## 3 Methodological Background: LMI-Based Control and Observer Design
### Robust State Observation  
Following the duality principle between control and observer synthesis, as described for example in [10; 26], leads to the counterparts of Theorem 1 and Corollary 1 according to the following Theorem 3, where the observer differential equation is given by Equation (11).  
**Theorem 3** ([10] Sufficient stability condition for robust state observer).: _The error dynamics of a robust state observer according to Equation (11) with the exactly known output matrix \(\mathbf{C}_{\mathrm{m}}\) are robustly asymptotically stable if the observer gain \(\mathbf{H}\) satisfies the bilinear matrix inequalities_  
\[\left(\mathbf{A}_{v}-\mathbf{HC}_{\mathrm{m}}\right)^{T}\cdot\mathbf{P}_{ \mathrm{O}}^{\prime}+\mathbf{P}_{\mathrm{O}}^{\prime}\cdot\left(\mathbf{A}_{v }-\mathbf{HC}_{\mathrm{m}}\right)\prec 0 \tag{27}\]  
_for some positive definite matrix \(\mathbf{P}_{\mathrm{O}}^{\prime}=\mathbf{P}_{\mathrm{O}}^{\prime}{}^{T}\succ 0\) at all polytope vertices \(v\in\{1,\ldots,n_{v}\}\) in (14)._  
As stated before, the matrix \(\mathbf{C}_{\mathrm{m}}\) in (27) does not necessarily have to be identical to the matrix \(\mathbf{C}\) to be fed back in the control law (12). Note that the prime symbol \((\cdot)^{\prime}\) in (27) has been introduced to indicate that the matrix \(\mathbf{P}_{\mathrm{O}}^{\prime}\) is typically not the same as \(\mathbf{P}_{\mathrm{O}}\) in the following corollary.  
**Corollary 3** ([10]).: _To allow for an efficient solution of the bilinear matrix inequalities in Theorem 3, after considering the duality between control and observer synthesis by evaluating the transposed of (27) according to_  
\[\left(\mathbf{A}_{v}-\mathbf{HC}_{\mathrm{m}}\right)\cdot\mathbf{P}_{\mathrm{O }}+\mathbf{P}_{\mathrm{O}}\cdot\left(\mathbf{A}_{v}-\mathbf{HC}_{\mathrm{m}} \right)^{T}\prec 0, \tag{28}\]  
_the linearizing change of variables_  
\[\mathbf{P}_{\mathrm{O}}=\mathbf{Q}_{\mathrm{O}}{}^{-1}\quad\text{and}\quad \mathbf{H}=\left(\mathbf{Y}_{\mathrm{O}}\mathbf{P}_{\mathrm{O}}\right)^{T} \tag{29}\]is introduced. Multiplying (27) from the left and right with the matrix \(\mathbf{Q}_{\mathrm{O}}\) leads to equivalent LMIs_  
\[\mathbf{A}_{v}^{T}\mathbf{Q}_{\mathrm{O}}+\mathbf{Q}_{\mathrm{O}}\mathbf{A}_{v} -\mathbf{C}_{\mathrm{m}}^{T}\mathbf{Y}_{\mathrm{O}}-\mathbf{Y}_{\mathrm{O}}^{T }\mathbf{C}_{\mathrm{m}}\prec 0 \tag{30}\]  
_for all polytope vertices \(v\in\{1,\ldots,n_{v}\}\) in (14)._  
Main Results: Iterative LMI Solution for a Combined Control and Observer Synthesis in the Presence of Stochastic Noise  
This section provides the main result of this paper. It consists of the description of an augmented stochastic differential equation model, comprising the controlled plant as well as the robust state observer. Based on this augmented model, optimality conditions for the combined computation of the control and observer gains are presented which allow us to minimize the size of the domain around the desired operating point for which stability cannot be proven due to the presence of stochastic actuator, process, and sensor noise. For a visualization of such domains in the frame of oscillation attenuation for boom cranes in marine applications, the reader is referred to [27].