158,308 research outputs found
New Results on Negative Imaginary Systems Theory with Application to Flexible Structures and Nano-Positioning
Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure control systems, and are often very sensitive to changes in environmental variables. These can lead to vibrational effects which limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extend by using force actuators combined with colocated measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received a great attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory. In this thesis, we provide a generalization and development of negative imaginary systems theory to include a wider class of systems.
In the generalization of NI systems theory, we provide a new negative imaginary definition that allows for flexible systems with free body motion. Also, we provide a new stability condition for a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative imaginary (SNI). This general stability result captures all previous NI stability results which have been developed.
This thesis also presents analytical tools for negative imaginary systems theory, which can be useful in the practical applications of the theory. Two methods that can be used for checking the negative imaginary property for a given system are presented. Also, methods for enforcing NI dynamics on mathematical system models to satisfy an NI Property are explored. A systematic method to design controllers for NI systems with guaranteed robust stability also is presented. A practical application of control system design for a three-mirror cavity locking system is presented in the end of the thesis
The Krein Matrix: General Theory and Concrete Applications in Atomic Bose-Einstein Condensates
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it
is especially important to locate not only the unstable eigenvalues (i.e.,
those with positive real part), but also those which are purely imaginary but
have negative Krein signature. These latter eigenvalues have the property that
they can become unstable upon collision with other purely imaginary
eigenvalues, i.e., they are a necessary building block in the mechanism leading
to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a
general theory for constructing a meromorphic matrix-valued function, the
so-called Krein matrix, which has the property of not only locating the
unstable eigenvalues, but also those with negative Krein signature. These
eigenvalues are realized as zeros of the determinant. The resulting finite
dimensional problem obtained by setting the determinant of the Krein matrix to
zero presents a valuable simplification. In this paper the usefulness of the
technique is illustrated through prototypical examples of spectral analysis of
states that have arisen in recent experimental and theoretical studies of
atomic Bose-Einstein condensates. In particular, we consider one-dimensional
settings (the cigar trap) possessing real-valued multi-dark-soliton solutions,
and two-dimensional settings (the pancake trap) admitting complex multi-vortex
stationary waveforms.Comment: 26 pages, 16 figures (revised version on April 18 2013
Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems
Sufficient and necessary conditions for the stability of positive feedback
interconnections of negative imaginary systems are derived via an integral
quadratic constraint (IQC) approach. The IQC framework accommodates
distributed-parameter systems with irrational transfer function
representations, while generalising existing results in the literature and
allowing exploitation of flexibility at zero and infinite frequencies to reduce
conservatism in the analysis. The main results manifest the important property
that the negative imaginariness of systems gives rise to a certain form of IQCs
on positive frequencies that are bounded away from zero and infinity. Two
additional sets of IQCs on the DC and instantaneous gains of the systems are
shown to be sufficient and necessary for closed-loop stability along a homotopy
of systems.Comment: Submitted to Automatica, A preliminary version of this paper appeared
in the Proceedings of the 2015 European Control Conferenc
Effect of the Vacuum Energy Density on Graviton Propagation
It is known that the value L of the vacuum energy density affects the
propagation equation for gravitons: A mass term appears in the propagation
equation, such that m^2=-L. As a consequence, the polarization states of
gravitons also change. This effect of the L-term has been confirmed by recent
calculations in a curved background, which is the only proper setting, since
solutions of the classical Einstein equations in the presence of a L-term
represent a space with constant curvature. A real value for the mass (when L<0)
will show up as a slight exponential damping in the gravitational potential,
which is however strongly constrained by astronomical data. The consequences of
an imaginary mass (for L>0) are still unclear; on general grounds, one can
expect the onset of instabilities in this case. This is also confirmed by
numerical simulations of quantum gravity which became recently available. These
properties gain a special interest in consideration of the following. (1) The
most recent cosmological data indicate that L is positive and of the order of
0.1 J/m^3. Is this value compatible with a stable propagation of gravitons? (2)
The answer to the previous question lies perhaps in the scale dependence of the
effective value of L. L may be negative at the small distance/large energy
scale at which the quantum behavior of gravitational fields and waves becomes
relevant. Furthermore, local contributions to the vacuum energy density (in
superconductors in certain states, and in very strong static electromagnetic
fields) can change locally the sign of L, and so affect locally the propagation
and the properties of gravitons. The graviton wavefunction, for different
values of the parameters, may be characterized by superluminal phase velocity
or by unitarity only in imaginary valued time.Comment: CP699, Space Technology and Applications International Forum-STAIF
2004, proceedings published by AIP and edited by M.S. El-Gen
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
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