Stable adaptive systems
Feedback Control Theory
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This paper deals with the identification of linear time—varying continuous—time systems whose parameters have bounded first-order time-derivative under square-integrable output noise. The proposed parameter updating algorithm is one of σ-modification type with Covariance Matrix. The Covariance Matrix updating rule is performed by using a modification with respect to standard previous estimation schemes used for time-invariant plants. Such a modification consists of including, apart from the standard ones in least-squares estimation with forgetting factor, two extra additive weighted terms involving the identity matrix as well as the second power of such a matrix with the appropriate signs. That parameter estimation strategy ensures that the norm of the covariance matrix is always finitely upper-bounded without using adaptation freezing techniques over a prescribed norm threshold. The main role played by the σ-modification relies on introducing a penalty for large values of the covariance inverse and the parametrical error. Furthermore, in the most general formalism setting, the knowledge of absolute upper-bounds on either the parameter vector or its time-derivative are not requested. The estimation scheme guarantees the asymptotic convergence of the prediction error to zero and that of the estimates to the true parameters under standard properties of persistent excitation of the input and controllability of both plant and filter realizations in the case when the plant is noise-free and either time-invariant after finite time asymptotically time-invariant. If the plant is slowly time-varying and the covariance matrix is updated with the standard least-squares rule, the variations of the integrals of the squares of the estimation vector and its time-derivative norms vary not faster than uniformly linearly with the interval lengths considered for integration. Furthermore, the overall changes are arbitrarily small over any finite time-interval provided that the time-derivative of the true parameter vector and the correction coefficients in the covariance updating rule are arbitrarily small.