58 research outputs found
Resilient Leader-Follower Consensus to Arbitrary Reference Values
The problem of consensus in the presence of misbehaving agents has
increasingly attracted attention in the literature. Prior results have
established algorithms and graph structures for multi-agent networks which
guarantee the consensus of normally behaving agents in the presence of a
bounded number of misbehaving agents. The final consensus value is guaranteed
to fall within the convex hull of initial agent states. However, the problem of
consensus tracking considers consensus to arbitrary reference values which may
not lie within such bounds. Conditions for consensus tracking in the presence
of misbehaving agents has not been fully studied. This paper presents
conditions for a network of agents using the W-MSR algorithm to achieve this
objective.Comment: Accepted for the 2018 American Control Conferenc
Determining r- and (r,s)-Robustness of Digraphs Using Mixed Integer Linear Programming
There has been an increase in the use of resilient control algorithms based
on the graph theoretic properties of - and -robustness. These
algorithms guarantee consensus of normally behaving agents in the presence of a
bounded number of arbitrarily misbehaving agents if the values of the integers
and are sufficiently large. However, determining an arbitrary graph's
robustness is a highly nontrivial problem. This paper introduces a novel method
for determining the - and -robustness of digraphs using mixed integer
linear programming; to the best of the authors' knowledge it is the first time
that mixed integer programming methods have been applied to the robustness
determination problem. The approach only requires knowledge of the graph
Laplacian matrix, and can be formulated with binary integer variables. Mixed
integer programming algorithms such as branch-and-bound are used to iteratively
tighten the lower and upper bounds on and . Simulations are presented
which compare the performance of this approach to prior robustness
determination algorithms.Comment: arXiv admin note: text overlap with arXiv:1810.01784. AUTHOR NOTES:
This is the extended preprint version of the article accepted by Automatica.
The file arXiv:1810.01784 is the conference version of this articl
Chebyshev Approximation and Higher Order Derivatives of Lyapunov Functions for Estimating the Domain of Attraction
Estimating the Domain of Attraction (DA) of non-polynomial systems is a
challenging problem. Taylor expansion is widely adopted for transforming a
nonlinear analytic function into a polynomial function, but the performance of
Taylor expansion is not always satisfactory. This paper provides solvable ways
for estimating the DA via Chebyshev approximation. Firstly, for Chebyshev
approximation without the remainder, higher order derivatives of Lyapunov
functions are used for estimating the DA, and the largest estimate is obtained
by solving a generalized eigenvalue problem. Moreover, for Chebyshev
approximation with the remainder, an uncertain polynomial system is
reformulated, and a condition is proposed for ensuring the convergence to the
largest estimate with a selected Lyapunov function. Numerical examples
demonstrate that both accuracy and efficiency are improved compared to Taylor
approximation.Comment: 8 pages, 7 figures, IEEE Conference on Decision and Control, 201
Determining r-Robustness of Digraphs Using Mixed Integer Linear Programming
Convergence guarantees of many resilient consensus algorithms are based on
the graph theoretic properties of - and -robustness. These algorithms
guarantee consensus of normally behaving agents in the presence of a bounded
number of arbitrarily misbehaving agents if the values of the integers and
are sufficiently high. However, determining the largest integer for
which an arbitrary digraph is -robust is highly nontrivial. This paper
introduces a novel method for calculating this value using mixed integer linear
programming. The method only requires knowledge of the graph Laplacian matrix,
and can be formulated with affine objective and constraints, except for the
integer constraint. Integer programming methods such as branch-and-bound can
allow both lower and upper bounds on to be iteratively tightened.
Simulations suggest the proposed method demonstrates greater efficiency than
prior algorithms.Comment: Accepted to 2019 American Control Conferenc
Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization
This paper proposes novel gradient-flow schemes that yield convergence to the
optimal point of a convex optimization problem within a \textit{fixed} time
from any given initial condition for unconstrained optimization, constrained
optimization, and min-max problems. The application of the modified gradient
flow to unconstrained optimization problems is studied under the assumption of
gradient-dominance. Then, a modified Newton's method is presented that exhibits
fixed-time convergence under some mild conditions on the objective function.
