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Journal Articles Mathematics of Control, Signals, and Systems Year : 2014

## Growth rates for persistently excited linear systems

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Yacine Chitour
Fritz Colonius
• Function : Author
Mario Sigalotti

#### Abstract

We consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ where $\alpha$ belongs to a given class of persistently exciting signals. We seek maximal $\alpha$-uniform stabilisation and destabilisation 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 the pair $(A,B)$ verifies a certain Lie bracket generating condition, 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 sole assumption of controllability of the pair $(A,B)$.

### Dates and versions

hal-00851671 , version 1 (16-08-2013)
hal-00851671 , version 2 (07-10-2013)

### Identifiers

• HAL Id : hal-00851671 , version 2
• ARXIV :
• DOI :

### Cite

Yacine Chitour, Fritz Colonius, Mario Sigalotti. Growth rates for persistently excited linear systems. Mathematics of Control, Signals, and Systems, 2014, 26 (4), pp.589-616. ⟨10.1007/s00498-014-0131-0⟩. ⟨hal-00851671v2⟩

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