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On global asymptotic stability of SPR adaptive systems without persistent excitation

We present a sufficient condition for global asymptotic stability of linear time-varying systems of the form x = Ax + Bφ<sup>⊺</sup> (t)θ, θ = −φ(t)C<sup>⊺</sup>x with strictly positive real transfer function W (s) = C<sup>⊺</sup> (sI − A)<sup>−1</sup> B and the vector φ(t) not satisfying the well-known persistent excitation condition. It is also shown that the criterion is optimal in some well-defined sense — making the condition “almost” necessary as well. This class of systems arise in many control applications including system identification and adaptive control and, to the best of the authors' knowledge, no necessary and sufficient condition for global asymptotic stability has been reported.