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Analysis of Sensations by Ernst Mach

By Ernst Mach

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6). 2 below, plays a s i g n i f i c a n t part in the analysis of chapter 5. 1, the following d e f i n i t i o n of s t a b i l i t y along the pass is expressed in terms of the existence of f i n i t e bounds on the scalars He and l a as a ~ + ~. I t s e f f e c t i v e action is to demand that the r a t e of approach of the output sequence to the limit p r o f i l e has a guaranteed geometric upper bound independent of pass length for ~ > a o. 2: The. 57) o Despite i t s well defined physical meaning, t h i s d e f i n i t i o n is not in appropriate form f o r the derivation of s t a b i l i t y c r i t e r i a .

Suppose, therefore, that the variables Uj(t), Xj(t) and Yj(t), j _> 1, have been suitably extended from [O,a] to [O,+m) and let the same symbols denote these extensions. Then the Laplace transforms, o r ' s transforms', are defined as follows. 2~3(t ) + ... ~X(t,z) : . ~C(3(t) + ... 76) = ~l(t) ~ 2 ( t ) + z-2 ~ 3 ( t ) + ... respectively where ~ denotes the Laplace transform with respect to the along the pass variable t. 76) are contained in the following result. ]]p denotes any suitable vector norm.

37) and ~l(t) . . . l(t) ~ O. 40) which has no solution i f y~(O) is not in the range of P(z), i . e . zI - La cannot be surjective if [P(z) l = O. 4 is asymptotically stable i f , and only i f , a l l eigenvalues of the mxm matrix D1 l i e in the open unit c i r c l e in the complex plane. 4) is independent of the system matrices and, in p a r t i c u l a r , independent of the eigenvalues of A. 3. 4 holds and a strongly convergent sequence (Uk}k>1 is applied. 29) by t h e i r strong limits.

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