# Comparison of Nonlinear and Linear Controllers for Magnetic Levitation System
## 3 Control Design
### Robust Pole Placement--Pole Region Approach
#### 3.1.2 \(D_{r}\) Regions and \(D_{r}\) Stability  
Robust pole-placement belongs to efficient tools for uncertain system control design. So called \(D_{R}\) region approach, introduced in [9] provides computationally tractable LMI condition which can be used for a robust controller design for uncertain polytopic systems. We start with definition of basic notions. A definition of \(D_{R}\) region in the complex plain as set of points conforming to the specified matrix inequality is taken from [9].  
\[D_{R}=\{z\in C:R_{11}+R_{12}z+R_{12}^{T}z^{*}+R_{22}zz^{*}<0\}. \tag{11}\]  
To get convex formulation transformable to LMI, we assume that \(R_{22}\) is positive definite (semidefinite). A generalized-\(D_{R}\) stability notion can be defined for a specific \(D_{R}\) domain determined for system poles. Matrix \(A_{CL}(.)\) is said to be \(D_{R}\) stable if and only if all its eigenvalues lie in the \(D_{R}\) region defined by (11). Standard Lyapunov stability condition can be then generalized for the \(D_{R}\) stability in the form of matrix inequality as summarized in the following lemma.  
**Lemma 1** ([9]).: _Closed-loop matrix \(A_{CL}(\xi)\in R^{n\times n}\) is \(D_{R}\) stable if and only if there exists a positive definite matrix \(P(\xi)\in R^{n\times n}\) such that_  
\[R_{11}\otimes P(\xi)+R_{12}\otimes(P(\xi)A_{CL}(\xi))+R_{12}^{T}\otimes(A_{CL} (\xi)^{T}P(\xi))+R_{22}\otimes(A_{CL}^{T}(\xi)P(\xi)A_{CL}(\xi))<0. \tag{12}\]  
It is important to note that the \(D_{R}\) stability condition (12) can be for state feedback control directly converted to LMI which significantly reduces computational effort.  
Standard domains for stable poles and matrices \(R_{ij}\) defining the corresponding \(D_{R}\) regions (11) can be listed as:  
1. open left-half plane of the complex plane, \(R_{11}=0\), \(R_{12}=1\), \(R_{22}=0\) (continuous-time systems),
2. interior of the unit circle, \(R_{11}=-1\), \(R_{12}=0\), \(R_{22}=1\) (discrete-time systems),
3. shifted left-half plane of the complex plane corresponding to stability degree \(\lambda\), \(R_{11}=2\lambda,R_{12}=1,R_{22}=0\) (continuous-time systems),
4. interior of the circle centered in [0, 0] with radius \(r=1/sqrt(\alpha)\) corresponding to stability degree \(sqrt(\alpha)\), \(R_{11}=-1/\alpha\), \(R_{12}=0\), \(R_{22}=1\) (discrete-time systems),
5. interior of the convex cone with vertex angle \(2\varphi\), corresponding to the relative damping (given by a ratio of the imaginary and real part of the complex pole in continuous-time systems) \[R_{11}=R_{22}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix},R_{12}=\begin{bmatrix}sin(\varphi)&cos(\varphi)\\ -cos(\varphi)&sin(\varphi)\end{bmatrix}.\]  
The domains (I) and (II) correspond to basic stability, domains (III) and (IV) to stability degree, which influences the speed of system response and also robustness, domain (V) determines the relative damping, belonging to important performance indices since it is related to oscillations of the system response. Recall that the stability degree defines the distance to the standard stability border, for continuous-time systems it given by the biggest real part of system poles--see (III); for discrete time system it is given by the inverse of radius of the circle where all system poles lie--see (IV).