# A Data-Driven Model for Aerodynamic Loads on Road Vehicles Exposed to Gusty Bora-Like Winds
## 4 Modeling Framework  
The experimental results are further employed to develop a simple data-driven modeling approach, which is expected to predict the time history of unsteady aerodynamic loads on vehicles in transient winds. To simulate unsteady wind forces on the generic van model, the following linear time invariant (LTI) model ([PERSON] [11]) is assumed,  
\[ \begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B} \mathbf{u}(t),\\ F(t)=\mathbf{c}^{\mathrm{T}}\mathbf{x}(t)+\mathbf{d}^{\mathrm{T}}\mathbf{u}(t), \end{array}  \tag{1}\]  
where \(\mathbf{A}\in\Re^{n\times n}\), \(\mathbf{B}\in\Re^{n\times 2}\), \(\mathbf{c}\in\Re^{n}\) and \(\mathbf{d}\in\Re^{2}\) are state matrix, input matrix, output vector and feed-through vector, respectively. Time-dependent vector \(\mathbf{x}(t)\in\Re^{n}\) is the state vector, \(F(t)\) is the wind-induced force and \(\mathbf{u}(t)= \begin{pmatrix}u(t)&e(t)\end{pmatrix} ^{\mathrm{T}}\in\Re^{2}\) is the input vector comprising of wind velocity \(u(t)\) and error channel \(e(t)\). The primary motivation for choosing the LTI model for modeling periodic cross-wind loads on a vehicle is its inherent simplicity. Due to the fact that the wind is simulated in laboratory conditions, its spectrum exhibits discrete frequency content. This is unlike bora-like winds in real conditions, where the spectrum is continuous. Indeed, by calculating a discrete Fourier transform of the wind velocity \(u(t)\), as seen in Figure 4, dominant harmonic signals are clearly identified at frequencies 0.166 Hz, 0.5 Hz, 0.834 Hz and 1.165 Hz, as that is commonly the case with the ideal signals, where the harmonic components have peaks at 3, 5 and 7 times larger frequencies than the first low-frequency peak. In Figure 4, the velocity time series is reported in the form of unfiltered amplitude spectrum of longitudinal velocity fluctuations. Therefore, by calculating discrete Fourier transforms for the unsteady aerodynamic forces experienced by the van model, harmonic signals are observed at the same frequencies, i.e., at frequencies close to 0.166 Hz, 0.5 Hz, 0.834 Hz and 1.165 Hz. As an illustration, the discrete Fourier transform of the unsteady side force is presented in Figure 5.  
This provides additional justification for selecting a linear model, i.e., it is well known that the output of a LTI system fed with harmonic signal at a given frequency is also harmonic, with exactly the same frequency content. Thus, this indicates that the measured harmonic signals at the van location, yielding the four discrete frequencies specified above, are due to harmonic wind velocity oscillations only. This serves as a basis for subsequent system identification, i.e., estimation of the unknown model parameters \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{c}\) and \(\mathbf{d}\). Figure 4: The unfiltered amplitude spectrum of longitudinal velocity fluctuations (discrete Fourier transform of the measured wind velocity): (**a**) model A and (**b**) model B. Prior to this, however, the measured data are pre-processed in the following fashion: Depending upon the transducer position on the vehicle, the measured data also contain signals due to vehicle-induced unsteady flow and vortex shedding, as well as the noise. Such signals are not a direct consequence of wind variations and they present a significant difficulty in system identification. Hence, four band-pass filters centered at the characteristic frequencies are applied to the measured wind force data in order to filter out the spurious signals. For that purpose, second-order digital peaking filters with appropriate bandwidths are used. As a result, wind force data with the desired frequency content are obtained and used in model parameter calculations. This filtering approach generally allows for removing the less relevant content of wind fluctuations. Finally, the model parameters **A**, **B**, **c** and **d** are estimated using the subspace identification algorithm ([PERSON] and [PERSON] [11]).