358 research outputs found
A polynomial approach to the realization of J-lossless behaviours
In this paper, a class of behaviours known as J-lossless behaviours is introduced, where J is a symmetric two-variable polynomial matrix. For a certain J, it is shown that the resulting set of J-lossless behaviours are SISO behaviours such that for each of such behaviours, there exists a quadratic differential form which is positive for nonzero trajectories of the behaviour and whose derivative is equal to the product of the input variable and the derivative of the output variable. Earlier, Van der Schaft and Oeloff had considered a specific form of realization for such behaviours that plays an important role in their model reduction procedure. In our paper, we give a method of computation of a state space realization from a transfer function of such a behaviour in the same form as considered by Van der Schaft and Oeloff, using polynomial algebraic methods. Apart from being useful in enlarging the scope of the model reduction procedure of Van der Schaft and Oeloff, we show that our method of realization also has application in the synthesis of lossless mechanical systems with given transfer functions using springs and masses
A polynomial approach to the realization of J-lossless behaviors
We consider the problem of realizing lossless behaviours with respect to the supply rate equal to the scalar product of the input and of the derivative of the output variables. Using polynomial algebraic method we devise a realization procedure which, starting from an image representation, yields the same state representation used by van der Schaft and Oeloff in the context of model reduction. We also apply the insights derived from this realization procedure to the synthesis of lossless mechanical systems with given transfer functions using springs and masses
Dissipative systems: uncontrollability, observability and RLC realizability
The theory of dissipativity has been primarily developed for controllable
systems/behaviors. For various reasons, in the context of uncontrollable
systems/behaviors, a more appropriate definition of dissipativity is in terms
of the dissipation inequality, namely the {\em existence} of a storage
function. A storage function is a function such that along every system
trajectory, the rate of increase of the storage function is at most the power
supplied. While the power supplied is always expressed in terms of only the
external variables, whether or not the storage function should be allowed to
depend on unobservable/hidden variables also has various consequences on the
notion of dissipativity: this paper thoroughly investigates the key aspects of
both cases, and also proposes another intuitive definition of dissipativity.
We first assume that the storage function can be expressed in terms of the
external variables and their derivatives only and prove our first main result
that, assuming the uncontrollable poles are unmixed, i.e. no pair of
uncontrollable poles add to zero, and assuming a strictness of dissipativity at
the infinity frequency, the dissipativities of a system and its controllable
part are equivalent. We also show that the storage function in this case is a
static state function.
We then investigate the utility of unobservable/hidden variables in the
definition of storage function: we prove that lossless autonomous behaviors
require storage function to be unobservable from external variables. We next
propose another intuitive definition: a behavior is called dissipative if it
can be embedded in a controllable dissipative {\em super-behavior}. We show
that this definition imposes a constraint on the number of inputs and thus
explains unintuitive examples from the literature in the context of
lossless/orthogonal behaviors.Comment: 26 pages, one figure. Partial results appeared in an IFAC conference
(World Congress, Milan, Italy, 2011
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
A theory of passive linear systems with no assumptions
This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordWe present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.This research was conducted in part during a Fellowship supported by the Cambridge Philosophical Society , http://www.cambridgephilosophicalsociety.org
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
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