Optimal control of Formula One car energy recovery systems

Abstract

The utility of orthogonal collocation methods in the solution of optimal control problems relating to Formula One racing is demonstrated. These methods can be used to optimise driver controls such as the steering, braking and throttle usage, and to optimise vehicle parameters such as the aerodynamic down force and mass distributions. Of particular interest is the optimal usage of energy recovery systems (ERSs). Contemporary kinetic energy recovery systems are studied and compared with future hybrid kinetic and thermal/heat ERSs known as ERS-K and ERS-H, respectively. It is demonstrated that these systems, when properly controlled, can produce contemporary lap time using approximately two-thirds of the fuel required by earlier generation (2013 and prior) vehicles.

Keywords:

• numerical optimal control
• orthogonal collocation
• pseudo-spectral methods
• lap-time simulation
• Formula One car modelling
• energy recovery systems

Acknowledgements

This work was supported by the UK Engineering and Physical Sciences Research Council.

Notes

1. denotes the set of n-dimensional real vectors.

2. H( · ) = 0 when the argument is nonpositive, with H( · ) = 1, otherwise. The Heaviside step function is not differentiable at the origin and so is approximated by a smooth function such as in which ε is ‘small’.

3. Optimal control problems with costs and dynamics that are linear in the control usually involve switching strategies. In these problems, the control takes on limit values dictated by a switching function that is derivable from the minimum principle; this is the well-known bang-bang principle. In the case that the switching function is zero for a finite time, the problem has a singular arc and a bang-singular-bang strategy results (see Kirk, 1970, p. 246).

4. This is defined as a number between 0 and 1; with 0 representing a completely discharged and 1 a fully charged ES.