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Commande et stabilité des systèmes commutés : Application Fluid Power

Abstract : This work focuses on the control and stability analysis of an electro-pneumatic system, i.e. a linear pneumatic cylinder controlled by two servo valves regulating the mass flow entering each chamber of the actuator. The general problem is motivated by the appearance of stick-slip on theelectro-pneumatic system, hardly taken into account by the current studies in automatic control. This problem, encountered throughout the years, concerns all mono- and multidimensional linear and non-linear controls systems studied at the laboratory. In pneumatic cylinders, the phenomenon consists in a displacement of the rod a while after it has come to a rest ; this is due to the fact that the force acting on the rod initially becomes smaller that the threshold which is necessary for a motion, and then this threshold is overcome later on. In this case, stick-slip is caused by the presence of dry friction and by the pressure dynamics in the chambers, which continue to evolve (integrating the net incoming mass flow from the servovalves) even after the rod has stopped. The first part of this thesis proposes a nonlinear switching control law in order to avoid stick-slip on pneumatic cylinder, taking into account with the variations of dry friction that may occur at any time causing this phenomenon. This technique is implemented and its effectiveness is recognized. The greatest part of this thesis deals with the stability analysis of the pneumatic cylinder with its switched control law. The presence of dry friction and the application of a switched control law requires an appropriate method for approaching the stability analysis ; this method is based on considering the closed-loop system as belonging to a class of switched systems called piecewise affine systems (PWA). The main difficulty in this approach lies in obtaining adequate Lyapunov functions for proving stability, which turns into an optimization problem under LMI constraints (Linear Matrix Inequality) using the S-procedure. In order to analyze the stability of a PWA system, a first method is proposed allowing the computation of a piecewise quadratic Lyapunov function through an optimization problem under LMI constraints. The methods takes into account, in contrast to conventional methods, that the states might converge not to a single point but to a set of equilibrium points. The proposed approach allows also the study of robustness with respect to parametric variations in the system. A second method is also proposed for the construction of a type of Lyapunov functions called piecewise polynomial, using the “sum of squares” and “power transformation” techniques. This approach proposes less conservative sufficient conditions than those imposed by the piecewise quadratic Lyapunov functions, yielding a more succesfull stability test when for PWA systems featuring sliding modes and parametric variations. In fact, on PWA systems with discontinuous dynamics (which can generate sliding phenomena), piecewise quadratic Lyapunov functions might prove ineffective to prove the stability. Therefore, the results on piecewise quadratic Lyapunov functions are extended in order to compute piecewise polynomial Lyapunov functions of higher order, by solving an optimization problem under LMI constraints. These functions are more general and allow less conservative conditions compared to those formerly developed in the literature. Both of these methods have been applied to the stability analysis of the set of equilibrium points of the pneumatic cylinder, considering first a friction model in saturation form and then a model in relay form with a discontinuous dynamics. The application of the methods is successful, i.e. the robust stability is proven under dry friction threshold variations, with possibility of sliding modes.
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  • HAL Id : tel-01988971, version 1

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Omar Ameur. Commande et stabilité des systèmes commutés : Application Fluid Power. Automatique / Robotique. Ecole Centrale Lyon, 2015. Français. ⟨tel-01988971⟩

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