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2,400 |
An exact analysis of stable allocation
|
math.OC
|
Shapley and Scarf introduced a notion of stable allocation between traders
and indivisible goods, when each trader has rank-ordered each of the goods. The
purpose of this note is to prove that the distribution of ranks after
allocation is the same as the distribution of search distances in uniform
hashing, when the rank-orderings are independent and uniformly random.
Therefore the average sum of final ranks is just $(n+1)H_n-n$, and the standard
deviation is O(n). The proof involves a family of interesting one-to-one
correspondences between permutations of a special kind.
|
math
|
2,401 |
Two-way rounding
|
math.OC
|
Given $n$ real numbers $0\leq x_1,...,x_n<1$ and a permutation~$\sigma$ of
$\{1,...,n\}$, we can always find $\xbar_1,...,\xbar_n\in\{0,1\}$ so that the
partial sums $\xbar_1+... +\xbar_k$ and $\xbar_{\sigma 1}+... +\xbar_{\sigma
k}$ differ from the unrounded values $x_1+... + x_k$ and $x_{\sigma 1}+...
+x_{\sigma k}$ by at most $n/(n+1)$, for $1\leq k\leq n$. The latter bound is
best possible. The proof uses an elementary argument about flows in a certain
network, and leads to a simple algorithm that finds an optimum way to round.
|
math
|
2,402 |
Control of nonlinear underactuated systems
|
math.OC
|
In this paper we introduce a new method to design control laws for non-linear
underactuated systems. Our method produces an infinite dimensional family of
control laws, whereas most control techniques only produce a finite dimensional
family. These control laws each come with a natural Lyapunov function. The
inverted pendulum cart is used as an example. In addition, we construct an
abstract system which is open loop unstable and cannot be stabilized using any
linear control law, and demonstrate that our method produces a stabilizing
control law.
|
math
|
2,403 |
Further results on controllability of recurrent neural networks
|
math.OC
|
This paper studies controllability properties of recurrent neural networks.
The new contributions are:
(1) an extension of the result in the previous paper "Complete
controllability of continuous-time recurrent neural networks" (Sontag and
Sussmann) to a slightly different model, where inputs appear in an affine form,
(2) a formulation and proof of a necessary and sufficient condition, in terms
of local-local controllability, and
(3) a complete analysis of the 2-dimensional case for which the hypotheses
made in previous work do not apply
|
math
|
2,404 |
Feedback Stabilization over Commutative Rings: Further study of the coordinate-free approach
|
math.OC
|
This paper is concerned with the coordinate-free approach to control systems.
The coordinate-free approach is a factorization approach but does not require
the coprime factorizations of the plant. We present two criteria for feedback
stabilizability for MIMO systems in which transfer functions belong to the
total rings of fractions of commutative rings. Both of them are generalizations
of Sule's results in [SIAM J. Control Optim., 32-6, 1675-1695(1994)]. The first
criterion is expressed in terms of modules generated from a causal plant and
does not require the plant to be strictly causal. It shows that if the plant is
stabilizable, the modules are projective. The other criterion is expressed in
terms of ideals called generalized elementary factors. This gives the
stabilizability of a causal plant in terms of the coprimeness of the
generalized elementary factors. As an example, a discrete finite-time delay
system is considered.
|
math
|
2,405 |
A Tuner that Accelerates Parameters
|
math.OC
|
We propose a tuner, suitable for adaptive control and (in its discrete-time
version) adaptive filtering applications, that sets the second derivative of
the parameter estimates rather than the first derivative as is done in the
overwhelming majority of the literature. Comparative stability and performance
analyses are presented.
|
math
|
2,406 |
Extremal Optimization: Methods derived from Co-Evolution
|
math.OC
|
We describe a general-purpose method for finding high-quality solutions to
hard optimization problems, inspired by self-organized critical models of
co-evolution such as the Bak-Sneppen model. The method, called Extremal
Optimization, successively eliminates extremely undesirable components of
sub-optimal solutions, rather than ``breeding'' better components. In contrast
to Genetic Algorithms which operate on an entire ``gene-pool'' of possible
solutions, Extremal Optimization improves on a single candidate solution by
treating each of its components as species co-evolving according to Darwinian
principles. Unlike Simulated Annealing, its non-equilibrium approach effects an
algorithm requiring few parameters to tune. With only one adjustable parameter,
its performance proves competitive with, and often superior to, more elaborate
stochastic optimization procedures. We demonstrate it here on two classic hard
optimization problems: graph partitioning and the traveling salesman problem.
|
math
|
2,407 |
Games, predictions, interactivity
|
math.OC
|
This short note is devoted to the unraveling of the hidden interactivity of
ordinary games which is an artefact of predictions of the behaviour of other
players by the fixed player and describes deviations of their real behaviour
from such predictions. A method to improve the predictions is proposed.
Applications to the strategical analysis of interactive games are also briefly
specified.
|
math
|
2,408 |
A Parameterization of Stabilizing Controllers over Commutative Rings
|
math.OC
|
We present a parameterization of the stabilizing controllers over commutative
rings. In the classical case, that is, in the case where there exist the
right-/left-coprime factorizations of the given plant, the stabilizing
controllers can be parameterized by the method called
``Youla-Kucera-parameterization''. However, it is known that there exist models
in which some stabilizable transfer matrices do not have their
right-/left-coprime factorizations. In such models, we cannot employ the
Youla-Kucera-parameterization directly. Our method of this paper can be applied
to even such models.
|
math
|
2,409 |
Feedback Stabilization over Commutative Rings with no Right-/Left-Coprime Factorizations
|
math.OC
|
Anantharam showed in 1985 the existence of a model in which some stabilizable
plants do not have its right-/left-coprime factorizations. In this paper, we
give a condition of the nonexistence of the right-/left-coprime factorizations
of stabilizable plants as a generalization of Anantharam's result. As examples
of the models which satisfy the condition, we present two models. We illustrate
the construction of a stabilizing controller of stabilizable single-input
single-output plants of such models.
|
math
|
2,410 |
A method for computing quadratic Brunovsky forms
|
math.OC
|
In this paper, for continuous, linearly-controllable quadratic control
systems with a single input, an explicit, constructive method is proposed for
studying their Brunovsky forms, initially studied in [W. Kang and A. J. Krener,
Extended quadratic controller normal form and dynamic state feedback
linearization of nonlinear systems, SIAM Journal on Control and Optimization,
30:1319-1337, 1992]. In this approach, the computation of Brunovsky forms and
transformation matrices and the proof of their existence and uniqueness are
carried out simultaneously. In addition, it is shown that quadratic
transformations in the aforementioned paper can be simplified to prevent
multiplicity in Brunovsky forms. This method is extended for studying discrete
quadratic systems. Finally, computation algorithms for both continuous and
discrete systems are summarized, and examples demonstrated.
