The ball and beam system is a very simple and powerful control system problem. The easy construction of this system combined with its challenging control design requirement makes it one of the most favorable example models for control engineers. The model contains a horizontal beam which can pivot about its center; a DC Motor whose shaft is connected to the center of the beam; and a ball that can freely roll on top of the beam. The basic idea is to accurately tilt the beam about its center, using the motor, to indirectly control the position of the ball that freely rolls on the beam. In this thesis, the L1 adaptive control technique is considered for precise positioning of the rolling ball on the beam. Two different architectures of L1 adaptive control namely, the L1 adaptive state feedback control and the L1 adaptive output feedback control are designed and verified in simulation. L1 adaptive control guarantees transient performance and robustness in presence of fast adaptation without introducing or enforcing persistence of excitation.

Research into observer design for non-linear discrete-time systems has produced many design methods. There is no general design method however and that provides the motivation to search for a new simple and realizable design method. In this thesis, an observer for non-linear discrete-time systems is designed using the sliding mode technique. The equation of the observer error is split into two parts; the first part being a linearized model of the system and the second part an uncertain vector. The sliding mode technique is introduced to eliminate the uncertainty caused by the uncertain vector in the observer error equation. By choosing the sliding surface and the boundary layer, the observer error is attracted to the sliding surface and stays within the sliding manifold. Therefore, the observer error converges to zero. The proposed technique is applied to two cases of observers for nonlinear discrete-time systems. The second case is chosen to be a particular practical system, namely the non-linear discrete-time ball and beam system. The simulations show that the sliding mode technique guarantees the convergence of the observer error for both systems.

In this thesis different control methods are applied to the beam and ball system. Test setup for the previous thesis is handled, circuit assemblies and hardware redesigned. As the system is controlled by the control law partitioning method by a computer, discrete time system model is created. The controllability and the observability of the system are analyzed and a nonlinear controller by using control law partitioning in other words computed torque is designed. State feedback control algorithm previously designed is repeated. In case of calculating the non measurable state variables two different reduced order observers are designed for these two different controllers, one for control law partitioning controller and the other for state-feedback controller. Two controller methods designed for the thesis study are tested in the computer environment using modeling and simulation tools (Also a different controller by using sliding mode controller is designed and tested in the computer environment using simulation tools). A controller software program is written for the designed controller algorithms and this software is tested on the test setup. It is observed that the system is stable when we apply either of the control algorithms.