The Robust Control of Negative Imaginary Systems

Abstract

Collocated actuators and sensors on a lightly damped flexible structure always lead to an alternating property of the poles and zeros near the imaginary axis. The frequency response associated with this behaviour is referred to as a negative imaginary (NI) frequency response and has attracted sufficient interest to result in the development of a theory of negative imaginary systems. It is understood that the NI property guarantees the robust stability of a wide class of SISO systems in the face of parameter perturbation. In addition, the positive feedback interconnection of an NI system and an SNI system is known to be robustly stable if a certain DC gain condition is satisfied. Taking advantage of this feedback interconnection property, NI systems theory has found application in many areas such as control of large flexible space structures, aircraft wings, robotic manipulators, nano positioning, gantry cranes and active bridge stabilization, to name a few. One of the challenges in applying NI systems theory to control problems is forming the feedback structure that takes advantage of the robust stability properties of NI systems. It would be ideal if, for a given system, we could rearrange the system into a robust control framework with a plant of reduced order and where the unmodeled dynamics are captured as uncertainty with the NI or strictly NI (SNI) property, via a feedback interconnection. In this thesis, we develop the theory of negative imaginary systems directed towards a complete framework for robust control. The definition of a negative imaginary transfer function was shown to allow for undesirable unstable modes when applied to a non-minimal system. We generalise this definition to a negative imaginary realisation that is not necessarily minimal. Building upon this, we offer new analysis of the stability of NI/SNI systems and extend existing theory to include singular NI systems. In addition, we develop results on the perturbation of negative imaginary systems and their stability. Using the developed NI perturbation theory, we present necessary and sufficient conditions for synthesising a state feedback controller that results in an SNI closed-loop transfer function with a prescribed degree of stability. Closed-loop NI and SNI systems under state feedback do not have an arbitrary degree of stability, we analytically show the degree of stability a given SISO system can achieve. When the full system states are not available for feedback, we may use output feedback to render a closed-loop system NI or SNI. In addressing the NI output-feedback control problem, we present sufficient conditions for deriving a dynamic controller via output feedback. Moreover, toward the SNI output-feedback control problem, we present both sufficient and necessary conditions for synthesising a controller. Four methods of modelling systems to fit within an NI robust control framework are offered. Each model can be rearranged to fit a robust control framework with a guaranteed stable uncertainty with the NI or SNI property. We conclude with a practical application of one of the aforementioned modelling techniques for the control of a flexible cantilever with a collocated sensor and actuator. Show more