|
math
|
2,411 |
Mathematical Problems in the Control of Underactuated Systems
|
math.OC
|
In this paper we will discuss problems and techniques related to
underactuated systems. We give a mathematical formulation of several problems
arising from applications, review some standard and new techniques, and pose
some interesting and challenging open questions.
|
math
|
2,412 |
Input-Output-to-State Stability
|
math.OC
|
This work explores Lyapunov characterizations of the input-output-to-state
stability (IOSS) property for nonlinear systems. The notion of IOSS is a
natural generalization of the standard zero-detectability property used in the
linear case. The main contribution of this work is to establish a complete
equivalence between the input-output-to-state stability property and the
existence of a certain type of smooth Lyapunov function. As corollaries, one
shows the existence of ``norm-estimators'', and obtains characterizations of
nonlinear detectability in terms of relative stability and of finite-energy
estimates.
|
math
|
2,413 |
Remarks regarding the gap between continuous, Lipschitz, and differentiable storage functions for dissipation inequalities appearing in $H_\infty$ control
|
math.OC
|
This paper deals with the regularity of solutions of the Hamilton-Jacobi
Inequality which arises in H-infinity control. It shows by explicit
counterexamples that there are gaps between existence of continuous and locally
Lipschitz (positive definite and proper) solutions, and between Lipschitz and
continuously differentiable ones. On the other hand, it is shown that it is
always possible to smooth-out solutions, provided that an infinitesimal
increase in gain is allowed.
|
math
|
2,414 |
A Convex Maximization Problem: Discrete Case
|
math.OC
|
We study a specific convex maximization problem in n-dimensional space. The
conjectured solution is proved to be a vertex of the polyhedral feasible
region, but only a partial proof of local maximality is known. Integer
sequences with interesting patterns arise in the analysis, owing to the number
theoretic origin of the problem.
|
math
|
2,415 |
A Convex Maximization Problem: Continuous Case
|
math.OC
|
We study a specific convex maximization problem in the space of continuous
functions defined on a semi-infinite interval. An unexplained connection to the
discrete version of this problem is investigated.
|
math
|
2,416 |
Elementary Factors and Reduced Minors for Linear Systems over Commutative Rings
|
math.OC
|
In 1994, Sule presented the necessary and sufficient conditions of the
feedback stabilizability of systems over unique factorization domains in terms
of elementary factors and in terms of reduced minors. Recently, Mori and Abe
have generalized his theory over commutative rings. They have introduced the
notion of the generalized elementary factor, which is a~generalization of the
elementary factor, and have given the necessary and sufficient condition of the
feedback stabilizability. In this paper, we present two generalization of the
reduced minors. Using each of them, we state the necessary and sufficient
condition of the feedback stabilizability over commutative rings. Further we
present the relationship between the generalizations and the generalized
elementary factors.
|
math
|
2,417 |
Matching control laws for a ball and beam system
|
math.OC
|
This note describes a method for generating an infinite-dimensional family of
nonlinear control laws for underactuated systems. For a ball and beam system,
the entire family is found explicitly.
|
math
|
2,418 |
Matching and digital control implementation for underactuated systems
|
math.OC
|
This note describes two problems related to the digital implementation of
control laws in the infinite dimensional family of matching control laws,
namely state estimation and sampled data induced error. The entire family of
control laws is written for an inverted pendulum cart. Numerical simulations
which include sampled data and a state estimator are presented for one of the
control laws in this family.
|
math
|
2,419 |
The fractional - order controllers: Methods for their synthesis and application
|
math.OC
|
This paper deals with fractional-order controllers. We outline mathematical
description of fractional controllers and methods of their synthesis and
application. Synthesis method is a modified root locus method for
fractional-order systems and fractional-order controllers. In the next section
we describe how to apply the fractional controller on control systems.
|
math
|
2,420 |
The Calibration Method for Free Discontinuity Problems
|
math.OC
|
The calibration method is used to identify some minimizers of the
Mumford-Shah functional. The method is then extended to more general free
discontinuity problems.
|
math
|
2,421 |
Matching, linear systems, and the ball and beam
|
math.OC
|
A recent approach to the control of underactuated systems is to look for
control laws which will induce some specified structure on the closed loop
system. This basic idea is used in several papers already. In this paper, we
will describe one matching condition and an approach for finding all control
laws that fit the condition. After an analysis of the resulting control laws
for linear systems, we will present the results from an experiment on a ball
and beam system.
|
math
|
2,422 |
The modelling and analysis of fractional-order control systems in discrete domain
|
math.OC
|
This paper deals with fractional-order controlled systems and
fractional-order controllers in the discrete domain. The mathematical
description by the fractional difference equations and properties of these
systems are presented. A practical example for modelling the fractional-order
control loop is shown and obtained results are discussed in conclusion.
|
math
|
2,423 |
Neural Stabilization/Excitation Control of a High-Order Power System by Adaptive Feedback Linearization
|
math.OC
|
This paper discusses the systematic design of an adaptive feedback
linearizing neurocontroller for a high-order model of the synchronous
machine/infinite bus power system. The power system is first modelled as an
input-output nonlinear discrete-time system approximated by two neural
networks. The approach allows a simple linear pole-placement controller (which
is itself not a neural network) to be designed. The control law is specified
such that the controller adaptively calculates an appropriate feedback
linearizing control law at each sampling instant by utilizing plant parameter
estimates provided by the neural system model. The control system also adapts
itself on-line. This avoids the requirement for exact knowledge of the power
system dynamics and full state measurement as well as other difficulties
associated with implementing analytical input-output feedback linearizing
control for a complex power system model. Furthermore, a departure is made from
the `ad hoc' manner in which many neural controllers have been designed for
power systems; the approach used here has foundations in control theoretic
concepts of adaptive feedback linearization and pole-placement control design.
Simulation results demonstrate the performance of this controller for a
representative example of a single-machine/infinite bus power system
configuration under various operational conditions.
|
math
|
2,424 |
Output Feedback Control for Stabilizable and Incompletely Observable Nonlinear Systems
|
math.OC
|
This paper introduces a new approach for output feedback stabilization of
SISO systems which, unlike most of the techniques found in the literature, does
not use high-gain observers and control input saturation to achieve separation
between the state feedback and observer designs. Rather, we show that by using
nonlinear observers, together with a projection algorithm, the same kind of
separation principle is achieved for a larger class of systems, namely
stabilizable and incompletely observable plants. Furthermore, this new approach
avoids using knowledge of the inverse of the observability mapping, which is
needed by most techniques in the literature when controlling general
stabilizable systems.
|
math
|
2,425 |
Output Feedback Control of Jet Engine Stall and Surge Using Pressure Measurements
|
math.OC
|
The problem of controlling surge and stall in jet engine compressors is of
fundamental importance in preventing damage and lengthening the life of these
components. In this paper, we use the Moore-Greitzer mathematical model to
develop an output feedback controller for these two instabilities (only one of
the three states is measurable). This problem is particularly challenging since
the system is not completely observable and, hence, none of the output feedback
control techniques found in the literature can be applied to recover the
performance of a full state feedback controller. However, we show how to
successfully solve it by using a novel output feedback approach for the
stabilization of general stabilizable and incompletely observable systems.
