Controlled Invariant Distributions and Differential Properties

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• 刊名：Lecture Notes in Control and Information Sciences
• 出版年：2017
• 出版时间：2017
• 年：2017
• 卷：470
• 期：1
• 页码：3-19
• 全文大小：208 KB
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• 作者单位：Arjan van der Schaft (6)
  
  6. Johann Bernoulli Institute for Mathematics and Computer Science, Jan C. Willems Center for Systems and Control, University of Groningen, Groningen, The Netherlands
  
• 丛书名：Nonlinear Systems
• ISBN：978-3-319-30357-4
• 刊物类别：Engineering
• 刊物主题：Control Engineering
  Vibration, Dynamical Systems and Control
  
• 出版者：Springer Berlin / Heidelberg
• 卷排序：470

文摘

The theory of (controlled) invariant (co-)distributions is reviewed, emphasizing the theory of liftings of vector fields, one-forms and (co-)distributions to the tangent and cotangent bundle. In particular, it is shown how invariant distributions can be equivalently described as invariant submanifolds of the tangent and cotangent bundle. This naturally leads to the notion of an invariant Lagrangian subbundle of the Whitney sum of tangent and cotangent bundle, which amounts to a special case of the central equation of contraction analysis. The interconnection of the prolongation of a nonlinear control system (living on the tangent bundle of the state space manifold) with its Hamiltonian extension (defined on the cotangent bundle) is shown to result in a differential Hamiltonian system. The invariant submanifolds of this differential Hamiltonian system corresponding to Lagrangian subbundles are seen to result in general differential Riccati and differential Lyapunov equations. The established framework thus yields a geometric underpinning of recent advances in contraction analysis and convergent dynamics.