Page 221 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics
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220   Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics


                           Differentiating s 2 along (14.14)and (14.15), we have

                                        s
                                 s ˙ 2  =¨ 1 + α˙ 1
                                             s
                                                                           s       (14.17)
                                    = F φ 	 F (h(x,t) + b 0u −¨y d ) + H 1 (x,e,t) + α˙ 1
                                                                    s
                                    = F φ 	 F (ξ(y d ,s 1 ,s 2 ) + b 0u −¨y d ) + α˙ 1
                        where the non-linear function ξ(y d ,s 1 ,s 2 ) is derived as
                                                             H 1 (x,e,t)
                                          ξ(y d ,s 1 ,s 2 ) = h(x,t) +  .          (14.18)
                                                              F φ 	 F
                           Since ξ(y d ,s 1 ,s 2 ) is not easy to calculate precisely, an NN will be used
                        to cope with this unknown function. Hence, there exists an ideal weight
                        vector W so that the non-linear function ξ(y d ,s 1 ,s 2 ) can be expressed as
                                 ∗
                                           ξ(y d ,s 1 ,s 2 ) = W  ∗T φ(X) + ε      (14.19)


                                                                        T   5
                        where the input vector of NN is X = y d , ˙y d , ¨y d ,s 1 ,s 2  ∈ R .
                           In the following, a non-singular terminal sliding mode funnel control
                        approach is developed for tracking control of the motor servo system (14.7).
                        To make s 2 converge to zero within a finite time, the non-singular terminal
                        sliding mode manifold is designed as

                                                 q/p
                                               s
                                             β|˙ 2 | sgn(s 2 ) + s 2 = 0           (14.20)
                        where β> 0, p and q arepositiveodd integers with p < q.
                           Substituting (14.17)into(14.20) and using (14.19), the controller is
                        designed as



                               1          T                   1       1   p/q
                         u =−     −¨ y d + ˆ W φ(X) + μsgn(s 2 ) +  α˙ s 1 + |s 2 | sgn(s 2 )
                               b 0                          F φ 	 F   β
                                                                                   (14.21)
                        where ˆ W is the estimate of the unknown NN weight W and μ is the
                                                                            ∗
                                                                        T
                        upper bound of the NN approximation error ε and ˜ W φ(X),where ˜ W =
                        W − ˆ W is the NN weight estimation error.
                           ∗
                           The adaptive law for updating ˆ W is given by
                                                 ˙
                                                 ˆ W =  è( X)s 2                   (14.22)

                        where   is a positive definite and diagonal matrix.
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