110 research outputs found
Optimal Control of Two-Player Systems with Output Feedback
In this article, we consider a fundamental decentralized optimal control
problem, which we call the two-player problem. Two subsystems are
interconnected in a nested information pattern, and output feedback controllers
must be designed for each subsystem. Several special cases of this architecture
have previously been solved, such as the state-feedback case or the case where
the dynamics of both systems are decoupled. In this paper, we present a
detailed solution to the general case. The structure of the optimal
decentralized controller is reminiscent of that of the optimal centralized
controller; each player must estimate the state of the system given their
available information and apply static control policies to these estimates to
compute the optimal controller. The previously solved cases benefit from a
separation between estimation and control which allows one to compute the
control and estimation gains separately. This feature is not present in
general, and some of the gains must be solved for simultaneously. We show that
computing the required coupled estimation and control gains amounts to solving
a small system of linear equations
Empirical Model Reduction of Controlled Nonlinear Systems
In this paper we introduce a new method of model reduction for nonlinear systems
with inputs and outputs. The method requires only standard matrix computations, and
when applied to linear systems results in the usual balanced truncation. For nonlinear
systems, the method makes used of the Karhunen-Lo`eve decomposition of the state-space,
and is an extension of the method of empirical eigenfunctions used in fluid dynamics. We
show that the new method is equivalent to balanced-truncation in the linear case, and
perform an example reduction for a nonlinear mechanical system
Positive Forms and Stability of Linear Time-Delay Systems
We consider the problem of constructing Lyapunov functions for linear
differential equations with delays. For such systems it is known that
exponential stability implies the existence of a positive Lyapunov function
which is quadratic on the space of continuous functions. We give an explicit
parametrization of a sequence of finite-dimensional subsets of the cone of
positive Lyapunov functions using positive semidefinite matrices. This allows
stability analysis of linear time-delay systems to be formulated as a
semidefinite program.Comment: journal version, 14 page
Warm-started wavefront reconstruction for adaptive optics
Future extreme adaptive optics (ExAO) systems have been suggested with up to 10^5 sensors and actuators. We analyze the computational speed of iterative reconstruction algorithms for such large systems. We compare a total of 15 different scalable methods, including multigrid, preconditioned conjugate-gradient, and several new variants of these. Simulations on a 128×128 square sensor/actuator geometry using Taylor frozen-flow dynamics are carried out using both open-loop and closed-loop measurements, and algorithms are compared on a basis of the mean squared error and floating-point multiplications required. We also investigate the use of warm starting, where the most recent estimate is used to initialize the iterative scheme. In open-loop estimation or pseudo-open-loop control, warm starting provides a significant computational speedup; almost every algorithm tested converges in one iteration. In a standard closed-loop implementation, using a single iteration per time step, most algorithms give the minimum error even in cold start, and every algorithm gives the minimum error if warm started. The best algorithm is therefore the one with the smallest computational cost per iteration, not necessarily the one with the best quasi-static performance
Optimal Dorfman Group Testing For Symmetric Distributions
We study Dorfman's classical group testing protocol in a novel setting where
individual specimen statuses are modeled as exchangeable random variables. We
are motivated by infectious disease screening. In that case, specimens which
arrive together for testing often originate from the same community and so
their statuses may exhibit positive correlation. Dorfman's protocol screens a
population of n specimens for a binary trait by partitioning it into
nonoverlapping groups, testing these, and only individually retesting the
specimens of each positive group. The partition is chosen to minimize the
expected number of tests under a probabilistic model of specimen statuses. We
relax the typical assumption that these are independent and indentically
distributed and instead model them as exchangeable random variables. In this
case, their joint distribution is symmetric in the sense that it is invariant
under permutations. We give a characterization of such distributions in terms
of a function q where q(h) is the marginal probability that any group of size h
tests negative. We use this interpretable representation to show that the set
partitioning problem arising in Dorfman's protocol can be reduced to an integer
partitioning problem and efficiently solved. We apply these tools to an
empirical dataset from the COVID-19 pandemic. The methodology helps explain the
unexpectedly high empirical efficiency reported by the original investigators.Comment: 20 pages w/o references, 2 figure
A scheme for robust distributed sensor fusion based on average consensus
We consider a network of distributed sensors, where each sensor takes a linear measurement of some unknown parameters, corrupted by independent Gaussian noises. We propose a simple distributed iterative scheme, based on distributed average consensus in the network, to compute the maximum-likelihood estimate of the parameters. This scheme doesn't involve explicit point-to-point message passing or routing; instead, it diffuses information across the network by updating each node's data with a weighted average of its neighbors' data (they maintain the same data structure). At each step, every node can compute a local weighted least-squares estimate, which converges to the global maximum-likelihood solution. This scheme is robust to unreliable communication links. We show that it works in a network with dynamically changing topology, provided that the infinitely occurring communication graphs are jointly connected
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