On the Guaranteed Nonsingularity of a Class of Banded Matrices for Optimal Control Generation

The contribution of the present paper is to further extend the generality of the method presented in [Driessen et al., [2]], for model-based minimum-time trajectory generation for dynamic systems with equality and inequality path constraints. In the above reference, the guaranteed nonsingularity of the sub-problem’s banded-matrix (footnote 2 therein) was stated for the case where no constraint involved both inputs and states. In the last sentence here we are referring to the modified-Gauss-Newton (MGN) matrix from footnote 2 of [Driessen et al., [2]]. We are simultaneously referring to the matrix in the left hand side of equation (⁸) of the present paper; the two differ only in a re-ordering of the rows and columns. Herein, this matrix will be referred to as the MGN matrix irrespective of row/column-ordering which has no bearing on the singularity/nonsingularity of a matrix. While it is often the case that no constraint contains both inputs and states, it is not the case for some problems. The contribution of the present paper is the proof that the above-mentioned banded MGN matrix is guaranteed nonsingular for absolutely any constraints, with or without mixed terms, and is furthermore still assured to provide the descent property.