Synthesis of absolutely stabilizing PID controllers and its application to a ball and wheel system

We consider the problem of synthesizing proportional-integral-derivative (PID) controllers that absolutely stabilize a given Lur´e system. Based on the circle criterion and the stability characterization of the strictly positive real property, sufficient conditions for the existence of stabilizing PID controllers are given in terms of simultaneous stabilization of complex polynomials. The results from the earlier work are then used to solve the resulting complex polynomial stabilization problem. For a fixed proportional gain, and by sweeping over a variable, the set of the stabilizing integral and derivative gain values can be determined constructively using linear-programming techniques. The proposed synthesis method is used to design a stabilizing PID controller for the ball and wheel system and the experimental results are also presented.