|
math
|
2,426 |
Modelling and analysis of fractional-order regulated systems in the state space
|
math.OC
|
In this paper we present the mathematical description and analysis of a
fractional-order regulated system in the state space. A little historical
background of our results in the analysis and synthesis of the fractional-order
dynamical regulated systems is given. The methods and results of simulations of
the fractional-order system described by a state space equation equivalent to
three-member fractional-order differential equation with a fractional-order
$PD^{\delta}$ regulator are then presented. The possibility of investigating
the stability of such systems is also considered.
|
math
|
2,427 |
Control for a class of nonlinear systems with a time-varying structure
|
math.OC
|
In this paper we present a direct adaptive control method for a class of
uncertain nonlinear systems with a time-varying structure. We view the
nonlinear systems as composed of a finite number of ``pieces,'' which are
interpolated by functions that depend on a possibly exogenous scheduling
variable. We assume that each piece is in strict feedback form, and show that
the method yields stability of all signals in the closed-loop, as well as
convergence of the state vector to a residual set around the equilibrium, whose
size can be set by the choice of several design parameters. The class of
systems considered here is a generalization of the class of strict feedback
systems traditionally considered in the backstepping literature. We also
provide design guidelines based on L-infinity bounds on the transient.
|
math
|
2,428 |
The modelling and analysis of fractional-order control systems in frequency domain
|
math.OC
|
This paper deals with fractional-order controlled systems and
fractional-order controllers in the frequency domain. The mathematical
description by fractional transfer functions and properties of these systems
are presented. The new ways for modelling of fractional-order systems are
illustrated with a numerical example and obtained results are discussed in
conclusion.
|
math
|
2,429 |
Partial synchronicity and the (max,+) semiring
|
math.OC
|
In this paper we illustrate how non-stochastic (max,+) techniques can be used
to describe partial synchronization in a Discrete Event Dynamical System. Our
work uses results from the spectral theory of dioids and analyses (max,+)
equations describing various synchronization rules in a simple network. The
network in question is a transport network consisting of two routes joined at a
single point, and our Discrete Events are the departure times of transport
units along these routes. We calculate the maximum frequency of circulation of
these units as a function of the synchronization parameter. These functions
allow us further to determine the waiting times on various routes, and here we
find critical parameters (dependent on the fixed travel times on each route)
which dictate the overall behavoiur. We give explicit equations for these
parameters and state the rules which enable optimal performance in the network
(corresponding to minimum waiting time).
|
math
|
2,430 |
The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling
|
math.OC
|
This paper introduces a class of linear programming examples which cause the
simplex method to cycle indefinitely and which are the simplest possible
examples showing this behaviour. The structure of examples from this class
repeats after two iterations. Cycling is shown to occur for both the most
negative reduced cost and steepest edge column selection criteria. In addition
it is shown that the EXPAND anti-cycling procedure of Gill et al.is not
guaranteed to prevent cycling.
|
math
|
2,431 |
Iterated Local Search
|
math.OC
|
This is a survey of "Iterated Local Search", a general purpose metaheuristic
for finding good solutions of combinatorial optimization problems. It is based
on building a sequence of (locally optimal) solutions by: (1) perturbing the
current solution; (2) applying local search to that modified solution. At a
high level, the method is simple, yet it allows for a detailed use of
problem-specific properties. After giving a general framework, we cover the
uses of Iterated Local Search on a number of well studied problems.
|
math
|
2,432 |
Self-scaled barrier functions on symmetric cones and their classification
|
math.OC
|
Self-scaled barrier functions on self-scaled cones were introduced through a
set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the
construction of long-step interior point algorithms. This paper provides firm
foundation for these objects by exhibiting their symmetry properties, their
intimate ties with the symmetry groups of their domains of definition, and
subsequently their decomposition into irreducible parts and algebraic
classification theory. In a first part we recall the characterisation of the
family of self-scaled cones as the set of symmetric cones and develop a
primal-dual symmetric viewpoint on self-scaled barriers, results that were
first discovered by the second author. We then show in a short, simple proof
that any pointed, convex cone decomposes into a direct sum of irreducible
components in a unique way, a result which can also be of independent interest.
We then show that any self-scaled barrier function decomposes in an essentially
unique way into a direct sum of self-scaled barriers defined on the irreducible
components of the underlying symmetric cone. Finally, we present a complete
algebraic classification of self-scaled barrier functions using the
correspondence between symmetric cones and Euclidean Jordan algebras.
|
math
|
2,433 |
Integral-Input-Output to State Stability
|
math.OC
|
A notion of detectability for nonlinear systems is discussed. Within the
framework of ``input to state stability'' (ISS), a dual notion of ``output to
state stability'' (OSS), and a more complete detectability notion,
``input-output to state stability'' (IOSS) have appeared in the literature.
This note addresses a variant of the IOSS property, using an integral norm to
measure signals, as opposed to the standard supremum norm that appears in ISS
theory.
|
math
|
2,434 |
Decomposition Algorithms for Stochastic Programming on a Computational Grid
|
math.OC
|
We describe algorithms for two-stage stochastic linear programming with
recourse and their implementation on a grid computing platform. In particular,
we examine serial and asynchronous versions of the L-shaped method and a
trust-region method. The parallel platform of choice is the dynamic,
heterogeneous, opportunistic platform provided by the Condor system. The
algorithms are of master-worker type (with the workers being used to solve
second-stage problems, and the MW runtime support library (which supports
master-worker computations) is key to the implementation. Computational results
are presented on large sample average approximations of problems from the
literature.
|
math
|
2,435 |
Network Flow Optimization for Restoration of Images
|
math.OC
|
The network flow optimization approach is offered for restoration of
grayscale and color images corrupted by noise. The Ising models are used as a
statistical background of the proposed method. The new multiresolution network
flow minimum cut algorithm, which is especially efficient in identification of
the maximum a posteriori estimates of corrupted images, is presented. The
algorithm is able to compute the MAP estimates of large size images and can be
used in a concurrent mode. We also describe the efficient solutions of the
problem of integer minimization of two energy functions for the Ising models of
gray-scale and color images.
|
math
|
2,436 |
On the lambda-equations for matching control laws
|
math.OC
|
We discuss matching control laws for underactuated systems. We previously
showed that this class of matching control laws is completely charactarized by
a linear system of first order partial differential equations for one set of
variables followed by a linear system of first order PDEs for a second set of
variables. Here we derive a new first order system of partial differential
equations that encodes all compatibility conditions for the lambda-equations.
We give four examples illustrating different features of matching control laws.
