Control of an Under-Sensed and Under-Actuated Linear Inverted Pendulum

In the mechanical system control, under-actuation, meaning the situation when the number of actuators is less than the number of position variables, is commonly seen; whereas full state feedback or full position feedback is often utilized, implying that the number of sensors used is at least equal to the number of position variables. What if less sensors are to be used? In such a situation, we say the system is under-sensed. To control an under-sensed system, we need to know where to place the limited number of sensors for the ease of control and how to design the controller effectively under incomplete position information. This paper gives a case study on controlling an under-sensed and under-actuated system. An under-sensed and under-actuated linear (USUAL) inverted pendulum, which has only one position sensor and one force actuator, is investigated. For this USUAL inverted pendulum, we first address the sensor placement problem by analyzing the dependence of system zeros on the sensor location. After a reasonable sensor location is chosen, we then study the issue of controller design. The optimally robust stabilizing controller, which minimizes $H$ ∞ norm of the Gang of Four transfer matrix, is shown to be effective.