PendCon

                Designing PID controllers 

                for a weakly damped

                or unstable plant















Main Characteristics of the Plant

In contrast to the rat e or position control experiments for the motor, the plant is now nonlinear. So, the students are confronted with a new topic, namely linearization. In order to apply the linear methods taught in a first course, the plant has to be linearized. For the hanging pendulum, a weakly damped PT2 system results whereas the inverted pendulum is described by an unstable PT2 system with one positive and one negative pole. Possible experiments for the students are the analysis of step responses and the study of frequency response for low frequencies, for frequencies at or near the eigenfrequency of the pendulum and for large frequencies.


Controller Design

Various controllers are feasible. The simplest one is a differentiating controller (damping injection) which prestabilizes the pendulum. By this way, the students learn that control can be used to improve the dynamical properties of the plant with respect to the needs of a human operator. A PD controller can be used for tracking, but leads a stationary tracking error. The tracking error is avoided by a PID controller, which can be found by pole placement. It is also possible to ask the students to design a PID controller which compensates the weakly damped pole pair and to let them compare the tracking and the disturbance rejection properties of the controller. This leads to a better understanding of the concept of internal stability. For the inverted pendulum, pole placement is also possible or, alternatively, to find a controller which compensates The stable pole and is additionally designed such that the stable pole is mirrored at the imaginary axis for the closed- loop system. For the pendulum, all these methods are easily done by simple manually performed computations and no assistance of SW tools is needed.


Summary

The Pendulum Experiment is ideally suited for the students to get a basic understanding of fundamental concepts of control theory like step response, frequency response, linearization, stability, tracking,  disturbance rejection, stationary control error, design by pole placement and others.