## 5 Simulation Results
### Trajectory Tracking Control  
As a second application for the proposed LMI-based parameterization of the control laws (10), Case 1, and (12), Case 2, and its comparison with the LQG design, consider the following four scenarios:  
* \(\mathbf{G}_{\mathrm{P}}=10^{-2}\cdot\mathbf{I}\), \(g_{u}=2\cdot 10^{-2}\,\mathrm{m}\), \(g_{\mathrm{m}}=10^{-2}\,\mathrm{rad}\approx 0.573\,\mathrm{deg}\);
* \(\mathbf{G}_{\mathrm{P}}=10^{-2}\cdot\mathbf{I}\), \(g_{u}=0\,\mathrm{m}\), \(g_{\mathrm{m}}=10^{-2}\,\mathrm{rad}\approx 0.573\,\mathrm{deg}\);
* \(\mathbf{G}_{\mathrm{P}}=10^{-2}\cdot\mathbf{I}\), \(g_{u}=0\,\mathrm{m}\), \(g_{\mathrm{m}}=5\cdot 10^{-2}\,\mathrm{rad}\approx 2.865\,\mathrm{deg}\);
* \(\mathbf{G}_{\mathrm{P}}=10^{-2}\cdot\mathbf{I}\), \(g_{u}=0\,\mathrm{m}\), \(g_{\mathrm{m}}=10\cdot 10^{-2}\,\mathrm{rad}\approx 5.730\,\mathrm{deg}\). Here, we assume tracking of a continuous reference trajectory along the unstable branch of the previously investigated bifurcation diagram. Figure 5 presents a comparison of all three control approaches with an ideal tracking behavior in a noise-free setting. Obviously, the control effort of the LMI-based output feedback control is the best in all scenarios. This is achieved by minimizing the domain of non-provable stability properties and by feeding back only the first two components of the estimated state vector. To achieve accurate tracking of the reference trajectory \(y_{\mathrm{d}}\) in Figure 6, the feedforward control signal \(u_{\mathrm{ff}}\) in (10) and (12) was chosen as a signal that is piecewise constant for  
Figure 5: Comparison of the control effort for the LMI-based full state feedback controller (Case 1), the LMI-based output feedback controller (Case 2), and the LQG technique with the ideal noise-free control signal. intervals of \(100\,\mathrm{ms}\). It is determined in the sense of a P-type iterative learning control approach by the update rule  
\[u_{\text{ff},i+1}(t_{k})=u_{\text{ff},i}(t_{k})+S_{k}\cdot(y_{\text{d}}(t_{k})- \hat{y}_{i}(t_{k})) \tag{67}\]  
with the initialization  
\[u_{\text{ff},1}(t_{k})=0, \tag{68}\]  
where \(u_{\text{ff},i}(t)=u_{\text{ff},i}(t_{k})\) generally holds for \(t_{k}\leq t<t_{k+1}\). The learning gain \(S_{k}\), weighting the difference between the observed system output \(\hat{y}_{i}(t_{k})=\mathbf{C}_{\text{m}}\cdot\hat{\mathbf{x}}_{i}(t_{k})\) in the iteration \(i\) and the reference signal, is given by  
\[S_{k}=\Big{(}\mathbf{C}_{\text{m}}\cdot(-\mathbf{A}_{k}+\mathbf{B}_{k}\cdot \mathbf{K})^{-1}\cdot\mathbf{B}_{k}\Big{)}^{-1}, \tag{69}\]  
where \(\mathbf{K}\) is either the LQG controller's gain, the full state feedback gain in (10), or \(\mathbf{K}=\big{[}\mathbf{K}_{0}\quad 0\big{]}\) for the partial state feedback (12); \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) result from evaluating the system matrices in (61) for the current state estimate \(\hat{\mathbf{x}}_{i}(t_{k})\) at the point of time \(t_{k}\) during the epoch \(i\) (also denoted as _trial_ or _pass_ in the literature on iterative learning control). Figure 6: Comparison of the output tracking behavior as well as the state reconstruction accuracy for the proposed LMI-based output feedback controller (Case 2) after the 10 th iteration. For the first iteration \(i=1\), the controller  
\[u_{i}(t)=u_{\text{ff},i}(t)-k_{1}\cdot(x_{1}(t)-y_{\text{d}}(t))-k_{2}x_{2}(t)-k_{ 3}x_{3}(t), \tag{70}\]  
is identical to (66) introduced in the previous subsection due to the initialization (68). In addition to the accurate trajectory tracking for the final iteration \(i=10\), Figure 6 shows the strong suppression of measurement noise by the LMI-based observer and the fact that the actual trajectories for the state \(x_{1}\) and the corresponding estimate \(\hat{x}_{1}\) practically cannot be distinguished within the graphical resolution. Finally, Figures 7 and 8 show the noise-insensitive convergence of the control signals and root mean square (RMS) output tracking errors for all four scenarios S1-S4.