On strong consistency of a class of recursive stochastic Newton-Raphson type algorithms with application to robust linear dynamic system identification

Abstract

The recursive stochastic algorithms for estimating the parameters of linear discrete-time dynamic systems in the presence of disturbance uncertainty has been considered in the paper. Problems related to the construction of min-max optimal recursive algorithms are demonstrated. In addition, the robustness of the proposed algorithms has been addressed. Since the min-max optimal solution cannot be achieved in practice, an approximate optimal solution based on a recursive stochastic Newton-Raphson type procedure is suggested. The convergence of the proposed practically applicable robustified recursive algorithm is established theoretically using the martingale theory. Both theoretical and experimental analysis related to the practical robustness of the proposed algorithm are also included.

Analysis of a class of adaptive robustified predictors in the presence of noise uncertainty

“…This is, in fact, an efficient robustness requirement. On the other hand, the derivative Ψ of the H-function in (17), the socalled influence function in qualitative robustness [13,14], is given by…”

“…This is, in turn, a resistant robustness requirement. Thus, the choice of the Huber's loss function in (17) provides for resistant robustness, along with an efficient robustness property. Application of the Robbins-Monro approach [11,13] results in a stochastic gradient algorithm for estimating unknown vectors of parameters θ of an adaptive robust predictor, where the gradient of criterion defined by Eq.…”

“…In relation (19) the non-linear function ( ) ( ) (15), (17). The form of the stochastic gradient algorithm is [11,13,17] ( ) ( ) ( ) (…”

“…The convergence property of the proposed adaptive robust predictor can be investigated using the Martingale theory [2,16,17,18]. The basic convergence result is the lemma of Neveu [18].…”

“…The unknown parameters of the proposed adaptive robust predictor are estimated in each step by applying a recursive algorithm of the stochastic gradient type [11,13]. The convergence of the adaptive robustified predictor algorithm is established theoretically using the Martingale theory [2,15,16,17]. Starting from a linear singleinput/single-output ARMAX system representation, it has been shown that the proposed adaptive robust one-step ahead prediction converges, in the Cesaro sense, to the optimal prediction of the system output.…”