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

ISBN-10: 1402007116

ISBN-13: 9781402007118

ISBN-10: 1461302617

ISBN-13: 9781461302612

Many units (we say dynamical structures or just platforms) 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 manage the enter with the intention to keep watch over the output, that's for acquiring a wanted output. the sort of mechanism, the place the enter is changed in accordance with the output measured, is termed suggestions. The learn and layout of such automated procedures is named keep watch over conception. As we are going to see, the time period process embraces any equipment and keep watch over idea has a wide selection of purposes within the genuine global. keep an eye on idea is an interdisci plinary area on the junction of differential and distinction equations, method idea and data. additionally, the answer of a regulate challenge contains many themes of numerical research and ends up in many attention-grabbing computational difficulties: linear algebra (QR, SVD, projections, Schur supplement, dependent matrices, localization of eigenvalues, computation of the rank, Jordan general shape, Sylvester and different equations, platforms of linear equations, regulariza tion, etc), root localization for polynomials, inversion of the Laplace remodel, computation of the matrix exponential, approximation concept (orthogonal poly nomials, Pad6 approximation, persevered fractions and linear fractional transfor mations), optimization, least squares, dynamic programming, and so on. So, keep an eye on idea can be a. sturdy excuse for offering a variety of (sometimes unrelated) problems with numerical research and the tactics for his or her answer. This booklet isn't really a ebook on control.

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T, to)]-T. So, the notion of duality means that, although their physical interpretation is quite different, the concepts of controllability and observability are mathematically equivalent. If two systems, defined by the triplets (A,B,C) and (AI,BI' Cd, are algebraically equivalent (that is Al = TAT-I, BI = TB and C1 = CT-l) then C(A 1 ,Bt} O(A 1 ,C1 ) 7. = = T· C(A, B) O(A,C)·T- 1 • Realization The notions of controllability and observability are related to realization theory. Realization theory consists of determining a system having a given rational transfer function matrix G that is, equivalently, transforming a given input u into a given output y.

It means that some state variables are not affected by the input or are not visible in the output. 10 needs the (numerical) computation of the rank of a matrix. This is a difficult numerical problem which involves the use of the SVD. 14 Let us illustrate the concepts of controllability and observability by another example. We consider the system x~ = 2Xl + 3X2 + U , x~ = -2X2. Clearly, X2(t} = exp( -2t}X2(O}. So, it is impossible, by changing u, to obtain X2(t,} = x, for a given value of x" and the system is not completely controllable.

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. 445-450]. 1 Poles We assume that the eigenvalues AI, . . , An of the matrix A are distinct (the case of multiple eigenvalues is more complicated, see [53]).

### Computational Aspects of Linear Control by Claude Brezinski (auth.), Claude Brezinski (eds.)

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