| The inverted pendulum system has long been considered an intriguing problem for control theory and its applications. The stabilization of such a system is a primary challenge for researchers in this field because the degree of difficulty of each problem depends on the type of the chosen system and the feasibility of the controller designed for that specific system. In this research, the real-time single control input stabilization of a triple link inverted pendulum system, a highly nonlinear and unstable system, is experimentally and theoretically investigated and achieved. At first, a general nonlinear dynamic model is developed. Thereafter, the major causes of instability are revealed through extensive computer simulations and experimental observations. Hardware improvements and modification of the traditional state space control algorithm, along with experience gained from real-time stabilization of a double link inverted pendulum system, lead to a successful single input control stabilization of a custom built triple link inverted pendulum about its upright position. Although the linear design process results in a stable system, the control law is neither influenced by the nonlinear dynamics nor by the physical limitations of the real system. Hence, a nonlinearly constrained optimization technique is employed as an alternative solution to the problem. The stability criterion and the system's limitations are formulated as infinite dimensional nonlinear inequality constraints and a cost function of the linear gains is minimized for a short interval of time. The new set of gains obtained from the numerical optimization is in the neighborhood of the gains found experimentally. A small difference between those two gain sets still exists due to the fact that the nonlinear simulation model can approximate, but is unable to duplicate, the physical system's behavior. |
