[论文解读] Optimal trajectory tracking

Jakob Löber|arXiv (Cornell University)|Jan 1, 2015

Advanced Thermodynamics and Statistical Mechanics参考文献 108被引用 5

一句话总结

本文提出了一种非线性仿射控制系统的最优轨迹跟踪框架，通过识别一类‘可精确实现的轨迹’——其控制输入可显式推导以实现完美跟踪。通过采用线性化假设并将正则化参数视为小扰动，该方法揭示了非线性最优控制中的潜在线性结构，从而为包括机械系统和FitzHugh-Nagumo模型在内的广泛系统类提供了精确解析解。

ABSTRACT

This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of so-called exactly realizable trajectories is proposed. For exactly realizable desired trajectories exists a control signal which enforces the state to exactly follow the desired trajectory. This approach does not only yield an explicit expression for the control signal in terms of the desired trajectory, but also identifies a particularly simple class of nonlinear control systems. Systems in this class satisfy the so-called linearizing assumption and share many properties with linear control systems. For example, conditions for controllability can be formulated in terms of a rank condition for a controllability matrix analogously to the Kalman rank condition for linear time invariant systems. Furthermore, exactly realizable trajectories arise as solutions to unregularized optimal control problems. Based on that insight, the regularization parameter is used as the small parameter for a perturbation expansion. This results in a reinterpretation of affine optimal control problems with small regularization term as singularly perturbed differential equations. The small parameter originates from the formulation of the control problem and does not involve simplifying assumptions about the system dynamics. Combining this approach with the linearizing assumption, approximate and partly linear equations for the optimal trajectory tracking of arbitrary desired trajectories are derived. For vanishing regularization parameter, the state trajectory becomes discontinuous and the control signal diverges. On the other hand, the analytical treatment becomes exact and the solutions are exclusively governed by linear differential equations. Thus, the possibility of linear structures underlying nonlinear optimal control is revealed. This fact enables the derivation of exact analytical solutions to an entire class of nonlinear trajectory tracking problems with affine controls. This class comprises, among others, mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model with a control acting on the activator.

研究动机与目标

• 开发一种用于非线性仿射控制系统最优轨迹跟踪的解析框架。
• 识别一类满足线性化假设的系统，使得最优控制存在精确解析解。
• 将小正则化最优控制问题重新解释为奇异摄动系统，以推导近似解。
• 建立类似于线性系统的可控性与输出可实现性的条件。
• 在机械系统和FitzHugh-Nagumo模型上展示该方法，表明在正则化参数趋近于零时可实现精确可解性。

提出的方法

• 引入‘可精确实现轨迹’的概念——即可通过显式控制输入完美跟踪的期望轨迹。
• 应用线性化假设，利用投影算子 P 和 Q 将状态方程分解为解耦分量。
• 推导出类似于线性时不变系统中Kalman秩条件的可控性条件。
• 将具有小正则化的最优控制问题重新解释为奇异摄动系统，使用正则化参数 ϵ 作为小参数。
• 通过匹配渐近展开法推导内层（边界层）和外层（常规）解。
• 通过在 ϵ → 0 极限下匹配内层与外层展开，构建状态与控制信号的复合解。

实验结果

研究问题

• RQ1 在何种条件下，非线性仿射控制系统中的期望轨迹可被精确跟踪？
• RQ2 如何通过将正则化参数视为小扰动来简化最优控制问题的结构？
• RQ3 可精确实现轨迹与无正则化最优控制问题之间存在何种关系？
• RQ4 满足线性化假设的非线性系统在多大程度上继承了线性系统的性质，例如基于秩的可控性？
• RQ5 能否通过利用奇异摄动结构，为非线性系统推导出最优轨迹跟踪的解析解？

主要发现

• 对于满足线性化假设的系统，可精确实现轨迹存在，从而可为控制输入提供显式解析表达式。
• 此类系统的可控性由一个可控性矩阵类比的秩条件决定，与线性系统的Kalman秩条件一致。
• 在正则化消失的极限下（ϵ → 0），状态轨迹变为不连续，控制信号发散，但解析处理变为精确，且受线性微分方程支配。
• 该方法为一类非线性最优控制问题（包括一维机械系统和具有仿射控制的FitzHugh-Nagumo模型）提供了精确解析解。
• 奇异摄动方法可推导出任意期望轨迹的近似解，且在 ϵ → 0 时收敛于精确解。
• 该框架揭示，当将正则化参数视为小扰动时，非线性最优控制问题可能具有潜在的线性结构。