2.1. Problem Formulation

3.1. The Equivalent Subsystems Method

3.2. QFT Design

6.

Calculation of QFT bounds.

Every plant in the  ${\mathsf{\omega }}_{\mathrm{i}}$  template as well as the compensator can be expressed in their respective polar form:

$$\mathrm{P}\left(\mathrm{j}{\mathsf{\omega }}_{\mathrm{i}}\right)=\mathrm{p}{\mathrm{e}}^{\mathrm{j}\mathsf{\theta }}=\mathrm{p}\mathrm{⦟}\mathsf{\theta }\mathrm{C}\left(\mathrm{j}{\mathsf{\omega }}_{\mathrm{i}}\right)=\mathrm{c}{\mathrm{e}}^{\mathrm{j}\mathsf{\phi }}=\mathrm{c}\mathrm{⦟}\mathsf{\phi }$$

(26)

QFT bounds are calculated from the control specifications and considering model uncertainty. By substituting the polar forms (26) into individual selected performance specification types, they can be rearranged into quadratic inequalities in the form:

$${\mathrm{l}}_{{\mathsf{\omega }}_{\mathrm{i}}}^{\mathrm{k}}\left(\mathrm{p},\mathsf{\varphi },{\mathsf{\delta }}_{\mathrm{k}},\mathsf{\theta }\right)=\mathrm{a}{\mathrm{c}}^2+\mathrm{b}\mathrm{c}+\mathrm{f}\ge 0$$

(27)

where the coefficients  $\mathrm{a},\mathrm{b},\mathrm{f}$  are functions of  $\mathrm{p},\mathsf{\varphi },\mathsf{\theta }$  and  ${\mathsf{\delta }}_{\mathrm{k}}$ .

Using an appropriate algorithm [19], the quadratic inequalities (27) are solved and translated into a set of curves in the Nichols diagram for each frequency and specification type. Thus, at each frequency  ${\mathsf{\omega }}_{\mathrm{i}}$  and for each specification  $\mathrm{k}$ there is a bound  ${\mathrm{B}}_{\mathrm{k}}\left(\mathrm{j}{\mathsf{\omega }}_{\mathrm{i}}\right)$  which can be dashed or solid, depending on whether the area satisfying the bound is above or below the line.

7.

Calculation of intersection of the QFT bounds and checking for their compatibility.

If more than one performance specification is considered, it is necessary to find the intersection of all bounds at each frequency.

8.

Controller design by loop shaping in the Nichols diagram.

After plotting the bounds in the Nichols diagram, it is sufficient to deal just with a single (nominal plant) to find a controller that meets the bounds. Hence, QFT provides elegant and practical solution by integrating information associated with the model uncertainty and all control specifications in a set of simple curves. The loop shaping design is carried out by adding poles and zeros until the nominal loop lies near its bounds (optimally on the top of the bounds at each frequency).

The QFT compensator has a general form:

$$\mathrm{C}\left(\mathrm{s}\right)={\mathrm{k}}_{\mathrm{C}}\frac{{\prod }_{\mathrm{i}=1}^{{\mathrm{n}}_{\mathrm{r}\mathrm{z}}}\left(\left(\frac{\mathrm{s}}{{\mathrm{z}}_{\mathrm{i}}}\right)+1\right){\prod }_{\mathrm{i}=1}^{{\mathrm{n}}_{\mathrm{c}\mathrm{z}}}\left(\left(\frac{{\mathrm{s}}^2}{{\mathsf{\omega }}_{\mathrm{n}\mathrm{i}}^2}\right)+\left(\frac{2{\mathsf{\xi }}_{\mathrm{i}}}{{\mathsf{\omega }}_{\mathrm{n}\mathrm{i}}}\right)\mathrm{s}+1\right)}{{\mathrm{s}}^{\mathrm{r}}{\prod }_{\mathrm{j}=1}^{{\mathrm{m}}_{\mathrm{r}\mathrm{p}}}\left(\left(\frac{\mathrm{s}}{{\mathrm{p}}_{\mathrm{j}}}\right)+1\right){\prod }_{\mathrm{j}=1}^{{\mathrm{m}}_{\mathrm{c}\mathrm{p}}}\left(\left(\frac{{\mathrm{s}}^2}{{\mathsf{\omega }}_{\mathrm{n}\mathrm{j}}^2}\right)+\left(\frac{2{\mathsf{\xi }}_{\mathrm{j}}}{{\mathsf{\omega }}_{\mathrm{n}\mathrm{j}}}\right)\mathrm{s}+1\right)},$$

(28)

where  $\mathrm{z}$  denotes the zeros,  $\mathrm{p}$  the poles,  ${\mathsf{\omega }}_{\mathrm{n}}$  is the natural frequency,  $\mathsf{\xi }$  is the relative damping and  ${\mathrm{k}}_{\mathrm{C}}$  is the gain.

9.

Prefilter design.

A prefilter is used to solve the reference tracking problem.

10.

Analysis and validation.

This step includes:

-

frequency domain analysis of each specification for all the significant plants within the model uncertainty,

-

time domain simulation for linear and nonlinear system.

5.1. Description of the Controlled Plant

5.2. Control Problem Statement

5.3. Robust Decentralized Controller Design

I. 

Modelling the uncertain MIMO system

The linearized model of the uncertain plant based on parameter values in Table 1 is:

$$\mathrm{P}\left(\mathrm{s}\right)=\left[\begin{array}{cc}\frac{0.747{\mathsf{\gamma }}_1}{62.044\mathrm{s}+1}& \frac{0.747\left(1-{\mathsf{\gamma }}_2\right)}{\left(22.992\mathrm{s}+1\right)\left(62.044\mathrm{s}+1\right)}\\ \frac{0.949\left(1-{\mathsf{\gamma }}_1\right)}{\left(29.995\mathrm{s}+1\right)\left(90.031\mathrm{s}+1\right)}& \frac{0.949{\mathsf{\gamma }}_2}{90.031\mathrm{s}+1}\end{array}\right].$$

(37)

When considering the minimum phase plant configuration, the two uncertain parameters  ${\mathsf{\gamma }}_1,$   ${\mathsf{\gamma }}_2\in <0, 1>$ are interrelated according to (33). The set of realizations of the uncertain plant was generated by gridding  ${\mathsf{\gamma }}_1,$   ${\mathsf{\gamma }}_2\in <0.3, 0.9>$ and considering all their combinations that meet (33). The number of all feasible combinations is  $\mathrm{N}=10$ . The corresponding combinations specifying the considered uncertainty region are depicted in Figure 4. The nominal model was selected as (37) evaluated in  ${\mathsf{\gamma }}_1=$   ${\mathsf{\gamma }}_2=0.7.$ 

II. 

Generation of uncertain equivalent subsystems

III. 

Calculation of QFT templates of equivalent subsystems

IV. 

Independent design of SISO local controllers  $C_i\left(s\right)=\mathrm{1,2}$ for uncertain equivalent subsystems

V. 

Analysis in the frequency and time domains