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

The estimation law of unknown parameters vector ${\theta}$ is proposed for one class of nonlinearly parametrized regression equations $y\left( t \right) = \Omega \left( t \right)\Theta \left( \theta \right)$. We restrict our attention to parametrizations that are widely obtained in practical scenarios when polynomials in $\theta$ are used to form $\Theta \left( \theta \right)$. For them we introduce a new 'linearizability' assumption that a mapping from overparametrized vector of parameters $\Theta \left( \theta \right)$ to original one $\theta$ 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 $\theta$ 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.