The last example is a system with two unactuated degrees of freedom that admits
only basic solutions to the matching equations. There are systems with many
matching control laws where only basic solutions are potentially useful. We
introduce a rank condition indicating when this is likely to be the case.
|
math
|
2,437 |
On the Boundary Control of Systems of Conservation Laws
|
math.OC
|
The paper is concerned with the boundary controllability of entropy weak
solutions to hyperbolic systems of conservation laws. We prove a general result
on the asymptotic stabilization of a system near a constant state. On the other
hand, we give an example showing that exact controllability in finite time
cannot be achieved, in general.
|
math
|
2,438 |
Verification Theorems for Hamilton-Jacobi-Bellman equations
|
math.OC
|
We study an optimal control problem in Bolza form and we consider the value
function associated to this problem. We prove two verification theorems which
ensure that, if a function $W$ satisfies some suitable weak continuity
assumptions and a Hamilton-Jacobi-Bellman inequality outside a countably
$\mathcal H^n$-rectifiable set, then it is lower or equal to the value
function. These results can be used for optimal synthesis approach.
|
math
|
2,439 |
A Small-Gain Theorem with Applications to Input/Output Systems, Incremental Stability, Detectability, and Interconnections
|
math.OC
|
A general ISS-type small-gain result is presented. It specializes to a
small-gain theorem for ISS operators, and it also recovers the classical
statement for ISS systems in state-space form. In addition, we highlight
applications to incrementally stable systems, detectable systems, and to
interconnections of stable systems.
|
math
|
2,440 |
Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization
|
math.OC
|
The paper is concerned with patchy vector fields, a class of discontinuous,
piecewise smooth vector fields that were introduced in AB to study feedback
stabilization problems. We prove the stability of the corresponding solution
set w.r.t. a wide class of impulsive perturbations. These results yield the
robusteness of patchy feedback controls in the presence of measurement errors
and external disturbances.
|
math
|
2,441 |
Stability Rates for Patchy Vector Fields
|
math.OC
|
The paper is concerned with the stability of the set of trajectories of a
vector field, in the presence of impulsive perturbations. Patchy vector fields
are discontinuous, piecewise smooth vector fields that were introduced in AB to
study feedback stabilization problems. For patchy vector fields in the plane,
with polygonal patches in generic position, we show that the distance between a
perturbed trajectory and an unperturbed one is of the same order of magnitude
as the impulsive forcing term.
|
math
|
2,442 |
Asymptotic cauchy gains: Definitions and small-gain principle
|
math.OC
|
A notion of "asymptotic Cauchy gain" for input/output systems, and an
associated small-gain principle, are introduced.
A Lyapunov-like characterization allows the computation of these gains for
state-space systems, and the formulation of sufficient conditions insuring the
lack of oscillations and chaotic behaviors in a wide variety of cascades and
feedback loops.
|
math
|
2,443 |
Vibrational control in H_infinity problems
|
math.OC
|
We consider the application of the theory of vibrational control to
H_infinity-problems. We study the possibility of introduction of high-frequency
parametric vibrations in order to decrease the minimal attainable value of the
H_infinity-norm. We prove the existence of the stabilizing solution of the
Riccati equation with quickly oscillating coefficients. This solution is found
using the averaging technique as a series of the small parameter.
|
math
|
2,444 |
Measurement to Error Stability: a Notion of Partial Detectability for Nonlinear Systems
|
math.OC
|
In previous work the notion of input to state stability (ISS) has been
generalized to systems with outputs, yielding a number of useful concepts. When
considering a system whose output is to be kept small (i.e. an error output),
the notion of input to output stability (IOS) arises. Alternatively, when
considering a system whose output is meant to provide information about the
state (i.e. a measurement output), one arrives at the detectability notion of
output to state stability (OSS). Combining these concepts, one may consider a
system with two outputs, an error and a measurement. This leads naturally to a
notion of partial detectability we call measurement to error stability (MES).
This property characterizes systems in which the error signal is detectable
through the measurement signal.
This paper provides a partial Lyapunov characterization of the MES property.
A closely related property of stability in three measures (SIT) is introduced,
which characterizes systems for which the error decays whenever it dominates
the measurement. The SIT property is shown to imply MES, and the two are shown
to be equivalent under an additional boundedness assumption. A nonsmooth
Lyapunov characterization of the SIT property is provided, which yields the
partial characterization of MES. The analysis is carried out on systems
described by differential inclusions -- implicitly incorporating a disturbance
input with compact value-set.
|
math
|
2,445 |
Numerical Models for the Simulation of the Fractional-Order Control Systems
|
math.OC
|
This contribution deals with the creation of numerical models for the
simulation of the dynamic characteristics of fractional-order control systems
and their comparison with analytical models. We give the results of the
comparison of dynamic properties in fractional- and integer-order systems with
a controller, designed for an integer-order system as the best approximation to
given fractional-order system. Other open questions are pointed out, which
should be answered in this area of research.
|
math
|
2,446 |
Identification of Fractional-Order Dynamical Systems
|
math.OC
|
This contribution deals with identification of fractional-order dynamical
systems. We consider systems whose mathematical description is a three-member
differential equation in which the orders of derivatives can be real numbers.
We give a discretization method and a numerical solution of differential
equations of this type. An experimental method of identification is given which
is based on evaluation of transfer characteristics. This is a combination of
the method of derivatives of transfer characteristics and of the method of
passive search. The verification was performed on systems with known parameters
and also on a laboratory object.
|
math
|
2,447 |
State-Space Controller Design for the Fractional-Order Regulated System
|
math.OC
|
In this paper we will present a mathematical description and analysis of a
fractional-order regulated system in the state space and the state-space
controller design based on placing the closed-loop poles on the complex plane.
Presented are the results of simulations and stability investigation of this
system.
|
math
|
2,448 |
Fractional-Order State Space Models
|
math.OC
|
In this paper we will present some alternative types of mathematical
description and methods of solution of the fractional-order dynamical system in
the state space. We point out the difference in the true sense of the name
"state" space for the integer-order and fractional-order system and the
importance of the initialization function for the fractional-order system. Some
implications concerning the state feedback control theory are discussed.
