Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems

Abstract

This paper addresses robust stability, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance analysis for linear time-invariant parameter-dependent (LTIPD) systems, in which the state-space matrices are polynomially parameter-dependent, using polynomially parameter-dependent Lyapunov functions (PDLFs). Our results are derived via multiple "slack variable" approach, which has previously been proposed for the non- negativity check of polynomial functions and it has been proved that the conservatism is no more than SOS approach. Our derived conditions are only sufficient conditions for the original problems; however, they encompass the previously proposed methods using single slack variables. We illustrate the effectiveness of our methods with randomly generated numerical examples compared to the methods using single slack variables.