# Optimal Control Method of Semi-Active Suspension System and Processor-in-the-Loop Verification
## 1 Introduction  
Reducing the forces affecting the vehicle body through the suspension system improves ride comfort. Suspension systems also enhance the road-holding ability by increasing the physical contact point between the tire and the surface of the road [1, 2, 3, 4]. Road-holding and ride comfort parameters show a trade-off behavior [5]. This means that improving one of them declines the other. In the design of traditional suspension systems, elements of the system are determined by establishing a balance between these parameters. In view of controllability, passive, semi-active, and active suspension systems are the configuration for the suspension systems. Passive systems, which contain fixed-value components, cannot be controllable. Semi-active suspension systems (SASSs) usually contain magnetorheological (MR), electrorheological or variable orifice-type damping elements. These damping elements provide an adjustable damping ratio. Active suspension systems and SASS try to suppress the conflict between road holding and ride comfort by arranging unsprung mass acceleration, suspension deflection, and tire deflection by using several control algorithms [2, 4, 6]. SASS is widely preferred due to its low cost and low power consumption [7, 8]. The most important suspension element that provides this advantage is the damper. An MR damper is a controllable semi-active damping device used for vibration mitigation in several fields, such as seismic isolation, knee prosthesis, weapon recoil damper, enginemounts, seat suspension, and especially automotive suspension systems. In controllable suspension systems, the effectiveness of the controller on the suspension performance is quite high, as well as the damper. The proportional-integral-derivative (PID) controller is commonly preferred in engineering applications, owing to its cost effectiveness, ease of implementation, resilience, and low cost compared to the other methods. However, just a few studies on PID controllers have been carried out in the context of the SASS applications because regulating three controller coefficients (proportional (\(K_{d}\)), integral (\(K_{i}\)), and derivative (\(K_{d}\)) coefficients) requires a significant duration. The primary critical issue to obtain proper consistency on road holding and ride comfort is designing a robust controller with an appropriate optimization method. The proper design of the PID controller is possible with the appropriate gains. A common but time-consuming approach to determining these gains is to use a trial-and-error method [9]. This method involves adjusting the parameters in small increments and observing the system's response to define the optimal values. Alternatively, there are various tuning techniques, including the Ziegler-Nichols method, Cohen-Coon method, etc. On the other hand, using a linear controller structure to control nonlinear systems may not be effective in providing maximum performance for all states of the system. Therefore, there is a need to optimize the controller parameters over the entire operating range. Different performance parameters can be merged in controller optimization as an objective function. Optimizing the controller parameters is crucial to obtaining the global best values in the objective function [10]. For this reason, optimization algorithms, for instance, particle swarm optimization (PSO), gradient descent, genetic algorithm, and ant colony optimization, have been used to adjust these parameters. PSO is a population and stochastic-based optimization method stimulated by the social roles of fish and bird swarms [11]. This optimization method is used in many real-world problems thanks to its flexibility and easy adaptation to the problems. Furthermore, it also has a strong global search ability and robust performance [12]. MR damper models are in two main categories as parametric and non-parametric. Damper behavior can be characterized using a parametric model, such as a modified Bouc-Wen model.