### Diagonal and Recursive Parameter Estimation for Black-box Systems with Bounded Inputs and Outputs

#### Abstract

Estimation theory is a branch of stochastic and signal processing that deals with estimating the parameter values based on an observable known signal as a random variable. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the observable known signal. An estimator attempts to approximate the unknown parameters using the stochastic signal. In the estimation theory it is assumed that the output signal is random with the probability depending on the interest parameters. The estimation takes the measured observable signal as an input and produces an estimation of internal unknown gains. It is also preferable to derive an estimation that exhibits optimality, achieving minimum average error over some class, for example, an unbiased minimum variance as estimation. This paper presents the development of an optimal stochastic estimator for a black-box system in a m−dimensional space, observing noise with an unknown dynamics model. The results are described in a state space, with a discrete stochastic estimator and noise characterization. The results are obtained by an algorithm to construct the diagonal form for the state space system. Thus, the matrix is estimated in probability considering the distribution function. The estimation technique is used on the instrumental variable based on a gradient stochastic matrix. This kind of matrix contribution is optimal in the probability sense. This is a new technique for an instrumental variable tool, and a diagonalization process avoiding the calculation of pseudo-inverse matrices is presented with a linear computational complexity O(j) and j as the diagonal matrix dimension. The results show that it is possible to reconstruct the observable signal with a probability approximation. The advantages with respect to traditional solutions are focused on estimating the matrix contribution on line with a linear complexity.