Inexact Solves in Interpolatory Model Reduction

We investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem. We give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods. We show that when a Petrov-Galerkin framework is employed for the inexact solves, the associated reduced order model is an exact interpolatory model for a nearby full-order system; thus demonstrating backward stability. We also give evidence that for \h2-optimal interpolation points, interpolatory model reduction is robust with respect to perturbations due to inexact solves. Finally, we demonstrate the effecitveness of direct use of inexact solves in optimal ${\mathcal H}_2$ approximation. The result is an effective model reduction strategy that is applicable in realistically large-scale settings.Comment: 42 pages, 5 figure

Approximation optimale H2 par systèmes structurés à retards entrées/sorties

In this paper, the H 2 optimal approximation of a n y × n u transfer function G(s) by a finite dimensional systemĤ d (s) including input/output delays, is addressed. The underlying H 2 optimal-ity conditions of the approximation problem are firstly derived and established in the case of a poles/residues decomposition. These latter form an extension of the tangential interpolatory conditions , presented in [1, 2] for the delay-free case, which is the main contribution of this paper. Secondly, a two stage algorithm is proposed in order to practically obtain such an approximation

Interpolatory H2\mathcal{H}_2-optimality Conditions for Structured Linear Time-invariant Systems

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. We show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.Comment: 20 page

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