Presented are the results of simulations.
|
math
|
2,449 |
A remark on the converging-input converging-state property
|
math.OC
|
Suppose that an equilibrium is asymptotically stable when external inputs
vanish. Then, every bounded trajectory which corresponds to a control which
approaches zero and which lies in the domain of attraction of the unforced
system, must also converge to the equilibrium. This "well-known" but
hard-to-cite fact is proved and slightly generalized here.
|
math
|
2,450 |
Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems
|
math.OC
|
This paper addresses the time-optimal control problem for a class of control
systems which includes controlled mechanical systems with possible dissipation
terms. The Lie algebras associated with such mechanical systems enjoy certain
special properties. These properties are explored and are used in conjunction
with the Pontryagin maximum principle to determine the structure of singular
extremals and, in particular, time-optimal trajectories. The theory is
illustrated with an application to a time-optimal problem for a class of
underwater vehicles
|
math
|
2,451 |
On the output-input stability property for multivariable nonlinear control systems
|
math.OC
|
We study the recently introduced notion of output-input stability, which is a
robust variant of the minimum-phase property for general smooth nonlinear
control systems. The subject of this paper is developing the theory of
output-input stability in the multi-input, multi-output setting. We show that
output-input stability can be viewed as a combination of two system properties,
one related to detectability and the other to left-invertibility. For systems
affine in controls, we provide a necessary and sufficient condition for
output-input stability, which relies on Hirschorn's nonlinear structure
algorithm.
|
math
|
2,452 |
Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem
|
math.OC
|
We consider the homogenized linear feasibility problem, to find an $x$ on the
unit sphere, satisfying $n$ line ar inequalities $a_i^Tx\ge 0$. To solve this
problem we consider the centers of the insphere of spherical simpl ices, whose
facets are determined by a subset of the constraints. As a result we find a new
combinatorial algor ithm for the linear feasibility problem. If we allow
rescaling this algorithm becomes polynomial. We point out that the algorithm
solves as well the more general convex feasibility problem. Moreover numerical
experiments s how that the algorithm could be of practical interest.
|
math
|
2,453 |
Gradient algorithms for finding common Lyapunov functions
|
math.OC
|
This paper is concerned with the problem of finding a quadratic common
Lyapunov function for a family of stable linear systems. We present gradient
iteration algorithms which give deterministic convergence for finite system
families and probabilistic convergence for infinite families.
|
math
|
2,454 |
Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
|
math.OC
|
We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.
|
math
|
2,455 |
How to Evolve Safe Control Strategies
|
math.OC
|
Autonomous space vehicles need adaptive control strategies that can
accommodate unanticipated environmental conditions. The evaluation of new
strategies can often be done only by actually trying them out in the real
physical environment. Consequently, a candidate control strategy must be deemed
safe--i.e., it won't damage any systems--prior to being tested online. How to
do this efficiently has been a challenging problem.
We propose using evolutionary programming in conjunction with a formal
verification technique (called model checking) to evolve candidate control
strategies that are guaranteed to be safe for implementation and evaluation.
|
math
|
2,456 |
Exact Feedback Linearization of Stochastic Control Systems
|
math.OC
|
This paper studies exact linearization methods for stochastic SISO affine
controlled dynamical systems. The systems are defined as vectorfield triplets
in Euclidean space. The goal is to find, for a given nonlinear stochastic
system, a combination of invertible transformations which transform the system
into a controllable linear form. Of course, for most nonlinear systems such
transformation does not exist.
We are focused on linearization by state coordinate transformation combined
with feedback. The difference between Ito and Stratonovich systems is
emphasized. Moreover, we define three types of linearity of stochastic systems
-- g-linearity, sigma-linearity, and g sigma-linearity.
Six variants of the stochastic exact linearization problem are studied. The
most useful problem -- the Ito-g sigma linearization is solved using the
correcting term, which proved to be a very useful tool for Ito systems. The
results are illustrated on a numerical example solved with help of symbolic
algebra.
|
math
|
2,457 |
A Remarkable Property of the Dynamic Optimization Extremals
|
math.OC
|
At the core of optimal control theory is the Pontryagin maximum principle -
the celebrated first order necessary optimality condition - whose solutions are
called extremals and which are obtained through a function called Hamiltonian,
akin to the Lagrangian function used in ordinary calculus optimization
problems. A remarkable property of the extremals is that the total derivative
with respect to time of the corresponding Hamiltonian equals the partial
derivative of the Hamiltonian with respect to time. In particular, when the
Hamiltonian does not depend explicitly on time, the value of the Hamiltonian
evaluated along the extremals turns out to be constant (a property that
corresponds to energy conservation in classical mechanics). We present a
generalization of the above property. As applications of the new relation,
methods for obtaining conserved quantities along the Pontryagin extremals and
for characterizing problems possessing given constants of the motion are
obtained.
|
math
|
2,458 |
Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems
|
math.OC
|
We consider the Lagrange problem of optimal control with unrestricted
controls and address the question: under what conditions we can assure optimal
controls are bounded? This question is related to the one of Lipschitzian
regularity of optimal trajectories, and the answer to it is crucial for closing
the gap between the conditions arising in the existence theory and necessary
optimality conditions. Rewriting the Lagrange problem in a parametric form, we
obtain a relation between the applicability conditions of the Pontryagin
maximum principle to the later problem and the Lipschitzian regularity
conditions for the original problem. Under the standard hypotheses of
coercivity of the existence theory, the conditions imply that the optimal
controls are essentially bounded, assuring the applicability of the classical
necessary optimality conditions like the Pontryagin maximum principle. The
result extends previous Lipschitzian regularity results to cover optimal
control problems with general nonlinear dynamics.
|
math
|
2,459 |
The Convergence of the Extended Kalman Filter
|
math.OC
|
We demonstrate that the extended Kalman filter converges locally for a broad
class of nonlinear systems. If the initial estimation error of the filter is
not too large then the error goes to zero exponentially as time goes to
infinity. To demonstrate this, we require that the system be $C^2$ and
uniformly observable with bounded second partial derivatives.
|
math
|
2,460 |
Quantized control via locational optimization
|
math.OC
|
This paper studies state quantization schemes for feedback stabilization of
control systems with limited information. The focus is on designing the least
destabilizing quantizer subject to a given information constraint. We explore
several ways of measuring the destabilizing effect of a quantizer on the
closed-loop system, including (but not limited to) the worst-case quantization
error. In each case, we show how quantizer design can be naturally reduced to a
version of the so-called multicenter problem from locational optimization.
Algorithms for solving such problems are discussed. In particular, an iterative
solver is developed for a novel weighted multicenter problem which most
accurately represents the least destabilizing quantizer design.
|
math
|
2,461 |
Multistability in Monotone I/O Systems, Preliminary Report
|
math.OC
|
We extend the setup in our previous paper to deal with the case in which more
than one steady state may exist in feedback configurations. This provides a
foundation for the analysis of multi-stability and hysteresis behaviour in high
dimensional feedback systems.
|
math
|
2,462 |
Smoothed analysis of algorithms
|
math.OC
|
Spielman and Teng introduced the smoothed analysis of algorithms to provide a
framework in which one could explain the success in practice of algorithms and
heuristics that could not be understood through the traditional worst-case and
average-case analyses. In this talk, we survey some of the smoothed analyses
that have been performed.
|
math
|
2,463 |
A Globally Convergent LCL Method for Nonlinear Optimization
|
math.OC
|
For optimization problems with nonlinear constraints, linearly constrained
Lagrangian (LCL) methods sequentially minimize a Lagrangian function subject to
linearized constraints. These methods converge rapidly near a solution but may
not be reliable from arbitrary starting points. The well known example \MINOS\
has proven effective on many large problems. Its success motivates us to
propose a globally convergent variant. Our stabilized LCL method possesses two
important properties: the subproblems are always feasible, and they may be
solved inexactly. These features are present in \MINOS only as heuristics.
