## 5 Simulation Results
### Tracking the Unstable Branch of the Bifurcation Diagram  
Figure 4a shows the numerically computed equilibria of the system (58) in dependence of piecewise constant inputs \(u=x_{3}\). The hysteresis behavior of the open-loop system can be clearly seen so that starting with angles of approximately \(-110^{\circ}\), an increase of \(x_{3}\) leads to a jump in the angle \(x_{1}\) (transition from the open to the filled green triangle at some \(x_{3}>0\) in Figure 4), while successively reducing \(x_{3}\) leads to a jump between two different angles for some \(x_{3}<0\) (red triangles in Figure 4). Figure 4a can be interpreted as a bifurcation diagram in the phase space, where the branch between both open green and red triangles is unstable and is not visible in an open-loop operation of the system.  
However, this branch can be visualized by using the closed-loop control strategy as shown in Figure 4b,c with  
\[u=-k_{1}\cdot(x_{1}-y_{\text{d}})-k_{2}x_{2}-k_{3}x_{3}, \tag{66}\]  
where \(y_{\text{d}}\) is a piecewise constant reference signal and \(k_{3}=0\) holds in the second scenario due to the choice of \(\mathbf{C}\) according to (56) for the LMI-based controller (12).  
For the sake of comparison with a standard approach (which, however, does not strictly guarantee closed-loop stability due to the parameter dependency of the system model), an LQG approach has been implemented additionally. It makes use of a continuous-time steady-state Extended Kalman Filter. Both LQG and LMI approaches are parameterized with \(\mathbf{G}_{\text{p}}=10^{-2}\cdot\mathbf{I}\), \(g_{\text{u}}=0\), and \(g_{\text{m}}=10^{-2}\) in this subsection, where \(\mathbf{C}_{\text{m}}\) is chosen according to (57).  
The controller optimization in the LQG case was performed with \(\mathbf{Q}=\mathbf{I}\) and \(R=1\) for the point-valued parameters \(a_{21}=14.4\) and \(a_{23}=-17\). These values were chosen to reflect the strongest instability of the open-loop plant for the control synthesis. Due to the state dependency of the system matrix in (61), this parameterization does not provide a strict proof of stability. In contrast, the LMI approach provides a strict proof of stability and considerably reduced deviations in transient operation from the unstable branch of the bifurcation diagram compared to the LQG design. To limit the observer's eigenvalues so that a simulation with a fixed step size of \(10^{-3}\) is admissible, the heuristically chosen penalty term \(5\cdot 10^{-5}\cdot\|\mathbf{Y}_{\text{O}}\|_{2}\) was added onto \(J\), defined in (41).  
All simulations were performed for piecewise constant signals for \(y_{\text{d}}\) (hold time of 5 s) with increments of approximately \(1.41^{\circ}\). If only the terminal states of each step are plotted (cf. Figure 3(d)), the LMI-case reconstructs the exact unstable branch with deviations of less than \(10^{-4}\), which is slightly better than in the LQG case. Obviously, the LMI-based solution approach does not only provide a strict proof of stability of the observer-based control structure but in addition, it leads to a significant reduction of the control effort, resulting also in much smoother values for \(x_{3}\) in Figure 3(c). The reduction of the control effort is investigated in more detail in the following example.  
Figure 4: Guidance of the [PERSON] catastrophe machine along the unstable branch of equilibria in the \((x_{3},x_{1})\)-plane; comparison of an observer-based full state feedback control parameterized by the LQG technique with the LMI-based parameterization of (12).