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9 records

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DOE OSTI2021

Generalized Toeplitz–Hankel matrices and their application to a layered electron gas

We extend the standard result for the eigenspectrum of the Toeplitz matrix C ij = e –κ|i–j| with 0 ≤ i,j ≤ N and κ ϵ C to a combination of a Toeplitz matrix and a Hankel matrix. We apply this result to find the plasma modes of a layered assembly of a 2-dimensional electron gas. We find a sum rule relating the geometric mean of the frequencies of the plasma modes to the determinant of this Toeplitz matrix, for which an analytical expression is obtained. Here, we apply the same technique to the generalized case when the layers are not evenly spaced, where the corresponding matrix is not a Toeplitz-Hankel combination. Despite this fact, it is possible to find properties of the eigenspectrum, and the eigenmodes are localized to a few layers instead of extending across the system.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

The geometry of the modular bootstrap

Abstract We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves onR + ⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above$$ {\Delta }_{\textrm{gap}}^{\ast } $$ ∆ gap ∗ , where$$ \frac{c-1}{12}<{\Delta}_{\textrm{gap}}^{\ast }<\frac{c}{12} $$ c − 1 12 < Δ gap ∗ < c 12 , all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

Physics↗

DOE OSTI2022

Online data-enabled predictive control

We develop an online data-enabled predictive (ODeePC) control method for optimal control of unknown systems, building on the recently proposed DeePC (Coulson et al., 2019). Our proposed ODeePC method leverages a primal-dual algorithm with real-time measurement feedback to iteratively compute the corresponding real-time optimal control policy as system conditions change. The proposed ODeePC conceptual-wise resembles standard adaptive system identification and model predictive control (MPC), but it provides a new alternative for the standard methods. ODeePC is enabled by computationally efficient methods that exploit the special structure of the Hankel matrices in the context of DeePC with Fast Fourier Transform (FFT) and primal-dual algorithm We provide theoretical guarantees regarding the asymptotic behavior of ODeePC, and we demonstrate its performance through numerical examples.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

The geometry of the partial realization problem

It is shown that the space of sequences of length n which have an extrapolation of McMillan degree k, and no extrapolations of lower McMillan degree can be given the structure of a differentiable manifold. This approach makes the proof of certain known results on the partial realization problem quite straightforward and makes it possible to establish some important new results as well. A key tool is the fact, proven here, that the set of n by a real Hankel matrices of rank r is a manifold with r+1 connected components.

Brockett, R. W.↗

NASA NTRS1983

Analysis of structural perturbations in systems via cost decomposition methods

It has long been common practice to analyze linear dynamic systems by decomposing the total response in terms of individual contributions which are easier to analyze. Examples of this philosophy include the expansion of transfer functions using: (1) the superposition principle, (2) residue theory and partial fraction expansions, (3) Markov parameters, Hankel matrices, and (4) regular and singular perturbations. This paper summarizes a new and different kind of expansion designed to decompose the norm of the response vector rather than the response vector itself. This is referred to as "cost-decomposition' of the system. The notable advantages of this type of decomposition are: (a) easy application to multi-input, multi-output systems, (b) natural compatibility with Linear Quadratic Gaussian Theory, (c) applicability to the analysis of more general types of structural perturbations involving inputs, outputs, states, parameters. Property (c) makes the method suitable for problems in model reduction, measurement/actuator selections, and sensitivity analysis.

Skelton, R. E.↗

NASA NTRS1988

System identification in the microcomputer environment

An implementation of an algorithm for system identification, the eigenvalue realization algorithm (ERA), on a microcomputer is reported. Test results are presented to show that accurate structural identification can be performed in a microcomputer environment. It is also shown that the memory limits (640 K) of small machines are generally adequate for solving large scale problems (e.g., Hankel matrices with 15,000 elements), with run times being on the order of several minutes.

Garcia, Ephrahim↗

DOE OSTI2019

A Themodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

We develop a thermodynamically consistent, fractional visco-elastoplastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco- elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix-vector operations for each time-step, allowing us to reduce the computational complexity from O(N 3 ) to O(N 2 ) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

36 MATERIALS SCIENCE↗

NASA NTRS1996

Balanced Placement Of Vibration Sensors And Actuators

Mathematical approach that involves Hankel singular values leads to method for estimation of optimal placement of relatively few vibration sensors and actuators in multivariable flexible structure. Approach extends and complements previous developments regarding approximate decomposition, to Hankel singular values, of singular values of controllability and observability grammian matrices of multivariable flexible structure. Mathematical constructs also described in "Designing An Approximately Balanced LQG Compensator" (NPO-19000).

Gawronski, Wodek↗

NASA NTRS1989

Model reduction for flexible structures - Test data approach

A reduced model of a system is obtained by truncating part of its state variables. Hankel singular values and component costs determine which component is deleted or retained in the reduced model. In this paper Hankel singular values and component costs of a flexible structure are obtained from the resonance test data, rather than from the system matrices. Test data, besides system dynamics, include also actuators and sensors dynamics. For this reason, the reduced model obtained from test data can be far from the optimal one. In this paper the reconstruction of the flexible structure indices from the joint actuator-sensor-flexible structure indices is discussed.

Gawronski, Wodek↗