# Learning Output Reference Model Tracking for Higher-Order Nonlinear Systems with Unknown Dynamics +
## 4 Validation Case Studies
### IMF-AVI on the Nonlinear TITOAS Aerodynamic System  
The ORM tracking problem on the more challenging TITOAS angular position control [50] (Figure 3) is aimed next. The azimuth (horizontal) motion behaves as an integrator while the pitch (vertical) positioning is affected differently by the gravity for the up and down motions. Coupling between the two channels is present. A simplified deterministic continuous-time state-space model of this process is given as two coupled state-space sub-systems [12; 46]:  
Figure 2: Convergence results of the linearly paramaterized iterative model-free approximate Value Iteration (LP-IMF-AVI) for the linear process example.  
\[\begin{cases}\dot{\omega}_{h}=(sat(U_{h})-M_{h}(\omega_{h}))/2.5\cdot 10^{-5},\\ \dot{K}_{h}=0.216F_{h}(\omega_{h})cos(\alpha_{v})-0.058\Omega_{h}+0.0178 sat(U_{v })cos(\alpha_{v}),\\ \Omega_{h}=K_{k}/(0.0238 cos^{2}(\alpha_{v})+3\cdot 10^{-3}),\\ \dot{\omega}_{h}=\Omega_{h},\end{cases} \tag{33}\] \[\begin{cases}\dot{\omega}_{v}=(sat(U_{v})-M_{v}(\omega_{v}))/1.6 3\cdot 10^{-4},\\ \Omega_{v}=\frac{1}{0.03}\left(\begin{matrix}0.2F_{v}(\omega_{v})-0.0127\Omega _{v}-0.0935 ina\alpha_{v}+\\ -9.28\cdot 10^{-6}\Omega_{v}|\omega_{v}|+4.17\cdot 10^{03}sat(U_{h})-0.05 cos \alpha_{v}+\\ -0.021\Omega_{h}^{2}sina_{v}cos\alpha_{v}-0.093 sin\alpha_{v}+0.05\end{matrix} \right)\\ \dot{\alpha}_{v}=\Omega_{v},\end{cases}\]  
where \(sat()\) is the saturation function on \([-1;1]\), \(U_{h}=u_{1}\) is the azimuth motion control input, \(U_{v}=u_{2}\) is the vertical motion control input, \(\alpha_{h}(rad)=y_{1}\in[-\pi,\pi]\) is the azimuth angle output, \(\alpha_{v}(rad)=y_{2}\in[-\pi/2,\pi/2]\) is the pitch angle output, other states being described in [11; 48]. The nonlinear static characteristics obtained by polynomial fitting from experimental data are for \(\omega_{v},\omega_{h}\in(-4000;4000)\)[46]:  
\[\begin{array}{rcl}M_{v}(\omega_{v})&=&9.05\times 10^{-12}\omega_{v}^{3}+2.7 6\times 10^{-10}\omega_{v}^{2}+1.25\times 10^{-4}\omega_{v}+1.66\times 10^{-4}, \\ F_{v}(\omega_{v})&=&-1.8\times 10^{-18}\omega_{v}^{5}-7.8\times 10^{-16}\omega_{v}^{4 }+4.1\times 10^{-11}\omega_{v}^{3}+2.7\times 10^{-8}\omega_{v}^{2}\\ &&+3.5\times 10^{-5}\omega_{v}-0.014,\\ M_{h}(\omega_{h})&=&5.95\times 10^{-13}\omega_{h}^{3}-5.05\times 10^{-10}\omega_{h} ^{2}+1.02\times 10^{-4}\omega_{h}+1.61\times 10^{-3},\\ F_{h}(\omega_{h})&=&-2.56\times 0^{-20}\omega_{h}^{5}+4.09\times 10^{-17}\omega_{h} ^{4}+3.16\times 10^{-12}\omega_{h}^{3}-7.34\times 10^{-9}\omega_{h}^{2}\\ &&+2.12\times 10^{-5}\omega_{h}+9.13\times 10^{-3}.\end{array} \tag{34}\]  
A zero-order hold on the inputs and a sampler on the outputs of (33) lead to an equivalent MP discrete-time model of sampling time \(T_{s}=0.1s\) and of relative degree 1 (one), suitable for input-state data collection  
\[P:\begin{cases}\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_{k},\mathbf{u}_{k}),\\ \mathbf{y}_{k}=\mathbf{g}(\mathbf{x}_{k})=[\alpha_{h,k},\alpha_{v,k}]^{\top}, \end{cases} \tag{35}\]  
where \(\mathbf{x}_{k}=[\omega_{h,k},\Omega_{h,k},\alpha_{h,k},\omega_{v,k},\Omega_{v, k},\alpha_{v,k}]^{\top}\in\mathbb{R}^{6}\) and \(\mathbf{u}_{k}=[u_{k,1},u_{k,2}]^{\top}\in\mathbb{R}^{2}\). The process' dynamics will not be used for learning the control in the following.