# Optimal Control Method of Semi-Active Suspension System and Processor-in-the-Loop Verification
## 2 Materials and Methods
### Controller Design and Optimization  
The PID controller is preferred due to the reasons mentioned before for the MR damper control. The basic control diagram for the SASS is shown in Figure 9. Three control terms can be explained: the proportional expression (P) is in relation to the current error, the integral expression (I) is contingent upon the cumulative past errors, and the derivative expression (D) entails a prognostication of future errors. The controller output is regulated to maintain zero error condition. The error used for the controller is as follows:  
\[e(t)=0-ba \tag{5}\]  
where \(ba\) is the sprung mass acceleration for a ride comfort index. The PID controller fundamental equation is  
\[u(t)=K_{p}e(t)+K_{i}\int e(t)dt+K_{d}\frac{de(t)}{dt} \tag{6}\]  
The control signal \(u(t)\) is the current for the MR damper. These coefficients were determined with the optimization method. A stochastic optimization algorithm, PSO, developed by [PERSON] and [PERSON] [48], is based on the behavior of searching for potential food in a specific area of bird or fish swarms' particles. This algorithm is commonly utilized in problems that involve determining optimal particle placement within a given search space. In the context of SASS, the PSO algorithm employs an objective function that encompasses conflicting performance measures, such as ride comfort, road holding, and suspension deflection. The primary aim of employing an optimization algorithm is to optimize the system's performance metrics by minimizing or maximizing them. By considering the particles' position and velocity, the swarm collaboratively identifies the target location, resembling the search for the global best. Through multiple iterations, the PSO algorithm strives to achieve the optimal value for the swarm of particles, resulting in enhanced system performance. When applied to a PID controller, the tuning parameters (\(K_{p}\), \(K_{i}\), and \(K_{d}\)) represent the particles within the swarm, while the performance criteria serve as the objective function for the algorithm. In suspension design, three key performance criteria are essential: ride comfort, road holding, and handling. When designing the controller, the primary requirement lies in ride comfort, which is assessed through sprung mass acceleration. Providing comfort requires dampening the vibrations to which passengers are subjected [49]. External factors, such as road disturbances, can cause a lack of contact between the tires and the road. Therefore, minimizing tire deflection, which affects road holding, is crucial. Additionally, the reduction in suspension deflection is vital, particularly concerning the structural characteristics of the  
Figure 8: Experimental test setup. vehicle. Failure to adhere to this value results in a decline in ride comfort, road holding, handling and an increase in suspension system wear and fatigue. The aim is to minimize the cost function for optimization when taking into account the quarter car model given in Figure 1 and Equation (1), model variables in Table 2. The optimization is handled to supply control objectives, to minimize sprung mass acceleration for ride comfort, tire deflection for road holding, and suspension deflection for vehicle wear and handling. The cost function to be minimized is written to supply suspension system performance as  
\[J=w_{1}\int_{0}^{t}z_{s}^{2}dt+w_{2}\int_{0}^{t}t_{d}^{2}+w_{3}\int_{0}^{t}s_{ d}^{2} \tag{7}\]  
where the optimization period \([0,t]\) refers to the road disturbance period. The first term of Equation (7) is sprung mass acceleration (\(z_{s}\)), mentioned as a state variable \(\dot{x}_{1}\). The second term is tire deflection (\(t_{d}\)), which is about the road holding capability and is calculated as  
\[t_{d}=z_{u}-z_{r}=x_{3}-z_{r} \tag{8}\]  
The last term is suspension deflection and is calculated as  
\[s_{d}=z_{s}-z_{u}=x_{1}-x_{3} \tag{9}\]  
Another key parameter for the suspension system is the dynamic load coefficient (DLC).