Time Optimal Control of Four-in-Wheel-Motors Driven Electric Cars

Abstract

The paper deals with the time optimal control of automatically driven electric cars in a test path under state and input constraints. The problem can be formulated as a dynamic nonlinear optimal control problem (DNOCP). Keywords 4WD electric vehicle · Time optimal control · Direct multiple shooting · Nonlinear programming · Optimal force distribution · Model predictive control IntroductionPursuit of optimal behavior in technical and other (biological, economical etc.) type of systems is a common goal of human research. In this area dynamic and static optimization problems are of different complexity. Especially, the difficulty increases if some of the optimization variables are integer or binary valued as in the case of combustion engine driven cars where the gear shift in the control is an integer variable. Solution of such MIOCP (mixed-integer optimal control) problem was presented in an earlier work [1]. Similar problems were discussed in the literature for cars described by ordinary (ODE) or differential-algebraic (DAE) equations in [2-4] considering fixed or moving time interval or based on moving horizon predictive approaches [5,6].There are several techniques to solve dynamic nonlinear optimal control (DNOCP) problems: dynamic programming approach by solving the Hamilton-Jacobi-Bellmann equation, indirect methods (known as first optimize, then discretize) and direct methods (first discretize, then optimize). The latter group of techniques can be applied to large-scale optimal control problems. In direct methods the continuous time infinite dimensional DNOCP problem is first discretized and reformulated to a finite-dimensional static nonlinear program (NLP).A professional software package to solve the original continuous time problem may be MUSCOD-II [7], however it is not an open software. Hence, the novel professional and open optimization system OPTI [8] was chosen, which has also an interface to AMPL modeling language to formulate optimum problems at high level and has better performance than MAT-LAB Optimization Toolbox.A novel method will be presented to solve the time optimal control of electric cars first using a single track dynamic model, i.e. the time optimal control of two-in-wheel-motors driven (2WD) cars, in order to decrease the complexity of the optimization problem. In the discussion the contact forces between tire and road will be modeled using Pacejka's magic formulas which assure more accurate modeling than using cornering stiffnesses and linearization. The reason is that time optimality results in large acceleration/deceleration forces leaving the linear domain of the slip angle and force characteristics.

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