Toward Efficient Calculation of Inverses in Control Allocation for Safety-Critical Applications

Many control allocation algorithms require the calculation of (pseudo)inverses of control effectiveness matrices, also referred to as a B matrix, which for nonlinear systems might change over time. Such cases would require an online calculation of the respective inverses. Storage of all possible, offline precalculated inverses might exceed available memory sizes in common aircraft applications. This is especially relevant for systems with a high number of control effectors, like novel aircraft configurations. Several control allocation algorithms exist that require updates of the matrix to be inverted, the considered example being Redistributed Scaled Pseudoinverse. Within the Redistributed Scaled Pseudoinverse algorithm, the control allocation problem is solved iteratively by sequentially removing the columns of the B matrix that belong to saturated effectors. An approach using the Sherman–Morrison formula is presented in this study, which calculates the inverses based on recursive updates. This proposed approach has the following advantages over conventional Redistributed Scaled Pseudoinverse algorithm: reduced computational load and ease of protection against run-time errors. These make it a candidate for use in the context of safety-critical applications. The approach gives promising results and shows significant decrease of computational time. However, specific numerical challenges require additional investigations.