Robust stabilisation of uncertain nonlinear systems with matched and unmatched uncertainties

This paper considers the problem of robust stabilisation of uncertain nonlinear systems with both matched and unmatched uncertainties. A new robust controller is proposed to guarantee practical or asymptotic stability under non-vanishing or vanishing unmatched uncertainties. First, the system uncertainties are decomposed optimally into the matched and unmatched portions with an objective of the minimum norm for the unmatched part. Next, instead of a robust control design based on only the matched part of uncertainties commonly used in the previous works, a robust controller is synthesised based on both matched and mismatched portions of the uncertainties. It is shown that in uncertain systems whose state cannot simultaneously stay in a so-called null set and diverge to infinity, the proposed controller guarantees stability no matter how large the input-unrelated unmatched uncertainties are. Numerical examples illustrate that the proposed controller is more robust against uncertainties, and offers improved damping and stabilisation features.