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On the use of gradual dense-sparse discretizations in receding horizon control

机译：关于渐进式稀疏离散化在后退水平控制中的应用

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摘要

A key factor to success in implementations of real time optimal control, such as receding horizon control (RHC), is making efficient use of computational resources. The main trade-off is then between efficiency and accuracy of each RHC iteration, and the resulting overall optimality properties of the concatenated iterations, that is, how closely this represents a solution to the underlying infinite time optimal control problem (OCP). Both these issues can be addressed by adapting the RHC solution strategy to the expected form of the solution. Using gradual dense-sparse (GDS) node distributions in direct transcription formulations of the finite time OCP solved in each RHC iteration is a way of adapting the node distribution of this OCP to the fact that it is actually part of an RHC scheme. We have previously argued that this is reasonable, because the near future plan must be implemented now, but the far future plan can and will be revised later. In this paper, we investigate RHC applications where the asymptotic qualitative behavior of the OCP solution can be analyzed in advance. For some classes of systems, explicit exponential convergence rates of the solutions can be computed. We establish such convergence rates for a class of control affine nonlinear systems with a locally quadratic cost and propose to use versions of GDS node distributions for such systems because they will (eventually) be better adapted to the form of the solution. The advantages of the GDS approach in such settings is illustrated with simulations.
机译：成功实现实时最佳控制（例如后视水平控制（RHC））的关键因素是有效利用计算资源。然后，主要的权衡是在每次RHC迭代的效率和准确性与所连接的迭代的最终总体最优性之间进行的权衡，也就是说，这在多大程度上代表了对底层无限时间最优控制问题（OCP）的解决方案。通过使RHC解决方案策略适应解决方案的预期形式，可以解决这两个问题。在每次RHC迭代中求解的有限时间OCP的直接转录公式中使用渐进式稀疏（GDS）节点分布是一种使该OCP的节点分布适应于它实际上是RHC方案一部分这一事实的一种方式。我们以前曾认为这是合理的，因为必须立即执行近期计划，但是将来计划可以并且以后将进行修订。在本文中，我们研究了RHC应用，其中可以预先分析OCP解决方案的渐近定性行为。对于某些类型的系统，可以计算解决方案的显式指数收敛速度。我们为一类具有局部二次成本的控制仿射非线性系统建立了这样的收敛速度，并提议针对此类系统使用GDS节点分布的版本，因为它们（最终）将更好地适应解决方案的形式。通过模拟说明了在这种情况下GDS方法的优势。

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