# Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics +
## 4 Validation Case Studies
### Initial Controller with Model-Free VRFT  
An initial model-free multivariable IO controller is first found using model-free VRFT, as described in [11; 24; 32]. The ORM is \(\mathbf{M}(z)=diag(M_{1}(z),M_{2}(z))\) where \(M_{1}(z),M_{2}(z)\) are the  
Figure 3: The two-inputs–two-outputs aerodynamic system (TITOAS) experimental setup [50].  
discrete-time counterparts of \(M_{1}(s)=M_{2}(s)=1/(3s+1)\) obtained for a sampling period of \(T_{s}=0.1\) s. The VRFT prefilter is chosen as \(\mathbf{L}(z)=\mathbf{M}(z)\). A pseudo-random binary signal (PRBS) of amplitude \([-0.1;0.1]\) is used on both inputs \(u_{k,1},u_{k,2}\) to open-loop excite the pitch and azimuth dynamics simultaneously, as shown in Figure 4. The IO data \(\{\tilde{\mathbf{u}}_{k},\tilde{\mathbf{y}}_{k}\}\) is collected with low-amplitude zero-mean inputs \(u_{k,1},u_{k,2}\), in order to maintain the process linearity around the mechanical equilibrium, such that to fit the linear VRFT design framework (c) 2019 IEEE [12]).  
An un-decoupling linear output feedback error diagonal controller with the parameters computed by the VRFT approach is [12; 46]  
\[\mathbf{C}(z,\mathbf{\theta}) =\begin{bmatrix}P_{11}(z)\,/(1-z^{-1})&0\\ 0&P_{22}(z)\,/(1-z^{-1})\end{bmatrix}, \tag{36}\] \[P_{11}(z) =2.9341-5.8689z^{-1}+3.9303z^{-2}-0.9173z^{-3}-0.0777z^{-4},\] \[P_{22}(z) =0.6228-1.1540z^{-1}+0.5467z^{-2},\]  
where the parameter vector \(\mathbf{\theta}\) groups all the coefficients of \(P_{11}(z),P_{22}(z)\). Controller (36) is obtained for \(\mathbf{\theta}\) as the least squares minimizer of \(l_{VR}(\mathbf{\theta})=\sum_{k=1}^{N}\|\tilde{\mathbf{u}}_{k}^{L}-\mathbf{C}(z, \mathbf{\theta})\tilde{\mathbf{e}}_{k}^{L}\|_{2}^{2}\) where \(\tilde{\mathbf{u}}_{k}^{L}=\mathbf{L}(z)\tilde{\mathbf{u}}_{k}=\mathbf{L}(z)[ \tilde{u}_{k,1},\tilde{u}_{k,2}]^{\top},\tilde{\mathbf{e}}_{k}^{L}=\mathbf{L} (z)\tilde{\mathbf{e}}_{k}=\mathbf{L}(z)[\tilde{e}_{k,1},\tilde{e}_{k,2}]^{\top}\), \([\tilde{e}_{k,1},\tilde{e}_{k,2}]^{\top}=(\mathbf{M}^{-1}(z)-\mathbf{I}_{2} )[\tilde{y}_{k,1},\tilde{y}_{k,2}]^{\top}\). Here, \(l_{VR}(\mathbf{\theta})\) is an approximation of the c.f. \(l_{MR}^{\infty}\) from (5) obtained for \(\gamma=1\). The controller (36) will then close the feedback control loop as in \(\mathbf{u}_{k}=\mathbf{C}(z,\mathbf{\theta})(\mathbf{r}_{k}-\mathbf{y}_{k})\).  
Notice that, by formulation, the VRFT controller tuning aims to minimize the undiscounted (\(\gamma=1\)) \(l_{MR}^{\infty}\) from (5), but via the output feedback controller (36) that processes the feedback control error \(\mathbf{e}_{k}=\mathbf{r}_{k}-\mathbf{y}_{k}\). The same goal to minimize (5) is pursued by the subsequent IMF-AVI design of a state-feedback controller tuning for the extended process. Nonlinear (in particular, linear) state-feedback controllers can also be found by VRFT as shown in [24; 32], to serve as initializations for the IMF-AVI, or possibly, even for Polt-like algorithms. However, should this not be necessary, IO feedback controllers are much more data-efficient, requiring significantly less IO data to obtain stabilizing controllers.