## 5 Simulation Results
### Modeling  
Depending on the location \(\bar{x}\), which is assumed to be fixed, this simple mechanical setup with the connecting elements of the variable lengths \(L_{1}\) and \(L_{2}\), which are given by linear springs with stiffness \(k\), can show nonlinear behavior including hysteresis and chaotic motion [18]. Both springs have the nominal length \(L_{0}\).  
The first goal of designing a partial state feedback controller (using only the measurable angle \(x_{1}\) and its derivative \(x_{2}\), where both are smoothened and estimated by a state observer) is to realize a stable transition between two bifurcation points at which the open-loop dynamics change between asymptotic stability and instability (represented by a jump-like motion between two significantly different angles). Second, a tracking controller is designed, which allows for following a reference trajectory \(y_{\text{d}}\) for the angle \(x_{1}\).  
According to the discussion above, the controller is parameterized with the output matrix  
\[\mathbf{C}=\begin{bmatrix}1&0&0\\ 0&1&0\end{bmatrix}, \tag{56}\]  
while  
\[\mathbf{C}_{\text{m}}=\begin{bmatrix}1&0&0\end{bmatrix} \tag{57}\]  
represents the sensor's relation to the state vector.  
The equations of motion of this system  
\[\dot{x}_{1} =x_{2}\] \[\dot{x}_{2} =\frac{k}{mR}\cdot\left(\frac{L_{1}-L_{0}}{L_{1}}a-\frac{L_{2}-L_ {0}}{L_{2}}\bar{x}\right)\sin(x_{1})-dx_{2}+\frac{k}{mR}\cdot\frac{L_{2}-L_{0 }}{L_{2}}\cdot\cos(x_{1})\cdot x_{3} \tag{58}\] \[\dot{x}_{3} =\frac{1}{T}\cdot(-x_{3}+u)\]  
with the velocity-proportional damping term \(dx_{2}\) and the state-dependent spring lengths  
\[L_{1}=\sqrt{\left(R\cos(x_{1})+a\right)^{2}+\left(R\sin(x_{1})^{2}\right)} \tag{59}\]  
and  
\[L_{2}=\sqrt{\left(\bar{x}-R\cos(x_{1})\right)^{2}+\left(x_{3}-R\sin(x_{1})^{2 }\right)} \tag{60}\]  
can be derived on the basis of [PERSON]'s equations of second kind. For this derivation, it is assumed that the total mass \(m\) of the system is located at a single point on the edge of the rotating disc.  
Figure 3: Application scenario: Zeeman catastrophe machine.  
For the LMI-based design procedure, the system model is re-written into the quasi-linear form  
\[\dot{\mathbf{x}}=\begin{bmatrix}0&1&0\\ a_{21}&-d&a_{23}\\ 0&0&-\frac{1}{T}\end{bmatrix}\cdot\mathbf{x}+\begin{bmatrix}0\\ 0\\ \frac{1}{T}\end{bmatrix}\cdot u \tag{61}\]  
of a polytopic uncertainty system model with \(\mathbf{A}_{v}\) and \(n_{v}=4\), where  
\[a_{21}=\frac{k}{mR}\cdot\left(\frac{L_{1}-L_{0}}{L_{1}}a-\frac{L_{2}-L_{0}}{L_ {2}}\tilde{x}\right)\frac{\sin(x_{1})}{x_{1}}\in[-5.2\;;\;14.4] \tag{62}\]  
and  
\[a_{23}=\frac{k}{mR}\cdot\frac{L_{2}-L_{0}}{L_{2}}\cdot\cos(x_{1})\in[-256\;; \;-17]. \tag{63}\]  
These intervals have been determined for the parameters \(T=0.05\), \(k=5\), \(m=0.1\), \(R=0.15\), \(L_{0}=0.1\), \(a=0.3\), \(\tilde{x}=0.23\), \(d=0.5\) and the worst-case operating domains  
\[x_{1}\in[-265\;;\;-95]\cdot\frac{\pi}{180} \tag{64}\]  
and  
\[x_{3}\in[-0.2\;;\;0.2]. \tag{65}\]  
All of these parameters and state variables are assumed to be given in terms of corresponding SI units.  
In the following simulations, the nonlinear model is used to represent the controlled plant and observer. The observer gain \(\mathbf{H}\) is designed according to Section 4 with a point-valued system matrix \(\mathbf{A}_{\text{nom}}\) given by the nominal values of the quasi-linear form. Note that this is only one solution of the parameter-dependent observer matrix \(\mathbf{A}(\xi)\). This simplification is possible under the assumption that both tracking and state estimation errors are small. In general, however, the stability of the nonlinear observer can be proven by checking the stability of all \((2n_{\mathbf{x}})^{2}\) combinations of the vertex matrices \(\mathbf{A}_{v}\) of the controlled plant to each extremal system matrix \(\mathbf{A}_{v}\) of the observer.