Building upon this method, a novel technique for solving convex optimization
problems with linear equality constraints that yields convergence to the
optimal point in fixed time is developed. More specifically, constrained
optimization problems formulated as min-max problems are considered, and a
novel method for computing the optimal solution in fixed-time is proposed using
the Lagrangian dual. Finally, the general min-max problem is considered, and a
modified scheme to obtain the optimal solution of saddle-point dynamics in
fixed time is developed. Numerical illustrations that compare the performance
of the proposed method against Newton's method, rescaled-gradient method, and
Nesterov's accelerated method are included to corroborate the efficacy and
applicability of the modified gradient flows in constrained and unconstrained
optimization problems.Comment: 15 pages, 11 figure
Finite-Time Stability of Switched and Hybrid Systems with Unstable Modes
In this work, we study finite-time stability of switched and hybrid systems
in the presence of unstable modes. We present sufficient conditions in terms of
multiple Lyapunov functions for the origin of the system to be finite time
stable. More specifically, we show that even if the value of the Lyapunov
function increases in between two switches, i.e., if there are unstable modes
in the system, finite-time stability can still be guaranteed if the finite time
convergent mode is active long enough. In contrast to earlier work where the
Lyapunov functions are required to be decreasing during the continuous flows
and non-increasing at the discrete jumps, we allow the Lyapunov functions to
increase \emph{both} during the continuous flows and the discrete jumps. As
thus, the derived stability results are less conservative compared to the
earlier results in the related literature, and in effect allow the hybrid
system to have unstable modes. Then, we illustrate how the proposed finite-time
stability conditions specialize for a class of switched systems, and present a
method on the synthesis of a finite-time stabilizing switching signal for
switched linear systems. As a case study, we design a finite-time stable output
feedback controller for a linear switched system, in which only one of the
modes is controllable and observable. Numerical example demonstrates the
efficacy of the proposed methods
r-Robustness and (r,s)-Robustness of Circulant Graphs
There has been recent growing interest in graph theoretical properties known
as r- and (r,s)-robustness. These properties serve as sufficient conditions
guaranteeing the success of certain consensus algorithms in networks with
misbehaving agents present. Due to the complexity of determining the robustness
for an arbitrary graph, several methods have previously been proposed for
identifying the robustness of specific classes of graphs or constructing graphs
with specified robustness levels. The majority of such approaches have focused
on undirected graphs. In this paper we identify a class of scalable directed
graphs whose edge set is determined by a parameter k and prove that the
robustness of these graphs is also determined by k. We support our results
through computer simulations.Comment: 6 pages, 6 figures. Accepted to 2017 IEEE CD
A Hybrid Approach to Persistent Coverage in Stochastic Environments
This paper considers the persistent coverage of a 2-D manifold that has been
embedded in 3-D space. The manifold is subject to continual impact by intruders
which travel at constant velocities along arbitrarily oriented straight-line
trajectories. The trajectories of intruders are estimated online with an
extended Kalman filter and their predicted impact points contribute normally
distributed decay terms to the coverage level. A formal hybrid control strategy
is presented that allows for power-constrained 3-D free-flyer agents to
persistently monitor the domain, track and intercept intruders, and
periodically deploy from and return to a single charging station on the
manifold. Guarantees on intruder interception with respect to agent power
lifespans are formally proven. The efficacy of the algorithm is demonstrated
through simulation.Comment: Extended version of work conditionally accepted for publication in
Automatic
Motion planning and Collision Avoidance using Non-Gradient Vector Fields. Technical Report
This paper presents a novel feedback method on the motion planning for
unicycle robots in environments with static obstacles, along with an extension
to the distributed planning and coordination in multi-robot systems. The method
employs a family of 2-dimensional analytic vector fields, whose integral curves
exhibit various patterns depending on the value of a parameter lambda. More
specifically, for an a priori known value of lambda, the vector field has a
unique singular point of dipole type and can be used to steer the unicycle to a
goal configuration. Furthermore, for the unique value of lambda that the vector
field has a continuum of singular points, the integral curves are used to
define flows around obstacles. An almost global feedback motion plan can then
be constructed by suitably blending attractive and repulsive vector fields in a
static obstacle environment. The method does not suffer from the appearance of
sinks (stable nodes) away from goal point. Compared to other similar methods
which are free of local minima, the proposed approach does not require any
parameter tuning to render the desired convergence properties. The paper also
addresses the extension of the method to the distributed coordination and
control of multiple robots, where each robot needs to navigate to a goal
configuration while avoiding collisions with the remaining robots, and while
using local information only. More specifically, based on the results which
apply to the single-robot case, a motion coordination protocol is presented
which guarantees the safety of the multi-robot system and the almost global
convergence of the robots to their goal configurations. The efficacy of the
proposed methodology is demonstrated via simulation results in static and
dynamic environments
Decentralized Goal Assignment and Safe Trajectory Generation in Multi-Robot Networks via Multiple Lyapunov Functions
This paper considers the problem of decentralized goal assignment and
trajectory generation for multi-robot networks when only local communication is
available, and proposes an approach based on methods related to switched
systems and set invariance. A family of Lyapunov-like functions is employed to
encode the (local) decision making among candidate goal assignments, under
which a group of connected agents chooses the assignment that results in the
shortest total distance to the goals. An additional family of Lyapunov-like
barrier functions is activated in the case when the optimal assignment may lead
to colliding trajectories, maintaining thus system safety while preserving the
convergence guarantees. The proposed switching strategies give rise to feedback
control policies that are computationally efficient and scalable with the
number of agents, and therefore suitable for applications including
first-response deployment of robotic networks under limited information
sharing. The efficacy of the proposed method is demonstrated via simulation
results and experiments with six ground robots
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