DYNAMIC OUTPUT FEEDBACK CONTROLLER SYNTHESIS FOR NEGATIVE IMAGINARY SYSTEMS AND APPLICATION TO VIBRATION ATTENUATION

Abstract

In the Single-Input-Single-Output (SISO) setting, Negative Imaginary (NI) systems refer to systems with negative frequency response for all positive frequencies. The NI theory was motivated by the study of inertial systems with collocated position sensor and force actuator. This type of systems can have nonminimum phase zero, be improper and have up to a relative degree of two. Thus, stability analysis and controller synthesis using passivity theorem may not be applicable. Moreover, using traditional design techniques such as $ \mathcal{H}_\infty $ and $ \mathcal{H}_2 $ to control this kind of systems may lead to conservative results and poor performance due to the oscillatory nature of the systems and the effect of the unmodelled dynamics on the closed-loop system performance. The NI property can be observed in a number of physical systems such as cantilever beams, large space structures, robotic manipulators, gantry crane systems, vibration shock absorbers, series elastic actuators and mechatronic systems. This, in addition to the simple internal stability result of interconnected NI systems, which depends only on the DC loop gain, has made the theory appealing for practical engineering applications. Example of some of these applications include vibration control of lightly-damped flexible structure, motion control of robotic arm, the control of a DC servo motor, nano-positioning control of an atomic force microscope, consensus control of multi-agent systems, to mention but a few. Due to the vast areas of application of the NI theory, controller synthesis techniques that achieve some transient performance level, such as prescribed decay rate, becomes imperative. This thesis therefore focuses on the synthesis technique of such controllers. A set of Linear Matrix Inequalities (LMIs) conditions are proposed for the synthesis of a dynamic output feedback controller. These LMIs render the closed-loop system to be NI and ensure that the DC gain condition for closed-loop internal stability is satisfied. The proposed conditions also ensures that the closed-loop system has prescribed decay rate via the $ \alpha -$ pole placement technique. The thesis then introduces a new class of NI systems called the Linear Time-Varying (LTV) NI systems. First, a time domain definition is provided, before a state-space characterization is proposed using Linear Differential Matrix Inequalities (LDMIs). Furthermore, a specialized case of the LTV NI systems called the Linear Parameter-Varying (LPV) NI systems are introduced. The state-space characterization of the LPV NI systems is provided using LMIs. Finally, a set of sufficient conditions for the stability of two asymptotically stable LTV NI systems is proposed and the result is shown to specialise to the LPV NI systems case. Also, this thesis introduces a robust controller synthesis technique using the principles of Internal Model Control (IMC) and NI theory. In the proposed technique, the design of the Youla parameter is cast as an NI controller synthesis problem. Two different synthesis methods are provided, both of which are sufficient conditions. One is an LMI-based approach and the other is a frequency domain approach. Finally, the thesis deals with hardware validation of the proposed LMI-based IMC NI controller synthesis technique. The efficacy of the technique is demonstrated on a flexible cantilever beam with collocated position sensor and force actuator. The controller is designed to attenuate the vibration of the flexible structure caused by external disturbances.

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Alexander Lanzon (Supervisor)