Concepciones y Definiciones Modernas sobre Discapacidad Intelectual

x(kT + T ) = A_dx(kT ) + w(kT ) (4.64) y(kT ) = C_dx(kT ) + ν(kT ) (4.65) The white noise assumption of the noise terms is an important requirement for the described system identification method. If the input u(kT ) contains dominant fre-quency components in addition to white noise, then the white noise assumption becomes invalid. Consequently, these dominant frequency components will be in-cluded in the eigenfrequencies of the system and will appear as poles of the state matrix A_d. This can cause the system identification results to become unclear or inaccurate due to the mixing of poles that represent the stochastic response of the system with spurious poles that represent the dominant frequencies originating from the input. Only those poles representing the stochastic response are of interest.

4.4.4 Identification of Stochastic State Space Models

The stochastic state space model described by Equations (4.64) and (4.65) can be solved using the Stochastic Subspace Identification (SSI) technique [127], which involves directly fitting a parametric model to the output time series data from the measurement transducers. Numerous algorithms that implement SSI are available, and one of the more recent algorithms is the Numerical Algorithm for Subspace State Space System Identification (abbreviated by the algorithum author to N4SID) [176]. The data-driven identification scheme used in the algorithm can provide

reliable state space models for complex, multivariable, dynamic systems using only the measured outputs.

Subspace system identification algorithms generally consist of two steps. The first step estimates the state vector x(kT ) by using two linear algebra tools: QR decomposition and singular value decomposition (SVD) [113]. Once the states are known, the identification of the unknown matrices, A_d and C_d, is achieved via a linear least-squares solution [95].

The modal parameters of the system are then determined via the standard eigen-value problem;

A_d= ΨΛΨ⁻¹ (4.66)

where Ψ ∈ C^2n×2n are the discrete time eigenvectors arranged in columns and Λ = 

µ_i⁻¹

∈ C^2n×2n is a diagonal matrix containing the discrete time com-plex eigenvalues. For under-damped systems, the structural modes are represented by complex conjugate pairs of eigenvalues (µ_i, µ_i+1^∗) and corresponding eigenvec-tors. The complex conjugate pair of discrete time eigenvalues can be expressed as eigenfrequencies f_i and damping ratios ζ_i by first converting to continuous time eigenvalues λ_i = ln(µ_i)/∆T , and then using the expressions established in Equations (4.35) and (4.36), resulting in the following relationships [95];

f_i = |λⁱ|

2π , ζ_i = −Re(λⁱ)

|λi| (4.67)

The estimated state vector x(kT ) does not generally correspond to a physical mean-ing. Therefore the eigenvectors Ψ of the state vector are converted to physical values by multiplying by the output matrix C_d, and the mode shapes are given by;

Φ = C_dΨ (4.68)

Implementing the SSI technique requires determining the number of parameters in the model, which is an essential but challenging process. The number of parame-ters in the model is also referred to as the model order. This model order establishes the state space dimension — the dimensions of the state matrix A_d. If the order chosen is too small, then the dynamics and noise characteristics of the system can-not be accurately modelled. Alternatively, selecting a model order that is too high can lead to significant uncertainties of the model parameters.

Obtaining an estimate of the order can be achieved by doubling the number of peaks in a frequency plot of a non-parametric spectrum-driven estimate, such as the peak picking technique or from the complex mode indication function. The number of peaks counted is doubled because each corresponds to a pair of complex conjugate eigenvalues, and therefore the model order is twice the number of resonance

frequen-cies observed in the spectral plot. In practice, the main focus is the extraction of modal parameters as opposed to finding the order that produces the best model.

For this reason, the most suitable approach is to view model results generated from multiple orders that range from a theoretical minimum, such as doubling the number of resonance peaks, to an over-specified order.

