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Concept# Nonlinear control

Résumé

Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the "plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output.
Control theory is divided into two branches. Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called linear time invariant (LTI) systems. Th

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Théorie du contrôle

En mathématiques et en sciences de l'ingénieur, la théorie du contrôle a comme objet l'étude du comportement de systèmes dynamiques paramétrés en fonction des trajectoires de leurs paramètres.

Automatique

L’automatique est une science qui traite de la modélisation, de l’analyse, de l’identification et de la commande des systèmes dynamiques. Elle inclut la cybernétique au sens étymologique du terme, e

Automation

L'automation consiste à utiliser les services d'un logiciel dans une application informatique.
L'automation peut donc être considérée comme une procédure d'automatisation.
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ME-523: Nonlinear Control Systems

Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Divers exemples illustrent la théorie (exercices papier crayon et simulations à l'ordinateur).

EE-715: Optimal control

This doctoral course provides an introduction to optimal control covering fundamental theory, numerical implementation and problem formulation for applications.

ME-524: Advanced control systems

This course covers some theoretical and practical aspects of robust and adaptive control. This includes H-2 and H-infinity control in model-based and data-driven framework by convex optimization, direct, indirect and switching adaptive control. The methods are implemented in a hands-on lab.

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This thesis is about modelling, design and control of Miniature Flying Robots (MFR) with a focus on Vertical Take-Off and Landing (VTOL) systems and specifically, micro quadrotors. It introduces a mathematical model for simulation and control of such systems. It then describes a design methodology for a miniature rotorcraft. The methodology is subsequently applied to design an autonomous quadrotor named OS4. Based on the mathematical model, linear and nonlinear control techniques are used to design and simulate various controllers along this work. The dynamic model and the simulator evolved from a simple set of equations, valid only for hovering, to a complex mathematical model with more realistic aerodynamic coefficients and sensor and actuator models. Two platforms were developed during this thesis. The first one is a quadrotor-like test-bench with off-board data processing and power supply. It was used to safely and easily test control strategies. The second one, OS4, is a highly integrated quadrotor with on-board data processing and power supply. It has all the necessary sensors for autonomous operation. Five different controllers were developed. The first one, based on Lyapunov theory, was applied for attitude control. The second and the third controllers are based on PID and LQ techniques. These were compared for attitude control. The fourth and the fifth approaches use backstepping and sliding-mode concepts. They are applied to control attitude. Finally, backstepping is augmented with integral action and proposed as a single tool to design attitude, altitude and position controllers. This approach is validated through various flight experiments conducted on the OS4.

Many real-world systems are intrinsically nonlinear. This thesis proposes various algorithms for designing control laws for input-affine single-input nonlinear systems. These algorithms, which are based on the concept of quotients used in nonlinear control design, can break down a single-input system into cascade of smaller subsystems of reduced dimension. These subsystems are well defined for feedback-linearizable systems. However, approximations are required to handle non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages. During the forward stage, an equivalence relationship is defined to isolate the states that are not directly affected by the input, which reduces the dimension of the system. The resulting system is an input-affine single-input system controlled by a pseudo-input which represents a degree of freedom in the algorithm. The pseudo-input is a complementary state required to complete the diffeomorphism. This procedure is repeated (n − 1) times to give a one-dimensional system, where n is the dimension of the system. The backward stage begins with the one-dimensional system obtained at the end of the forward stage. It iteratively builds the control law required to stabilize the system. At every iteration, a desired profile of the pseudo-input is computed. In this next iteration, this desired profile is used to define an error that is driven asymptotically to zero using an appropriate control law. The quotient method is implemented through two algorithms, with and without diffeomorphism. The algorithm with diffeomorphism clearly depicts the dimension reduction at every iteration and provides a clear insight into the method. In this algorithm, a diffeomorphism is synthesized in order to obtain the normal form of the input vector field. The pseudo-input is the last coordinate of the new coordinate system. A normal projection is used to reduce the dimension of the system. For the algorithm to proceed without any approximation, it is essential that the last coordinate appears linearly in the projection of the transformed drift vector field. Necessary and sufficient conditions to achieve linearity in the last coordinate are given. Having the pseudo-input appearing linearly enables to represent the projected system as an input-affine system. Hence, the whole procedure can be repeated (n−1) times so as to obtain a one-dimensional system. In the second algorithm, a projection function based on the input vector field is defined that imitates both operators, the push forward operater and the normal projection operator of the previous algorithm. Due to the lack of an actual diffeomorphism, there is no apparent dimension reduction. Moreover, it is not directly possible to separate the drift vector field from the input vector field in the projected system. To overcome this obstacle, a bracket is defined that commutes with the projection function. This bracket provides the input vector field of the projected system. This enables the algorithm to proceed by repeating this procedure (n−1) times. As compared with the algorithm with diffeomorphism, the computational effort is reduced. The mathematical tools required to implement this algorithm are presented. A nice feature of these algorithms is the possibility to use the degrees of freedom to overcome singularities. This characteristic is demonstrated through a field-controlled DC motor. Furthermore, the algorithm also provides a way of approximating a non-feedback-linearizable system by a feedback-linearizable one. This has been demonstrated in the cases of the inverted pendulum and the acrobot. On the other hand, the algorithm without diffeomorphism has been demonstrated on the ball-on-a-wheel system. The quotient method can also be implemented whenever a simulation platform is available, that is when the differential equations for the system are not available in standard form. This is accomplished numerically by computing the required diffeomorphism based on the data available from the simulation platform. Two versions of the numerical algorithm are presented. One version leads to faster computations but uses approximation at various steps. The second version has better accuracy but requires considerably more computational time.

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