# Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics +
## 3 Solution to the ORM Tracking Problem
For example, with NNs, adding more layers and more neurons, enhances the approximation capability and theoretically reduces the residuals in (24). **Theorem 2**. : _Let the sequences \(\{\tilde{\mathbf{C}}_{j}(\mathbf{x}_{k}^{E})\}\) and \(\{\tilde{Q}_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\}\) evolve as in (24), (25), the sequences \(\{\mathbf{C}_{j}(\mathbf{x}_{k}^{E})\}\) and \(\{Q_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\}\) evolve as in (13), (14). Initialize \(\tilde{Q}_{0}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})=Q_{0}(\mathbf{x}_{k}^{E}, \mathbf{u}_{k})=0,\forall(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\) and let A3 hold. Then_  
\[\underline{\psi}Q_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\tilde{Q}_{j}( \mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\overline{\psi}Q_{j}(\mathbf{x}_{k}^{ E},\mathbf{u}_{k}) \tag{27}\]  
Proof. : (c) 2019 IEEE [12]) First, the development proceeds by induction for the left inequality. For \(j=0\) it is clear that \(\underline{\psi}Q_{0}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\tilde{Q}_{0}( \mathbf{x}_{k}^{E},\mathbf{u}_{k})\). For \(j=1\), (13) produces \(Q_{1}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})=v(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\) and left-hand side of (26) reads \(\min_{\mathbf{u}}\{\underline{\psi}v(\mathbf{x}_{k}^{E},\mathbf{u}_{k})+0\} \leq\tilde{Q}_{1}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\). Then \(\underline{\psi}Q_{1}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\tilde{Q}_{1}( \mathbf{x}_{k}^{E},\mathbf{u}_{k})\). Next assume that  
\[\underline{\psi}Q_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\tilde{Q}_{j}( \mathbf{x}_{k}^{E},\mathbf{u}_{k}) \tag{28}\]  
holds at iteration \(j\). Based on (28) used in (26), it is valid that  
\[ \begin{split}\min_{\mathbf{u}}\{\underline{\psi}v(\mathbf{x}_{k}^ {E},\mathbf{u}_{k})+\gamma\underline{\psi}Q_{j}(\mathbf{x}_{k+1}^{E},\mathbf{u })\}\leq\\ \min_{\mathbf{u}}\{\underline{\psi}v(\mathbf{x}_{k}^{E},\mathbf{u} _{k})+\gamma\tilde{Q}_{j}(\mathbf{x}_{k+1}^{E},\mathbf{u})\}\leq\tilde{Q}_{j+1 }(\mathbf{x}_{k}^{E},\mathbf{u}_{k}).\end{split}  \tag{29}\]  
Notice from (29) that  
\[ \begin{split}\min_{u}\{\underline{\psi}v(\mathbf{x}_{k}^{E}, \mathbf{u}_{k})+\gamma\underline{\psi}Q_{j}(\mathbf{x}_{k+1}^{E},\mathbf{u })\}=\underline{\psi}\min_{u}\{v(\mathbf{x}_{k}^{E},\mathbf{u}_{k})+\gamma Q_ {j}(\mathbf{x}_{k+1}^{E},\mathbf{u})\}\\ \stackrel{{\eqref{eq:13}}}{{=}}\underline{\psi}Q_{j+1}( \mathbf{x}_{k}^{E},\mathbf{u}_{k})\end{split}  \tag{30}\]  
From (29), (30) it follows that \(\underline{\psi}Q_{j+1}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\leq\tilde{Q}_{j+1 }(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\) proving the left side of (27) by induction. The right side of (27) is shown similarly, proving Theorem 2. _Comment 6_. : (c) 2019 IEEE [12]) Theorem 2 shows that the trajectory of \(\{\tilde{Q}_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\}\) closely follows that of \(\{Q_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\}\) in a bandwidth set by \(\underline{\psi},\overline{\psi}\). It does not ensure that \(\{\tilde{Q}_{j}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})\}\) converges to a steady-state value, but in the worst case, it oscillates around \(Q^{*}(\mathbf{x}_{k}^{E},\mathbf{u}_{k})=lim_{j\to\infty}Q_{j}(\mathbf{x}_{k}^ {E},\mathbf{u}_{k})\) in a band that can be made arbitrarily small by using powerful approximators. By minimizing over \(\mathbf{u}_{k}\) both sides of (27), similar conclusions result for the controller sequence \(\{\tilde{\mathbf{C}}_{j}(\mathbf{x}_{k}^{E})\}\) that closely follows \(\{\mathbf{C}_{j}(\mathbf{x}_{k}^{E})\}\). In the following Section, the IMF-AVI is validated on two illustrative examples. The provided theoretical analysis supports and explains the robust learning performance of the nonlinearly parameterized IMF-AVI with respect to the linearly parameterized one.