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Harib et al. Intell Robot 2022;2(1):37-71  https://dx.doi.org/10.20517/ir.2021.19                                                              Page 41

                                                                       n
                            n
               where ξ(t) ∈ R  is a bounded time-dependant disturbance, x ∈ R  is the extended system state vector, y ∈
                n
                                                   m
                                                                                N
               R  is the controlled system output, u ∈ R  is the control input and Ψ ∈ R  is the known N-dimensional
                                                                                           m
               regressor vector. We assume (A , B, B , C ) are known and A  is Hurwitz. y  ∈ R  in this case is a
                                                    ref
                                           ref
                                                 ref
                                                                       ref
                                                                                    cmd
                                                                                      N×m
               bounded command for y. Λ ∈ R m×m  is a diagonal positive definite matrix and Κ ∈ R  is a constant matrix,
               where both matrices represent the matched uncertainties of the system. In addition, we assume that,
               and that the disturbance upper bound ξ  ≥ 0 is known and constant.
                                                max
               The control goal is bounded tracking of the reference model dynamics,









                                                               m
               driven by a bounded time-dependant command y  ∈ R .
                                                         cmd
               Based on Equation (1), the control input is selected as,







               where K ∈ R N×m  is the matrix of adaptive parameters. If we substitute Equation (4) into Equation (1), we
               get,






               where,






               is the matrix of estimation errors. The tracking error is e = x - x . Subtracting the reference model dynamics
                                                                    ref
               in Equation (3) from that of Equation (1) yields the tracking error dynamics,







               The Lyapunov function candidate is selected,
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