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Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems

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2 Author(s)
Manitius, A. ; Université de Montréal, Montreal, Canada ; Triggiani, R.

New sufficient conditions for function space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded systems as a special case of an abstract equation in Hilbert space R^{n}\times L_{2}([- h, 0], R^{n}) (denoted as M_{2} }). For systems of type \cdot{x}(t)=A_{0}x(t)+A_{1}x(t-h)+Bu(t) , it is shown that most of controllability properties are described by a certain polynomial matrix P(\lambda ) , whose columns can be generated by an algorithm comparing A_{0}^{i}B,A_{0}^{i} B and mixed powers of A0and A1multiplied by B. It is shown that the M2-approximate controllability of the system is guaranteed by certain triangularity properties of P(\lambda ) . By using the Luenberger canonical form, it is shown that the system is M2-approximately controllable if the pair (A_{1},B) is controllable and if each of the spaces spanned by columns of [B,A_{1}B,... ,A_{1}^{j}B], j=O...n-1 , is invariant under transformation A0. Other conditions of this type are also given. Since the M2-approximate controllability implies controllability of all the eigenmodes of the system, the feedback stabilizability with an arbitrary exponential decay rate is guaranteed under hypotheses leading to M2-approximate controllability. Some examples are given.

Published in:

Automatic Control, IEEE Transactions on  (Volume:23 ,  Issue: 4 )