Fractional Variational Integration and Optimal Control

Lead Research Organisation: University of Oxford
Department Name: Engineering Science

Abstract

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.

Planned Impact

Optimisation and optimal control problems arise in many areas of engineering that are of national and international importance. These include the process industries, aerospace and the new emerging hybrid powertrains associated with the automotive industries. This project aims to develop a structure-preserving modelling methodology for dissipative electro-mechanical systems and combine it with a numerical optimal control problem solver. Since the system structure is taken into account, the developed methods and algorithms yield accurate solutions that will preserve many of the classical system symmetries associated with Noether's theorem. The developed methods, and the resulting software package, will be beneficial to a number of industries including, for example, the automotive, railway and aerospace sectors. The project's contribution will have a broad range applicability and impact.
 
Description One of the key limitations of Lagrangian and Hamiltonian mechanics is their inability to treat dissipation using variational means. While this grant is still on going, we have made substantial contributions to the use of fractional calculus in the treatment of dissipation within stationary action type frameworks. An extension of the approach to optimal control problems including dissipation has been developed. These works have been been presented at conferences and published in the conference proceedings. A journal publication including more results on the resulting simulation methods based on the fractional derivative approach is currently under review. The developed theory of fractional calculus has been applied for the representation of advection diffusion equations (submitted as a conference paper). Also the role of symmetries for such fractional systems has been investigated by proving a fractional Noether theorem which ensures the existence of preserved quantities in the system which will be also exactly preserved by the developed numerical scheme (journal article in preparation).
Furthermore, the team submitted two journal publications on the use of inerter-based transmission lines as a means of mimicking linear (first publication which is accepted in IEEE TAC and will appear in April 2020) and non-linear (second publication und review) dissipation. We regard this as an interesting and technically meaningful advance in the field. Furthermore, the team submitted a journal publication on a more general approach of embedding a dissipative system into a larger conservative systems with special emphasis on the Hamiltonian side as well which is important for the coherence of the approach. In addition, the team submitted two journal publication on optimal control applications in astrodynamics (the restricted three body problem with drag and Hill's problem with drag). Currently, the team is working on higher order integration methods to allow for more accurate simulation methods of the underlying systems which is of high importance in (particularly) astrodynamical applications. One conference paper has been published on the numerical order calculation of the fractional variational integrators, another journal publication on higher order approaches is in preparation.
Exploitation Route These results are expected to impact the development of optimal control solvers for mechanical systems, and to be of interest to numerical analysts interested in numerical integration algorithms.
Sectors Aerospace, Defence and Marine,Education,Transport