# Hybrid Particle Swarm Optimization Genetic LQR Controller for Active Suspension
## 1 Introduction  
Vehicle suspension systems are the entire support systems consisting of springs and dampers between the car body and tires, and play a major role in modern vehicles. Suspension systems can isolate the vibrations created by road surfaces, which guarantee ride comfort and handling stability [1; 2; 3]. Generally speaking, according to the forms of damper, vehicle suspension systems are divided into three categories: passive suspension systems, semi-active suspension systems, and active suspension systems [4; 5]. The passive suspension systems are assembled with irrevocable dampers. Semi-active suspension systems are constructed with various dampers. Thus, the passive suspension and semi-active suspension are not able to change with road conditions, and, they are limited in improving the performance of ride comfort and handling stability [6]. Compared with the passive and semi-active suspension, pneumatic or hydraulic actuators replace the dampers in active suspension systems, which can proactively generate the force to absorb and deliver the load produced by vehicle vibration [7]. Hence, active suspension systems possess better performance in ride comfort and road holding capability. Due to the excellent characteristics of active suspension, many control algorithms were applied in active suspension controllers during the past few decades [8; 9; 10; 11; 12; 13; 14; 15; 16]. [PERSON] et al. [8] proposed a time-delay feedback controller to reduce suspension vibration. [PERSON] et al. [9] carried out the sliding mode control in active suspension systems with data acquisition delay. [PERSON] et al. [10] applied the integrated fuzzy wheelbase preview control to heavy-duty vehicles. In addition, many advanced control methods have attracted considerable interest, such as adaptive control, fuzzy logic control, NN control,and artificial intelligence, which have been studied by many scholars [11; 12; 13]. Recently, the model predictive control (MPC) attracted many scholars, drawing attention to vehicle control [14; 15; 16]. MPC is an optimization-based control method whose objective function uses relative errors [17]. In recent years, the linear quadratic regulator (LQR) control strategy received increasing attention, as it can easily realize the closed-loop control [18; 19; 20; 21; 22]. Compared with other control methods, LQR has a simple mathematical treatment and easily constructs objective function. The performance indexes of LQR are explicit physical parameters (for example, sprung mass acceleration, the deflection used in this paper). LQR is also a multi-objective optimization algorithm that can achieve the overall optimality. Moreover, it is easy to realize closed-loop control; thus, it is widely used in the vibration system [23; 24]. At the same time, the LQR control method is one of the most important issues in active suspension control theory and is widely used. LQR control is a control algorithm that faces multiple objectives, which can make the system's comprehensive performance work the best. [PERSON]. [18] proposed the LQR-based power train control method for fuel cell hybrid vehicles. [PERSON]. [19] investigated the LQR controller for the carlike robot. Ref. [20] used the PID-LQR control method to improve vehicle ride comfort. The key to the LQR control method lies in the selection of the weight matrices Q and R. Therefore, different approaches have arisen for the optimization of the weight matrix of the controller. However, there exists a shortcoming in the LQR control system, in which the weight coefficients of matrix Q and matrix R are determined by experience, which cannot guarantee the optimal performance. The traditional method can only rely on empirical selection, and the process is too cumbersome; it could not obtain the optimal solution [25; 26; 27]. Currently, the most commonly used methods include Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Recent studies have also mentioned an artificial fish algorithm.