The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method --- nonlinearity-subtracted DMD (NSDMD) --- that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.
Benjamín Herrmann, Katherine Cao, S. Brunton et al.· 0 citations
Future aircraft with increasingly flexible high aspect ratio wings are more vulnerable to gust and turbulence encounters. Active control technologies are therefore required to mitigate the effects of atmospheric disturbances and reduce structural sizing loads. However, the achievable load alleviation performance is constrained by system limitations such as time delays, parasitic dynamics, actuator limits, and sensor noise. In this context, model predictive control systems offer strong potential, as they can address these limitations. This paper presents the design and evaluation of such a model predictive gust load alleviation controller for a flexible test wing. The aeroelastic simulation model is based on a modal description of the structural dynamics and aerodynamic strip theory, with its parameters identified from ground vibration and wind tunnel tests. A Kalman filter is designed to estimate structural loads and non-measurable quantities including generalized structural coordinates and wind disturbances from highly noisy wind tunnel measurements. Preview information of upcoming gusts is provided to the controller, enabling feedforward control to compensate for time delays. The formulation can account for actuator limits and maximum allowable loads, ensuring effective operation within the system boundaries. To reduce the computational effort of the controller, Laguerre functions and an efficient soft output constraint formulation are employed. The resulting control system is evaluated in virtual wind tunnel tests based on the identified model with gust encounters of varying frequency. Further, the effects of degraded actuator limits and failure cases are investigated. Particular emphasis is placed on encounters with short and load-critical gusts, where the controller achieves good load alleviation performance despite restrictive system limitations.
Leif Rieck, Benjamín Herrmann, O. Luderer et al.· CEAS Aeronautical Journal· 0 citations