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By Cornelius T. Leondes

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Example text

Then, according to [17], Eqs. »(t) = * , 0 ; S t < ; s } ) = l . -martingale after time s for all F e CQ (U6). Denote by T(S,X; u) the following exit time: [inf {t > s: CusJt) Φ D T(S,X;W):= CusJs) = XED} where 0 if CuStX(s) = χ φ D [ oo if C" x(t) e D (44) for all ί > 5, (s, x) e [0, oo) x D>6 The quantity T(S, X; W) is the first exit time after s of ζ" x from D. Also, define the following class of admissible feedback strategies: U:= {U = (VE1,VE2,VP1,VP2)E U0: sup (s,x)e[0,oo)*D Elxz(s,x;u)< oo}, (45) where E" x denotes the expectation operator with respect to P" x.

37) By introducing the quasi full-state estimate x(t) = FTxc(t) G Un, so that μχ(ή = x(t) and xe(t) = Φχ(ή e U"u, we can write (37) as x(t) = μΑμχ(ή + ßQ*Vil(y(t) - Cx(t)). (38) Note that although the implemented estimator (37) has the state xe(f) e M"u (38) can be viewed as a quasi full-order estimator whose geometric structure is entirely dictated by the projection μ. Specifically, error inputs Q^V^iyit) — Cx(t)) are annihilated unless they are contained in [ ^ ( μ ) ] 1 = 3#(μΎ). Hence the observation subspace of the estimator is precisely @(μΎ).

Sections V and VI present techniques based explicitly on the finite element method and capable of identifying mass and stiffness distributions. Then, in Section VII a Rayleigh-Ritz type parameter identification technique is discussed, which permits the identification of mass and stiffness distributions on the basis of prior knowledge of modal information such as natural frequencies. In Section VIII a procedure for identifying natural frequencies and mode shapes of self-adjoint systems is presented.