An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems
An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems
复制标题
对稳定性算子理论方法的代数解释。
DOI：
10.1007/s10440-005-6697-2
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
A. Quadrat
中科院分区：
文献类型：
--
作者：
A. Quadrat
本文的目的是表明分数理想方法 [23, 26] 和算子理论方法 [4, 6, 8, 9, 33, 34] 之间存在对偶性来解决稳定性问题。特别是，这种对偶性有助于我们理解算子理论方法如何反映系统的代数性质，反之亦然。就模块而言，我们描述了内部稳定工厂或允许（弱）互质因式分解的工厂的域和图。此外，我们证明内部稳定性意味着被控对象的图和稳定控制器的图是全局信号空间的直接加数。这些结果概括了[6,8,9,33,34]中获得的结果。最后，我们展示了一类信号空间，在该信号空间上，内部稳定性相当于将闭环系统的误差 e1 和 e2 映射到输入 u1 和 u2 的线性算子的有界逆的存在。
The purpose of this paper is to show that a duality exists between the fractional ideal approach [23, 26] and the operator-theoretic approach [4, 6, 8, 9, 33, 34] to stabilization problems. In particular, this duality helps us to understand how the algebraic properties of systems are reflected by the operator-theoretic approach and conversely. In terms of modules, we characterize the domain and the graph of an internally stabilizable plant or that of a plant which admits a (weakly) coprime factorization. Moreover, we prove that internal stabilizability implies that the graph of the plant and the graph of a stabilizing controller are direct summands of the global signal space. These results generalize those obtained in [6, 8, 9, 33, 34]. Finally, we exhibit a class of signal spaces over which internal stabilizability is equivalent to the existence of a bounded inverse for the linear operator mapping the errors e1 and e2 of the closed-loop system to the inputs u1 and u2.