H2\mathcal{H}_2 Pseudo-Optimal Reduction of Structured DAEs by Rational Interpolation

In this contribution, we extend the concept of $\mathcal{H}_2$ inner product and $\mathcal{H}_2$ pseudo-optimality to dynamical systems modeled by differential-algebraic equations (DAEs). To this end, we derive projected Sylvester equations that characterize the $\mathcal{H}_2$ inner product in terms of the matrices of the DAE realization. Using this result, we extend the $\mathcal{H}_2$ pseudo-optimal rational Krylov algorithm for ordinary differential equations to the DAE case. This algorithm computes the globally optimal reduced-order model for a given subspace of $\mathcal{H}_2$ defined by poles and input residual directions. Necessary and sufficient conditions for $\mathcal{H}_2$ pseudo-optimality are derived using the new formulation of the $\mathcal{H}_2$ inner product in terms of tangential interpolation conditions. Based on these conditions, the cumulative reduction procedure combined with the adaptive rational Krylov algorithm, known as CUREd SPARK, is extended to DAEs. Important properties of this procedure are that it guarantees stability preservation and adaptively selects interpolation frequencies and reduced order. Numerical examples are used to illustrate the theoretical discussion. Even though the results apply in theory to general DAEs, special structures will be exploited for numerically efficient implementations

Structure-preserving tangential interpolation for model reduction of port-Hamiltonian Systems

Port-Hamiltonian systems result from port-based network modeling of physical systems and are an important example of passive state-space systems. In this paper, we develop the framework for model reduction of large-scale multi-input/multi-output port-Hamiltonian systems via tangential rational interpolation. The resulting reduced-order model not only is a rational tangential interpolant but also retains the port-Hamiltonian structure; hence is passive. This reduction methodology is described in both energy and co-energy system coordinates. We also introduce an $\mathcal{H}_2$-inspired algorithm for effectively choosing the interpolation points and tangential directions. The algorithm leads a reduced port-Hamiltonian model that satisfies a subset of $\mathcal{H}_2$-optimality conditions. We present several numerical examples that illustrate the effectiveness of the proposed method showing that it outperforms other existing techniques in both quality and numerical efficiency

Synthèse et validation d'une lois de contrôle de charge a deux degrés de liberté

International audienceThe design and assessment of a two degree of freedom gust load allevi-ation control system for a business jet aircraft is presented in this paper. The two degrees of freedom are a disturbance estimator to compute the incoming gusts as well as a feedback control law to mitigate the estimated disturbance to reduce the aircraft loads. To facilitate the estimator design, high order, infinite models of the structural and aerodynamic aircraft dynamics are approximated by low order models using advanced model reduction techniques. For the robust disturbance estimator design an innovative approach relying on nullspace based techniques together with non-linear optimizations is proposed. Time delays, originating from the aerodynamics modeling, the discrete control loop, and the sensor and actuator dynamics, play a key role in the stability and performance assessment of a gust load alleviation controller. Thus, a novel analytical analysis method is presented to explicitly evaluate the influence of these time delays on the closed loop. Finally, the developed tool-chain is applied to a fly-by-wire business jet aircraft. The resulting two degree of freedom gust load alleviation system is verified in a simulation campaign using a closed loop, non-linear simulator of the aircraft

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Discrete representations of random signals

"July 14, 1961." "Submitted to the Department of Electrical Engineering, M.I.T., August 22, 1960, in partial fulfillment of the requirements for the degree of Doctor of Science."Bibliography: p. 85-87.Army Signal Corps Contract DA 36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.Kenneth L. Jordan, Jr

A Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television

Cataloged from PDF version of article.Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory. © 2007 IEEE

H₂ model reduction for diffusively coupled second-order networks by convex-optimization

This paper provides an $H_2$ optimal scheme for reducing diffusively coupled second-order systems evolving over undirected networks. The aim is to find a reduced-order model that not only approximates the input-output mapping of the original system but also preserves crucial structures, such as the second-order form, asymptotically stability, and diffusive couplings. To this end, an $H_2$ optimal approach based on a convex relaxation is implemented to reduce the dimension, yielding a lower order asymptotically stable approximation of the original second-order network system. Then, a novel graph reconstruction approach is employed to convert the obtained model to a reduced system that is interpretable as an undirected diffusively coupled network. Finally, the effectiveness of the proposed method is illustrated via a large-scale networked mass-spring-damper system

Statistical Communication Theory

Contains forward and reports on nine research projects