The new algorithm has been implemented in \Matlab, with the option to use
either the \MINOS or \SNOPT Fortran codes to solve the linearly constrained
subproblems. Only first derivatives are required. We present numerical results
on a nonlinear subset of the \COPS, \CUTE, and HS test-problem sets, which
include many large examples. The results demonstrate the robustness and
efficiency of the stabilized LCL procedure.
|
math
|
2,464 |
Gauge Symmetries and Noether Currents in Optimal Control
|
math.OC
|
We extend the second Noether theorem to optimal control problems which are
invariant under symmetries depending upon k arbitrary functions of the
independent variable and their derivatives up to some order m. As far as we
consider a semi-invariance notion, and the transformation group may also depend
on the control variables, the result is new even in the classical context of
the calculus of variations.
|
math
|
2,465 |
On the problem of global optimisation of a multivariable function
|
math.OC
|
One of the actual problems in the field of numerical optimisation, as is well
known, is the problem of the search for the global extremum of a multivariate
function [1-9,13,14,17-21]. Various versions of the random search methods
[6,8,9] are considered to be the most reliable to solve the problem of global
optimisation. In this work we present the little-known methods of Halton and
LP-search, which has been proved as one of the best practical solutions of the
global optimisation problem.
|
math
|
2,466 |
A Semidefinite Representation for some Minimum Cardinality Problems
|
math.OC
|
Using techniques developed in [Lasserre02], we show that some minimum
cardinality problems subject to linear inequalities can be represented as
finite sequences of semidefinite programs. In particular, we provide a
semidefinite representation of the minimum rank problem on positive
semidefinite matrices. We also use this technique to cast the problem of
finding convex lower bounds on the objective as a semidefinite program.
|
math
|
2,467 |
Quasi-Invariant Optimal Control Problems
|
math.OC
|
We study in optimal control the important relation between invariance of the
problem under a family of transformations, and the existence of preserved
quantities along the Pontryagin extremals. Several extensions of Noether
theorem are provided, in the direction which enlarges the scope of its
application. We formulate a more general version of Noether's theorem for
optimal control problems, which incorporates the possibility to consider a
family of transformations depending on several parameters and, what is more
important, to deal with quasi-invariant and not necessarily invariant optimal
control problems. We trust that this latter extension provides new
possibilities and we illustrate it with several examples, not covered by the
previous known optimal control versions of Noether's theorem.
|
math
|
2,468 |
The Lax conjecture is true
|
math.OC
|
In 1958 Lax conjectured that hyperbolic polynomials in three variables are
determinants of linear combinations of three symmetric matrices. This
conjecture is equivalent to a recent observation of Helton and Vinnikov.
|
math
|
2,469 |
Matrosov's theorem using a family of auxiliary functions: an analysis tool to aid time-varying nonlinear control design
|
math.OC
|
We present a new result on uniform attractivity of the origin for nonlinear
time-varying systems. Our theorem generalizes Matrosov's theorem which extends,
in a certain manner, Krasovskii-LaSalle invariance principle to the case of
general nonlinear time-varying systems. We show the utility of our theorem by
addressing a control problem of port interconnected driftless systems. The
latter includes as special case, the control of chained-form nonholonomic
systems which has been extensively studied in the literature.
|
math
|
2,470 |
A catalog of inverse-kinematics planners for underactuated systems on matrix groups
|
math.OC
|
This paper presents motion planning algorithms for underactuated systems
evolving on rigid rotation and displacement groups. Motion planning is
transcribed into (low-dimensional) combinatorial selection and
inverse-kinematics problems. We present a catalog of solutions for all
underactuated systems on $\SE{2}$, $\SO{3}$ and $\SE{2}\times\real$ classified
according to their controllability properties.
|
math
|
2,471 |
Exploring the capability and limits of the feedback mechanism
|
math.OC
|
Feedback is a most important concept in control systems, its main purpose is
to deal with internal and/or external uncertainties in dynamical systems, by
using the on-line observed information. Thus, a fundamental problem in control
theory is to understand the maximum capability and potential limits of the
feedback mechanism. This paper gives a survey of some basic ideas and results
developed recently in this direction, for several typical classes of uncertain
dynamical systems including parametric and nonparametric nonlinear systems,
sampled-data systems and time-varying stochastic systems.
|
math
|
2,472 |
Global Asymptotic Controllability Implies Input to State Stabilization
|
math.OC
|
We study nonlinear systems with observation errors. The main problem
addressed in this paper is the design of feedbacks for globally asymptotically
controllable (GAC) control affine systems that render the closed loop systems
input to state stable with respect to actuator errors. Extensions for fully
nonlinear GAC systems with actuator errors are also discussed.
|
math
|
2,473 |
Positive forward rates in the maximum smoothness framework
|
math.OC
|
In this article we present a non-linear dynamic programming algorithm for the
computation of forward rates within the maximum smoothness framework. The
algorithm implements the forward rate positivity constraint for a
one-parametric family of smoothness measures and it handles price spreads in
the constraining dataset. We investigate the outcome of the algorithm using the
Swedish Bond market showing examples where the absence of the positive
constraint leads to negative interest rates. Furthermore we investigate the
predictive accuracy of the algorithm as we move along the family of smoothness
measures. Among other things we observe that the inclusion of spreads not only
improves the smoothness of forward curves but also significantly reduces the
predictive error.
|
math
|
2,474 |
Digital Fractional Order Controllers Realized by PIC Microprocessor: Experimental Results
|
math.OC
|
This paper deals with the fractional-order controllers and their possible
hardware realization based on PIC microprocessor and numerical algorithm coded
in PIC Basic. The mathematical description of the digital fractional -order
controllers and approximation in the discrete domain are presented. An example
of realization of the particular case of digital fractional-order PID
controller is shown and described.
|
math
|
2,475 |
Comparison of the methods for discrete approximation of the fractional-order operator
|
math.OC
|
In this paper we will present some alternative types of discretization
methods (discrete approximation) for the fractional-order (FO) differentiator
and their application to the FO dynamical system described by the FO
differential equation (FDE). With analytical solution and numerical solution by
power series expansion (PSE) method are compared two effective methods - the
Muir expansion of the Tustin operator and continued fraction expansion method
(CFE) with the Tustin operator and the Al-Alaoui operator. Except detailed
mathematical description presented are also simulation results. From the Bode
plots of the FO differentiator and FDE and from the solution in the time domain
we can see, that the CFE is a more effective method according to the PSE
method, but there are some restrictions for the choice of the time step. The
Muir expansion is almost unusable.
