There are some systems that are yet to be modelled using certain methods. One of them is Rotary Inverted Pendulum (RIP) on a wheeled cart which is yet to be modeled using the bond graph technique. Therefore, this work explored the bond graph technique for this system. Using this technique, which uses the concept of energy (power) transfer between elements in a system, the system was modeled. Then, the state space equations of the system, which give the first-order differential equations, were derived. It was observed that the system has five state variables because of the five integrally causal storage elements.

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.

A SMC for the ball-on-sphere system was designed in this work. The mathematical system’s model was first derived and a SMC was designed. Then, Lyapunov’s method was used to test for the convergence on the sliding surface, and convergence of the system's states to the sliding surface was guaranteed. To reduce chattering, a super twisting SMC was designed. A controller that is linear was first given to the system and the simulation results showed that, while there is disturbance, achieving origin’s asymptotic stability is not viable. A SMC was then applied next, while there is disturbance, origin’s asymptotic stability was attained in finite-time. Then a 2nd order SMC was applied and the results showed faster origin’s asymptotic stability in finite time. Therefore, the real effect of applying a 2nd order SMC is faster asymptotic stability of the origin1. To reduce chattering, a ST SMC was applied and the chattering was observed to be reduced efficiently.

Linear Quadratic Regulator (LQR) is one of the optimal control methods that continue to gain popularity. This paper designed an optimal LQR controller to control the system of the ball-on-sphere. System equations were derived and due to the nonlinearity of the system, the equations were linearized. After that, the coefficient matrices of the system dynamics were derived. Given some initial conditions, the response was simulated and controlled close to the desired values. An improvement of about 87% was achieved and the performance of the controller was observed to be good based on the simulation results. The results showed that LQR controller is one of the best optimal control methods because of its high performance improvement.