6 Identification System for Structural Health Monitoring in Buildings 33 Elements of matrices in (6.4) are related to the properties of each floor. An important advantage of (6.3) is that the solution of the system does not require any coordinate transformation. Matrix distribution is given by = 1 h2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 · · · 0 1 −2 1 0 · · · 0 0 1 −2 1 · · · 0 . . . . . . . . . . . . . . . . . . 0 0 · · · 1 −2 1 0 0 · · · 0 1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , b = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 . . . 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.5) u =[u0,u1,u2, . . . ,un] T, ˙u =[˙u0, ˙u1, ˙u2, . . . , ˙un] T, ¨u =[¨u0, ¨u1, ¨u2, . . . , ¨un] T (6.6) and h=H/(n+1) for uniformly spaced floors. Considering that the acceleration in the boundary condition is measurable and equivalent to the seismic excitation, ¨u0 = ¨ug. The entries u0 and ˙u0 ∈ R+ within vectors u and ˙u ∈ R p×1 in (6.6) describe the displacement and velocity of the ground, respectively. While the remaining floors ui and ˙ui withi =1, 2, . . . ,n represent the respective measurements in each floor. Finally ¨u ∈Rp×1 is the absolute acceleration vector. 6.3 Identification System Assuming that in a real building parameters β 2 M y η 2 M are unknown, in this section an identification system is designed to estimate them in real time. Let the discretized equation (6.3) be rewritten as1 z =β 2 M u+η 2 M ˙u, with z =(¨u−b¨ug) (6.7) redefining β 2 M u =ψβ 2 V =ψ β (6.8) η 2 M ˙u = ˙ψη 2 V = ˙ψ η (6.9) the elements of the matrices β 2 M y η 2 M are used to build the vectors β 2 V y η 2 V ∈ R p×1. Hence, u y ˙u are now matrices with dimension Rp×p. For ease of notation, they are renamed as ψy ˙ψand defined by β = 0 β 2 1 β 2 2 β 2 3 . . . β 2 n−1 β 2 n T (6.10) ψ = 1 h2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 · · · 0 u0 −2u1 u2 0 0 · · · 0 0 u1 −2u2 u3 0 · · · 0 0 0 u2 −2u3 u4 · · · 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 · · · un−2 −2un−1 un 0 0 0 · · · 0 un−1 −un ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.11) η = 0 η 2 1 η 2 2 η 2 3 . . . η 2 n−1 η 2 n T (6.12) 1This identification system was first used in [10].
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