## 2 Observer-Based Control of Systems with Polytopic Uncertainty: A Cautionary Tale  
As an introductory warning example, consider the unstable linear plant  
\[\dot{x}(t)=a\cdot x(t)+u(t) \tag{1}\]  
with the interval parameter \(a\in[a]=[a\ ;\ \overline{a}]=[1\ ;\ 3]\). It is desired to parameterize a linear feedback controller \(u(t)=-k\cdot\hat{x}(t)\), where \(\hat{x}(t)\) is a filtered state estimate resulting from the observer  
\[\hat{x}(t)=a_{\text{nom}}\cdot\hat{x}(t)+u(t)+h\cdot(y_{\text{m}}(t)-\hat{x}( t)). \tag{2}\]  
Here, \(y_{\text{m}}(t)=x(t)\) represents the measured plant output and \(a_{\text{nom}}\) is a fixed point-valued parameter taken from the interval \([a]\). **Remark 1**. : _Because this example should purely visualize the loss of validity of the classical separation principle of control and observer synthesis due to the bounded uncertainty of the parameter \(a\in[a]\), stochastic process and sensor noise are not considered in this section. Noise will be accounted for systematically in Sections 3 and 4._  
A classical control synthesis for the system model (1) would assume that \(\hat{x}(t)\approx x(t)\) holds. Then, the closed-loop dynamics result in  
\[\dot{x}(t)=(a-k)\cdot x(t), \tag{3}\]  
which become asymptotically stable if the controller gain \(k\) is chosen so that  
\[k>\overline{a}=3 \tag{4}\]  
holds. Similarly, the observer error dynamics with \(e(t)=x(t)-\hat{x}(t)\) would classically be stated as  
\[\dot{e}(t)=(a-h)\cdot e(t), \tag{5}\]  
under the assumption of an identical parameter \(a\) in both the plant and observer model, where  
\[h>\overline{a}=3 \tag{6}\]  
would ensure stability. When considering that \(a_{\text{nom}}\) in the observer differential Equation (2) is some point value taken from the interval \([\underline{a}\ ;\ \overline{a}]\), the resulting combined system dynamics turn into  
\[ \begin{bmatrix}\dot{x}(t)\\ \dot{\hat{x}}(t)\end{bmatrix} = \begin{bmatrix}a&-k\\ h&a_{\text{nom}}-k-h\end{bmatrix} \cdot \begin{bmatrix}x(t)\\ \hat{x}(t)\end{bmatrix} =\mathbf{\tilde{A}}(a,a_{\text{nom}},k,h)\cdot \begin{bmatrix} x(t)\\ \hat{x}(t),\end{bmatrix}  \tag{7}\]for which the control and observer gains \(k\) and \(h\) need to be chosen so that all eigenvalues of \(\mathbf{\dot{A}}(a,a_{\mathrm{nom}},k,h)\) are guaranteed to lie strictly within the open left complex half-plane for all possible realizations of \(a\in[a]\). Due to the simple structure of the scalar system model with a single uncertain parameter, the optimal choice (with respect to maximizing the admissible domains for the gains \(k\) and \(h\)) is obvious; it would be \(a_{\mathrm{nom}}=\overline{a}=3\). However, in more general settings, such simple statements are typically not possible. At least for systems with a small number of uncertain parameters, as well as controller and observer gains, the choice could be assisted analytically by the parameter space approach from [8]. For more complex models, only numerical techniques are helpful for determining the admissible domains of stabilizing controller and observer gains \(k\) and \(h\), respectively. For the academic example in this section, a numerical analysis of the stability domains is shown in Figure 1. Domains of robustly stabilizing gains are highlighted in a light gray color, while gains with unstable eigenvalues for at least one \(a\in[a]\) are highlighted in dark gray. Clearly, setting \(a_{\mathrm{nom}}\) to a value that is significantly smaller than \(\overline{a}\) (e.g., to the interval midpoint \(\frac{1}{2}\cdot(a+\overline{a})\) in Figure 1) requires much larger gains for \(k\) and \(h\) than the settings in Figure 1-d. The loss of the validity of the separation principle of control and observer design can be seen clearly from the fact that the boundary between stable and unstable realizations is a curved line that is not parallel to the coordinate axes \(k\) and \(h\) in Figure 1. The separation principle would only hold in cases where the stability boundaries for \(k\) and \(h\) are fully decoupled.