With the state-space method, many controllers can be designed optimally. LQR and LQG are two of these controllers. These two controllers are covered much in the literature. Despite this, not many works cover the ball-on-sphere system. Therefore, the research designed optimal LQR and LQG controllers for the system of ball-on-sphere and did a comparative analysis between the two. System dynamics were first investigated and the mathematical model was derived. After that, the system was linearized and then the state-space representation was obtained. Using this representation, the two controllers were designed and applied to the system for control. The control was done based on the specified desired system performance. Finally, the controllers' performances were analyzed and compared. Results obtained showed that both controllers met the desired system performance. With θ_x is 87.14% and θ_y is 86.43% less than their respective unregulated settling times, LQR satisfied the at least 80% performance requirement more than LQG. For LQG, θ_x is 82.35% and θ_y is 82.95% less than their respective unregulated settling times. These values are less than that of LQR. It was also observed that minimizing the total control energy leads to maximizing the total transient energy but LQG maximizes the total transient energy more than LQR. Another finding was that all states played role in regulating the controller to the desired system performance. Without regulation, LQG was found to be more efficient than LQR but in general, LQR is more efficient than LQG because, in LQG, settling time (of ball's angles) of less than 1.00 sec could not be realized. LQR is 4.79% and 3.48% more efficient than LQG in x and y directions, respectively, for the ball’s angles settling time. This research is significant because it is the first to design and do a comparative analysis of LQR and LQG controllers for the ball-on-sphere system. Therefore, bridging the existing gap in the literature is the value of this research.

Control system plays a critical function as one of the essential bedrocks of contemporary social development. Differential equations are time-based equations. The analysis of these equations according to time-domain, is what the theory of modern control is based on. It uses a state-space method which allows direct design in the time-domain. With the state-space method, many controllers can be designed optimally. LQG is one of these controllers. This controller is covered much in the literature. Despite this, not many works cover the ball-on-sphere system. Therefore, the research designed an optimal LQG controller for the system of ball-on-sphere. System dynamics were first investigated and the mathematical model was derived. After that, the system was linearized and then the state-space representation was obtained. Using this representation, the controller was designed and applied to the system for control. The control was done based on the specified desired system performance. Finally, the controller's performance was analyzed. Results obtained showed that the controller met the desired system performance. The controller satisfied the at least 80% performance requirement with θ_x is 82.35% and θ_y is 82.95% less than their respective unregulated settling times. It was also observed that minimizing the total control energy leads to maximizing the total transient energy. Another finding was that all states played role in regulating the controller to the desired system performance. Unfortunately, a settling time (of the ball's angles) of less than 1.00 sec could not be realized. The realized performance is 2.35% and 2.95% more than the desired performance in x and y directions, respectively, for the ball’s angles settling time. This research is significant because it is the first to design an LQG controller for the ball-on-sphere system. Therefore, bridging the existing gap in the literature is the value of this research.