By Ciprian Foias, Hitay Özbay, Allen Tannenbaum
Since its inception within the early Nineteen Eighties, H( optimization conception has develop into the regulate method of selection in strong suggestions research and layout. the aim of this monograph is to offer, in an academic style, a self contained operator theoretic method of the H( keep watch over for disturbed parameter platforms, that's, platforms which admit endless dimensional country areas. Such structures come up for difficulties modelled by means of partial differential equations or that have time delays. along with elucidating the math of H( regulate, wide therapy is given to its actual and engineering underpinnings. The options given within the ebook are rigorously illustrated by way of benchmark difficulties: an risky method with a time hold up which comes from the regulate of the X-29, and the keep watch over of a Euler-Bernoulli versatile beam with Kelvin-Voigt damping.
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Extra info for Robust Control of Infinite Dimensional Systems: Frequency Domain Methods
ROBUST STABILITY If P. = P m then Wm and IV. 1o) Our assumptions on the nominal plant imply that ]P(jw) l - IN1 (jw)N2(jw)]. This means that we can see W,~N1N2 as an additive uncertainty bound, and ~ as a multiplicative uncertainty bound. hr~, then whether we consider multiplicative or additive perturbations does not make any difference. 2 (On the weights): We assume that Wcl, W c]-1 , (WreN2), (W,,~N2) -1, are rational functions in ~ , and the additive uncertainty weight is given by = WmN1N . 11) We further assume that there exist wo and K _> 2, such that IW,n(jw)l > K for all w > Wo.
This assumption is crucial in our solution of the 7~¢¢ control problems, since the complexity of computations depend on the order of rod(Z). , md is inner). The theory presented here does not need this assumption, but we will use it to avoid technical details involving "outer factor absorption" problem for which we refer to . e. a is chosen such that P(s) does not have any pole or zero at s = a). l_e_hls), hi "- 21n(~), h > 0. 5. So, the plant has only one pole in the closed right half plane.
Then we claim that ¢(s) := 1 - [ ( 1 - ( b - ~ ) s ) / ( 1 - ( ' ~ ) s ] 1/2 1 -Jr [(1 - - ( b - ~ ) s ) / ( 1 -- ( @ ) 8 1 1 / 2 is a conformal equivalence 4 : G ---* D such that 4(0) = 0. It is a general fact, for given simply connected region G with 0 E G, a conformal equivalence of G with D which maps 0 to 0 is unique up to rotation of D, that is if 4 (0 : G ~ D i = 1,2, are two conformal equivalences with 6(0(0) = 0, then exp (j0)4 0) = 4 (2) for some 0 E [0, 2~r). If we then require that ¢ have real coefficients (such a conformal equivalence will always exist in all the cases of interest to us), then $ is uniquely determined.
Robust Control of Infinite Dimensional Systems: Frequency Domain Methods by Ciprian Foias, Hitay Özbay, Allen Tannenbaum