Linear Output Regulation for Unknown Stable Systems With Uncertain Minimum-Phase Actuators

Abstract

Consider a linear stable system - described by the transfer function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> of unknown parameters, unknown order, and unknown relative degree - under a linear exosystem generating biased multi-sinusoidal references and/or disturbances with at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> different known frequencies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\omega _{i}$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$i=1,\ldots,q$</tex-math></inline-formula> . It has been recently established that a linear regulator with minimal order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(2q+1)$</tex-math></inline-formula> exists under the knowledge of the positive or negative signs of: i) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(0)$</tex-math></inline-formula> ; ii) either <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Re (P(j\omega _{i}))$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Im (P(j\omega _{i}))$</tex-math></inline-formula> , for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$i=1,\ldots,q$</tex-math></inline-formula> [ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$j$</tex-math></inline-formula> is the imaginary unit]. This technical note explores the case in which the measurable input <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula> to the aforementioned system is provided by an unknown linear actuator. It is actually shown that the regulator design can be naturally extended to such a scenario, provided that the actuator process is minimum-phase, of known relative degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho \geq 1$</tex-math></inline-formula> and with known sign of the high-frequency gain.