5. Modeling of the time-delay adaptive neural network and the delayed plant

The nonlinear delayed unknown plant and the neural network are given as:

$$aD\_t^a \mathbf{x}\_p = \mathbf{f}\_\mathbf{p} \left(\mathbf{x}\_p(t-\tau)\right) + \mathbf{g}\_\mathbf{p} \left(\mathbf{x}\_p(t)\right)\mathbf{u}\_\mathbf{p},$$

$$aD\_t^a \mathbf{x}\_n = \mathbf{A}(\mathbf{x}) + \mathbf{W}^\* \Gamma\_\mathbf{z}[\mathbf{x}(t-\tau)] + w\_{per} + \Omega\_\mathbf{u}$$

where xp, <sup>f</sup> <sup>p</sup> <sup>∈</sup> <sup>n</sup>, <sup>u</sup> <sup>∈</sup> <sup>m</sup>, gp <sup>∈</sup> nxn. And <sup>f</sup> <sup>p</sup>, is unknown and gp <sup>¼</sup> <sup>I</sup>, <sup>A</sup> ¼ �λI, with Γ Lypschitz function, W <sup>∗</sup> are the fixed weigths but unknown from the neural networks, which minimize the modeling error.

Theorem: We will show that epand en tend to zero and therefore e tends to zero, that is, the neural network follows the plant.

For this proposal, we first define the modeling error between the neural network and the plant: ep ¼ xp � xn, whose derivative in the time is

$$aD\_t^a e\_p = aD\_t^a \varkappa\_p - aD\_t^a \varkappa\_n \tag{31}$$

Adding and subtracting, to the right hand side from (34) the terms <sup>W</sup>^ <sup>Γ</sup>zð Þ xrx tð Þ � <sup>τ</sup> , <sup>α</sup><sup>p</sup> t,W^ � �

$$
\hbar \mathbf{a} D\_t^a e\_p = \mathbf{A}(e\_p) - \tilde{W} \Gamma\_\mathbf{z} [\mathbf{x}\_n(t-\tau)] + \hat{W} \mathcal{O}(e\_p(t-\tau)) + \tilde{u}\_p \tag{32}
$$
