### Discrete -Time Negative Imaginary Systems

Authors: Augusto Ferrante, Alexander Lanzon, Lorenzo Ntogramatzidis

Journal: Automatica

Publication Date: 02 March, 2017

School of: Electrical and Electronic Engineering

### In Abstract

**Towards new nano-positioning feedback controllers**

Robust vibration attenuation and control of flexible structures are notoriously challenging problems because these systems are often modelled using high order linear models with highly resonant dynamics. An acute problem in the control of such systems is that unmodelled dynamics (known as spillover dynamics) and imperfect modelling of resonant frequencies or damping levels of the highly resonant modes can severely degrade the performance of control systems or even lead to instability if the controller is not designed to be robust against these discrepancies. Negative imaginary systems theory has emerged as a powerful way to handle these fundamental difficulties.

Prominent examples of negative imaginary systems include nano-positioning controllers where fast/precise nano-resolution positional movements are actuated via co-located piezoelectric force transducers. Other examples of negative imaginary systems include the control of flexible dynamics in air vehicles, large space structures, dual-stage disk drives, etc.

Now, a team of researchers from The University of Manchester, Padova University and Curtin University, have brought continuous-time negative imaginary systems theory to discrete-time for the first time. The results are important because they enable discrete-time analysis and control of negative imaginary systems which can be used in future in digital electronic devices such as microcontrollers.

- The nomenclature “Negative Imaginary system” arises from the fact that these systems have a frequency response that has negative imaginary part for all positive frequencies.
- Did you know that small variations in the natural frequencies of resonant modes in highly resonant systems can give rise to large modelling discrepancy?
- Negative imaginary systems theory is quickly gaining international recognition as a key methodology for nano-positioning feedback control, and some of its advantages have been demonstrated in atomic force microscopy.
- The theory of negative imaginary systems was first pioneered at The University of Manchester, in collaboration with The University of New South Wales, in 2008 – see DOI: 10.1109/tac.2008.919567.