The Inverted Pendulum on a Cart is a classical control theory problem that helps understand the importance of feedback control systems for a coupled plant. In this study, a custom built pendulum system is coupled with a linearly actuated cart and a control system is designed to show the stability of the pendulum. The three major objectives of this control system are to swing up the pendulum, balance the pendulum in the inverted position (i.e. $180^\circ$), and maintain the position of the cart. The input to this system is the translational force applied to the cart using the rotation of the tires. The main objective of this thesis is to design a control system that will help in balancing the pendulum while maintaining the position of the cart and implement it in a robot. The pendulum is made free rotating with the help of ball bearings and the angle of the pendulum is measured using an Inertial Measurement Unit (IMU) sensor. The cart is actuated by two Direct Current (DC) motors and the position of the cart is measured using encoders that generate pulse signals based on the wheel rotation. The control is implemented in a cascade format where an inner loop controller is used to stabilize and balance the pendulum in the inverted position and an outer loop controller is used to control the position of the cart. Both the inner loop and outer loop controllers follow the Proportional-Integral-Derivative (PID) control scheme with some modifications for the inner loop. The system is first mathematically modeled using the Newton-Euler first principles method and based on this model, a controller is designed for specific closed-loop parameters. All of this is implemented on hardware with the help of an Arduino Due microcontroller which serves as the main processing unit for the system.

Proportional-Integral-Derivative (PID) controllers are a versatile category of controllers that are commonly used in the industry as control systems due to the ease of their implementation and low cost. One problem that continues to intrigue control designers is the matter of finding a good combination of the three parameters - P, I and D of these controllers so that system stability and optimum performance is achieved. Also, a certain amount of robustness to the process is expected from the PID controllers. In the past, many different methods for tuning PID parameters have been developed. Some notable techniques are the Ziegler-Nichols, Cohen-Coon, Astrom methods etc. For all these techniques, a simple limitation remained with the fact that for a particular system, there can be only one set of tuned parameters; i.e. there are no degrees of freedom involved to readjust the parameters for a given system to achieve, for instance, higher bandwidth. Another limitation in most cases is where a controller is designed in continuous time then converted into discrete-time for computer implementation. The drawback of this method is that some robustness due to phase and gain margin is lost in the process. In this work a method of tuning PID controllers using a loop-shaping approach has been developed where the bandwidth of the system can be chosen within an acceptable range. The loop-shaping is done against a Glover-McFarlane type ℋ∞ controller which is widely accepted as a robust control design method. The numerical computations are carried out entirely in discrete-time so there is no loss of robustness due to conversion and approximations near Nyquist frequencies. Some extra degrees of freedom owing to choice of bandwidth and capability of choosing loop-shapes are also involved and are discussed in detail. Finally, comparisons of this method against existing techniques for tuning PID controllers both in continuous and in discrete-time are shown. The results tell us that our design performs well for loop-shapes that are achievable through a PID controller.