Negative Imaginary Property Research Articles

On non-proper negative imaginary systems

This paper studies non-proper negative imaginary systems. First, the concept of negative imaginary transfer functions that may have poles at the origin and infinity is introduced. Then, a generalized lemma is presented to provide a sufficient condition to characterize the non-proper negative imaginary properties of systems. The generalized lemma is given in terms of complex variable s in the quarter domain of analyticity. Compared to previous results, our result removes the symmetric restriction. Also, a new relationship is established between (lossless) negative imaginary and (lossless) positive real transfer function matrices by using a minor decomposition. The results in this paper give us the possibility to address non-proper and non-symmetric descriptor systems with negative imaginary frequency response. Several examples are presented to illustrate the results.

STABILITY ANALYSIS AND VIBRATION CONTROL OF A CLASS OF NEGATIVE IMAGINARY SYSTEMS

This paper presents stability analysis and vibration control of a class of negative imaginary systems. A flexible manipulator that moves in a horizontal plane is considered and is modelled using the finite element method. The system with two poles at the origin is shown to possess negative imaginary properties. Subsequently, an integral resonant controller (IRC) which is a strictly negative imaginary controller is designed for the position and vibration control of the system. Using the IRC, the closed-loop system is observed to be internally stable and simuation results show that satisfactory hub angle response is achieved. Furthermore, vibration magnitudes at the resonance modes are suppressed by 48 dB.

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

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.

Negative Imaginary Lemmas for Descriptor Systems

This technical note studies the negative imaginary properties of descriptor linear systems based on state-space realizations. Under the assumption of a minimal realization, necessary and sufficient conditions are established to characterize the negative imaginary properties of descriptor systems in terms of linear matrix inequalities with equality constraints. In particular, a negative imaginary lemma, a strict negative imaginary lemma and a lossless negative imaginary lemma are developed. A multiple-input and multiple-output RLC circuit network is used as an illustrative example to validate the developed theory.

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.

Stability Analysis of Switched Systems with 'Mixed'-Negative Imaginary Property

This paper discusses the stability of feedback systems in which both plant and controller are switched. Switched systems considered here have all their subsystems satisfying the 'mixed'-negative imaginary property. A definition for dissipativity (for switched systems) is proposed, and dissipative switched systems are shown to be stable (under certain conditions). Switched systems with 'mixed'-negative imaginary property are shown to be dissipative and conditions for stability are derived. As an illustration of the results, a switched controller is designed for a nanopositioning stage, which has a 'mixed'-negative imaginary frequency response function. Simulations show that the closed loop is stable and the designed controller damps the resonances satisfactorily.

A negative imaginary approach to the actuation of a guitar string

A feedback control approach to damp the resonances of a guitar string is considered. A guitar string is tightly stretched and fastened at both its ends. This string is placed in a magnetic field and set to vibrations by passing current through it. A displacement sensor is placed such that the actuator and the sensor are collocated. The linear dynamics characterizing the input/output (supplied current/measured displacement) relationship is identified and a model is fit. It is observed that the model satisfies the negative imaginary condition. An Integral Resonant Controller (IRC) is designed, by exploiting the negative imaginary characteristics, to damp the first five modes of the string. Signal conditioning circuitry associated with the sensor could introduce significant time delays that destroy the negative imaginary property. Here, an IRC is designed even when the system does not exhibit negative imaginary characteristics. The effectiveness these control designs are experimentally validated.

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.

Spectral characterization and enforcement of negative imaginariness for descriptor systems

Systems with counterclockwise input–output dynamics (or negative imaginary transfer functions) arise in various applications such as the modeling of flexible mechanical structures or electrical circuits when certain kinds of measurements are taken. In this paper we introduce descriptor systems with such an additional structure. We state various of their properties and prove algebraic characterizations of negative imaginariness in terms of spectral conditions of certain structured matrix pencils. For this purpose we also analyze particular boundary cases which are characterized by properties of a structured Kronecker canonical form. Finally, we describe a method which can be used to restore the negative imaginary property in case that it is lost. This happens, e.g., when a system with theoretically negative imaginary transfer function is obtained by, e.g., model order reduction methods, linearization, or other approximations. The method is illustrated by numerical examples.

Spectral Conditions for the Negative Imaginary Property of Transfer Function Matrices

AbstractThis paper derives some necessary and sufficient conditions for linear time invariant systems to have the negative imaginary property in both the single-input-single-output as well as the multi-input-multi-output cases. The conditions for a system to be negative imaginary are described in terms of spectral conditions obtained for a given transfer function matrix.

Feedback Control of Negative-Imaginary Systems

This paper investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negative-imaginary systems is used to reveal the robustness properties of multi-input, multi-output (MIMO) positive-position feedback controllers and related types of controllers for flexible structures. The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics.