We consider control from the perspective of competitive analysis. Unlike much prior work on learning-based control, which focuses on minimizing regret against the best controller selected in hindsight from some specific class, we focus on designing an online controller which competes against a clairvoyant offline optimal controller. A natural performance metric in this setting is competitive ratio, which is the ratio between the cost incurred by the online controller and the cost incurred by the offline optimal controller. Using operator-theoretic techniques from robust control, we derive a computationally efficient state-space description of the the controller with optimal competitive ratio in both finite-horizon and infinite-horizon settings. We extend competitive control to nonlinear systems using Model Predictive Control (MPC) and present numerical experiments which show that our competitive controller can significantly outperform standard $H_2$ and $H_{\infty}$ controllers in the MPC setting.
Supplementary Material to: Passive Controller Realization of a Biquadratic Impedance with Double Poles and Zeros as a Seven-Element Series-Parallel Network for Effective Mechanical Control
A biology-inspired approach to the positive integral control of positive systems -- the antithetic, exponential, and logistic integral controllers
使用非负控制输入的正系统的积分控制是在生物化学、流行病学和生态学中出现的重要问题。一个直接的解决方案是在控制器和系统之间使用ON-OFF非线性。然而，该解决方案仅在控制器在计算机系统中实现时可用。当情况并非如此时，如在生物学中，需要探索替代方法。基于最近在生物系统控制方面的研究，我们提出了一种基于所谓的\emph{antithetic integral controller}和两个\emph{positively regularized integral controllers}（所谓的\emph{exponential integral controller}和\emph{logistic integral controller}）的使用非负控制的正系统的积分控制的理论。对于所有这些控制器，我们建立了几个定性的结果，我们连接到标准的结果积分控制。我们还获得了额外的结果是特定的类型的控制器。例如，我们表明一个有趣的结果规定，如果的对偶积分控制器的增益是适当的选择，然后局部稳定的平衡点的闭环系统不依赖于耦合参数的选择，一个额外的参数特定于此控制器。相反，我们还表明，如果耦合参数是适当的选择，那么无论增益的值的闭环系统的平衡点是局部稳定的。对于指数积分控制器，我们可以证明闭环系统平衡点的局部稳定性与控制器的增益和系统的增益无关。稳定性仅取决于控制器的指数速率，同样是特定于这种类型的控制…
