Optimization of Fixed-Order Controllers Using Exact Gradients

Finding good controller settings that satisfy complex design criteria is not trivial. This is also the case for the simple three parameter pid controller. One strategy is to formulate the design problem into an optimization problem. However, when using gradient-based optimization, the algorithm often fails to converge to the optimal solution. In many cases this is a result of inaccuracies in the estimation of the gradients. In this paper we derive exact gradients for the problem of optimizing performance (iae) with constraints on robustness (Ms, Mt). Using the exact gradients, we demonstrate increased accuracy and better convergence than with forward finite differences. The results may be easily extended to other objectives and fixed-order controllers, including Smith Predictor and pidf controllers.