Fractional Variational Integration and Optimal Control

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Large-scale optimal control problems can be solved numerically by approximating the original (continuous-time) problem with a nonlinear programming problem (NLP). This approximation (or transcription) process is more or less accurate, depending on the integration algorithm used to discretise the system dynamics and cost, and the associated mesh density (integration step size) employed. While this procedure is well established in general terms, there are a number of issues that limit the scope and utility of this approach. These include the size, sparsity and conditioning of the resulting NLP, the accuracy of the integration algorithm used to discretise the problem. We will focus particularly on optimal control problems relating to electro-mechanical systems, and will seek to exploit the special structure of these problems in order to improve their solubility. The standard approach to the solution of these problems is to use separate modelling and optimal control solution phases, with the optimal control problem then solved with general-purpose software. In the case of (electro-)mechanical systems this approach has a number of drawbacks that we seek to remedy: (i) mechanical modelling and optimal control are based on closely related variational principles, but this common heritage is not exploited by the standard solution process; (ii) general-purpose numerical integration algorithms can destroy the geometric structure of (electro-)mechanical system models. In the context of conservative systems inappropriate integration schemes may produce numerical dissipation and destroy other conserved quantities such the system's momentum. We propose to ensure that this does not occur through the use of variational, or symplectic integration, schemes. (iii) Since the special structure of mechanical models is not exploited in the standard approach, the number of decision variables is needlessly doubled. This follows from the appearance of the generalised velocities and the generalised positions in the system's state vector. In discrete mechanical models (models based on a discrete Euler-Lagrange equation) the generalised velocities are expressible in terms of time differences in the generalised positions and thus can be eliminated from the problem. For electro-mechanical systems our proposal is to combine the modelling and optimisation phases into a single whole using symmetry-preserving integration schemes.The derivation of the equations of motion using classical variational methods, such as stationary action, is limited to lossless systems. The main idea in this proposal is to extend existing variational modelling and integration approaches to electro-mechanical systems with dissipation. Dissipation includes such things as resistive losses, damping, aerodynamic losses, hysteresis losses and friction. To ensure a purely variational formulation we will make use of fractional calculus to model the dissipative terms in the system. The concept of modelling dissipation such as linear friction, or resistance by means of fractional derivatives was introduced in the late 1990s. However, its use to develop structure-preserving variational integration and optimal control methods for dissipative electro-mechanical systems is a completely new field of research and the main objective in this proposal. The developed methods and algorithms will be used to solve 'industrial strength' application problems from the automotive sector. Electro-mechanical systems appear in many areas of the automotive industry with optimal control problems arise in areas such as hybrid powertrain control. Once workable theory and software has been developed, they will be evaluated in two automotive projects. The first will be conducted with a Formula One team, while the second will be conducted with a leading engine and powertrain manufacturer.