## 3 Controller Designs for the Suspension System  
These controllers designed for the suspension system aim to increase car handling and passenger comfort. In addition, it is aimed to reduce the vibrations that occur in passive suspension systems by using the proposed control methods and designing a robust system. While the controller was being designed, the equations obtained for the system's control were prepared in the previous section. The system's control aims to design a controller that achieves minimum error value by increasing car handling and passenger comfort. The suspension system is controlled using linear quadratic regulator (LQR), PSO-based fuzzy logic control (FLC), and PSO-based fuzzy linear quadratic regulator (fuzzy-LQR) control methods. The most important problem during a system's control is determining the optimal control law that minimizes the determined performance index under various economic and safety constraints. The linear quadratic regulator (LQR) control method, which is one of the optimal control methods, is a method that uses the state-space model of the system [42]. It is a type of controller based on the principle of complete state feedback. The main objective of optimal control is to obtain control signals that result in a system satisfying certain physical constraints while overfulfilling (maximizing or minimizing) a selected performance criterion or cost function. The LQR control method is widely used because it is simple, optimal, and robust [43,44]. The input of the LQR control method is \(u=-K\times x\), where K represents the feedback control input and x represents the states of the system. Control input is chosen to minimize the following cost function, which is determined using state space equations. \[J=\frac{1}{2}\int_{0}^{t}\Big{(}x^{T}(t)Qx+u^{T}Ru\Big{)}dt \tag{11}\]  
The purpose of the control here is to minimize the integral of the squared power index. The matrices \(Q\) and \(R\) are the weight matrices here. \(Q\) is a positive semidefinite symmetric matrix and \(R\) is a positive definite matrix (\(Q\geq 0,R>0\)). In optimal control, the control vector \(u\), \(x\), indicates the system states and Equation (11) is quadratic concerning both \(x\)(t) and \(u\)(t). The optimal feedback input \(K\) is determined using the following equation. \[K=R^{-1}B^{T}P \tag{12}\]  
Here, \(B\) represents the input matrix of the system. The positive definite matrix \(p\) value is determined using the Riccati equation. Here A stands for the state matrix. \[A^{T}P+PA-PBR^{-1}B^{T}P+Q=0 \tag{13}\]  
Another ASS control method is the fuzzy logic control (FLC) method. The FLC algorithm is a method invented by [PERSON] [45]. An FLC algorithm consists of five stages. In the first stage, the input variables are converted into a fuzzy set and, in the second stage, rule tables, membership functions, and a rule base are formed. This rule base consists of a set of IF-THEN rules derived from the verbal statements of experts with knowledge of the system. In the third stage, the inference mechanism performs the inference process of these rules on the system and provides fuzzy outputs. In the fourth stage, the membership functions and the range of fuzzy sets are defined in the database. In the last stage, defuzzification, a fuzzy set is converted into a net value for output. In this paper, the boundary values of the membership functions of the FLC algorithm were obtained using particle swarm optimization (PSO), one of the swarm-based optimization algorithms [46,47]. The mean square error (MSE) was chosen as the PSO algorithm's goal function to reduce system-related faults. The Mamdani inference approach was applied to the FLC method that was used for system control. For all input and output values used, triangle-type membership functions are also preferred.