The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. It is unstable without control. The process is non linear and unstable with one input signal and several output signals. It is hence obvious that feedback of the state of the pendulum is needed to stabilize the pendulum. The aim of the study is to stabilize the pendulum such that the position of the carriage on the track is controlled quickly and accurately. The problem involves an arm, able to move horizontally in angular motion, and a pendulum, hinged to the arm at the bottom of its length such that the pendulum can move in the same plane as the arm. The conventional PID controller can be used for virtually any process condition. This makes elimination the offset of the proportional mode possible and still provides fast response. In this paper, we have modelled the system and studied conventional controller and LQR controller. It is observed that the LQR method works better compared to conventional controller.

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Adaptive cruise control (ACC) is an improved form of cruise vehicle where the vehicle will automatically follow the preceding vehicle at a desired speed keep a safe distance between the vehicles in all traffic scenarios. Additional to this the ACC vehicle guarantees safe vehicle tracking, good fuel efficiency by avoiding unwanted acceleration and breaking and high driver comfort by reducing the jerk to minimum. The designed controllers controls and optimizes the acceleration, engine power and breaking so that the required objectives are achieved. A two level hierarchical control architecture is used in which the nonlinear dynamics of the vehicle is compensated in the lower level controllerwere throttle power is controlled using a PID controller and brake pressure is controlled using a feedback controller. Upper level controller gives the required input commands to the lower level controller. Upper level controller is designed using a model predictive control (MPC) which generated a control signal determining the longitudinal acceleration based on the inter-vehicular distance and velocity. MPC generates a cost function considering minimal tracking and minimum velocity deviations. Other objects of ACC like the driver comfort, fuel efficiency and car following abilities will be taken as constraints of MPC. Simulations are done using a Laguerre function methods of MPC and is compared with normal MPC method. Simulations are done for different traffic scenarios and the results are compared for the design objectives.

Ball and beam system is a benchmark system to study various control algorithms. It consists of a beam that can be tilted by a DC servo motor and a ball rolling back and forth on the top of the beam. The main reason for its acceptance in labs across the globe is its property that it is open loop unstable. It can serve as a conventional tool for implementation of many classical and modern control system design method. In the paper, the theory of H-infinity control is used to design a control system for the ball and beam system. The theory of Robust control is extended to design a robust PID controller. Controller parameter values for PID controller i.e. gains are found from the robust controller designed.

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