Determination of Bandwidth of Q-Filter in Disturbance Observers to Guarantee Transient and Steady State Performance Under Measurement Noise

Q-filter-based disturbance observer (DOB) is one of the most widely used robust controller due to its design simplicity. Such simplicity arises from that reducing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex>, which is the time constant of low pass filters, not only ensures robust stability but also enhances nominal performance recovery—ability to recover the trajectory of nominal closed-loop system. However, in contrast to noise-free setup, excessively small <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> can rather damage the nominal performance recovery under measurement noise. That is, minimizing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> is no longer immediately guaranteeing nominal performance recovery. Motivated by this observation, this paper concentrates on determination of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> to guarantee transient and steady state performance. This analysis uses Lyapunov method utilizing the coordinate transformation inspired by the analysis based on singular perturbation theory. As a result, we present an affordable noise level and open interval for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> that guarantees both the required performances. The analysis can also lead to theoretical demonstration on that excessively reducing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tau$</tex> gives assurance to achieve target performance only for noise-free case.