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By van Maanen A.

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2 Prior and experimental information In this section, the main concepts of SM H∞ identification are introduced. Causal, discrete-time, single-input single-output, linear time-invariant, 2 BIBO stable, possibly distributed parameter, dynamical systems are consid. ered. Any such system S is uniquely determined by its impulse response hS = . S ∞ hk k=0 , whose power-series representation of the transfer function S (z) = . ∞ S k H∞ (D) defined as H∞ (D) = k=0 hk z is an element of the Hardy space . { f : D → C | f analytic in D and f ∞ = supz∈D |f (z)| < ∞}, where D = {z ∈ C : |z| < 1} is the open unit disk.

Moreover, since V ∗ ⊆ Ker C we have C(0, x2 , 0, x4 )t = C2 x2 + C4 x4 = 0 for any choice of x2 and x4 , and thus C2 = C4 = 0. To better display the structure of the matrix C, let us write Y = W CS ∗ for some direct sum∗ ∗ mand W. We have dimCS = dimCS = dim(Im T3 ) since Im T3 has no intersection with Ker C, and we can write, accordingly, C= C11 0 0 0 , C21 0 C23 0 with C23 square and nonsingular. The structure of C can be made even simpler if we make the following technical assumption (which is always satisfied over a field).

This property is important when the input set of Σd is split into a control input subset and several so-called failure input subsets. By analyzing the residual output, failure inputs, which are unknown inputs, may be detected and recognized. Therefore, the problem of constructing a residual generator is related to the problem of state observation in presence of unknown inputs. The chapter is organized as follows. In Section 2 we describe briefly the relationship between delay-differential systems and systems with coefficients in a ring and we recall some notions and preliminary results of the geometric approach for system with coefficients in a ring.

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Preliminary Evidence of Internal Motion in the Spiral Nebula Messier 101 (1916)(en)(5s) by van Maanen A.

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