A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doubly coprime factorization of the system, while the recent system-level and input-output parametrizations require no doubly coprime factorization, but a set of equality constraints for achievable closed-loop responses. In this article, we present explicit affine mappings among Youla, system-level, and input-output parameterizations. Two direct implications of these affine mappings are: 1) any convex problem in the Youla, system-level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks, including the convex system-level synthesis; 2) the condition of quadratic invariance is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of either Youla, system-level, or input-output parameters.

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关于Youla、系统级和Input__utput参数化的等价性


内部稳定控制器的凸参数化是许多控制器合成过程的基础。著名的 Youla 参数化依赖于系统的双互质因式分解，而最近的系统级和输入输出参数化不需要双互质因式分解，而是需要一组可实现闭环响应的等式约束。在本文中，我们提出了 Youla、系统级和输入输出参数化之间的显式仿射映射。这些仿射映射的两个直接含义是：1）Youla、系统级或输入输出参数中的任何凸问题都可以在这些框架中的任何其他一个框架中等效地凸地表述，包括凸系统级综合； 2) 二次不变性的条件对于经典分布式控制问题来说是充分且必要的，以允许在 Youla、系统级或输入输出参数方面进行等效凸重构。