Monotonous Parameter Estimation of One Class of Nonlinearly Parameterized Regressions without Overparameterization

Abstract

The estimation law of unknown parameters vector $θ$ is proposed for one class of nonlinearly parametrized regression equations $y\left( t \right) = Ω\left( t \right)Θ\left( θ\right)$. We restrict our attention to parametrizations that are widely obtained in practical scenarios when polynomials in $θ$ are used to form $Θ\left( θ\right)$. For them we introduce a new 'linearizability' assumption that a mapping from overparametrized vector of parameters $Θ\left( θ\right)$ to original one $θ$ exists in terms of standard algebraic functions. Under such assumption and weak requirement of the regressor finite excitation, on the basis of dynamic regressor extension and mixing technique we propose a procedure to reduce the nonlinear regression equation to the linear parameterization without application of singularity causing operations and the need to identify the overparametrized parameters vector. As a result, an estimation law with exponential convergence rate is derived, which, unlike known solutions, (i) does not require a strict P-monotonicity condition to be met and a priori information about $θ$ to be known, (ii) ensures elementwise monotonicity for the parameter error vector. The effectiveness of our approach is illustrated with both academic example and 2-DOF robot manipulator control problem.