$L_{1}$-Based Adaptive Identification With Saturated Observations

It is well known that saturated output observations are prevalent in various practical systems and that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula>-norm is more robust than the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula>-norm-based parameter estimation. Unfortunately, adaptive identification based on both saturated observations and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula>-optimization turns out to be a challenging nonlinear problem, and has rarely been explored in the literature. Motivated by this and the need to fit with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula>-based index of prediction accuracy in, e.g., judicial sentencing prediction problems, we propose a two-step weighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula>-based adaptive identification algorithm. Under certain excitation conditions much weaker than the traditional persistent excitation condition, we will establish the global convergence of both the parameter estimators and the adaptive predictors. It is worth noting that our results do not rely on the widely used independent and identically distributed assumptions on the system signals, and thus, do not exclude applications to feedback control systems. We will demonstrate the advantages of our proposed new adaptive algorithm over the existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula>-based ones, through both a numerical example and a real-data-based sentencing prediction problem.