Hopf Bifurcations of Moore-Greitzer PDE Model with Additive Noise

The Moore-Greitzer partial differential equation (PDE) is a commonly used mathematical model for capturing flow and pressure changes in axial-flow jet engine compressors. Determined by compressor geometry, the deterministic model is characterized by three types of Hopf bifurcations as the throttle coefficient decreases, namely surge (mean flow oscillations), stall (inlet flow disturbances) or a combination of both. Instabilities place fundamental limits on jet-engine operating range and thus limit the design space. In contrast to the deterministic PDEs, the Hopf bifurcation in stochastic PDEs is not well understood. The goal of this particular work is to rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic stall bifurcation points in the presence of additive noise acting on the fast modes. We also show that the reduced-dimensional approximations (SDEs) contain multiplicative noise. Instability margins in the presence of uncertainties can be thus approximated, which will eventually lead to lighter and more efficient jet engine design

Model Analysis and Nonlinear Control of Air Compressors

RÉSUMÉ Pendant des décennies, les turbines à gaz ont été des dispositifs importants et fiables dans les domaines de la production d'énergie, de l'industrie pétrochimique, et de l'aéronautique. Ces machines utilisent les compresseurs centrifuges et axiaux qui se dégradent en présence d’instabilités aérodynamiques telles que le pompage et le décrochage tournant. Ces dernières limitent la performance et peuvent causer des sollicitations mécaniques importantes, une réduction de la durée de vie, du bruit et des vibrations. De plus, dans les compresseurs axiaux à vitesse variable (CAVV), les variations de vitesse affectent la stabilité des systèmes et peuvent entraîner le pompage et le décrochage tournant. Cela limite le taux de variation de vitesse et pénalise la performance. Le travail présenté dans cette thèse dresse premièrement l'analyse de bifurcation du modèle des CAVVs afin d’étudier l'impact de la dynamique de la vitesse sur la stabilité de points de fonctionnement efficaces. Ici, le taux de variation de vitesse (accélération) est défini comme un nouveau paramètre du modèle et une analyse détaillée de bifurcation numérique est fournie. Les résultats des simulations dans le domaine temporel valident non seulement l'analyse de bifurcation, mais élargissent aussi nos connaissances sur la réponse transitoire du modèle, qui est d’une importance majeure. L'analyse réalisée révèle que les variations de vitesse peuvent mener à un décrochage tournant entièrement développé ainsi qu’au décrochage temporaire mentionné précédemment. Les résultats montrent que les instabilités développées dépendent fortement du taux d'accélération. L'impact des autres paramètres du modèle, les vitesses initiale et finale, et la contribution des modes du décrochage sont également étudiés. Au niveau du contrôle, malgré toutes les réalisations présentées, la conception d’une commande robuste même pour des systèmes de compression axiaux à vitesse constante demeure encore un problème difficile. Ici, deux méthodes de commande non linéaires: le contrôle par modes glissants et le contrôle par passivité sont proposées pour résoudre ce problème de stabilité. Ces deux approches traitent de tous les aspects difficiles du sujet qui apparaissent dans la littérature : l'impact des perturbations externes, le manque de connaissance précise des paramètres du modèle, et l'absence d’un retour d’état complet.---------- ABSTRACT For decades, gas turbines have been important, widespread, and reliable devices in the field of power generation, petrochemical industry, and aeronautics. They employ centrifugal and axial compressors which suffer from aerodynamic instabilities, namely, surge and rotating stall. These performance limiting instabilities can cause component stress, lifespan reduction, noise, and vibration. Furthermore, in variable speed axial compressors (VSACs), speed variations affect the system stability and can lead to surge and rotating stall. This limits the rate of speed variations and results in important performance penalties. The present work firstly addresses the bifurcation analysis of VSACs’ model to investigate the impact of speed dynamics on the stability of efficient operating points. Here, the rate of speed variations (acceleration rate) is defined as a new parameter of the model and a detailed numerical bifurcation analysis is provided. The results of time-domain simulations not only validate the results of bifurcation analysis, but also broaden our knowledge about the transient response of the model, which is a matter of importance as well. The analysis reveals that speed variations can lead to a fully developed rotating stall as well as the previously reported temporary stall developments. The results show that the developed instabilities depend to a great extent on the acceleration rate. The impact of other key issues such as throttle gain, viscosity factor, initial speed, final speed, and the contribution of stall modes are also explored. From the control point of view, despite reported achievements, robust control design for compression systems remains a challenging problem. In this work, at first, two nonlinear approaches are proposed to tackle the stability problem of constant-speed axial compressors (CSACs). The first approach is a robust passivity-based control and the second one is a second order sliding mode control. The approaches tackle the challenging problems being addressed in the literature such as: the impact of external perturbations, the lack of detailed parameters knowledge, and the absence of full-state feedback. They drive the control from pressure and mass flow measurements and use throttle and close-coupled valve actuations. Finally, this study reports that these methods can be used in the case of VSACs by applying the required modifications to simultaneously control speed and instabilities. This simultaneous control design has been an open problem and the proposed method can improve the performance of VSACs

