Output-Feedback Control of Nonlinear Systems using Control Contraction Metrics and Convex Optimization

Control contraction metrics (CCMs) are a new approach to nonlinear control design based on contraction theory. The resulting design problems are expressed as pointwise linear matrix inequalities and are and well-suited to solution via convex optimization. In this paper, we extend the theory on CCMs by showing that a pair of "dual" observer and controller problems can be solved using pointwise linear matrix inequalities, and that when a solution exists a separation principle holds. That is, a stabilizing output-feedback controller can be found. The procedure is demonstrated using a benchmark problem of nonlinear control: the Moore-Greitzer jet engine compressor model.Comment: Conference submissio

Analytic Parameterization of Stabilizing Controllers for the Moore-Greitzer Compressor Model

This work presents an extension, simplification and application of a design procedure for dynamic output feedback design for systems with nonlinearities satisfying quadratic constraints (QC). Our method was motivated by the challenges of output feedback control design for the three-state Moore-Greitzer (MG) compressor model. The classical three-state MG model is a nonlinear dynamical system that is widely used in stall/surge analysis and control design. First, we find the parameter set of the stabilizing dynamic output feedback controllers for the surge subsystem by using conditions for stability of a transformed system and the associated matching conditions. Second, we choose the optimal control parameters from the stabilizing set with respect to different desired criteria. We show the set of parameters of the stabilizing controllers for the surge subsystem and the set of parameters of the stabilizing controllers with extended integral part for MG compressor. We present simplified sufficient conditions for stabilization, new constraints for the corresponding parameters and examples of optimal problem for the surge subsystem of the Moore-Greitzer compressor model. We discuss the degree of robustness and clarify an alternative proof of stability of the closed-loop system with the surge subsystem and the stabilizing dynamic output feedback controller without an integral state. In addition, we show the derivation of a quadratic function by using CVX

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

Global Stabilization for a Class of Coupled Nonlinear Systems with Application to Active Surge Control

We propose here a new procedure for output feedback design for systems with nonlinearities satisfying quadratic constraints. It provides an alternative for the classical observer-based design and relies on transformation of the closed-loop system with a dynamic controller of particular structure into a special block form. We present two sets of sufficient conditions for stability of the transformed block system and derive matching conditions allowing such a representation for a particular challenging example. The two new tests for global stability proposed for a class of nonlinear systems extend the famous Circle criterion applied for infinite sector quadratic constraints. The study is motivated and illustrated by the problem of output feedback control design for the well-known finite dimensional nonlinear model qualitatively describing surge instabilities in compressors. Assuming that the only available measurement is the pressure rise, we suggest a constructive procedure for synthesis of a family of robustly globally stabilizing feedback controllers. The solution relies on structural properties of the nonlinearity of the model describing a compressor characteristic, which includes earlier known static quadratic constraints and a newly found integral quadratic constraint. Performance of the closed-loop system is discussed and illustrated by simulations

Output-feedback semiglobal stabilization of stall dynamics for preventing hysteresis and surge in axial-flow compressors

This paper deals with the use of feedback control to prevent hysteresis and surge in axial-flow compressors. We present a dynamic output-feedback controller that semiglobally stabilizes every rotating stall equilibrium, and a range of axisymmetric equilibria of the Moore-Greitzer model for axial-flow compressors. The dynamic controller combines a two-state-variable-feedback backstepping controller from the literature with a nonlinear, reduced order, high-gain observer that estimates the mass flow through the compressor from measurements of the pressure rise across it. Given an equilibrium and a compact inner bound on the domain of attraction, we use Lyapunov techniques to compute an explicit lower bound on the observer gain such that the specified equilibrium is asymptotically stable for the closed-loop system, with a domain of attraction that contains the specified inner bound. We use a numerical example to illustrate how the inner bound on the domain of attraction can be specified so that the closed-loop compressor does not exhibit hysteresis and surge oscillations even in response to changes in the throttle setting that are dictated by large and sudden changes in the desired operating point. Simulation results are used to demonstrate the absence of hysteresis and surge in the closed-loop compressor dynamics