Due to the presence of measurement noise and other non-white-noise sources typical to ambient vibration measurements, non-structural modes will be included in the results set for each model order. Numerous methods are available to dis-cern structural from non-structural modes [128]. The stability plot [10] is one such method that compares modal properties for each identified mode at a certain model order with those obtained for the previous model order. If the modal property differ-ences are within pre-set limits, the identified mode is deemed stable. The stability limits are set for the change in frequency ∆f , change in damping ratios ∆ζ, and for the change in mode shape, which can be measured via the modal assurance criteria (MAC) described in Equation (4.43). The stability limits are defined as;

fi(p)− fⁱ(p + 1)

f_i(p) < ∆f (4.69)

ζ_i(p)− ζi(p + 1)

ζ_i(p) < ∆ζ (4.70)

(1− MAC(p, p + 1)) < ∆MAC (4.71) where p denotes the model order at which modal properties are identified. Using these comparisons and considering the nature of the noise processes, the identified non-structural modes will not show stability between consecutive model orders, and will be filtered from the identified modes.

Example: Stochastic Subspace Identification

Using computer code developed in Matlab and the System Identification Toolbox R2007b [169], the SSI technique is applied to data obtained from measurements recorded approximately two years after construction completion of Latitude tower.

The response time histories are displayed as part of the FDD example in Figure 4.3.

The stability limits are ∆f = 0.01, ∆ζ = 0.05, and ∆MAC = 0.99. For this example the model order has been increased by two at each iteration to improve the legibility of the stability plot displayed in Figure 4.8. The average normalised power spectral density of all input channels is also displayed as a guide to the stability plot.

The points in the stability plot (denoted by N) indicate a stable mode of vi-bration according to the simultaneous application of the limits described in Section 4.4.4 for natural frequency, damping ratio, and mode shape. If any of the three criteria are not satisfied the identified mode is not considered to be stable and is

0 0.5 1 1.5

Figure 4.8: Stability plot for SSI time domain system identification. Stability limits:

∆f = 0.01, ∆ζ = 0.05, and ∆MAC = 0.99. (N Stable SSI Model, — Power Spectral Density)

not plotted. The first three modes of vibration, between the frequencies 0.2 Hz and 0.5 Hz, are observed in both the SSI output and the spectral plot. They represent the fundamental modes of vibration for y-axis translation, x-axis translation, and torsion. The modes identified between the frequencies 0.7 Hz and 1.2 Hz represent the second modes of vibration for x-axis translation, y-axis translation, and torsion.

The natural frequency and damping ratio estimates for the first six modes of vi-bration are presented in Table 4.2. The standard deviation in the damping ratio estimate σ_ζ is reported as a percentage of the damping ratio. A comparison of the mode shapes for the SSI and FFD analysis produced a MAC value greater than 0.98 for all mode shapes, and therefore the mode shapes for the SSI are virtually identical to those displayed in Figure 4.7.

The stability plot in Figure 4.8 identifies three modes of vibration between 0.25 Hz and 0.30 Hz when the model order exceeds approximately 70. This addi-tional mode has a natural frequency of 0.279 Hz, which is approximately equal to the FDD estimate of 0.286 Hz. The intermittent nature of the stable orders for this mode indicates weak stability. As concluded for the FDD analysis, this mode is due to modal interference between the fundamental y-axis translation mode at f_y1 = 0.256 Hz and the fundamental x-axis translation mode at f_x1 = 0.293 Hz.

f ζ σ_ζ q¨_max

(Hz) (%) (%) (mg_n)

Translation X1 0.293 0.94 10.3 0.07

Translation Y1 0.256 0.91 5.5 0.13

Torsion T1 0.417 1.02 3.3 0.15

Translation X2 0.876 0.83 4.8 –

Translation Y2 1.010 1.18 3.4 –

Torsion T2 1.078 0.92 3.3 –

Table 4.2: Results for SSI time domain system identification.

The SSI derived mode shape of this additional mode is a superposition of the these two fundamental modes. The estimates for the dynamic characteristics of the funda-mental modes used the model orders above 70. By using these higher model orders, the results are less likely to be influenced by the presence of modal interference.