1,046 research outputs found
Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error
It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomenon, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the assimilation in the unstable subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with nonnegative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations, and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision, and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability, and filter boundedness. Additionally, we include a detailed introduction to the multiplicative ergodic theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors
Application of optimal control theory to the design of the NASA/JPL 70-meter antenna servos
The application of Linear Quadratic Gaussian (LQG) techniques to the design of the 70-m axis servos is described. Linear quadratic optimal control and Kalman filter theory are reviewed, and model development and verification are discussed. Families of optimal controller and Kalman filter gain vectors were generated by varying weight parameters. Performance specifications were used to select final gain vectors
Indirect methods for the numerical solution of ordinary linear boundary value problems
PhD thesisThis thesis is mainly concerned with indirect numerical solution
methods for linear two point boundary value problems. We
concentrate particularly on problems with separated boundary
conditions which have a 'dichotomy' property. We investigate
the inter-relationship of various methods including some which
have first appeared since the work for this thesis began. We
examine the stability of these methods and in particular we
consider circumstances in which the methods discussed give rise
to well conditioned decoupling transformations. Empirical
comparisons of some of the methods are described using a set
of test problems including a number of
ill conditioned problams.
'stiff' and marginally
In the past the main method of error estimation has been to
repaat the whole calculation. Here an alternative error
estimation technique is proposed and a related iterative
improvement method is considered. Although results for this are
not completely conclusive we think they justify the need for
further research on the method as it shows promise of being a
novel and reliable practical method of solving both well
conditioned and ill conditioned problems
Array algorithms for H-infinity estimation
In this paper we develop array algorithms for H-infinity filtering. These algorithms can be regarded as the Krein space generalizations of H-2 array algorithms, which are currently the preferred method for implementing H-2 biters, The array algorithms considered include typo main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H-infinity filters, as these conditions are built into the algorithms themselves. However, since H-infinity square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H-2 case, further investigation is needed to determine the numerical behavior of such algorithms
Aircraft adaptive learning control
The optimal control theory of stochastic linear systems is discussed in terms of the advantages of distributed-control systems, and the control of randomly-sampled systems. An optimal solution to longitudinal control is derived and applied to the F-8 DFBW aircraft. A randomly-sampled linear process model with additive process and noise is developed
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