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 work, 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.  This research has been supported by the Australian Research Council under the Discovery Projects and Laureate Fellowships funding schemes. The results of this work have been published in [Mohamed A. Mabrok, Abhijit G. Kallapur, Ian R. Petersen, and Alexander Lanzon. Generalizing negative imaginary systems theory to include free body dynamics: Control of highly resonant structures with free body motion. IEEE Transactions on Automatic Control, 59(10):2692—2707, October 2014. arXiv:1305.1079.]