# Are linear controllers inadequate for nonminimum phase systems?

Based on the following publication:

Kouro, S., Perez, M.A., Rodriguez, J., Llor, A.M. and Young, H.A., 2015. Model predictive control: MPC's role in the evolution of power electronics. IEEE Industrial Electronics Magazine, 9(4), pp.8-21.

Available here, it is said that:

MPC can be also used with nonminimum phase plants, which usually appear in rectifiers connected to the grid, avoiding linear controllers, which, in these cases, have a very restricted performance , , .

Dose it mean that linear controllers are not adequate for nonminimum phase systems? If yes, why?

## 1 Answer

A LTI system is nonminimum phase if it has one or more zeros in the right half plane. When using a LTI controller for feedback also then the bandwidth of this controller will be limited to roughly to smallest magnitude of zeros in the right half plane.

This is because a right half plane zero gives a phase drop and slope increase. But near the bandwidth a slope of minus one (time 20 dB per decade) with a phase of -90° is desired. A minimum phase zero, so located in the left hand plane, could be counteracted by a left hand plane pole in order to ensure the desired asymptote at the bandwidth. If this could also be done for a nonminimum phase zero, in which case a right half plane pole would need to be added. According to the Nyquist criterion this would then require a counterclockwise encirclement of the minus one point of the openloop Nyquist diagram. The magnitude and phase of such counterclockwise encirclement look very similar to a that of a right half plane pole and zero, however when encircling the minus one point you already cross the zero dB line and therefore also give an upper bound for the bandwidth.

So no, a nonminimum phase LTI system can be controller using a LTI controller for feedback, but the bandwidth will be limited.

• Thank you very much. But I did not understand if a nonlinear controller will make any clear advantage in this case. – Adams Jun 4 '18 at 1:13
• @Adams You can use nonlinear controllers for linear systems, for example variable gain control, which allows for better disturbance rejection if its magnitude/frequency content change over time. However I haven't looked into nonlinear control for nonminimum phase systems. But you question is about whether or not you can use linear control for nonminimum phase systems, which you can. – fibonatic Jun 4 '18 at 2:44