By Hidenori Kimura
1 Introduction.- 2 parts of Linear platforms Theory.- three Norms and Factorizations.- four Chain-Scattering Representations of the Plant.- five J-Lossless Conjugation and Interpolation.- 6 J-Lossless Factorizations.- 7 H-infinity keep an eye on through (J, J')-Lossless Factorization.- eight State-Space ideas to H-infinity keep watch over Problems.- nine constitution of H-infinity keep an eye on
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Additional info for Chain-scattering approach to h[infinity] control
Let ei-1 be an escape situation of co E P~', cow-1 = co~_~ be the respective escape component and fii the first return situation of coi-1 after ei-1. If fii = n, then E~(a) = 0 for each a C cow-1 and the lemma is obvious because (col-l) C coi-1 So, assume that fii < n. Then, fii is necessarily an essential return situation. 17. Therefore, on constructing the set f~,, the interval cow-1 is split into disjoint intervals w,~,k. Now, each com,k belonging to P~, verifies the next lemma, whose proof will be postponed to the end of this chapter.
M,,k~). By taking A sufficiently large so that m i2 _< e~o 2 8 Irn~l2 ... Irn, I2 _< 28e~0(1"*~t+'+1"~'1). On the other hand, since Irn~l > A, it follows that s A _< Irnll + ... + lrn~l. Hence, ~1 M 71~(M) <_ps(M)2A-~M e ~oM <_p~(M)e~o , where p~(M) is the number of solutions of the equation M = Im~l+ ... + Ims l. 6. ESTIMATES OF THE EXCLUDED SET 51 If s _> 2, use the Stirling's formula to get ps(M) < ~ M~- 1) ] < s-1 s- <- - ~ M + s-1 1 (s_l)(M+s_l)_l_ 1 1-M+s-1 M+s-1 " Since (s - 1) (M + s - 1) -1 < sM -1 < A -1 approaches zero as A --, oo, we may take p~(M) <_3e~M, as long as A is large enough.
Thus, since the derivative along an orbit is the product of the derivatives on each iterate, one expects to compensate the small derivatives near the critical point with those on the points away from it. Nevertheless, this compensation will only be possible if the orbit has too many points far from the critical point. This fact holds if we assume a third hypothesis called free assumption. To set this hypothesis we shall introduce some notions. , ~,/+1 -- 1}, that will be called returns, binding periods and free periods, respectively.
Chain-scattering approach to h[infinity] control by Hidenori Kimura