# LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design
## 2 Preliminaries and Discrete-Time Pole Region Problem Formulation
### \(D_{R}\) Regions for a Robust Pole Placement  
This subsection is devoted to the description of pole regions appropriate for LMI based controller design. The closed loop system poles determine not only CLS stability, but also other performance specifications. Standard requirements on pole position include damping and stability degree. These indices directly influence the corresponding CLS overshoot, rise time, settling time, decay rate, or natural frequency, which belong to the key performance measures. When the uncertain system is considered, it is impossible to prescribe the exact pole position. In this case, a robust pole placement approach determines a whole region in a complex plane where all the CLS poles should be placed. Since LMI framework offers efficient computational tools for robust control design for linear uncertain systems, significant effort has been made to reformulate various robust control problems as LMI. Therefore, in pole placement controller design problem, LMI description of the prescribed pole region is a crucial issue. The general concept of convex pole regions appropriate for LMI formulation was presented in Reference [7], as summarized below. **Definition 1**. : _(\(D_{R}\)--stability), [7]\(D_{R}\) region is defined as a subset to a complex plane \(C\)_  
\[D_{R}=\Big{\{}z\in C:R_{11}+R_{12}z+R_{12}^{T}z^{*}+R_{22}zz^{*}<0\Big{\}}, \tag{4}\]  
_where \(R_{11}=R_{11}^{T}\in R^{d\times d}\), \(R_{12}\in R^{d\times d}\), \(R_{22}=R_{22}^{T}\in R^{d\times d}\)._  
For \(R_{22}\geq 0\), the \(D_{R}\) region can be equivalently described by the LMI condition. **Remark 1**. : _It should be noted that:_  
* \(D_{R}\) _regions are symmetric with respect to the real axis of complex plane;_
* _Matrix_ \(A\in R^{n\times n}\) _is_ \(D_{R}\)_--stable if and only if all its eigenvalues lie in the corresponding_ \(D_{R}\) _region;_
* _Intersection of_ \(D_{R}\) _regions is again_ \(D_{R}\) _region (due to convexity)._  
For a linear system, system poles are given by the eigenvalues of the system matrix \(A\). The next Theorem provides a basic condition for a matrix \(A\) to have poles in the \(D_{R}\) region described by Equation (4). **Theorem 1**. : [7]\(A\) _matrix \(A\in R^{n\times n}\) is \(D_{R}\)-stable for the defined \(D_{R}\) region (4) in a complex plane, if and only if there exists a symmetric positive definite matrix \(P\) such that_  
\[R_{11}\otimes P+R_{12}\otimes(PA)+R_{12}^{T}\otimes(A^{T}P)+R_{22}\otimes(A^{ T}PA)<0. \tag{5}\]  
Inequality (5) can be interpreted as a generalization of the well-known Lyapunov stability condition for linear systems. Standard \(D_{R}\) regions corresponding to the specified performance are recalled below together with the respective matrices \(R_{11},R_{12},R_{22}\) for both continuous and discrete-time systems. While in Figures 1 and 2 (stability and prescribed stability degree) both continuous-time domains and their discrete-time counterparts are represented by convex regions, this is no more the case when the relative damping (given by a ratio of the imaginary and real part of the pole) is prescribed. While the corresponding pole region for a continuous-time case is the convex interior of the cone, as shown in Figure 3a, the discrete-time counterpart is the nonconvex interior of the logarithmic spiral shown in Figure 3b. Since it has a heart shape, it can be also called _cardioid_. Since the latter region is nonconvex, it cannot be described as a \(D_{R}\) region, Equation (4). To convert the nonconvex cardioid interior into the \(D_{R}\) region description, inner approximations can be used. The existing simple inner approximations, see References [8; 15], use an inner circle or ellipse, as depicted in Figure 4, which can be further combined with the half plane. Note that, due to characteristics of any \(D_{R}\) region, the center of a corresponding circle or ellipse lies on the real axis. Figure 1: Stable system pole regions.