System-level, input-output and new parameterizations of stabilizing controllers, and their numerical computation
System-level, input-output and new parameterizations of stabilizing controllers, and their numerical computation
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DOI：
10.1016/j.automatica.2022.110211
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Yang Zheng;Luca Furieri;M. Kamgarpour;Na Li
中科院分区：
文献类型：
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作者：
Yang Zheng;Luca Furieri;M. Kamgarpour;Na Li
众所周知，内部稳定控制器C stab集是非凸的，但它允许使用某些闭环映射进行凸表征：经典的结果是Youla参数化，最近的两个概念是系统级参数化（SLP）和输入-输出参数化（IOP）。在本文中，我们讨论了新的凸参数化的存在性，并讨论了每种参数化在不同场景下的潜在权衡。我们的主要贡献有：(1)我们发现只有四组稳定的闭环转移矩阵等价于内部稳定性：其中一组用于SLP，另一组用于IOP，另外两组是新的，从而导致C stab的两个新的凸参数化。(2)我们在施加有限脉冲响应（FIR）近似后研究了这些参数化的性质，揭示了在给定FIR约束下，IOP具有最佳的逼近C stab的能力。(3)这四种参数化不需要对对象进行先验的双素数分解，而是施加了一组等式约束。然而，在浮点算术计算和/或实现中，这些等式约束永远不会完全得到满足。我们证明了IOP对于开环稳定对象具有数值鲁棒性，即等式约束中的小错配不会影响闭环稳定性；但在实践中，直接实施IOP将无法稳定开环不稳定系统。已知SLP在状态反馈情况下具有数值鲁棒性；在这里，我们表明，即使当对象是开环稳定时，四块SLP控制器的数值鲁棒性也需要逐个分析。
It is known that the set of internally stabilizing controller C stab is non-convex, but it admits convex characterizations using certain closed-loop maps: a classical result is the Youla parameterization, and two recent notions are the system-level parameterization (SLP) and the input–output parameterization (IOP). In this paper, we address the existence of new convex parameterizations and discuss potential tradeoffs of each parameterization in different scenarios. Our main contributions are:(1) We reveal that only four groups of stable closed-loop transfer matrices are equivalent to internal stability: one of them is used in the SLP, another one is used in the IOP, and the other two are new, leading to two new convex parameterizations of C stab.(2) We investigate the properties of these parameterizations after imposing the finite impulse response (FIR) approximation, revealing that the IOP has the best ability of approximating C stab given FIR constraints.(3) These four parameterizations require no a priori doubly-coprime factorization of the plant, but impose a set of equality constraints. However, these equality constraints will never be satisfied exactly in floating-point arithmetic computation and/or implementation. We prove that the IOP is numerically robust for open-loop stable plants, in the sense that small mismatches in the equality constraints do not compromise the closed-loop stability; but a direct IOP implementation will fail to stabilize open-loop unstable systems in practice. The SLP is known to enjoy numerical robustness in the state feedback case; here, we show that numerical robustness of the four-block SLP controller requires case-by-case analysis even when the plant is open-loop stable.