# Optimal Control Method of Semi-Active Suspension System and Processor-in-the-Loop Verification
## 2 Materials and Methods
### Characterization and Modeling of the MR Damper  
MR fluids are magnetic field-dependent fluids used in a semi-active damping system. MR fluids usually consist of magnetic particles dispersed in a carrier fluid. In this study, we used the MR damper, which was developed and described in [47]. The structure and prototype of the MR damper are shown in Figure 2, and the technical specifications are given in Table 3.  
A series of tests were performed to identify the MR damper's damping characteristics under harmonic excitation on the Roehrig 20 VS shock absorber test device as shown in Figure 3. For damping force measurement, tests were carried out by applying 3.18 Hz sinusoidal excitation and \(\pm 25\) mm displacement, taking into account the JASO C602:2001 standard. Tests were repeated for coil currents of 0 A, 0.2 A, 0.8 A, 1.4 A, and 2 A. The force-displacement F(X) and the force-velocity F(V) were achieved from the damping tests and can be investigated in detail in [47]. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{2}{c}{**State Variables**} & \multicolumn{2}{c}{**Outputs**} & \multicolumn{2}{c}{**Inputs**} \\
**Variables** & **Description** & **Variables** & **Description** & **Variables** & **Description** \\ \hline \(x_{1}=z_{s}\) & Sprung mass displacement & \(y_{1}=z_{s}\) & Sprung mass displacement & \(u=F_{MR}\) & MR damper force \\ \(z_{2}=\hat{z}_{s}\) & Sprung mass velocity & \(z_{2}=\hat{z}_{s}\) & Sprung mass velocity & \(d=z_{r}\) & Road input (as disturbance) \\ \(x_{3}=z_{u}\) & Unsprung mass displacement & \(y_{3}=z_{u}\) & Unsprung mass displacement & \\ \(x_{4}=\hat{z}_{u}\) & Unsprung mass velocity & \(y_{4}=\hat{z}_{u}\) & Unsprung mass velocity & \\ \hline \end{tabular}
\end{table} 
Table 2: State-space model parameters of SASS. Non-parametric modeling uses a black box to determine the damper dynamics based on experience and measurable data, eliminating the need for a damper structure. This non-parametric approach is thought to map the relationship between the tested damper's inputs and output data. A non-parametric MR damper model was developed by training a feedforward ANN for five different currents. Figure 3: The Roehrig 20 VS shock absorber test device. \begin{table}
\begin{tabular}{c c} \hline \hline
**Component and Specification** & **Value** \\ \hline Stroke & 180 mm \\ MR fluid type & Carbonyl iron-based \\ Coil inductance & 5.74 mH \\ Coil resistance & 34.45 \(\Omega\) \\ MR fluid gap & 1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: State-space model parameters of SASS. Figure 2: Structure and prototype of the MR damper. The inputs for this network included the coil current, velocity, and displacement of the MR damper. The damping force of the MR damper was chosen as the output of the network. The characterization tests of the MR damper were performed for five different current values, and a total of 6078 data were obtained for the non-parametric model. These data were used for obtaining the ANN model with the MATLAB Neural Network Toolbox. The test data were divided into three sections randomly. Then, 70% of the test data was for training, 15% of the data was for testing, and 15% was for validation of the network. The network was trained with the Levenberg-Marquardt algorithm, which evaluates performance by optimizing the weight and bias parameters, aiming to minimize training errors in order to attain the desired output. The best performance was obtained with a network structure of a single hidden layer containing 15 neurons with a tangent sigmoid activation function. A linear activation function was chosen for the output layer. The selection of the number of hidden layers, the number of neurons within these hidden layers, training epochs, and the choice of activation functions were determined through an empirical trial-and-error process.