Growth rates for persistently excited linear systems

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• 作者：Yacine Chitour (1) (2) (3)
  Fritz Colonius (4)
  Mario Sigalotti (3) (5)
  
• 关键词：Time ; varying systems ; Persistent excitation ; Uniform stabilization ; Maximal rate of exponential stability ; Lyapounov exponent
• 刊名：Mathematics of Control, Signals, and Systems (MCSS)
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：26
• 期：4
• 页码：589-616
• 全文大小：295 KB
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• 作者单位：Yacine Chitour (1) (2) (3)
  Fritz Colonius (4)
  Mario Sigalotti (3) (5)
  
  1. Laboratoire des Signaux et Syst猫mes, Sup茅lec, Gif-sur-Yvette, France
  2. Universit茅 Paris Sud, Orsay, France
  3. Team GECO, INRIA Saclay-脦le-de-France, Palaiseau, France
  4. Institut f眉r Mathematik, Universit盲t Augsburg, Augsburg, Germany
  5. CMAP, UMR 7641, 脡cole Polytechnique, Palaiseau, France
  
• ISSN：1435-568X

文摘

We consider a family of linear control systems \(\dot{x}=Ax+\alpha Bu\) on \(\mathbb {R}^d\) , where \(\alpha \) belongs to a given class of persistently exciting signals. We seek maximal \(\alpha \) -uniform stabilization and destabilization by means of linear feedbacks \(u=Kx\) . We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if there exists at least one \(K\) such that the Lie algebra generated by \(A\) and \(BK\) is equal to the set of all \(d\times d\) matrices, then the maximal rate of convergence of \((A,B)\) is equal to the maximal rate of divergence of \((-A,-B)\) . We also provide more precise results in the general single-input case, where the above result is obtained under the simpler assumption of controllability of the pair \((A,B)\) .