Fig. 1

Block diagrams illustrating the simplified model (upper plot) and the adaptive proportional-integral feedback model (lower plot). The upper plot depicts the block diagram of Eq. 2a and 2b, presenting a system (green block) with a negative feedback loop where the controller (orange block) parameter \(\theta (t)\) is influenced by z(t) and s. The lower plot represents the block diagram of Eq. 4a and 4b, illustrating the adaptive proportional-integral feedback \(sz(t)\big (lr(t)-y(t)\big )\), where sz(t) is the adaptive proportional-integral gain with two error terms \(r(t)-y(t)\) and \(lr(t)-y(t)\). The term d(t) represents the disturbance. Note that s has no impact on the control signal due to \(\mathbb {P}\)-invariance