Design of PID controllers using semi-infinite programming

The PID controller is widely used, and several methods have been proposed for choosing the controller parameters to achieve good performance. The controller tuning problem is set up as a semi-infinite program (SIP), with the integrated squared error (ISE) or the H∞ norm of the frequency domain error function (|𝐸(𝑠)|∞) as the objective function, and H∞ constraints for robustness and noise attenuation. Previous authors considered discrete points to enforce the H∞ constraints, however this is an outer approximation that does not guarantee a feasible point. When a feasible point can be found, it may require multiple iterations with a finer and finer discretisation. Here, the SIP is solved using a global optimisation algorithm. Several numerical experiments show that the proposed formulation converges quickly (<10 seconds) and gives sensible controller tuning values without the need to apply expert knowledge to the tuning problem. These results suggest that this is an attractive method for automated controller tuning.