# Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics +
## 4 Validation Case Studies
### Input-State-Output Data Collection  
ORM tracking is intended by making the closed loop CS match the same ORM \(\mathbf{M}(z)=diag(M_{1}(z),M_{2}(z))\). With the linear controller (36) used in closed-loop to stabilize the  
Figure 4: Open-loop input–output (IO) data from the two-inputs–two-outputs aerodynamic system (TITOAS) for Virtual Reference Feedback Tuning (VRFT) controller tuning [46].  
process, input-state-output data is collected for 7000 s. The reference inputs with amplitudes \(r_{k,1}\in[-2;2]\), \(r_{k,2}\in[-1.4;1.1]\) model successive steps that switch their amplitudes uniformly random at 17 s and 25 s, respectively. On the outputs \(u_{k,1},u_{k,2}\) of both controllers \(C_{11}(z),C_{22}(z)\), an additive noise is added at every 2 nd sample as an uniform random number in \([-1.6;1.6]\) for \(C_{11}(z)\) and in \([-1.7;1.7]\) for \(C_{22}(z)\). These additive disturbances provide an appropriate exploration, visiting many combinations of input-states-outputs. The computed controller outputs are saturated to \([-1;1]\), then sent to the process. The reference inputs \(r_{k,1},r_{k,2}\) drive the ORM [12; 46]:  
\[\begin{cases}x_{k+1,1}^{m}=0.9672x_{k,1}^{m}+0.03278r_{k,1},\\ x_{k+1,2}^{m}=0.9672x_{k,2}^{m}+0.03278r_{k,2},\\ \mathbf{y}_{k}^{m}=[y_{k,1}^{m},y_{k,2}^{m}]^{\top}=[x_{k,1}^{m},x_{k,2}^{m}]^ {\top}.\end{cases} \tag{37}\]  
Then the states of the ORM (also outputs of the ORM) are also collected along with the states and control inputs of the process, to build the process extended state (4). Let the extended state be:  
\[\mathbf{x}_{k}^{E}=[\underbrace{x_{k,1}^{m},x_{k,2}^{m}}_{(\mathbf{x}_{k}^{m}) ^{\top}},\underbrace{r_{k,1},r_{k,2}}_{\mathbf{r}_{k}^{\top}},(\mathbf{x}_{k} )^{\top}]^{\top}. \tag{38}\]  
Essentially, the collected \(\mathbf{x}_{k}^{E}\) and \(\mathbf{u}_{k}\) builds the transitions dataset \(D=\{(\mathbf{x}_{1}^{E},\mathbf{u}_{1},\mathbf{x}_{2}^{E}),...,(\mathbf{x}_{ \sigma 0000}^{E},\mathbf{u}_{70000},\mathbf{x}_{\sigma 0001}^{E})\}\) for \(N=70,000\), used for the IMF-AVI implementation. After collection, an important processing step is the data normalization. Some process states are scaled in order to ensure that all states are inside \([-1;1]\). The scaled process state is \(\tilde{\mathbf{x}}_{k}=[\omega_{l,k}/7200,25\cdot\Omega_{l,k},\alpha_{l,k}, \omega_{v,k}/3500,40\cdot\Omega_{v,k},\alpha_{v,k}]^{\top}\in\mathbb{R}^{6}\) and \(\mathbf{u}_{k}=[u_{k,1},u_{k,2}]^{\top}\in\mathbb{R}^{2}\). Other variables such as the reference inputs, the ORM states and the saturated process inputs already have values inside \([-1;1]\). The normalized state is eventually used for state feedback. Collected transition samples are shown in Figure 5 only for the process inputs and outputs, ORM's outputs and reference inputs, for the first 400 s (4000 samples) out of 7000 s [12].  
Note that the reference input signals \(r_{k,1},r_{k,2}\) used as sequences of constant amplitude steps for ensuring good exploration, do not have a generative model that obeys the [PERSON] assumption. To avoid this problem, the piece-wise constant reference input generative model \(\mathbf{r}_{k+1}=\mathbf{r}_{k}\) is employed by eliminating from the dataset \(D\) all the transition samples that correspond to switching reference inputs instants (i.e., when at least one of \(r_{k,1},r_{k,2}\) switches) (