## 3 Nonlinear State-Feedback VRFT for Approximate ORM Tracking Control Using Neural Networks  
An initial controller for the system (3) to achieve approximate ORM tracking employs the VRFT concept. Under assumptions _A1_ and _A2_, for tuning a nonlinear state-feedback controller, the designer may employ an input-state-output dataset of the form \(\left(\widetilde{\mathbf{u}}_{k},\widetilde{\mathbf{x}}_{k},\widetilde{ \mathbf{y}}_{k}\right)\), \(k=\overline{0,N-1}\), gathered from the process in an open-loop experiment lasting for \(N\) sample time steps, where persistently exciting \(\widetilde{\mathbf{u}}_{k}\) excites all the significant process dynamics. To achieve linear ORM tracking for a nonlinear process, a nonlinear state-feedback controller is more suitable than a linear one, being able to cope with the process nonlinearities. VRFT concept assumes that, if the controlled output \(\mathbf{y}_{k}\) produced in an open-loop experiment conducted on the stable process is both the control system's output and the ORM's output, then the closed-loop control system will match the reference model [4; 39; 40; 41; 42]. Let \(\widetilde{\mathbf{r}}_{k}=\mathbf{M}(z)^{-1}\widetilde{\mathbf{y}}_{k}\) be the virtual reference input that generates \(\widetilde{\mathbf{y}}_{k}\) when filtered through \(\mathbf{M}(z)\) which is assumed to be invertible with respect to the inverse filtering operation. It is called virtual since it is never set as a reference input to the closed-loop control system and it is only used in the offline controller tuning. The virtual states of the ORM are computable from Equation (2) as \(\widetilde{\mathbf{x}}_{k+1}^{-m}=\mathbf{f}^{m}(\widetilde{\mathbf{x}}_{k}^{ m},\widetilde{\mathbf{r}}_{k})\) serving to reconstruct the virtual extended state as \(\widetilde{\mathbf{x}}_{k}^{E}=\left[\widetilde{\mathbf{x}}_{k}^{m}\right]^{T }\left(\widetilde{\mathbf{x}}_{k}^{m}\right)^{T}\left(\widetilde{\mathbf{r}}_ {k}\right)^{T}\right]^{T}\) A controller that produces \(\widetilde{\mathbf{u}}_{k}\) when fed by \(\widetilde{\mathbf{x}}_{k}^{E}\) achieves the ORM tracking. VRFT translates the model reference tracking c.f. in Equation (5) to a controller identification c.f. A finite-time controller identification c.f. is [4]  
\[J_{VR}^{N}(\Theta)=\sum_{k=0}^{N-1}\left\|\widetilde{\mathbf{u}}_{k}-C( \widetilde{\mathbf{x}}_{k}^{E},\Theta)\right\|^{2}. \tag{6}\]  
Let the optimal controller parameter vector \(\Theta^{*}\) be the solution to the optimization problem \(\Theta^{*}=\underset{\Theta}{\operatorname{argmin}}J_{VR}^{N}(\Theta)\). _Theorem 2_ in [41] shows that if the controller parameterization is rich enough, then \(\Theta^{*}\) also minimizes \(J_{MR}^{\infty}\), proven for input-output models only. Motivated by [41], a formal proof is given as incentive for using state-feedback controllers tuned by nonlinear multi-input multi-output (MIMO) VRFT. Several other assumptions are considered:  
_A3_: The process (1) has an equivalent input-output form \(\mathbf{y}_{k}=\mathbf{P}(\mathbf{y}_{k-1},\ldots,\mathbf{y}_{k-ny},\mathbf{u }_{k-1},\ldots,\mathbf{u}_{k-nt})\), where \(ny\), \(nu\) are unknown process orders and the nonlinear map \(\mathbf{P}\) is invertible with respect to \(\mathbf{u}\), meaning that for given \(\mathbf{y}_{k}\), \(\mathbf{u}_{k}\) is recoverable as \(\mathbf{u}_{k-1}=\mathbf{P}^{-1}(\mathbf{y}_{k})\). Zero initial conditions are assumed at this point. Also, the ORM (2) has an equivalent input-output form \(\mathbf{y}_{k}^{m}=\mathbf{M}(\mathbf{y}_{k-1}^{m},\ldots,\mathbf{y}_{k-nt}^{ m},\mathbf{r}_{k-1},\ldots,\mathbf{r}_{k-nt})\) where \(nym\), \(nr\) are known ORM's orders, \(\mathbf{M}\) is a nonlinear invertible map with stable inverse, allowing the calculation of \(\mathbf{r}_{k-1}=\mathbf{M}^{-1}(\mathbf{y}_{k}^{m})\). Zero initial conditions are also assumed.