Michael Do¨hler, Falk Hille, Xuan-Binh Lam, Laurent Mevel and Werner Ru¨cker • For the Unweighted Principle Component algorithm of the data driven approach, the matrix H data p+1,q def =Y+ p+1Y−q T Y−q Y−q T −1 Y−q is defined. It enjoys the factorization property H data p+1,q =Op+1Xq (3) into matrix of observability and Kalman filter state sequence. As Hdata p+1,q is usually a very big matrix and difficult to handle, a thin RQ decomposition of the data matrices is done at first: Y−q Y+ p+1 =RQ= R11 0 R21 R22 Q1 Q2 . Then, Hdata p+1,q =R21Q1 follows and the subspace matrix is defined as H data,R p+1,q =R21, which enjoys also factorization property (3), but with a different matrix on the right side. See also [6] for further details. In what follows, the superscripts of the subspace matrix Hp+1,q are skipped, as the identification procedure is the same for the covariance and data driven approach. Now we want to obtain the eigenstructure of the system (1) from a given matrix Hp+1,q. The observability matrix Op+1 is obtained from a thin SVD of the matrix Hp+1,q and its truncation at the desired model order n: Hp+1,q =UΔ V T = (U1 U0) Δ1 0 0 Δ0 VT, Op+1 =U1 Δ 1/2 1 . (4) The observation matrixHis then found in the first block-row of the observability matrix Op+1. The state-transition matrix F is obtained from the shift invariance property of Op+1, namely O↑p(H,F)=Op(H,F) F, where O↑p(H,F) def =⎛ ⎜⎜⎜ ⎝ HF HF2 .. . HFp ⎞ ⎟⎟⎟ ⎠ . (5) Of course, for recoveringF, it is needed to assume that rank(Op)=dimF, and thus that the number p+1 of block-rows in Hp+1,q is large enough. The eigenstruc240
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