By Claude Brezinski (auth.), Claude Brezinski (eds.)

Many units (we say dynamical platforms or just structures) behave like black packing containers: they obtain an enter, this enter is remodeled following a few legislation (usually a differential equation) and an output is saw. the matter is to control the enter to be able to regulate the output, that's for acquiring a wanted output. this kind of mechanism, the place the enter is changed in keeping with the output measured, is termed suggestions. The examine and layout of such automated approaches is termed keep watch over thought. As we'll see, the time period method embraces any gadget and regulate idea has a wide selection of functions within the genuine international. keep an eye on concept is an interdisci plinary area on the junction of differential and distinction equations, process idea and facts. additionally, the answer of a keep watch over challenge includes many issues of numerical research and results in many attention-grabbing computational difficulties: linear algebra (QR, SVD, projections, Schur supplement, dependent matrices, localization of eigenvalues, computation of the rank, Jordan basic shape, Sylvester and different equations, structures of linear equations, regulariza tion, etc), root localization for polynomials, inversion of the Laplace rework, computation of the matrix exponential, approximation idea (orthogonal poly nomials, Pad6 approximation, persevered fractions and linear fractional transfor mations), optimization, least squares, dynamic programming, and so forth. So, keep watch over conception can also be a. strong excuse for offering numerous (sometimes unrelated) problems with numerical research and the tactics for his or her resolution. This e-book isn't really a publication on control.

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Each of them has its advantages and its drawbacks. The choice of a technique depends on the objectives to be fulfilled. Numerical considerations can also enter into the choice of a procedure according to the complexity of the problem. In this Section, we will describe some techniques for model reduction. For more details, see, for example, [86]. 13) where All E R kxk , B1 E R kxm and C 1 E Rpxk. Then, G is approximated by - G(s) = Cds! - All) -1 B 1· On this procedure, called truncation, see [96, pp.

22 A stable system with p = m is said to be passive if 1 "Is E (£, G(s) = G(s), 2 "Is E (£+ and "Ix E (£m, Re x* H(s)x ~ O. Passivity of a system means that it does not generate energy. 10. Poles and zeros In Section 9, we saw that the question of the stability of a linear system is connected to the localization of the eigenvalues of matrices or, equivalently, to the localization of the zeros of polynomials. We will now study this point in more details and derive some important consequences. In the presentation, we are mostly following [66, pp.

Extensions of the Pade approximation method to systems with uncertainties are given in [93,94]. Another algorithm for the solution of the partial realization problem, in polynomial description, is given in [23]. Related subjects are the factorization of Hankel matrices and the Euclidean algorithm; see [39] for a review in the scalar case m = p = 1. The topic is also connected to the matrix interpretation of formal orthogonal polynomials and Pade approximants [38, 13]. 8. Model reduction Model reduction consists in looking for a realization with a dimension k smaller than n.