Optimal performance tuning of a PI-controller for an integrator plant with uncertain parameters as a convex optimisation problem

In this paper, we consider the problem of tuning a PI-controller for optimal performance with an integrator plus delay plant with uncertain parameters. Here, the goal of performance tuning is to minimise the integrated absolute error (IAE) after system stimulation for either regulator or servo tuning. Therefore, tuning is formulated as an optimisation problem with controller parameters as variables and the IAE as objective function. We show that this optimisation problem is convex and thus an IAE minimum found by tuning is guaranteed to be the global minimum. To this end, we prove convexity of the IAE function by deriving a convex representation of the IAE´s domain for parameter tuning. This is shown to introduce convexity of the set of stabilising controller parameters as a necessary condition, which is proven to hold for the plant considered here independent of the plant parameter values. Readily available iterative algorithms can be utilised to solve the convex tuning problem and find global optimal controller parameters. This is demonstrated in a simulation example, using the Nelder-Mead Simplex algorithm for regulator tuning of a PI-controller for an integrator plus delay plant.