Robust control of a laboratory pendulum with uncertain high frequency dynamics

Abstract

This paper describes how modern methods on robust controller design can be applied to a simple laboratory experiment which may be used for education in an advanced course on control systems design. The pendulum dynamics are made "uncertain" by dividing the motor shaft into two parts and by combining these parts with a spring. Moreover, the dynamics of the pwm amplifier which drives the motor is completed by an electronic circuit such as an all-pass with a sufficiently high bandwidth. In the first step, a carefully designed PID controller is used to stabilize the nominal pendulum. If the spring is added, the system comes close to instability and the combination of spring and all-pass causes instability. A weighting scheme for performance and uncertainty is derived to analyze the robust stability and performance properties of the PID controller. In the second step, this weighting scheme is used to synthesize a μ controller by the D-K iteration. This controller robustly stabilizes the pendulum with the additional high frequency dynamics and has essentially the same performance properties as the PID controller for the nominal plant. It works for the very accurate simulation model as well as for the real hardware. By this way, it can be demonstrated to the students that these advanced methods bring a considerable progress even for simple control systems.