## 1 Introduction  
One of the most important advantages of the use of LMIs in the design of robust controllers and state observers is their capability to account for bounded parameter uncertainty by means of suitable (often polytopic) uncertainty models. In such a way, it becomes possible to include a guaranteed stability proof of the uncertain linear dynamic system directly in the design stage. Moreover, polytopic uncertainty models can also be employed to over-bound the influence of nonlinear state dependencies in the system and output equations if they can be reformulated in terms of a quasi-linear representation. In addition to the task of system stabilization, further optimizations of the closed-loop dynamics become possible, which include a reduction of sensitivity against external disturbances (commonly in an \(H_{\infty}\) sense) or the specification of admissible eigenvalue domains (so-called regions of \(\Gamma\)-stability, which serve among others as a representation for minimum damping ratios or bandwidth limitations). For general references about the theory and possible applications of LMIs in the frame of control and observer synthesis, the reader is referred to the works of [1, 2, 3, 4, 5, 6, 7, 8]. In addition, approaches for the assignment of admissible eigenvalue domains, partially with applications to the control and oscillation attenuation of mechanical systems with elastic spring elements, were considered recently in [9, 10], where a continuous-time setting was taken into account. For the discrete-time counterpart, cf. [11, 12]. In addition to continuous-time systems with an integer-order time derivative, LMI techniques have recently also become an active topic of research if robust controllers and observers are to be designed for fractional-order systems with uncertainty [13; 14; 15; 16]. As far as output feedback control procedures (instead of full state feedback controllers) are investigated in this framework, especially the work of [PERSON] [17] should be mentioned. However, in contrast to our paper, it did not deal with a simultaneous consideration of bounded parameter uncertainty, on the one hand, and stochastic input, system, and measurement noise, on the other hand. Besides the design of controllers with constant, state-independent gains, LMI design procedures were also developed in recent years to allow for robust gain adaptation schemes. Moreover, an example, where methods form the field of interval analysis [18; 19] were employed for an underlying reachability analysis, was published in [20]. Note that one of the attractive properties of LMIs is the existence of powerful software libraries that can be employed for a large variety of design tasks. In the current paper, the numerical implementation of the suggested LMI-based solution procedure makes use of Yalmip [21] as the user interface to MATLAB, while SeDuMi[22] is employed as the underlying solver. In addition to the aforementioned bounded uncertainties, most practical systems are also influenced by stochastic actuator, process, and sensor noise. Assuming linear dynamics, the combination of a linear quadratic regulator design with an optimal state estimation by means of Kalman filters (for additive Gaussian noise processes) can be seen as the best solution if a feedback of all state variables is desired. However, classical formulations for the solution of this design task assume perfectly known system, input, and output matrices [23; 24; 25]. This aspect directly motivates the goal to employ robust LMI-based design approaches (which are naturally suited for bounded uncertainty) also in a stochastic setting. An example, where such a kind of approach was developed, was published recently by [6]. There, the problem of control parameterizations was solved such that the output and input covariances (i.e., uncertainty on the closed-loop controlled states and actuator signals) fell below specific threshold values.