Controller Synthesis Free of Analytical Models: Three Term Controllers

The main focus of this paper is on direct data driven synthesis and design of controllers. We show that the complete set of stabilizing proportional-integral-derivative (PID) and first-order controllers for a finite dimensional linear time-invariant plant, possibly cascaded with a delay, can be calculated directly from the frequency response (Nyquist/Bode) data P(jomega) for omega epsi [0, infin) without the need of producing an identified analytical model. It is also shown that complete sets achieving guaranteed levels of performance measures such as gain margin, phase margin, and Hinfin norms can likewise be calculated directly from only Nyquist/Bode data. The solutions have important new features. For example it is not necessary to know the order of the plant or even the number of left half plane or right half plane poles or zeros. The solution also identifies, in the case of PID controllers an exact low frequency band over which the plant data must be known with accuracy and beyond which the plant information may be rough or approximate. These constitute important new guidelines for identification when the latter is to be used for control design. The model free approach to control synthesis and design developed here is an attractive complement to modern and post modern model based design methods which require complete information on the plant and generally produce a single optimal controller. A discussion is included, with illustrative example, of the sharp differences between model-free and model based approaches when computing sets of stabilizing controllers. For example, it is shown, that the identified model of a high order system can be non-PID stabilizable whereas the original data indicates it is PID stabilizable. The results given here are also a significant improvement over classical control loop-shaping approaches since we obtain complete sets of controllers achieving the design specifications. It can enhance fuzzy and neural approaches which are model free but cannot guarantee stability and performance. Finally, these results open the door to adaptive, model free, fixed order designs of real world systems.