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The set of controllable multi-input systems is generically convex
Mathematics of Control, Signals, and Systems ( IF 1.8 ) Pub Date : 2019-08-16 , DOI: 10.1007/s00498-019-0243-7
D. Hinrichsen , E. Oeljeklaus

In this paper, we investigate connectedness and convexity properties of the subspace \(\mathbf {L}_{n,m}^c(\mathbb {R})\) of controllable input pairs \((A,B)\in \mathbf {L}_{n,m}(\mathbb {R}):= \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\). We introduce three restricted convexity properties (“dense”, “almost sure” and “generic” convexity). In order to prove that the space \(\mathbf {L}_{n,m}^c(\mathbb {R})\) possesses these properties, we study the intersection of straight lines in \(\mathbf {L}_{n,m}(\mathbb {R})\) with the algebraic variety of uncontrollable input pairs in \(\mathbf {L}_{n,m}(\mathbb {R})\). While in the single-input case (\(m=1\)), the space \(\mathbf {L}_{n,1}^c(\mathbb {R})\) consists of two connected components, we prove that the space \(\mathbf {L}_{n,m}^c(\mathbb {R})\) is generically convex in the multi-input case. This is our main result. It directly implies Brockett’s theorem that \(\mathbf {L}_{n,m}^c(\mathbb {R})\) is pathwise connected if \(m\ge 2\). As another application, we derive the theorem of Hazewinkel and Kalman about the non-existence of continuous canonical forms for multi-input systems.



中文翻译:

可控制的多输入系统的集合通常是凸的

在本文中,我们研究了可控输入对\((A,B)\ in中子空间\(\ mathbf {L} _ {n,m} ^ c(\ mathbb {R})\)的连通性和凸性\ mathbf {L} _ {n,m}(\ mathbb {R}):= \ mathbb {R} ^ {n \ times n} \ times \ mathbb {R} ^ {n \ times} \)。我们介绍了三个受限的凸性(“密集”,“几乎确定”和“一般”凸性)。为了证明空间\(\ mathbf {L} _ {n,m} ^ c(\ mathbb {R})\)具有这些性质,我们研究\(\ mathbf {L} _ {n,m}(\ mathbb {R})\)\(\ mathbf {L} _ {n,m}(\ mathbb {R})\)中具有不可控制的输入对的代数形式。在单输入情况下(\(m = 1 \)),空格\(\ mathbf {L} _ {n,1} ^ c(\ mathbb {R})\)由两个相连的组件组成,我们证明了空间\(\ mathbf {L} _ {n,m} ^ c (\ mathbb {R})\)在多输入情况下通常是凸的。这是我们的主要结果。它直接暗示了Brockett定理,如果\(m \ ge 2 \)\(\ mathbf {L} _ {n,m} ^ c(\ mathbb {R})\)的路径连接。作为另一个应用程序,我们推导了Hazewinkel和Kalman关于多输入系统的连续规范形式不存在的定理。

更新日期:2019-08-16
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