A convex optimisation based approach to PI-controller tuning for time invariant integrator and low order lag plus delay plants

This paper addresses the problem of proportional integral (PI) feed-back-controller tuning to minimise control performance indices (CPI). We consider regulator tuning for stable and unstable low order lag and integrator plants with and without delay. Common CPIs such as the integrated absolute error (IAE) are used as objective functions to be minimised by parameter tuning. Many of these CPIs are convex functions if their domain can be shown to be convex, which guarantees that a CPI minimum found by tuning is the global minimum. We find a convex representation of the domain for fixed structure parameter tuning, by utilising the fact that CPIs merely characterise input-output behaviour of transfer functions, which allows for a relaxation of structural constraints. This is shown to introduce convexity of the set of stabilising controller parameters as a necessity and techniques to show this for the considered plants are proposed. Furthermore, a computational simple iterative algorithm that exploits convexity to solve the tuning problem to its global optimal controller parameters is shown. To illustrate the potential of the proposed approach, we show IAE minimal tuning of a PI controller for a simulated first order lag plus delay plant using the proposed algorithm and compare the results with classical tuning rules destined for IAE minimisation.