利用平方和进行长时间平均的昂贵控制及其在层流尾流中的应用
Expensive control of long-time averages using sum of squares and its application to a laminar wake flow
AI总结：
本文针对昂贵控制下的长时间平均成本控制问题，提出基于平方和与半定规划的凸化设计方法，并成功应用于抑制圆柱层流尾流涡脱落。
摘要：
本文提出了一种针对长时间平均成本控制的非线性状态反馈控制设计方法，其中假设控制代价是昂贵的。该方法基于平方和与半定规划技术，适用于右端项为状态变量和控制量的多项式函数的动力系统。其核心思想最早在（Chernyshenko et al., Phil. Trans. R. Soc. A, 372, 2014）中描述但未实现，即用同一平均量的上界优化来替代涉及长时间平均的成本函数这一困难优化问题。因此，控制器设计需要同时优化控制律和一个可调函数，该函数类似于Lyapunov函数。本文引入了一种方法，解决了此类优化中众所周知的固有非凸性问题。该方法基于控制是昂贵的这一形式假设，由此可推出最优控制量较小。由此得到的渐近优化问题是凸的。控制器中所有多项式系数的推导均以状态依赖的线性和双线性不等式的可解性条件形式给出。所提出的方法被应用于设计一个全信息反馈控制器，该控制器通过旋转振荡来抑制层流状态下圆柱绕流尾流中的涡脱落。文中报告了在受控尾流的降阶模型上和直接数值模拟中的控制结果。
英文摘要：
The paper presents a nonlinear state-feedback control design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semi-definite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented in (Chernyshenko et al., Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. The present paper introduces a method resolving the well-known inherent non-convexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported.
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