Damage detection in nonlinear systems using system augmentation and generalized minimum rank perturbation theory

A damage detection method is developed for nonlinear systems using model updating. The method uses a nonlinear discrete model of the system and the form of the nonlinearities to create an augmented linear model of the system. A modal analysis technique that uses forcing that is known but not prescribed is then used to solve for the modal properties of the augmented linear system after the onset of damage. Due to the specialized form of the augmentation, nonlinear damage causes asymmetrical damage in the updated matrices. Minimum rank perturbation theory is generalized so that it may be applied to the augmented system and can handle these asymmetrical damage scenarios. The method is demonstrated using numerical data from several nonlinear mass–spring systems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49014/2/sms5_5_037.pd

Dynamical systems : control and stability

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

Systems and control : 21th Benelux meeting, 2002, March 19-21, Veldhoven, The Netherlands

Book of abstract

Bibliography of Lewis Research Center technical publications announced in 1987

This compilation of abstracts describes and indexes the technical reporting that resulted from the scientific and engineering work performed and managed by the Lewis Research Center in 1987. All the publications were announced in the 1987 issues of STAR (Scientific and Technical Aerospace Reports) and/or IAA (International Aerospace Abstracts). Included are research reports, journal articles, conference presentations, patents and patent applications, and theses

System Augmentation and Optimal Feedback Auxiliary Signals for Interrogating Nonlinear Systems.

System augmentation is introduced and developed as a tool for modeling nonlinear systems in the context of a variety of applications including damage detection, system identification and sensing. The idea of creating augmented models enables the construction of higher dimensional models which are characterized by a specialized/augmented forcing. This specialized forcing enforces that an augmented model exactly follows a single trajectory of the nonlinear system when projected onto the subspace of the physical structure. Within the context of damage detection, the augmentation can be combined with a generalized minimum rank perturbation theory that is specifically developed to handle augmented systems. This model-based detection method uses the fact that often damage occurs in localized regions. That results in localized changes in the corresponding (model) matrices for the structure, which leads to perturbation matrices of minimum rank. Ideally one would have sensor information at all the nodes of a finite element model used for structural health monitoring. Practically, however, due to cost, weight and accessibility issues only a limited number of locations can be instrumented. Hence, an important, current challenge for damage detection is identifying multiple damages in complex structures using few sensors. This work develops a new integrated sensor placement and reduced order health assessment approach that can be applied to nonlinear structures. This method exploits the fact that the damageable regions (hot spots) of the system are often known in advance. The central advancement is an approach to expand the partial eigenvector information obtained from few sensors into the full space (of a detailed structural model) using the knowledge that damage is limited to the hot spots. Furthermore, most current approaches used for structural health monitoring are passive, while others are active in applying an auxiliary signal (excitation) to the structure. These current methods use predefined excitation signals (e.g. pulsed-waves, frequencysweeps). Such signals are designed offline and do not adapt to the response of the structure during its interrogation. In contrast, this work develops optimal feedback auxiliary signals. The feedback nature of these signals is a key enabling technique for enhancing sensitivity/selectivity, and leads to a new structural interrogation paradigm.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64813/1/kdsouza_1.pd