525 research outputs found
A Characterization of all Solutions to the Four Block General Distance Problem
All solutions to the four block general distance problem which arises in H^∞ optimal control are characterized. The procedure is to embed the original problem in an all-pass matrix which is constructed. It is then shown that part of this all-pass matrix acts as a generator of all solutions. Special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all-pass
transfer functions. As an application, necessary and sufficient conditions are found for the existence of an H^∞ optimal controller. Following that, a descriptor representation of all solutions is derived
Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks
Synchronization is an important and prevalent phenomenon in natural and
engineered systems. In many dynamical networks, the coupling is balanced or
adjusted in order to admit global synchronization, a condition called Laplacian
coupling. Many networks exhibit incomplete synchronization, where two or more
clusters of synchronization persist, and computational group theory has
recently proved to be valuable in discovering these cluster states based upon
the topology of the network. In the important case of Laplacian coupling,
additional synchronization patterns can exist that would not be predicted from
the group theory analysis alone. The understanding of how and when clusters
form, merge, and persist is essential for understanding collective dynamics,
synchronization, and failure mechanisms of complex networks such as electric
power grids, distributed control networks, and autonomous swarming vehicles. We
describe here a method to find and analyze all of the possible cluster
synchronization patterns in a Laplacian-coupled network, by applying methods of
computational group theory to dynamically-equivalent networks. We present a
general technique to evaluate the stability of each of the dynamically valid
cluster synchronization patterns. Our results are validated in an electro-optic
experiment on a 5 node network that confirms the synchronization patterns
predicted by the theory.Comment: 6 figure
Local Solutions of the Dynamic Programming Equations and the Hamilton Jacobi Bellman PDE
We present methods for locally solving the Dynamic Programming Equations
(DPE) and the Hamilton Jacobi Bellman (HJB) PDE that arise in the infinite
horizon optimal control problem. The method for solving the DPE is the discrete
time version of Al'brecht's procedure for locally approximating the solution of
the HJB. We also prove the existence of the smooth solutions to the DPE that
has the same Taylor series expansions as the formal solutions. Our procedure
for solving the HJB PDE numerically begins with Al'brecht's local solution as
the initial approximation and uses some Lyapunov criteria to piece together
polynomial estimates. The polynomials are generated using the method in the
Cauchy-Kovalevskaya Theorem.Comment: 115 pages, 9 figures, PhD dissertatio
Estimation for boundary-value descriptor systems
Cover title.Includes bibliographical references.Supported in part by the Air Force Office of Scientific Research. AFOSR-88-0032 Supported in part by the National Science Foundation. ECS-8700903 Supported in part by Institut National de Recherche en Informatique et en Automatique (INRIA), France.Ramine Nikoukhah ... [et al.]
Efficient positive-real balanced truncation of symmetric systems via cross-riccati equations
We present a highly efficient approach for realizing a positive-real balanced truncation (PRBT) of symmetric systems. The solution of a pair of dual algebraic Riccati equations in conventional PRBT, whose cost constrains practical large-scale deployment, is reduced to the solution of one cross-Riccati equation (XRE). The cross-Riccatian nature of the solution then allows a simple construction of PRBT projection matrices, using a Schur decomposition, without actual balancing. An invariant subspace method and a modified quadratic alternating-direction-implicit iteration scheme are proposed to efficiently solve the XRE. A low-rank variant of the latter is shown to offer a remarkably fast PRBT speed over the conventional implementations. The XRE-based framework can be applied to a large class of linear passive networks, and its effectiveness is demonstrated through numerical examples. © 2008 IEEE.published_or_final_versio
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