Application of the method of Controlled Lagrangians: from mechanical to electrical systems

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This thesis discusses the method of Controlled Lagrangians, a controller design method for nonlinear mechanical systems. This method, developed in the late 1990s, is based on the fact that the equations of motions of a mechanical system can be formulated using the Euler-Lagrange equations. The advantage of this method is that it provides, by design, an energy function for the closed loop equations, which can be used for stability analysis. The main vision inspiring this thesis is to transfer the method of Controlled Lagrangians to electrical systems and to develop a suitable structured procedure. The first part of this thesis presents the application of the method of Controlled Lagrangians to a mechanical crane system, whereby the controller design is described in detail. Especially, formulation of all design steps in matrix-vector form and the illustration with explicit calculations allows a broader audience to apply the method. A method for parameter tuning to optimize the performance of the closed loop system is presented, which uses the linearization of the closed loop equations. Since the structure of the closed loop equations is known, the linearization can be derived in a certain form, which makes it possible to specify the system matrix directly. The second part of this thesis analyzes the question how to apply the method of Controlled Lagrangians to electrical systems. Two theoretical frameworks are used, the Lagrange d'Alembert principle with constraints, and the Dirac structures, to develop a new form of the equations of motion for a certain class of electrical systems based on the Euler-Lagrange equations. An algorithm for the derivation of the equations of motion in this form is presented. The resulting new model equations are then used as a basis for further discussion of the controller design. The following shows that for the design of controllers for electrical systems, certain controller inputs allow a similar procedure as for the controller design for mechanical systems, and that new methods are needed for other controller inputs. Finally, the new modeling procedure developed in this thesis is contrasted with modeling approaches existing in the field of network theory.