|
math
|
2,476 |
Flexible Complementarity Solvers for Large-Scale Applications
|
math.OC
|
Discretizations of infinite-dimensional variational inequalities lead to
linear and nonlinear complementarity problems with many degrees of freedom. To
solve these problems in a parallel computing environment, we propose two
active-set methods that solve only one linear system of equations per
iteration. The linear solver, preconditioner, and matrix structures can be
chosen by the user for a particular application to achieve high parallel
performance. The parallel scalability of these methods is demonstrated for some
discretizations of infinite-dimensional variational inequalities.
|
math
|
2,477 |
A new conical internal evolutive LP algorithm
|
math.OC
|
In a previous paper, published in 1992, a primal conical LP algorithm with
exact finite coonvergence was presented. The underlying optimality condition
requires tangency of two sets (an affine space and a cone). In the algorithm
the two sets remain disjoint until the last step. This left open the
possibility of developing an internal algorithm in which, by the contrary, the
two sets keep intersecting each other. Such an algorithm along with a new
optimality condition is presented here. It is stressed that the results given
here complete the picture of the conical approach to LP in many other important
respect, as illustrated in detail in the introduction.
|
math
|
2,478 |
On the Constancy of the Pontryagin Hamiltonian for Autonomous Problems
|
math.OC
|
We provide a new, simpler, and more direct proof of the well known fact that
for autonomous optimal control problems the Pontryagin extremals evolve on a
level surface of the respective Pontryagin Hamiltonian.
|
math
|
2,479 |
A Harmonic Analysis Solution to the Static Basket Arbitrage Problem
|
math.OC
|
We consider the problem of computing upper and lower bounds on the price of a
European basket call option, given prices on other similar baskets. We focus
here on an interpretation of this program as a generalized moment problem.
Recent results by Berg & Maserick (1984), Putinar & Vasilescu (1999) and
Lasserre (2001) on harmonic analysis on semigroups, the K-moment problem and
its applications to optimization, allow us to derive tractable necessary and
sufficient conditions for the absence of static arbitrage between basket
straddles, hence between basket calls and puts.
|
math
|
2,480 |
Reachability problems for products of matrices in semirings
|
math.OC
|
We consider the following matrix reachability problem: given $r$ square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any $r\geq 2$ is
equivalent to the specialization to $r=2$. As an application of this result and
of a theorem of Krob, we show that when $r=2$, the vector and matrix
reachability problems are undecidable over the max-plus semiring
$(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.
|
math
|
2,481 |
The stochastic goodwill problem
|
math.OC
|
Stochastic control problems related to optimal advertising under uncertainty
are considered. In particular, we determine the optimal strategies for the
problem of maximizing the utility of goodwill at launch time and minimizing the
disutility of a stream of advertising costs that extends until the launch time
for some classes of stochastic perturbations of the classical Nerlove-Arrow
dynamics. We also consider some generalizations such as problems with
constrained budget and with discretionary launching.
|
math
|
2,482 |
Nonlinear internal models for output regulation
|
math.OC
|
In this paper we show how nonlinear internal models can be effectively used
in the design of output regulators for nonlinear systems. This result provides
a significant enhancement of the non-equilibrium theory for output regulation,
which we have presented in the recent paper entitled "Limit Sets, Zero
Dynamics, and Internal Models in the Problem of Nonlinear Output Regulation".
|
math
|
2,483 |
Further Results on Lyapunov Functions and Domains of Attraction for Perturbed Asymptotically Stable Systems
|
math.OC
|
We present new theorems characterizing robust Lyapunov functions and infinite
horizon value functions in optimal control as unique viscosity solutions of
partial differential equations. We use these results to further extend Zubov's
method for representing domains of attraction in terms of partial differential
equation solutions.
|
math
|
2,484 |
Bounded-From-Below Solutions of the Hamilton-Jacobi Equation for Optimal Control Problems with Exit Times: Vanishing Lagrangians, Eikonal Equations, and Shape-From-Shading
|
math.OC
|
We study the Hamilton-Jacobi equation for undiscounted exit time control
problems with general nonnegative Lagrangians using the dynamic programming
approach. We prove theorems characterizing the value function as the unique
bounded-from-below viscosity solution of the Hamilton-Jacobi equation which is
null on the target. The result applies to problems with the property that all
trajectories satisfying a certain integral condition must stay in a bounded
set. We allow problems for which the Lagrangian is not uniformly bounded below
by positive constants, in which the hypotheses of the known uniqueness results
for Hamilton-Jacobi equations are not satisfied. We apply our theorems to
eikonal equations from geometric optics, shape-from-shading equations from
image processing, and variants of the Fuller Problem.
|
math
|
2,485 |
Uniform stability of damped nonlinear vibrations of an elastic string
|
math.OC
|
Here we are concerned about uniform stability of damped nonlinear transverse
vibrations of an elastic string fixed at its two ends. The vibrations governed
by nonlinear integro-differential equation of Kirchoff type, is shown to
possess energy uniformly bounded by exponentially decaying function of time.
The result is achieved by considering an energy-like Lyapunov functional for
the system.
|
math
|
2,486 |
Geometrical and Numerical Design of Structured Unitary Space Time Constellations
|
math.OC
|
In this paper we propose constellations with suitable structure which allow
one to construct codes with excellent diversity using geometrical symmetry and
numerical methods. We also demonstrate how these structured constellations
out-perform currently existing constellations and explain why the proposed
constellation structure admit simple decoding algorithm: sphere decoding. The
presented design methods work for any dimensional constellation and for any
transmission rate. Moreover codes based on the proposed structure are very
flexible and can be optimized for any signal to noise ratio.
|
math
|
2,487 |
Pseudopolynomial algorithm for single machine scheduling (withdrawn)
|
math.OC
|
Withdrawn because of non-correctness. Would have implied too much to be true
:-|
|
math
|
2,488 |
Bond Market Completeness and Attainable Contingent Claims
|
math.OC
|
A general class, introduced in [Ekeland et al. 2003], of continuous time bond
markets driven by a standard cylindrical Brownian motion $\wienerq{}{}$ in
$\ell^{2},$ is considered. We prove that there always exist non-hedgeable
random variables in the space $\derprod{}{0}=\cap_{p \geq 1}L^{p}$ and that
$\derprod{}{0}$ has a dense subset of attainable elements, if the volatility
operator is non-degenerated a.e. Such results were proved in [Bj\"ork et al.
1997] in the case of a bond market driven by finite dimensional B.m. and marked
point processes. We define certain smaller spaces $\derprod{}{s},$ $s>0$ of
European contingent claims, by requiring that the integrand in the martingale
representation, with respect to $\wienerq{}{}$, takes values in weighted
$\ell^{2}$ spaces $\ell^{s,2},$ with a power weight of degree $s.$ For all $s >
0,$ the space $\derprod{}{s}$ is dense in $\derprod{}{0}$ and is independent of
the particular bond price and volatility operator processes.
