Designing of Adaptive Model-Free Controller Based on Output Error and Feedback Linearization

and Applied Analysis 3 A α , A β are matrices that will be updated adaptively with the following updating laws: ?̇? α (t) = −Q −1 ⋅ z ⋅ ξ T α ⋅ es, ?̇? β (t) = −Q −1 ⋅ z ⋅ ξ T β ⋅ es ⋅ u (t) (12) and finally the control law is described as follows: u (t) = 1 β (t) + β (x) (− [α (t) + ?̂? (x)] + v (t)) . (13) 2.2. Direct Adaptive Fuzzy Controller (DAFC). Control law u(t) in feedback linearization method can be obtained as follows: u (t) = 1 β (x) (−α (x) + v (t)) . (14) The goal is to estimate u(t), with consideration of basic fuzzy function as seen below: ?̂? (t) = z T