Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented

Stability of positive-position feedback controllers with low-frequency restrictions

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Integral Resonant Control for vibration damping and precise tip-positioning of a single-link flexible manipulator

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An analytical approach to integral resonant control of second-order systems

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A Survey of Riccati Equation Results in Negative Imaginary Systems Theory and Quantum Control Theory

This paper presents a survey of some new applications of algebraic Riccati equations. In particular, the paper surveys some recent results on the use of algebraic Riccati equations in testing whether a system is negative imaginary and in synthesizing state feedback controllers which make the closed loop system negative imaginary. The paper also surveys the use of Riccati equation methods in the control of quantum linear systems including coherent H∞ control.This work was supported by the Australian Research Council (ARC) under grants FL110100020 and DP160101121 and the Air Force Office of Scientific Research (AFOSR), under agreement number FA2386-16-1-4065

Modeling and Control of MEMS-based Multi-layered Prestressed Piezoelectric Cantilever Beam

Piezoelectric materials are the preferred smart materials for sensing and actuation in the form of micro and nano-engineering structures like beams and plates. Cantilever beams play a significant role as key components in atomic force microscopy and bio and chemical sensors. Adding an active layer such as lead zirconate titanate (PZT) thin-film to form smart cantilever beams with sensing and actuation capabilities is highly desirable to facilitate miniaturization, enhance performance and functionali- ties such as enabling on-chip high-speed parallel AFM. During the micro-fabrication process, residual stresses develop in the different layers of the cantilever beam, causes initial deflection. The residual stress in the different layers of the cantilever beam and the application of voltage to the PZT thin-film affects their dynamics. This the- sis investigates the dynamic behaviour and develops a control technique and a novel charge readout circuit to improve the performance of a micro-fabricated multi-layer prestressed piezoelectric cantilever beam as an actuator and a deflection sensor. Firstly, the fabrication process of a unimorph PZT cantilever beam is explained. A low thermal budget Ultra-high vacuum e-beam evaporated polysilicon thin-film (UHVEEpoly) process is used for the fabrication of multi-layered PZT cantilever beam in d31 mode. The sharp peaks at resonant frequencies in the frequency response of the PZT cantilever beam show very little damping and a large settling time of the cantilever beam. Secondly, the dynamic behaviour of the prestressed PZT cantilever beam is in- vestigated subjected to change in driving voltage. Experimental investigations show a shift in resonant frequencies of a PZT cantilever beam. However, there is no reported mathematical model that predicts the shift in resonance frequencies of a multi-layered prestressed piezoelectric cantilever beam subjected to a change in driving voltage. This work developed a mathematical model with experimental val- idation to estimate the shift in resonance frequencies of such cantilever beams with the change in the driving voltage. A very good agreement between the model predic- tions and experimental measurements for the frequency response of the cantilever beam at different driving voltages has been obtained. A novel linear formulation has been developed to predict the shift in resonance frequencies of the PZT can- i tilever beam for a wide range of driving voltages. The formulation shows that the shift in resonance frequencies of a multi-layered prestressed piezoelectric cantilever beam per unit of applied voltage is dependent on geometric parameters and material properties. Thirdly, a robust resonant controller has been designed and implemented to re- duce the settling time of a highly vibrating PZT cantilever beam. The controller design is based on a mixed negative-imaginary, passivity, and a small-gain approach. The motivation to design a resonant controller using the above-mentioned analyti- cal framework is its bandpass nature and the use of velocity feedback, as the charge collected from a vibrating PZT cantilever beam gives the velocity information of the beam. The proposed controller design results in finite gain stability for a pos- itive feedback interconnection between two stable linear systems with a large gain and phase margin. Experimental results demonstrate that the designed resonant controller is able to effectively damp the first resonant mode of a cantilever, signifi- cantly reducing settling time from 528 ms to 32 ms. The robustness of the designed resonant controller is tested against changes in the cantilever beam dynamics due to residual stress variation and or stress variation due to driving voltage. Finally, to facilitate the miniaturization of on-chip sensors and parallel high- speed AFM, a single layer of a PZT thin-film in a cantilever beam is used as a deflection sensor and an actuator instead of bulky optical deflection sensors. A novel charge readout circuit is designed for deflection sensing by capturing the electrical charge generated due to the vibration of the PZT beam. The signal-to-noise ratio and sensitivity analysis of the readout circuit shows similar results compared to the commercially available optical deflection sensors. Our work highlights very important aspects in the dynamic behaviour and perfor- mance of a multi-layered prestressed piezoelectric cantilever beam. The agreement between the proposed theoretical formulation and experimental investigations in modeling, control design, and a novel readout circuit will provide the platform for further the development and miniaturization of microcantilever-based technologies, including on-chip parallel HS-AFM

Negative imaginary theorem with an application to robust control of a crane system

This paper presents an integral sliding mode (ISM) control for a case of negative imaginary (NI) systems. A gantry crane system (GCS) is considered in this work. ISM is a nonlinear control method introducing significant properties of precision, robustness, stress-free tuning and implementation. The GCS model considered in this work is derived based on the x direction and sway motion of the payload. The GCS is a negative imaginary (NI) system with a single pole at the origin. ISM consist of two blocks; the inner block made up of a pole placement controller (NI controller), designed using linear matrix inequality for robustness and outer block made up of sliding mode control to reject disturbances. The ISM is designed to control position tracking and anti-swing payload motion. The robustness of the control scheme is tested with an input disturbance of a sine wave signal. The simulation results show the effectiveness of the control scheme