Nonlinear, Concave, Constrained Optimization in Six-Dimensional Space for Hybrid-Electric Powertrains 2023-01-0550

One of the building blocks of the Stellantis hybrid powertrain embedded control software computes the maximum and minimum values of objective functions, such as output torque, as a function of engine torque, hybrid motor torque and other variables. To test such embedded software, an offline reference function was created. The reference function calculates the ideal minimum and maximum values to be compared with the output of the embedded software. This article presents the offline reference function with an emphasis on mathematical novelties. The reference function computes the minimum and maximum points of a linear objective function as a function of six independent variables, subject to 42 linear and two nonlinear constraints. Concave domains, curved surfaces, disjoint domains and multiple local extremum points challenge the algorithm. As a theorem, the conditions and methods for running trigonometric calculations in 6D Euclidean space are presented. More complex geometric structures can be built by joining multiple triangles, adding one triangle at a time. Extensions to 4D, 5D, 7D, etc. are shown to be straightforward. Based on the rules of 6D trigonometry, a probing straight line is constructed. The overlap zone of the constraints is determined along a 6D probing straight line. The optimum point is found within the overlap zone. The algorithm iteratively changes the location and orientation of the straight line to find the optimum point. A formula for rotating any straight line by a given angle in 6D space is derived. The iteration with the probing straight line runs with 2.1 million starting points. In any iteration algorithm, the number of starting points must be determined. This is usually done based on intuition and experience. In this article, the minimum number of required iteration starting points is determined as a function of the required probability of finding the global optimum.