A simple condition in terms of $\ell^{s,2}$ norms is given on the volatility
operator processes, which implies if satisfied, that every element in
$\derprod{}{s}$ is attainable. In this context a related problem of optimal
portfolios of zero coupon bonds is solved for general utility functions and
volatility operator processes, provided that the $\ell^{2}$-valued market price
of risk process has certain Malliavin differentiability properties.
|
math
|
2,489 |
Sublevel sets and global minima of coercive functionals and local minima of their perturbations
|
math.OC
|
The aim of the present paper is essentially to prove that if $\Phi$ and
$\Psi$ are two sequentially weakly lower semicontinuous functionals on a
reflexive real Banach space and if $\Psi$ is also continuous and coercive, then
then following conclusion holds: if, for some $r > \inf_X \Psi$, the weak
closure of the set $\Psi^{-1}(]-\infty, r[)$ has at least $k$ connected
components in the weak topology, then, for each $\lambda > 0$ small enough, the
functional $\Psi + \lambda\Phi$ has at least $k$ local minima lying in
$\Psi^{-1}(]-\infty, r[)$.
|
math
|
2,490 |
Integral functionals on Sobolev spaces having multiple local minima
|
math.OC
|
In this paper, two multiplicity results about local minima of integrals of
the calculus of variations are established. The main tool used to prove them is
the theory developed in [B. Ricceri, Sublevel sets and global minima of
coercive functionals and local minima of their pertubations, math.OC/0402444].
|
math
|
2,491 |
Discrete Nonlinear Observers for Inertial Navigation
|
math.OC
|
We derive an exact deterministic nonlinear observer to compute the continuous
state of an inertial navigation system based on partial discrete measurements,
the so-called strapdown problem. Nonlinear contraction is used as the main
analysis tool, and the hierarchical structure of the system physics is
sytematically exploited. The paper also discusses the use of nonlinear
measurements, such as distances to time-varying reference points.
|
math
|
2,492 |
On the Strong Invariance Property for Non-Lipschitz Dynamics
|
math.OC
|
We provide a new sufficient condition for strong invariance for differential
inclusions, under very general conditions on the dynamics, in terms of a
Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the
multifunction, we assume a feedback realization condition that can in
particular be satisfied for measurable dynamics that are neither upper nor
lower semicontinuous.
|
math
|
2,493 |
Common Polynomial Lyapunov Functions for Linear Switched Systems
|
math.OC
|
In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$,
$x\in\R^n$, $u\in U$, and the problem of asymptotic stability for arbitrary
switching functions, uniform with respect to switching ({\bf UAS} for short).
We first prove that, given a {\bf UAS} system, it is always possible to build a
common polynomial Lyapunov function. Then our main result is that the degree of
that common polynomial Lyapunov function is not uniformly bounded over all the
{\bf UAS} systems. This result answers a question raised by Dayawansa and
Martin. A generalization to a class of piecewise-polynomial Lyapunov functions
is given.
|
math
|
2,494 |
On the existence of a common quadratic Lyapunov function for a rank one difference
|
math.OC
|
Suppose that A and B are real stable matrices, and that their difference A-B
is rank one. Then A and B have a common quadratic Lyapunov function if and only
if the product AB has no real negative eigenvalue. This result is due to
Shorten and Narendra, who showed that it follows as a consequence of the
Kalman-Yacubovich-Popov solution of the Lur'e problem. Here we present a new
and independent proof based on results from convex analysis and the theory of
moments.
|
math
|
2,495 |
Reconsidering Conflict and Cooperation
|
math.OC
|
An analysis of several important aspects of competition or conflict in games,
social choice and decision theory is presented. Inherent difficulties and
complexities in cooperation are highlighted. These have over the years led to a
certain marginalization of studies related to cooperation. The significant
richness of cooperation possibilities and the considerable gains which my lie
there hidden are indicated. Based on that, a reconsideration of cooperation is
suggested, as a more evolved form of rational behaviour. As one of the
motivations it is shown that the paradigmatic non-cooperative Nash equilibrium
itself rests on a strong cooperation assumption in the case of $n \geq 3$
players.
|
math
|
2,496 |
Generalized splines in R^n and optimal control
|
math.OC
|
We have found an inconsistency in our previous version of the paper
"Generalized splines in R^n and optimal control". We give a new-time-dependent
definition of spline curves in R^n which results from solving a non-autonomous
linear quadratic optimal control problem (P) where the matrix B(t) is assumed
to be rectangular with maximum rank. Nevertheless, our results are only valid
if B(t) is a square (nonsingular) matrix. This was pointed out to us by Andrey
Sarychev. We have proceeded with the necessary corrections.
%%%%%%%%%%%%%%%%%%
We give a new time-dependent definition of spline curves in R^n, which
extends a recent definition of vector-valued splines introduced by Rodrigues
and Silva Leite for the time-independent case. Previous results are based on a
variational approach, with lengthy arguments, which do not cover the
non-autonomous situation. We show that the previous results are a consequence
of the Pontryagin maximum principle, and are easily generalized using the
methods of optimal control. Main result asserts that vector-valued splines are
related to the Pontryagin extremals of a non-autonomous linear-quadratic
optimal control problem.
|
math
|
2,497 |
Optimal Control of Volterra Equations with Impulses
|
math.OC
|
We consider an optimal control problem for a system governed by a Volterra
integral equation with impulsive terms. The impulses act on both the state and
the control; the control consists of switchings at discrete times. The cost
functional includes both, an integrated cost rate (continuous part) and
switching costs at the discrete impulse times (discrete part). We prove
necessary optimality conditions of a form analogous to a discrete maximum
principle. For the particular case of a system governed by impulsive ordinary
differential equations, we obtain an impulsive maximum principle as a special
case of the necessary optimality conditions for impulsive Volterra equations.
|
math
|
2,498 |
Swarming Behavior of Multi-Agent Systems
|
math.OC
|
In this paper we consider a continuous-time anisotropic swarm model in
$n$-dimensional space with an attraction/repulsion function and study its
aggregation properties. It is shown that the swarm members will aggregate and
eventually form a cohesive cluster of finite size around the swarm center.
Moreover, the numerical simulations show that all agents will eventually enter
into and remain in a bounded region around the swarm center. The model is more
general than isotropic swarms and our results provide further insight into the
effect of the interaction pattern on individual motion in a swarm system.
|
math
|
2,499 |
Infinitesimal Characterizations for Strong Invariance and Monotonicity for Non-Lipschitz Control Systems
|
math.OC
|
We provide new infinitesimal characterizations for strong invariance of
multifunctions in terms of Hamiltonian inequalities and tangent cones. In lieu
of the standard local Lipschitzness assumption on the multifunction, we assume
a new feedback realizability condition that can in particular be satisfied by
control systems that are discontinuous in the state variable. Our realization
condition is based on H. Sussmann's unique limiting property, and allows a more
general class of feedback realizations than is allowed by the recent strong
invariance characterizations of Krastanov, Malisoff, and Wolenski. We also give
new nonsmooth monotonicity characterizations for control systems that may be
discontinuous in the state.
|
math
|
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