PhD thesis

New Results on Negative Imaginary Systems Theory with Application to Flexible Structures and Nano-Positioning

Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure control systems and are often very sensitive to changes in environmental variables. These can lead to vibrational effects that limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extent by using force actuators combined with colocated measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received great attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory. In this thesis, we provide a generalization and development of negative imaginary systems theory to include a wider class of systems. In the generalization of NI systems theory, we provide a new negative imaginary definition that allows for flexible systems with free body motion. Also, we provide a new stability condition for a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative imaginary (SNI). This general stability result captures all previous NI stability results that have been developed. This thesis also presents analytical tools for negative imaginary systems theory, which can be useful in the practical applications of the theory. Two methods that can be used for checking the negative imaginary property for a given system are presented. Also, methods for enforcing NI dynamics on mathematical system models to satisfy a NI property are explored. A systematic method to design controllers for NI systems with guaranteed robust stability also is presented. A practical application of control system design for a three-mirror cavity locking system is presented in the end of the thesis. The thesis is concluded by a conclusion chapter with a discussion of possible future work.

Full thesis