127,340 research outputs found
Path Following and Output Synchronization of Homogeneous Linear Time-Invariant Systems
This thesis examines two aspects of the path following control design problem for Linear
Time-Invariant (L.T.I.) systems assigned closed curves in their output space. In the first
part of the thesis we define a path following normal form for L.T.I. systems and study
structural properties related to this normal form. We isolate how unstable zero dynamics
alter the feasibility of using the path following normal form for control design. In the second
half of the thesis we consider a synchronized path following problem for a homogenous
multi-agent system and cast the problem as an instance of an output synchronization
problem to leverage recent results from the literature. It is desired that each individual
agent follow a specified path. The agents communicate with one another over an idealized
communication network to synchronize their positions along the path. The main result is
the construction of a dynamic feedback coupling that drives all the agents in the network
to their respective reference paths while simultaneously synchronizing their positions along
the path. Laboratory results are presented to illustrate the effectiveness of the proposed
approach
Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction
Extinction of an epidemic or a species is a rare event that occurs due to a
large, rare stochastic fluctuation. Although the extinction process is
dynamically unstable, it follows an optimal path that maximizes the probability
of extinction. We show that the optimal path is also directly related to the
finite-time Lyapunov exponents of the underlying dynamical system in that the
optimal path displays maximum sensitivity to initial conditions. We consider
several stochastic epidemic models, and examine the extinction process in a
dynamical systems framework. Using the dynamics of the finite-time Lyapunov
exponents as a constructive tool, we demonstrate that the dynamical systems
viewpoint of extinction evolves naturally toward the optimal path.Comment: 21 pages, 5 figures, Final revision to appear in Bulletin of
Mathematical Biolog
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
Optimal fluctuations and the control of chaos.
The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel-Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identied with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms
Performance bounds for optimal feedback control in networks
Many important complex networks, including critical infrastructure and
emerging industrial automation systems, are becoming increasingly intricate
webs of interacting feedback control loops. A fundamental concern is to
quantify the control properties and performance limitations of the network as a
function of its dynamical structure and control architecture. We study
performance bounds for networks in terms of optimal feedback control costs. We
provide a set of complementary bounds as a function of the system dynamics and
actuator structure. For unstable network dynamics, we characterize a tradeoff
between feedback control performance and the number of control inputs, in
particular showing that optimal cost can increase exponentially with the size
of the network. We also derive a bound on the performance of the worst-case
actuator subset for stable networks, providing insight into dynamics properties
that affect the potential efficacy of actuator selection. We illustrate our
results with numerical experiments that analyze performance in regular and
random networks
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