# LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design
## 1 Introduction  
Analysis and control of linear dynamic systems have reached the mature stage in control system theory. Control design strategies and techniques developed for linear dynamic systems can be often adopted, even for more complex cases in the presence of nonlinearities or uncertainties. The model uncertainties have to be considered due to the often unknown or varying parameters in real world applications. In such cases, the uncertainties can be included by the set (family) of system models. A controller which guarantees the required performance for the whole set of system models (for the uncertain system) is called a robust controller. Robust control has attracted notable interest of many authors in the past decades and several excellent books have been written [1; 2; 3; 4]. In practice, there are also requirements other than stability on the closed loop system's overall performance. The closed loop dynamic behaviour is determined by system poles (eigenvalues of a system matrix), therefore pole placement belongs to widely used controller design methods also studied in the robust control framework [5; 6; 7; 8; 9]. The corresponding controller then guarantees that the closed loop system poles have the predetermined values, thus shaping the closed loop system dynamics. Pole placement or robust pole placement is used in many applications, e.g., motion system control [10], servo-system control [11], or power system control [12; 13]. Though a vast amount of literature has been devoted to robust control and control algorithm design, e.g., References [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11], and various approaches have been developed both in frequency domain and in state space, there still remain open questions in this field. The important issue is computational tractability which also motivated linear matrix inequality (LMI) problem formulation and the use of the corresponding computationally efficient techniques that enable solving a large set of convex problemsin a polynomial time (see, e.g., Reference [1]). Thus, the LMI approach is advantageous for solving control problems with a convex formulation. Intensive research has been devoted to transforming the non-convex or NP-hard control problem into a convex optimization one in the LMI framework (e.g., [2]). Inner approximations belong to possible frequently used tools to reformulate a non-convex into a convex problem. The existing approaches either provide simple inner approximation (which can be too loose) (e.g., Reference [8]) or more precise ones. However, the latter require more computational effort, e.g., polynomial approximations based on iterative computations, [14]). The possibility of using the LMI formulation motivated a recent concept of the so called LMI and \(D_{R}\) pole regions, [7]. Thus, in robust pole placement controller design, it is advantageous to determine the required pole position by defining the corresponding LMI region. Determination of the required closed loop system (CLS) pole position depends on performance requirements. When the continuous-time system is considered, the pole regions related to the prescribed stability degree and relative damping (which belong to the basic performance indices closely connected, e.g., with overshoot, rise time, settling time, and decay rate) are convex and can be formulated by LMI or \(D_{R}\) regions [4; 5; 6; 9]. However, this is not the case for the discrete-time counterpart, where the prescribed relative damping corresponds to the non-convex pole region. There exist a few results on this topic, [8; 15], however the provided inner approximations can be too conservative. This motivated our recent work [16], where several alternatives for inner approximations were briefly outlined. In this paper we further develop the results from Reference [16] and concentrate on discrete-time \(D_{R}\) region concept for the prescribed damping, which can be arbitrarily combined with other convex pole region limitations (e.g., stability degree guaranteeing the prescribed decay rate).