Abstract
System identification can be viewed as a learning problem. After observing a random sample of inputs and the corresponding outputs, the aim is to learn a mapping from inputs to outputs given by an unknown control system. We study continous time linear control systems and calculate bounds to the Vapnik-Chervonenkis dimension which characterizes the difficulty of learning (in the Probably Approximately Correct sense). The bounds are in terms of the dimensions of the control system and the band-width of the inputs.
Abstract
As the amount of air traffic continues to rise, the current air traffic
control (ATC) system is becoming increasingly inefficient. Alternative
systems are being studied, among them the paradigm of "free flight".
This environment, based on user-preferred trajectories, requires conflict
detection and resolution on several levels. We present a model for
this controller on the lowest level, namely that of pairwise conflict
resolution. Two criteria for optimality of a resolution are considered.
In each case, solution trajectories are determined, and are proved optimal
using the Pontryagin Maximum Principle.
Abstract
This is the first of a series of talks about recent nonsmooth
versions of the maximum principle of optimal control theory. The first
lecture will deal with the issue of how to develop a theory of "generalized
differentials" at a point for maps that are not differentiable in the ordinary
sense. Naturally, the answer to this question depends very much on the
intended purpose of such a theory. Our specific goal is that of extending
the usual inverse function and implicit function theorems, valid for continuously
differentiable maps, especially the following evident corollary on transversal
intersections: if X, Y, Z are finite-dimensional real linear spaces, and
F, G are continuously differentiable maps from X to Z and from Y to Z sending
the origin to the origin are such that the sum of the images of their differentials
is the whole space Z, and the intersection of the images is at least one-dimensional,
then the intersection of the images of the maps contains a nontrivial curve.
Some known notions of generalized differential (such as Clarke's generalized
Jacobians and Warga's derivate containers) will be reviewed, and some new
ones will be proposed, leading to the desired extension for maps in a large
class that includes all Lipschitz maps, many continuous non-Lipschitz maps,
and even some important set-valued maps. The arguments depend crucially
on a theorem of Leray-Schauder on the existence of connected sets of zeros
of a homotopy. The relation of all this to the maximum principle will be
described in subsequent lectures, and will not be touched upon in this
talk.
Abstract
This is the second of a series of talks about recent nonsmooth versions of the maximum principle of optimal control theory. The first two lectures deal with the issue of how to develop a theory of ``generalized differentials'' at a point for maps that are not differentiable in the ordinary sense. In the second lecture, we will give the precise formulations of three such theories: J. Warga's ``derivate containers,'' and our ``semidifferentials'' and ``multidifferentials.'' We will then outline the proof of the corresponding open mapping and transversal intersection theorems. The argument is based on a combination of a technique due to Waszewski and a theorem of Leray-Schauder on the existence of connected sets of zeros of a homotopy,
Abstract
The purpose of the talk is to provide sufficient conditions for asymptotic
stabilization in probability of nonlinear control stochastic differential
systems by means of smooth state feedback laws. The technique used in this
work is based on the passive system approach for nonlinear stochastic differential
systems.
Abstract
We will discuss how hybrid systems naturally arise in various situations
when one is given a continuous-time system and might want to (or have to)
control it by some kind of discrete feedback. Particular examples of interest
include digital control, control of nonholonomic systems, and control of
systems with modeling uncertainty. Some questions motivated by the study
of hybrid control algorithms, such as stability problems for switched systems,
will be addressed.
Abstract
Adaptive critic designs (ACDs) have received increasing attention recently.
ACDs are defined as designs that approximate dynamic programming in the
general case, i.e., approximate optimal control over time in noisy, nonlinear
environments. There are many problems in practice which can be formulated
as cost maximization or minimization problems. Examples include error minimization,
energy minimization, profit maximization, and the like. Dynamic programming
is a very useful tool in solving these problems. However, it is often computationally
untenable to run dynamic programming due to the backward numerical process
required for its solutions, i.e., due to the ``curse of dimensionality''.
Over the years, progress has been made to circumvent the ``curse of dimensionality''
by building a system, called ``critic,'' to approximate the cost function
in dynamic programming. The idea is to approximate dynamic programming
solutions by using a function approximation structure such as a neural
network to approximate the cost function. Adaptive critic designs constitute
one of the most important research areas in computational intelligence.
Other terms used often as synonyms include Approximate Dynamic Programming
(controls engineers) and Reinforcement Learning (computer scientists).
This methodology is a very useful tool for building intelligent agents
in almost any environment. This talk will review the theorectical development
of Adaptive Critic Designs. It will mention a few successful applications.
Some problems for further investigation will also be discussed.
Abstract
Predictive control is a widely applied strategy for the synthesis of
controllers of large scale, MIMO systems. One reason for that
is the
possibility of taking explicitly into account constraints in the state
or
input variables which are virtually always present in practical applications.
The talk will review what are the available tools for the analysis of
stability and robustness of predictive control algorithms and will
present
some recent topics of research related to this field.
Abstract
There are many important differences between Riemannian and sub-Riemannian
geometry (in SR-geometry the tangent vector of an admissible curve
is
constrained to belong to a given distribution). One of them is the
existence
of abnormal minimizers. These curves depend intrisically on the
distribution
and not on the metric. Another difference is that the adherence of
the
conjugate locus of a point x contains x (the same is true for the cut
locus). As consequence of this, even spheres of small radius have
singularities. We will see on a particular situation, called the "Martinet
case", that in presence of abnormal minimizers the sphere of small
radius is
not sub-analytic.
First talk by Mikhail Krichman, "Detectability of nonlinear systems revisited"
Abstract
The notion of detectability deals with the question of estimating the true state of a system despite inability to measure all of the state's components (for example, when the system output provides a projection of a state on some subspace). It is well known that for a detectable linear system an observer can be built; and, assuming the system is also asymptotically controllable, it can be stabilized by dynamic feedback. The goal is to do something similar for nonlinear systems, in particular, one must first understand what "nonlinear detectability" is.
One approach was introduced by Sontag and Wang, who also obtained a Lyapunov characterization for their notion in case of autonomous systems. This talk will concern some steps recently made towards generalizing this result to systems with controls.
Second talk by Michael Malisoff, "Control-Lyapunov functions for restricted inputs"
Abstract
Smooth control-Lyapunov functions (clf's) provide foundations for much of current feedback control design. The theory of smooth clf's had its origins in a paper by Artstein of 1983. There it is shown that if there is clf with an appropriate small control property for a smooth control-affine system over Euclidean space with the origin as an equilibrium point relative to a suitable control set, then there is an almost smooth feedback law which is valued in that control set which globally stabilizes the system relative to the origin.
In this talk, we briefly review the theory of smooth clf's. We then exhibit explicit algebraic formulae for the feedback law described in Artstein's paper for the cases when the control set is itself a Euclidean space and for certain situations when the controls are valued in a Minkowski unit ball. These results are used to find approximate solutions to stabilization problems for the case of controls valued in the hypercube.
Back to Eduardo Sontag's Public Homepage.