Dear Leonhard,

I suggest you read some papers which I believe can guide you in your task.

-Nonminimum-Phase Zeros: Much to Do about Nothing-

www-personal.umich.edu/.../HoaggNonMinimumPha...

The paper is to illuminate the critical role of system zeros (UNDERSHOOT, OVERSHOOT, AND ZERO CROSSINGS) in control-system performance for the benefit of a wide audience both inside and outside the control systems community.

Finally a conceptual impediment to the acceptance of zero is the difficulty in understanding the ratio 1/0 is given.

-Method for Undershoot-Less Control of Non-Minimum Phase

https://arxiv.org/pdf/1401.0106

In this method first the non-minimum phase zero of the process is cancelled to an arbitrary degree by the proposed pre-compensator and then a classical controller is designed to control the series connection of these two systems.

Here are the files listed in attached.

Best regards

Hi Leonhard,

Before trying to solve the problem itself, I will suggest that we examine the precise challenge introduced by nonminimum phase zeros in the overall task of controller design.

Since an OL zero always tries to pull CL root loci towards itself, an OL nonminimum phase zero essentially contributes by additionally dragging the CL root loci to the unstable part of the pole-zero plane ! That is, a root locus branch that in absence of nonminimum phase zeros would have better stability properties, would tend to go unstable at lower gains if the OL nonminimum phase zeros are present !

The above is a rough description of the basic problem ! Obviously inclusion of "cancelling" unstable poles is not a solution, because they will right away bring with them additional bunch of unstable CL root branches !!

In my opinion, a good approach would be to attempt inclusion of significant additional minimum phase zeros so as to reverse the "drag towards instability" contributed to the root loci by nonminimum phase OL zeros. Of course, this has to be done with care, since realisation requirements may accordingly demand inclusion of additional stable poles as well.

Beyond this broad based suggestion, it is a bit difficult to suggest anything since we have not seen the actual pole zero scatter of your OL system ! But I can say with confidence that the above design approach, if done with adequate care, should give you a solution.

And nonlinear design methods may not have to be resorted to !!

With best wishes.

-Sanjay

Food for thought:

1. Eliminating unstable zeros by cancellation with unstable poles is

Not A Good Idea.

2. Theoretically, if the plant has an unequal number of inputs and outputs,

or if additional sensors (or actuators) can be added, then the plant

can be squared by combining outputs (or inputs) while placing the

zeros in the stable region. Then a controller is designed for the

squared plant.

3. The zeros are stabilized by solving the output-feedback pole-placement

problem for the inverse configuration.

4. Poles cannot always be placed using constant output feedback but may

require controller dynamics; hence, zero placement may require that

outputs (or inputs) be combined using dynamics.

5. Poles cannot always be placed by output feedback without cancellation

if the plant is not stabilizable and detectable, and zero placement can

have the same difficulty since it is pole placement for the inverse. The

cure is to modify plant physics or structure.