89,023 research outputs found
Stability Margin Scaling Laws for Distributed Formation Control as a Function of Network Structure
We consider the problem of distributed formation control of a large number of
vehicles. An individual vehicle in the formation is assumed to be a fully
actuated point mass. A distributed control law is examined: the control action
on an individual vehicle depends on (i) its own velocity and (ii) the relative
position measurements with a small subset of vehicles (neighbors) in the
formation. The neighbors are defined according to an information graph.
In this paper we describe a methodology for modeling, analysis, and
distributed control design of such vehicular formations whose information graph
is a D-dimensional lattice. The modeling relies on an approximation based on a
partial differential equation (PDE) that describes the spatio-temporal
evolution of position errors in the formation. The analysis and control design
is based on the PDE model. We deduce asymptotic formulae for the closed-loop
stability margin (absolute value of the real part of the least stable
eigenvalue) of the controlled formation. The stability margin is shown to
approach 0 as the number of vehicles N goes to infinity. The exponent on the
scaling law for the stability margin is influenced by the dimension and the
structure of the information graph. We show that the scaling law can be
improved by employing a higher dimensional information graph.
Apart from analysis, the PDE model is used for a mistuning-based design of
control gains to maximize the stability margin. Mistuning here refers to small
perturbation of control gains from their nominal symmetric values. We show that
the mistuned design can have a significantly better stability margin even with
a small amount of perturbation. The results of the analysis with the PDE model
are corroborated with numerical computation of eigenvalues with the state-space
model of the formation.Comment: This paper is the expanded version of the paper with the same name
which is accepted by the IEEE Transactions on Automatic Control. The final
version is updated on Oct. 12, 201
Formal Analysis of Linear Control Systems using Theorem Proving
Control systems are an integral part of almost every engineering and physical
system and thus their accurate analysis is of utmost importance. Traditionally,
control systems are analyzed using paper-and-pencil proof and computer
simulation methods, however, both of these methods cannot provide accurate
analysis due to their inherent limitations. Model checking has been widely used
to analyze control systems but the continuous nature of their environment and
physical components cannot be truly captured by a state-transition system in
this technique. To overcome these limitations, we propose to use
higher-order-logic theorem proving for analyzing linear control systems based
on a formalized theory of the Laplace transform method. For this purpose, we
have formalized the foundations of linear control system analysis in
higher-order logic so that a linear control system can be readily modeled and
analyzed. The paper presents a new formalization of the Laplace transform and
the formal verification of its properties that are frequently used in the
transfer function based analysis to judge the frequency response, gain margin
and phase margin, and stability of a linear control system. We also formalize
the active realizations of various controllers, like
Proportional-Integral-Derivative (PID), Proportional-Integral (PI),
Proportional-Derivative (PD), and various active and passive compensators, like
lead, lag and lag-lead. For illustration, we present a formal analysis of an
unmanned free-swimming submersible vehicle using the HOL Light theorem prover.Comment: International Conference on Formal Engineering Method
On resilient control of dynamical flow networks
Resilience has become a key aspect in the design of contemporary
infrastructure networks. This comes as a result of ever-increasing loads,
limited physical capacity, and fast-growing levels of interconnectedness and
complexity due to the recent technological advancements. The problem has
motivated a considerable amount of research within the last few years,
particularly focused on the dynamical aspects of network flows, complementing
more classical static network flow optimization approaches. In this tutorial
paper, a class of single-commodity first-order models of dynamical flow
networks is considered. A few results recently appeared in the literature and
dealing with stability and robustness of dynamical flow networks are gathered
and originally presented in a unified framework. In particular, (differential)
stability properties of monotone dynamical flow networks are treated in some
detail, and the notion of margin of resilience is introduced as a quantitative
measure of their robustness. While emphasizing methodological aspects --
including structural properties, such as monotonicity, that enable tractability
and scalability -- over the specific applications, connections to
well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201
A biased approach to nonlinear robust stability and performance with applications to adaptive control
The nonlinear robust stability theory of Georgiou and Smith [IEEE Trans. Automat. Control, 42 (1997), pp. 1200–1229] is generalized to the case of notions of stability with bias terms. An example from adaptive control illustrates nontrivial robust stability certificates for systems which the previous unbiased theory could not establish a nonzero robust stability margin. This treatment also shows that the bounded-input bounded-output robust stability results for adaptive controllers in French [IEEE Trans. Automat. Control, 53 (2008), pp. 461–478] can be refined to show preservation of biased forms of stability under gap perturbations. In the nonlinear setting, it also is shown that in contrast to linear time invariant systems, the problem of optimizing nominal performance is not equivalent to maximizing the robust stability margin
On Robustness in the Gap Metric and Coprime Factor Uncertainty for LTV Systems
In this paper, we study the problem of robust stabilization for linear
time-varying (LTV) systems subject to time-varying normalized coprime factor
uncertainty. Operator theoretic results which generalize similar results known
to hold for linear time-invariant (infinite-dimensional) systems are developed.
In particular, we compute an upper bound for the maximal achievable stability
margin under TV normalized coprime factor uncertainty in terms of the norm of
an operator with a time-varying Hankel structure. We point to a necessary and
sufficient condition which guarantees compactness of the TV Hankel operator,
and in which case singular values and vectors can be used to compute the
time-varying stability margin and TV controller. A connection between robust
stabilization for LTV systems and an Operator Corona Theorem is also pointed
out.Comment: 20 page
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations
The success of deep convolutional architectures is often attributed in part
to their ability to learn multiscale and invariant representations of natural
signals. However, a precise study of these properties and how they affect
learning guarantees is still missing. In this paper, we consider deep
convolutional representations of signals; we study their invariance to
translations and to more general groups of transformations, their stability to
the action of diffeomorphisms, and their ability to preserve signal
information. This analysis is carried by introducing a multilayer kernel based
on convolutional kernel networks and by studying the geometry induced by the
kernel mapping. We then characterize the corresponding reproducing kernel
Hilbert space (RKHS), showing that it contains a large class of convolutional
neural networks with homogeneous activation functions. This analysis allows us
to separate data representation from learning, and to provide a canonical
measure of model complexity, the RKHS norm, which controls both stability and
generalization of any learned model. In addition to models in the constructed
RKHS, our stability analysis also applies to convolutional networks with
generic activations such as rectified linear units, and we discuss its
relationship with recent generalization bounds based on spectral norms
Robustness properties of discrete time regulators, LOG regulators and hybrid systems
Robustness properites of sample-data LQ regulators are derived which show that these regulators have fundamentally inferior uncertainty tolerances when compared to their continuous-time counterparts. Results are also presented in stability theory, multivariable frequency domain analysis, LQG robustness, and mathematical representations of hybrid systems
Reduced Order Controller Design for Robust Output Regulation
We study robust output regulation for parabolic partial differential
equations and other infinite-dimensional linear systems with analytic
semigroups. As our main results we show that robust output tracking and
disturbance rejection for our class of systems can be achieved using a
finite-dimensional controller and present algorithms for construction of two
different internal model based robust controllers. The controller parameters
are chosen based on a Galerkin approximation of the original PDE system and
employ balanced truncation to reduce the orders of the controllers. In the
second part of the paper we design controllers for robust output tracking and
disturbance rejection for a 1D reaction-diffusion equation with boundary
disturbances, a 2D diffusion-convection equation, and a 1D beam equation with
Kelvin-Voigt damping.Comment: Revised version with minor improvements and corrections. 28 pages, 9
figures. Accepted for publication in the IEEE Transactions on Automatic
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