SIAM Journal on Control and Optimization, Volume 58, Issue 3, Page 1547-1578, January 2020.
Optimal controller synthesis is a bilinear problem and hence difficult to solve in a computationally efficient manner. We are able to resolve this bilinearity for systems with delay by first convexifying the problem in infinite dimensions, i.e., formulating the $H_\infty$-optimal state-feedback controller synthesis problem for distributed-parameter systems as a linear operator inequality, which a form of convex optimization with operator variables. Next, we use positive matrices to parameterize positive “complete quadratic” operators, allowing the controller synthesis problem to be solved using semidefinite programming (SDP). We then use the solution to this SDP to calculate the feedback gains and provide effective methods for real-time implementation. Finally, we use several test cases to verify that the resulting controllers are optimal to several decimal places as measured by the minimal achievable closed-loop $H_\infty$ norm, and as compared against controllers designed using high-order Padé approximations.

中文翻译：

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多时滞系统$ H_ \ infty $-最优控制器综合问题的凸解

SIAM控制与优化杂志，第58卷，第3期，第1547-1578页，2020年1月。
最优控制器综合是一个双线性问题，因此很难以计算有效的方式解决。通过首先在无限维上凸出问题，我们能够解决具有延迟的系统的这种双线性性，即，将分布参数系统的$ H_ \ infty $-最优状态反馈控制器综合问题公式化为线性算子不等式，带有运算符变量的凸优化的形式。接下来，我们使用正矩阵对正“完全二次”算子进行参数化，从而允许使用半定编程（SDP）解决控制器综合问题。然后，我们使用此SDP的解决方案来计算反馈增益，并为实时实施提供有效的方法。最后，