Linear Model Predictive Control (MPC) is a widely used method to control systems with linear dynamics. Efficient interior-point methods have been proposed which leverage the block diagonal structure of the quadratic program (QP) resulting from the receding horizon control formulation. However, they require two matrix factorizations per interior-point method iteration, one each for the computation of the dual and the primal. Recently though an interior point method based on the null-space method has been proposed which requires only a single decomposition per iteration. While the then used null-space basis leads to dense null-space projections, in this work we propose a sparse null-space basis which preserves the block diagonal structure of the MPC matrices. Since it is based on the inverse of the transfer matrix we introduce the notion of so-called virtual controls which enables just that invertibility. A combination of the reduced number of factorizations and omission of the evaluation of the dual lets our solver outperform others in terms of computational speed by an increasing margin dependent on the number of state and control variables.

中文翻译：

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$\mathcal{N}$IPM-MPC：一种用于模型预测控制的基于有效零空间方法的内点方法

线性模型预测控制 (MPC) 是一种广泛使用的方法来控制具有线性动力学的系统。已经提出了有效的内点方法，其利用由后退水平控制公式产生的二次规划 (QP) 的块对角线结构。然而，它们需要每次内点方法迭代进行两次矩阵分解，每一次用于计算对偶和原始。最近虽然提出了一种基于零空间方法的内点方法，它每次迭代只需要一次分解。虽然当时使用的零空间基导致密集的零空间投影，但在这项工作中，我们提出了一个稀疏的零空间基，它保留了 MPC 矩阵的块对角线结构。由于它基于传递矩阵的逆，我们引入了所谓的虚拟控件的概念，它实现了这种可逆性。分解次数的减少和对偶评估的省略相结合，使我们的求解器在计算速度方面优于其他求解器，增加的余量取决于状态和控制变量的数量。