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( 1) ( ) ( ) ( ) ( ) ( )c cc ck k kk k k= + + = += + = +x A x B p x Ax Bpy C x D p y Cx Dpɺ (1) where x, p, and y are the state (n x 1), control or input (r x 1) and output vector (m x 1), respectively. Ac, Bc, Cc and Dc are constant matrices representing a continuous-time system and A, B, C and D are for a discrete-time system. Primary objective of the ERA analysis is to reconstruct the discrete-time model identified by a triplet [A, B, and C] from the impulse response. The ERA starts with constructing a Hankel matrix which consists of pulse response samples. 1 22 3 11 1......(0). . . ....Y Y YY Y YH P QY Y Yββα βα α α β++ + − = = (2) where Pα = [C, CA, CA2, … ,CAα+shift-1]T is the observability matrix and Qβ = [B, AB, A2B, … , Aβ+shift-1B] is the controllability matrix and Yk indicates the system Markov parameter matrix. Since D=Y0, only the triplet needs to be reconstructed. However, the triplet is not determined uniquely [Juang 1994]. The Hankel matrix is constructed using time-shifted pulse response samples. The last row and column blocks are 10 times more time-shifted than normal time-shift size to calculate consistency indicators to be introduced later in this paper. The rank of the Hankel matrix is determined by checking the number of non-zero singular values. Relatively small or negligible singular values imply that they contain noise information as well as system information. Then the triplet for the discrete-time system can be identified as follows [Juang et al. 1985] 1/ 2 1/ 21/ 21/ 2(1)Tn n n nTn n rTm n n− −===A Σ R H S ΣB Σ S EC E R Σ⌢⌢⌢ (3) where Rn and Sn are formed by eliminating columns of R and S (i.e. Left and right eigenvectors of Hankel Matrix H(0) ) corresponding to relatively small or negligible singular values. It is noted that the relationships RnTRn=I and SnTSn=I are established. H(1) is a time-shifted Hankel matrix from H(0). EmT = [I 0] and ErT = [I 0] are used for selecting the system matrices B⌢ and C⌢ from extended controllability and observability matrices. However, in practical implementation, ERA analysis can contain a significant number of computational (e.g. nonphysical) modes due to nonlinearity, noise effect, and inadequate excitations, etc. Several consistency indicators are introduced to distinguish the nonphysical modes from identified modal characteristics through ERA analysis. NEW CONSISTENCY INDICATOR FOR FINDING TRUE PHYSICAL MODES The Consistent-Mode Indicator (CMI) was developed as a primary accuracy indicator of ERA to assess the consistency of the identified modes, which is a product of Extended Modal Amplitude Coherence (EMAC) and Modal Amplitude Collinearity (MPC) [Pappa et al. 1993]. Even though 50% CMI cut-off value is routinely used in practice, it allows some real modes with poor accuracy to be deleted assuming another ERA analysis is needed to identify the modes [Pappa 1999]. As noted, structural damping is associated with amplitude change of the ambient vibration, damage states and change of boundary frictions in real structures. In this section, an improved consistency indicator called CMI from oberservability matrix (CMI_O) is suggested in such a way that it is more consistent under expected environmental change. The CMI_O for the ith identified mode is defined as Ni ij ij=11CMI_O = EMAC_O MPC (%)N × ∑
(4) where MPC is the modal phase collinearity from Reference [Pappa et al. 1993]; EMAC_O is called as EMAC from observability matrix to be introduced in this section; i indicates the ith mode; j indicates the jth output point; and N indicates the total number of output points, that is, sensor locations. EMAC_O measures a predictability of free decaying mode shapes by the identified modal properties (e.g. natural frequency, mode shapes, damping ratio and modal participation factor). The actual modal amplitude at time step T0 corresponding to the last row of the Hankel matrix is calculated as follows 1/ 20 (T ) n n = Φ R Σ Ψ (5) where n R indicated the left eigenvector of the H(0) with only the last row block of Rn remained; Ψ is the eigenvector matrix of the identified system matrix ˆ A . Then the predicted modal amplitude from the identified results are calculated from observability matrix as follows 2 20 (T ) 'shift shift α α + − + −= = Φ CA Ψ C Λ⌢ ⌢ɶ (6) where shift is the number for time-shifting for the last row and column block of the Hankel matrix which is set to be 10 in this paper. Then the ratio of actual and predicted modal amplitude of the ith mode at the jth output point is calculated as 00 0000 00( ) for ( ) ( )( )( ) for ( ) ( )( )ijij ijijOijijij ijijTT TTZTT TT ≤ = > Φ Φ ΦΦΦΦ ΦΦɶɶɶɶ (7) In company with the modal amplitudes, phase angles, θij=tan-1(Im{ / ij ijΦ Φ ɶ }/Re{ / ij ijΦ Φ ɶ }) and −π ≤ θij ≤+ π , are also compared for the actual and predicted mode shapes as follows ( ) 1.0 /( / 4) for / 40.0 for otherwiseij ij OijW θ π θ π − ≤ = (8) where OijW are used as weighting factors. Then the EMAC from observability matrix for mode i and output point j is calculated as EMAC_O 100 (%)O Oij ij ijZ W = ⋅ × (9) The ERA in conjunction with NExT technique frequently assumes limited number of reference nodes. For example, one reference node is normally selected for calculation of the impulse response function in terms of cross-correlation functions.