Index Terms-Negative imaginary systems, flexible structures, free body motion.Abstract-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 piezoelectric actuator and sensor is presented.

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This paper discusses the application of a discrete-time extended Kalman filter (EKF) to the problem of estimating the decay time constant for a Fabry-Perot optical cavity for cavity ring-down spectroscopy (CRDS). The data for the estimation process is obtained from a CRDS experimental setup in terms of the light intensity at the output of the cavity. The cavity is held in lock with the input laser frequency by controlling the distance between the mirrors within the cavity by means of a proportional-integral (PI) controller. The cavity is purged with nitrogen and placed under vacuum before chopping the incident light at 25 KHz and recording the light intensity at its output. In spite of beginning the EKF estimation process with uncertainties in the initial value for the decay time constant, its estimates converge well within a small neighborhood of the expected value for the decay time constant of the cavity within a few ring-down cycles. Also, the EKF estimation results for the decay time constant are compared to those obtained using the Levenberg-Marquardt estimation scheme.

In this paper, a stabilization procedure that forces an uncertain system to be stable and satisfy the negative imaginary property is presented. The controller synthesis procedure is based on the negative imaginary lemma. As a result, the closed-loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty, such as in the case of unmodeled spill-over dynamics in a lightly damped flexible structure. A numerical example is presented to illustrate the usefulness of the proposed results.Index Terms-Negative imaginary systems, lightly damped systems, Riccati equations, uncertain system.

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A discrete-time robust extended Kalman filter (REKF) formulation for uncertain systems expressed in terms of a set-valued state estimator is described in this paper. The robust filter and Riccati equations are derived as an approximate solution to a reverse-time optimal control problem defining this set-valued state estimator. As presented, the uncertainties are modeled by a sum quadratic constraint (SQC) that takes into account both modeling uncertainties as well as uncertainties introduced from exogenous noise sources.