-time models, with a known bound for the approximation error introduced by state equations, has been solved using necessary conditions for saddle-point equilibrium of a two-person zero-sum linear quadratic game. OL and OLF control laws have been found, and an algorithm for their calculation has been proposed and tested.
1 citations
Journal Article•
TL;DR: In this article, a successive approximation algorithm which consists of two sequences of progressively finer stages of discretization, diagonalization method, with a prescribed number of iterations of the optimization algorithm carried in each stage.
Abstract: This paper continues the work of [3]. We determine successive approximation algorithm which consists of two sequences of progressively finer stages of discretization, diagonalization method, with a prescribed number of iterations of the optimization algorithm carried in each stage. The solutions of these problems are discretization strategies which minimize the time needed to reduce the initial cost-error by prescribed amount.
1 citations
References
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TL;DR: The algorithm can be used as a building block for solving other distributed graph problems, and can be slightly modified to run on a strongly-connected diagraph for generating the existent Euler trail or to report that no Euler trails exist.
13,828 citations
TL;DR: In this article, 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.
Abstract: 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. >
5,272 citations
Book•
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13 Feb 1986
TL;DR: Reading optimal control frank l lewis solution manual ebook pdf 2019 is extremely useful because you could get enough detailed information in the book technology has.
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3,133 citations
TL;DR: In this paper, a complete characterization of all rational functions that minimize the Hankel-norm is derived, and the solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations.
Abstract: The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.
2,980 citations
15 Jun 1988
TL;DR: In this article, simple state-space formulas are presented for a controller solving a standard H∞-problem, where the controller has the same state-dimension as the plant, its computation involves only two Riccati equations, and it has a separation structure reminiscent of classical LQG theory.
Abstract: Simple state-space formulas are presented for a controller solving a standard H∞-problem. The controller has the same state-dimension as the plant, its computation involves only two Riccati equations, and it has a separation structure reminiscent of classical LQG (i.e., H2) theory. This paper is also intended to be of tutorial value, so a standard H2-solution is developed in parallel.
2,875 citations