Control Systems Design of Nonlinear Systems and Negative Imaginary Systems

This project lies within the engineering field of control systems theory and applications. The aim of this project is to develop new controller design techniques for either general non-linear systems, or Negative Imaginary (NI) systems, or both. As these are two discrete tasks, they will be described separately.

Non-linear Systems

A common, simple method for controlling nonlinear systems is to first linearise the nonlinear system at an operating point, discarding the nonlinear elements, and to subsequently control the resultant linear system. While this method of controller design has proven fruitful, there is no measure of how accurate this linearized model of the system is and therefore how well the controller designed for the linear model will behave on the true non-linear system. Instead, controllers are designed for such systems with no acknowledgement of their non-linear behaviours.

Therefore, the student will investigate the effect of these (usually discarded) nonlinear elements on the linearised model, and use this to develop a more appropriate controller design method. The approach taken will be to robustly handle the nonlinear elements via feedback control. The novel engineering approach applied here is to treat the nonlinearities as perturbations to the linear system.

The result of this research would be to provide useful tools for practising control engineers to both determine the accuracy of their linearised models and their controllers, and to assist in designing controllers which produce more desirable results for such systems.

Negative Imaginary Systems

There has been much work over the last decade to analyse the properties of NI systems and to extend the definition of NI systems. Research in methods to synthesise controllers for these systems has also been carried out. However, these synthesis methods do not allow the control engineer the freedom to design the system to have the properties they desire. Therefore, this project aims to answer the following research question:

- How can the unique properties of negative imaginary systems be leveraged in designing a controller which has improved properties when compared to controllers designed using methods applicable to any system?

The approach taken will be as follows:

1. First, look at current controller design algorithms and methodologies which are applicable to all systems.
2. Using the properties of NI systems, attempt to modify the controller design methods in question, changing or relaxing assumptions and adding constraints based on the properties of NI systems.
3. This will result in a modified version of the current theories, applicable only to NI systems, which yield more desirable results than the generic theory would.
4. (If this proves infeasible) New theories will attempt to be developed from first principles to directly leverage the properties of NI systems.

The main type of real world system which exhibits the negative imaginary property are systems with force actuators and position sensors located in the same place ('co-located'). Examples of control problems with these properties include:
- vibration control of large space structures;
- nano-positioning systems, such as the control of atomic force microscopes.

This is an important area of engineering research because negative imaginary systems occur frequently in the world, so having more powerful tools to design controllers for these systems would be beneficial to practising control engineers.