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Tham khảo tài liệu 'adaptive control 2011 part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Adaptive Control for a Class of Non-affine Nonlinear Systems via Neural Networks 343 Assumption 4. The approximation error s is bounded as follows HI SN 15 where SN 0 is an unknown constant. Let M and N be the estimates respectively of M and N . Based on these estimates let Uad be the output of the NN Uad M ơ N Xm 16 Define M M M and N N N where we use notations Z diag M N Z diag M N Z diag M N for convenience. Then the following inequality holds T Z 7- 1 RIM IK 17 The Taylor series expansion of ơ NT Xnn for a given Xnn can be written as ở Ntx ở ỈỈTx ở NTx NTx O NTX 2 18 nn nnnn nn nn with Ở ở ỈNTXnn and Ở denoting its Jacobian O NTXnn 2 the term of order two. In the following we use notations Ở ơ NT Xnn Ở ở NTXnn . With the procedure as Appendix A the approximation error of function can be written as MTở NTxnn MíT ở NTxnn MT Ở Ở NTxm MíTỞ NTxnn O 19 and the disturbance term O can be bounded as M dKMH M kNTx- M 20 where the subscript F denotes Frobenius norm and the subscript 1 the 1-norm. Redefine this bound as IM PoMM ỈĨ Xnn 21 344 Adaptive Control where P max MH JIn f J M 11 and m 1 xnnMT0 llF T NTXm 1. Notice that p is an unknown coefficient whereas i9 is a known function. I 3.2 Parameters update law and stability analysis Substituting 14 and 16 into 13 we have T -kT MT d NT xm - Mt Td N Txm f - V -S s xm . 22 X nn s X nn X I r X nn R Using 19 the above equation can become T -kT MiT d-ở N MíTờ NT x f-S-V s. 23 X nn nn I r Theorem 1. Consider the nonlinear system represented by Eq. 2 and let Assumption 1-4 hold. If choose the approximation pseudo-control input f as Eq. 12 use the following adaptation laws and robust control law nrí T y 1 _ a 1 a 24 - Ảị Ạ I ị Y T F 1 tanh 7 Vr l . 1 tanh where F FT 0 R RT 0 are any constant matrices k1 0 and Y 0 are scalar design parameters ị is the estimated value of the uncertain disturbance term ị max p SN defining ị ị-ị with ị error of ị then guarantee that all signals in the system are uniformly bounded and that the .
