#### 3.1. Logarithmic Spirals Corresponding to the Prescribed Damping and Their Extreme Points

This subsection is devoted to the description of the domain in the complex plane, which represents the region corresponding to the prescribed damping both for continuous and discrete-time systems, [

15]. The conic domain depicted in

Figure 3a corresponds to a damping

$D=cos\phi .$ The smaller the angle

φ, the bigger the damping and less oscillating the system. The cone is described by its upper line, as follows:

and lower line, as follows:

The discrete-time counterpart can be received using a Z-transform of the continuous time poles. For a pair of conjugated poles

$s=a\pm ib$ we have after discretization with the sampling period

TConsidering Equations (10a) and (10b), we have

$b=ka$ and from Equation (11) we receive the upper and lower logarithmic spiral, respectively, as follows:

The parametric description of logarithmic spirals, Equations (12a) and (12b), corresponding in the discrete time domain to the prescribed damping, can be formulated as

where

$k=\mathrm{tg}\phi ,t=aT$, therefore,

$t\in \left(-\frac{\pi}{k},0\right)$.

Applying the rules for derivation of parametric equations, the extreme points can be found (for details see Reference [

15]), depicted in

Figure 5. The upper extreme point

${E}_{M}=\left[{x}_{M},{y}_{M}\right]$ is given by

The intersection point with the negative part of the real axis (

$y=0$), is as follows:

The extreme points coordinates ${x}_{0},{x}_{M},{y}_{M}$ are used to calculate parameters of all studied inner approximations.

#### 3.2. Inner Convex Approximations to the Nonconvex Domain Corresponding to the Prescribed Damping

In this subsection, we summarize the proposed inner convex approximations to the cardioid interior, as follows: Computation of their parameters, the corresponding figure, and matrices for ${D}_{R}$ region description. Recall, that all ${D}_{R}$ regions must be symmetric with respect to the real axis of the complex plane. Therefore, the centers of circle or ellipse always lie on the x-axis.

a) Inner circle

The inner circle approximation is proposed so that it has a maximal possible radius. Therefore, it is centered in $\left[{x}_{M},0\right]$ and the radius, $r$, is dependent on the shape of the spirals and calculated in Matlab as follows:

ak = xM − x0;

bk = yM;

r = min(ak, bk);

The ${D}_{R}$ region matrices (scalars) conforming to the general circle description, Equation (6), are given as:

R11 = xM^2 − r^2;

R12 = −xM;

R22 = 1;

Circle approximation is depicted in

Figure 6.

b) Inner ellipse

The inner ellipse is proposed so that it has maximal possible semi-axes. Analogously to the inner circle, it is centered in $\left[{x}_{M},0\right].$ The semi-axes are calculated as follows:

The ${D}_{R}$ region matrices (conforming to the general ellipse description, Equation (7)) can be calculated as follows:

The elliptic approximation is depicted in

Figure 7.

The next two options, c) and d), are appropriate for the case when poles with a nonnegative real part are required. Then it is advantageous to use the intersection of the right half plane with the inner circle or ellipse with bigger area than those in a) or b).

c) Right half-plane and circle

This option is suitable for angles bigger than 53°, when, for the extreme points given by Equations (14) and (15), it holds as follows: ${x}_{M}-{x}_{0}<{y}_{M}$. The corresponding circle is centered in $\left[{x}_{M},0\right]$ and its radius is set to $r={y}_{M}$. The resulting ${D}_{R}$ region is obtained as an intersection of the right half plane and the circle and the corresponding matrices are as follows:

The approximation is depicted in

Figure 8a.

d) Right half-plane and ellipse

The ellipse is centered in $\left[{x}_{M},0\right]$, as in the case of b), however, its x-semi-axis is increased, so that the ellipse reaches the intersection of the cardioid with the y-axis, denoted as y3. The resulting ${D}_{R}$ region is obtained as an intersection of the right half plane and the ellipse and the corresponding matrices, R11, R12, and R22, are calculated as follows:

y3 = exp (−pi/(2*tan(fi*pi/180)));

ak = xM*yM/sqrt (yM^2 − y3^2);

bk = yM;

R11 = [0 0 0; 0 − 1 − xM/ak; 0 −xM/ak −1];

R12 = [−1 0 0; 0 0 (1/ak − 1/bk)/2; 0 (1/ak + 1/bk)/2 0];

R22 = zeros (3, 3);

($\phi $ is represented by fi in the program).

Though the last approximation (intersection of the right half plane and ellipse) gets closer to the right-hand side border than the inner ellipse (compare

Figure 7b and

Figure 8b), there is still a significant area not included in the approximation. All the above approximations basically avoid the domain close to point [1, 0]. The next novel combination of the shifted cone and ellipse is proposed to also include this area close to the border, since it can be important to make control feasible.

e) Ellipse-cone

This approximation considers the intersection of an angle with the vertex in [1, 0] and the elliptic region. The crossing point

$[{x}_{e},{y}_{e}]$ lies on the cardioid (logarithmic spiral) and

${x}_{e}$ is considered as a parameter to be chosen. In

Figure 9, two variants for different choices of

${x}_{e}$ are shown.

The ellipse is centered in the middle of the cardioid x-axis, its x-semi-axis denoted as $ak$ is maximized, and the y-semi-axis $bk$ is derived so that it crosses $[{x}_{e},{y}_{e}]$, as follows:

and the corresponding ${D}_{R}$ region matrices are as follows:

The cone with the vertex in

$[{x}_{v},0]$ and inner angle

$2\gamma $ corresponds to

Figure 9 (

$\gamma $ is represented by

ga in the program). The respective

${D}_{R}$ region matrices for this shifted angle are as follows:

The ${D}_{R}$ region matrices describing the intersection of the ellipse and angle can be obtained by taking ${x}_{v}=1$ and merging the above matrices for the ellipse and cone, as follows:

Z = zeros (2, 2);

R11 = [R11e Z; Z R11v];

R12 = [R12e Z; Z R12v];

R22 = [R22e Z; Z R22v].

**Remark** **2.** It should be noted that any ${D}_{R}$ regions can be combined, so that their intersection is considered, as in the latter case, where the intersection of the ellipse and angle is described by the resulting ${D}_{R}$ region matrices. ${D}_{R}$ regions can be also combined with other performance criterion. Often, pole placement is considered together with H_{2} or H_{inf} minimization.

Comparison of all the above approximations is illustrated on the example in the next section, providing the results for laboratory magnetic levitation plant.