Abhijit G. Kallapur

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. 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.

A Discrete-Time Robust Extended Kalman Filter for Uncertain Systems With Sum Quadratic Constraints

This technical note outlines the formulation of a novel discrete-time robust extended Kalman filter for uncertain systems with uncertainties described in terms of sum quadratic constraints. The robust filter is an approximate set-valued state estimator which is robust in the sense that it can handle modeling uncertainties in addition to exogenous noise. Riccati and filter difference equations are obtained as an approximate solution to a reverse-time optimal control problem defining the set-valued state estimator. In order to obtain a solution to the set-valued state estimation problem, the discrete-time system dynamics are modeled backwards in time.

Spectral Conditions for Negative Imaginary Systems With Applications to Nanopositioning

The negative imaginary (NI) property is exhibited by many systems such as flexible structures with force actuators and position sensors and can be used to prove the robust stability of flexible structure control systems. In this paper, we derive methods to check for the NI and strict negative imaginary (SNI) properties in both the single-input single-output as well as multi-input multi-output cases. The proposed methods are based on spectral conditions on a corresponding Hamiltonian matrix obtained for a given system transfer function matrix. Under certain conditions, a given transfer function matrix satisfies the NI property if and only if the corresponding Hamiltonian matrix has no pure imaginary eigenvalues with odd multiplicity. It is also shown that a given transfer function matrix satisfies the SNI property if and only if the corresponding Hamiltonian matrix has no eigenvalues on the imaginary axis, except at the origin. The results of this paper are applied to check the NI property in two nanopositioning applications.

Pulsed quantum cascade laser-based CRDS substance detection: real-time detection of TNT

This paper presents experimental results from a pulsed quantum cascade laser based cavity ringdown spectrometer used as a high-throughput detection system. The results were obtained from an optical cavity with 99.8% input and output coupling mirrors that was rapidly swept (0.2s to 7s sweep times) between 1582.25 cm(-1) (6.3201μm) and 1697.00 cm(-1) (5.8928μm). The spectrometer was able to monitor gas species over the pressure range 585 torr to 1μtorr, and the analysis involves a new digital data processing system that optimises the processing speed and minimises the data storage requirements. In this approach we show that is it not necessary to make direct measurements of the ringdown time of the cavity to obtain the system dynamics. Furthermore, we show that correct data processing is crucial for the ultimate implementation of a wideband IR spectrometer that covers a range similar to that of commercial Fourier transform infrared instruments.

UAV Linear and Nonlinear Estimation Using Extended Kalman Filter

UAV state and parameter estimation for three dynamic models is presented using techniques of extended Kalman filter. Two nonlinear models and one linear model are used for the identification of a small unmanned aerial vehicle (UAV), all of which are derived from a common UAV coupled, dynamic model.

A new stability result for the feedback interconnection of negative imaginary systems with a pole at the origin

This paper is concerned with stability conditions for the positive feedback interconnection of negative imaginary systems. A generalization of the negative imaginary lemma is derived, which remains true even if the transfer function has poles on the imaginary axis including the origin. A sufficient condition for the internal stability of a feedback interconnection for NI systems including a pole at the origin is given and an illustrative example is presented to support the result.

A discrete-time robust extended Kalman filter

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.

Enforcing negative imaginary dynamics on mathematical system models

Flexible structures with collocated force actuators and position sensors lead to negative imaginary dynamics. However, in some cases, the mathematical models obtained for these systems, for example, using system identification methods may not yield a negative imaginary system. This paper provides two methods for enforcing negative imaginary dynamics on such mathematical models, given that it is known that the underlying dynamics ought to belong to this system class. The first method is based on a study of the spectral properties of Hamiltonian matrices. A test for checking the negativity of the imaginary part of a corresponding transfer function matrix is first developed. If an associated Hamiltonian matrix has pure imaginary axis eigenvalues, the mathematical model loses the negative imaginary property in some frequency bands. In such cases, a first-order perturbation method is proposed for iteratively collapsing the frequency bands whose negative imaginary property is violated and finally displacing the eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the negative imaginary dynamics. In the second method, direct spectral properties of the imaginary part of a transfer function are used to identify the frequency bands where the negative imaginary properties are violated. A pointwise-in-frequency scheme is then proposed to restore the negative imaginary system properties in the mathematical model.

A generalized negative imaginary lemma and Riccati-based static state-feedback negative imaginary synthesis

Abstract In this paper, we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition. Then, an algebraic Riccati equation method is given to determine if a system is negative imaginary. Also, a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property. The controller synthesis procedure is based on the proposed negative imaginary lemma. Using this procedure, 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 results.

Nonlinear estimation of ring-down time for a Fabry-Perot optical cavity

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.