<inline-formula> <tex-math notation="TeX">${\rm L}_{1}$</tex-math></inline-formula>-Adaptive Control: Stability, Robustness, and Interpretations

Abstract

An adaptive control design approach that involves the insertion of a strictly proper stable filter at the input of standard Model Reference Adaptive Control (MRAC) schemes has been proposed in the recent years. This approach was given the name L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Adaptive Control (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -AC) due to the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> bounds obtained for various signals. As part of the approach it is recommended to use very high adaptive gains for fast and robust adaptation. The purpose of this note is to analyze whether L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -AC provides any improvements to existing MRAC schemes by focusing on a simple plant whose states are available for measurement presented in [1]. Our analysis shows that the insertion of the proposed filter deteriorates the performance and robust stability margin bounds compared to standard MRAC, i.e., when the filter is removed. The use of high adaptive gains recommended in the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -AC approach may cause two major problems. First, it makes the differential equation of the adaptive law very stiff leading to possible numerical instabilities. Second, it makes the adaptive scheme less robust with respect to unmodeled dynamics.