Impulsive Optimal Control of Hybrid Finite-Dimensional Lagrangian Systems

The scope of this dissertation addresses numerical and theoretical issues in the impulsive control of hybrid finite-dimensional Lagrangian systems. In order to treat these aspects, a modeling framework is presented based on the measure-differential inclusion representation of the Lagrangian dynamics. The main advantage of this representation is that it enables the incorporation of set-valued force laws and control laws on acceleration and velocity level concisely. This property of the measure-differential inclusion representation renders the description of the hybrid behaviour of Lagrangian systems in the framework of set-valued control and force laws possible. Based on the MDI representation of Lagrangian dynamics the impactive blocking is analysed as a set-valued impulsive unbounded control law. The application to mechanical systems with impulsively-blockable degrees of freedom is presented. The numerical application of this set-valued control law is the formulation of a nonlinear programming (NLP) algorithm for underactuated mechanical systems with impulsively-blockable DOF. The natural numerical treatment of the measure-differential inclusion representation is based on Moreau's sweeping process. By applying this discretisation scheme together with an augmented Lagrangian based NLP method that performs the minimisations with a modified conjugate gradients method an optimisation scheme is presented. A numerical example is applied to the impulsive optimal control of a manipulator with one impulsively-blockable degrees of freedom. A further numerical method is introduced for the class of switching Lagrangian systems. This numerical method is a shooting method that performs the numerical integrations based on the sweeping process and the minimisations by making use of the augmented Lagrangian concept. The augmented Lagrangian is minimised by an optimisation method that relies on function value comparisons. This relatively easily implemented numerical method is applied to a wheeled robot which is a tenth-order dynamical system. This system has four different operating modes and time and control effort (quasi-) optimal trajectories are presented. The theoretical results of the dissertation include the statement and the derivation of necessary conditions for the impulsive optimal control of finite-dimensional Lagrangian systems. In this analysis, Lagrangian systems are considered on which the impulses are induced solely by the impulsive control action. The challenge in the derivation of these necessary conditions has been the concurrent discontinuity of state and costate on a Lebesgue negligible time instant. In order to tackle this problem, the instances of impulsive control action are considered as an internal boundaries on the time domain. By the introduction of the concepts of internal boundary variations and discontinuous transversality conditions by the author this problem is resolved and necessary conditions for mechanical systems in the first-order and second-order representations are derived. The discontinuous transversality conditions that result from the consideration of the internal boundary variations in the time domain are discovered and analysed by the author of the dissertation and are applied to the impulsive optimal control of Lagrangian systems.


ETH Dissertation 17760 Preprint



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