Transient Performance Improvement in Reduced-Order Model Reference Adaptive Control Systems

Model reference adaptive control based on a novel scalar update law

“…Many modification terms to the adaptive control law given by (5) are reported in the literature, see Reference 14 and references therein. In particular, Ristevski et al 14 recently presented an adaptive control law based on a scalar update law formed as $$$$ {u}_a(t)=-\hat{v}(t){\left\Vert \sigma \Big({x}_p(t)\right\Vert}_2\tanh \left({B}_p^T Pe(t){\mu}^{-1}\right), $$$$ where $$$ \hat{v}(t) $$$ satisfies the scalar update law predicated on the projection operator given by $$$$ {\displaystyle \begin{array}{ll}\dot{\hat{v}}(t)& =\gamma \mathrm{Proj}\kern.5em \left(\hat{v}(t),{\left\Vert \sigma \Big({x}_p(t)\right\Vert}_2{\left\Vert {B}_p^T Pe(t)\right\Vert}_2\right),\\ {}\hat{v}(0)& ={\hat{v}}_0.\end{array}} $$$$ This adaptive control law only needs to calculate one update law, $$$ \hat{v}(t) $$$. However, this adaptive controller ...…”

“…In general, the computational complexity of adaptive control law in the standard adaptive control is $$$ s\times m $$$ times that of ours. In addition, the adaptive control law in the works of Ristevski et al 14 and ours both rely on a scalar update law, whose calculations are all algebraic operations. However, compared with our adaptive control law, the adaptive control law proposed in the works of Ristevski et al 14 needs to calculate $$$ m $$$‐dimensional transcendental function $$$ \tanh \left({B}_p^T Pe(t){\mu}^{-1}\right) $$$.…”

“…For the adaptive control law proposed in the works of Ristevski et al, 14 we can obtain $$$$ {u}_a(t)=-\hat{v}(t){\left\Vert \sigma \Big({x}_p(t)\right\Vert}_2\tanh \left({\mu}^{-1}\left(3.125{e}_1+6.4091{e}_2\right)\right), $$$$ where $$$$ {\displaystyle \begin{array}{ll}\dot{\hat{v}}(t)& =\gamma \mathrm{Proj}\kern.5em \left(\hat{v}(t),{\left\Vert \sigma \Big({x}_p(t)\right\Vert}_2{\left\Vert {B}_p^T Pe(t)\right\Vert}_2\right),\\ {}\hat{v}(0)& ={\hat{v}}_0,\end{array}} $$$$ and the number of update law is also only 1.…”

“…Model reference adaptive control (MRAC) is an effective method that can handle this issue 6‐14 . Specifically, by designing a proper feedback controller with an adaptive compensation for the matched uncertainty, it can adjust adaptive parameters online such that the plant output asymptotically or approximately tracks a predesigned stable model reference output.…”

A Novel MRAC Scheme for Output Tracking

“…The model reference adaptive control (MRAC), which was originally introduced by Whitaker et al [10,11], is a widely used adaptive control technique. The essential feature of MRAC is to develop feedback controller structures and controller parameter-updating laws to ensure the asymptotic output or state tracking of an ideal reference model system, as well as closed-loop signals boundedness, despite the system parameters uncertainties [12][13][14]. Much effort has been dedicated to the development of MRAC theory.…”