# Nonlinear Control Design of a Half-Car Model Using Feedback Linearization and an LQR Controller
## 1 Introduction
[10] presented the analysis of a half-car model; the effect of road disturbances is reduced only in these roll and pitch motions. With the intention to ensure good force tracking, an inner PID-based hydraulic actuator forced feed back controlled loop is assimilated. Ref. [11] presented the technique of finite element method for the vibrational study analysis of a vehicle model with a light passenger for a half car model by considering a 4-DOF half-car model of a rigid body namely sprung mass acceleration, sprung mass pitch, front axle acceleration and rear axle acceleration. Road input profile was modelled by using height sensors and attendant software. Analysis of the three controllers e.g., the passive system, self-tunable fuzzy inference system controller and PID controller is presented in [12]. Ref. [13] presented the half car model with the support of PID controller technique. The authors concluded that all the existing work is based on a linearized model. Ref. [14] did a simulation of Half-Car model in Matlab Simulink and analyzed behavior of the suspension system by using a step signal. Ref. [15] presented a more detailed mathematical model of a suspension system for a half car model including the dynamics of driver, state feedback and output feedback controllers were proposed for this model and unwanted accelerations were minimized. Ref. [16] presented the vibrational analysis of a half car model with 2-DOF. Previous research occasionally considered the road disturbance effect and its tied effect on vehicle lateral response when concentrating on crosswise and vertical dynamics. An E-Class (SUV) level car model in CARSIM was used as a standard, along with and without road profiles [17]. Ref. [18] carried out the study of linear passive suspension system of half-car passive suspensions system using [PERSON]'s 2 law of motion. This validated half car Simulink model is also used to analyse RMS acceleration, i.e., Ride Comfort for different input road profiles. Ref. [19] studied the modelling,simulation and control of linear half-car suspension system for different control techniques using Matlab programming. However, the point to be noted here is that they did not take in to account the nonlinear effects, which play an important role in the modelling and hence, an accurate analysis cannot be carried out. Ref. [20] gave an innovative adaptive control scheme for a half-car model with an unknown dead-zone input. Ref. [21] accessed the adaptive sliding neural network control for a quarter car model with forceful uncertainties along with road disturbances. Ref. [22] addressed the problem of determining the optimal parameters of sky-hook damper in controlling stationary response of a half-car model crossing a rough road by comparing the control force using LQR with sky-hook damper suspension force. For a class of uncertain nonlinear systems, the problem of global output feedback stabilization is investigated [23], in which the function of output is time variant and continuous along with several time-delays in system stale. With zero dynamics of high-order nonlinear systems, finite-time stabilization issue is considered [24] as the systems deliberated are suffered from unmeasurable dynamic uncertainties as well as inherent ones. This paper [25] addressed the issue of global finite-time adaptive stabilization for a class of high order uncertain nonlinearized systems. A state feedback stabilizer along with an adaptive mechanism was designed by application of continuous domination to adaptive mode of the systems to be deliberated. Ref. [26] applied PID controller for ASS of a half vehicle model using MATLAB Simulink. Performance response of ASS was simulated and compared with a passive one. Ref. [27] suggested a black-box attuned simulation-based method for solving the nonlinear model predictive control problem by using a parameterized approach to control only the vertical dynamics of a half-car model fortified with semi-active (SA) suspension system.