“…Proof: Consider the Lyapunov function candidate (32) Using vector derivatives and following [20] and [32], the time derivative of (32) is (33) Note that and the th derivative of the output error is so that (34) and from (13), (26), (29), and (27), the equation shown at the bottom of the next page holds true. Also, from (15)- (18), (23), and (24), we have Substitute the aforementioned equation into (33) and assume that the ideal parameters are constant (which is achieved by having and in (17) and (18) represent approximation errors from the time-varying part of system dynamics) so that and and substitute (30) and (31) into (33) Notice that we did not consider a projection modification to the previous update laws.…”