# Nonlinear Control Design of a Half-Car Model Using Feedback Linearization and an LQR Controller
## 1 Introduction  
The suspension of a vehicle forms a very imperative system for an automobile, and one which assists in providing sustenance to the engine, body of vehicle, passengers and on the other hand it grips the tremors arising due to irregularity of the road surface. A suspension system is a combination of springs, dampers, and linkages that is responsible for connection between a vehicle body and its wheels. The springs and dampers that connect the axle to the chassis of the body play a significant role in minimizing the shocks and custody the chassis exaggerated to a minimum level. There are basically three types of vehicle suspension system, namely Passive suspension system (PSS), Semi-active suspension system (SASS), and Active suspension system (ASS). PSS is comprised of a conventional suspension that is springs and dampers, etc. In SASS, energy is only dissipated, it varies the firmness of shock absorbers to add energy to a limited extent. ASS comprises of an actuator that is responsible for adding energy to the system when desired. The primary purpose of a suspension system is tomanage a stable compromise among quality of ride, road handling and suspension limits. On one hand, to achieve good ride quality by reducing vertical acceleration, it is necessary to increase the suspension travel but on the hand to achieve good road handling suspension travel should be kept in limits. Therefore, the design of an ASS should be able to make a better compromise among the two trade-offs. The main problem of this research is to minimize the effects of road disturbances by maintaining a trade-off between good quality of ride, road handling and suspension limits of Half-Car model by use of a cost-effective technique with less complexity. This research used a Feedback Linearization technique and an LQR controller to solve the problem to a great extent. In the past, work has been done to solve this challenging problem but the mystery has not been solved fully. While trade-off between road handling or ride quality dominates in most of the methodologies, the rattle space trade-off can be more ultimate but it has not been overtly solved, although it may be more important. Mostly, tires in VASS were modelled as a linear conventional spring rebuking the tire dampness. Ignoring dampness, motion associated with the wheel of sprung and unsprung masses is disconnected and the vertical acceleration of the sprung mass prolongs modest which fallouts to the misleading consequences [1; 2]. The work presented in [3] described the link between the two models of LQG-optimal one DOF two DOF. Ref. [4] reviewed the techniques of application of optimal control to the active suspension, initializing from simple quarter-car models, 1-D followed by their half-car, 2-D and full-car, 3-D equivalents, while the primary focus is on Linear Quadratic (LQ) optimal control technique and active suspension. Ref. [5] presented a new adaptive control design technique of backstepping in VASS, which shows the application of road adaptive nonlinear control along with ASS in to a Half Car Model (HCM) using road-adaptive patterns. The work is based on the non linear filters which are quite involved from implementation perspective in real situation. Moreover, unsprung masses are not constant relying on the value of disturbance. Vertical and pitch displacements and accelerations are taken into the consideration although their importance is quite significant. Ref. [6] presented a control strategy of an ASS with having preview information. Ref. [7] did modelling and analysis of a half car model using a fuzzy logic controller. Ref. [8] further studied the possessions of the unsprung mass on the ASS. The carpet plots, the plots of r.m.s values of sprung and unsprung mass accelerations versus the suspension travel, were presented to give a clear comprehensive sight of the influence of several parameters on the system concerts. The mathematical model for a half car Vehicle Active Suspension System (VASS) was presented in [9]. Ref.