## 1 Introduction  
LMIs provide powerful tools for the design of guaranteed stabilizing controllers for exactly known (nominal) system models as well as for scenarios with bounded uncertainty in selected parameters. In both cases, the basic idea of using LMIs for control design (as well as for the dual task of state estimation) is the proof of asymptotic stability of the closed-loop dynamics by means of a suitable Lyapunov function candidate. Then, this stability proof can be included directly in the stabilizing control synthesis. The most important options for a control synthesis with the help of LMIs are the solution of a so-called pure feasibility problem (i.e., a controller guaranteeing asymptotic stability and, in the cases of bounded parameter uncertainty, achieving input-to-state stability) or extensions which deal with optimization problems (such as \(H_{2}\) and \(H_{\infty}\)). Furthermore, regions for the closed-loop eigenvalues can be specified so that minimum and maximum state variation rates during transient phases can be influenced systematically. Numerous references, such as [1, 2, 3, 4, 5, 6, 7], provide further information concerning techniques for an LMI-based control synthesis. For linear system models with uncertain, bounded parameters, parameter-independent Lyapunov function candidates can be used in many application scenarios. The resulting LMIs need to be specified for suitably chosen extremal system realizations of a polytopic uncertainty representation. Thereby, not only stability can be ensured but also restrictions on admissible eigenvalue domains (referred to as \(\Gamma\)-stability in [8]) can be guaranteed if the design task is solved successfully [5; 6]. If the assumption of a parameter-independent Lyapunov function leads to an unacceptably high level of conservativeness, augmented LMI representations can be introduced, which are based on parameter-dependent techniques (cf. [2; 3]). Besides pure parameter uncertainty, which reflects inevitable tolerances such as imprecisely known masses in mechanical systems, quasi-linear system models can also be handled with these techniques. In such cases, polytopic uncertainty representations are required that are composed of state-dependent system and input matrices, which are bounded by their element-wise defined worst-case realizations. In such a way, LMI techniques can also be employed for the design of robust controllers for nonlinear process models [9]. The statements above hold equally for the design of state observers with a Luenberger-like structure. Corresponding design criteria have been derived by exploiting the duality to control synthesis in [10]. However, the interconnection of independently designed controllers and state observers, where both of them need to be robust against bounded parameter uncertainty, is not ensured to be asymptotically stable. This issue is visualized in Section 2 with an academic example. As a consequence, it is inevitable to design controllers and observers simultaneously. For example, this problem is treated in [11; 12; 13] for discrete-time systems. The simultaneous controller and observer design leads to nonlinear matrix inequalities that can be solved in an iterative way. Moreover, most practical systems are subject to stochastic disturbances in the forms of actuator, process, and sensor noise. These systems can be controlled by an optimal observer-based LQG technique if the system model is linear and specified by precisely known, that is, point-valued system matrices. The LQG technique combines a linear quadratic regulator design with the optimal Gaussian (i.e., Kalman filter-based) state estimation. As summarized, for example, by [PERSON] in [7], the problem of control parameterizations that make sure that the noise-induced output and input covariances (i.e., uncertainty on the closed-loop system outputs and actuator signals) fall below specific threshold values, can be cast into LMIs.