This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilized to discretize continuous-time optimal control problems (OCPs) and formulate a nonlinear program (NLP), which guarantees the local convergence of a Newton-type method. A dedicated structure-exploiting algorithm (Riccati recursion) is proposed to perform a Newton-type method for the NLP efficiently because its sparsity structure is different from a standard OCP. The proposed method computes each Newton step with linear time-complexity for the total number of discretization grids as the standard Riccati recursion algorithm. Additionally, the computation is always successful if the solution is sufficiently close to a local minimum. Conversely, general quadratic programming (QP) solvers cannot accomplish this because the Hessian matrix is inherently indefinite. Moreover, a modification on the reduced Hessian matrix is proposed using the nature of the Riccati recursion algorithm as the dynamic programming for a QP subproblem to enhance the convergence. A numerical comparison is conducted with off-the-shelf NLP solvers, which demonstrates that the proposed method is up to two orders of magnitude faster. Whole-body optimal control of quadrupedal gaits is also demonstrated and shows that the proposed method can achieve the whole-body model predictive control (MPC) of robotic systems with rigid contacts.
翻译:本研究提出了一种高效的牛顿型方法,用于在特定模式序列下最佳控制交换系统。 一种基于网状精密的方法被用于将连续时间最佳控制问题( OCPs)分解, 并制定一个非线性程序( NLP), 保证牛顿型方法在当地趋同。 提议了一种专门的结构开发算法( Riccati 递归), 以高效的方式为NLP 执行牛顿型方法, 因为它的宽度结构结构不同于标准的 OCP 。 拟议的方法用直线时间兼容度来计算每个牛顿级步骤, 作为标准的 Riccati 递归回算法, 将离散电电网的总数计算为直线性时间兼容。 此外, 如果解决方案足够接近本地最小值方法, 计算总是成功的。 相反, 一般的四方形编程程序( QP) 解算法( Riccati) 解算器无法做到这一点, 因为Hesian 矩阵本身是不固定的。 此外, 提议对降低的赫萨提调矩阵进行修改, 用Riccati recultive contal- commal commission- paltra la lax lax lax lax lax lax 。 lax lax