This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. This paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power rebalancing and tracking of the DC optimal power flow points.

中文翻译：

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在线优化作为反馈控制器：稳定性和跟踪

本文开发并分析了基于反馈的在线优化方法，以将线性时不变（LTI）动力系统的输出调节至最优解。 随时间变化凸优化问题。该算法的设计基于连续时间的原始对偶动力学，经过适当修改以合并来自LTI动力学系统的反馈，并将其应用于近端增强拉格朗日函数。所得的闭环算法无需时动态系统中的（时变）扰动知识即可跟踪时变优化问题的解决方案。该分析利用积分二次约束来提供线性矩阵不等式（LMI）条件，以保证全局指数稳定性和有界跟踪误差。分析结果表明，在LTI动力学系统的动力学和算法之间有足够的时间尺度分隔时，可以始终满足LMI条件。本文还提出了一种改进的算法，该算法可以在约束较少的假设下跟踪约束优化问题的近似解轨迹。作为说明性示例，展示了所提出的算法用于输电系统，以压缩次级控制和三级控制之间的时间标度，并允许同时进行功率平衡和跟踪DC最佳功率流点。