3.3 Case of study: fuzzy observer

From the general structure of the fuzzy observer presented in Eq. (14) and the fuzzy Takagi-Sugeno model expressed in Eq. (9), the fuzzy observer for the 11-plate distillation column is expressed in Eq. (24):

3.2 Case of study: Takagi-Sugeno fuzzy model

Distillation - Modelling, Simulation and Optimization

when it is activated.

evaporate the mixture.

18

The state-space model of the case of study expressed in Eq. (10) contains non-

The premise variables considered in the fuzzy model are the reflux valve opening percentage (z<sup>3</sup> ¼ Rf ) due to its effect in all the states of the system; the light component composition in the condenser (z<sup>2</sup> ¼ x1Þ, where the distillate product is obtained; and the light component composition in the boiler ð Þ z<sup>1</sup> ¼ x<sup>11</sup> , where the original mixture is located; besides, the boiler is directly related to the heating power (one of the system input variables) that provides the energy required to

According to the steady-state dynamics of the distillation column, trapezoidaltype membership functions are chosen with two rules for each one, as expressed mathematically in Eqs. (21) and (22). This type of function is selected because it allows a greater range in the universe of discourse, where the belonging degree of

> b � z b � a

> z � a b � a

The number of subsystems of the Takagi-Sugeno fuzzy model is dependent on the number of combinations that the membership functions have; for the case of study considering three premise variables (z<sup>1</sup> ¼ x11, z<sup>2</sup> ¼ x1, and z<sup>3</sup> ¼ Rf ), each one

> Model 1 rule If z1ð Þt is M1, z2ð Þt is M3 and z3ð Þt is M5 Then x t \_ðÞ¼ A1x tð Þþ B1u tð Þ y ¼ C1x tð Þ Model 2 rule If z1ð Þt is M1, z2ð Þt is and M3 z2ð Þt is M6 Then x t \_ðÞ¼ A2x tð Þþ B2u tð Þ y ¼ C2x tð Þ Model 3 rule If z1ð Þt is M1, z2ð Þt is M4 andz3ð Þt is M5 Then x t \_ðÞ¼ A3x tð Þþ B3u tð Þ y ¼ C3x tð Þ

1, z , a

0, z . b

0, z , a

1, z . b

, a≤z≤b

, a≤z≤b

(21)

(22)

belonging is 1, which prevents an oscillation when the states stabilize:

8 >><

>>:

8 ><

>:

M1 ¼

M2 ¼

with two rules (zimax and zimin), the number of subsystems is 2<sup>3</sup> <sup>¼</sup> 8.

linearities in the A and B matrices since both are dependent on the states; in addition, the reflux input (Rf ) disturbs all the states of the system (compositions) Distillation - Modelling, Simulation and Optimization

$$\begin{aligned} \hat{\boldsymbol{x}}(t) &= \sum\_{i=1}^{8} h\_i(\mathbf{z}(t)) (A\_i \mathbf{x}(t) + B\_i \boldsymbol{u}(t) + K\_i(\boldsymbol{e}(t))) \\ \hat{\boldsymbol{y}} &= \sum\_{i=1}^{8} h\_i(\mathbf{z}(t)) (\mathbf{C}\_i \hat{\mathbf{x}}(t)) \end{aligned} \tag{24}$$

The fuzzy observer scheme proposed for the distillation column of the case study is shown in Figure 3, taking as inputs the heating power and the reflux action; the estimated states are the molar compositions in all the plates, and the measured outputs are the temperatures in the condenser; in plates 2, 4, 6, 8, and 10; and in the boiler.

Using the LMI system for eight rules, the LMI system for the fuzzy observer is obtained. The LMI's characteristics of each function are expressed in Eq. (25):

$$\begin{aligned} A\_1'P - C\_1'N\_1' + PA\_1 - N\_1C\_1 &< 0\\ A\_2'P - C\_2'N\_2' + PA\_2 - N\_2C\_2 &< 0\\ \vdots\\ A\_7'P - C\_7'N\_7' + PA\_7 - N\_7C\_7 &< 0 \end{aligned} \tag{25}$$

$$A\_8'P - C\_8'N\_8' + PA\_8 - N\_8C\_8 &< 0$$

<sup>K</sup><sup>1</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup>

Fuzzy Logic Modeling and Observers Applied to Estimate Compositions in Batch Distillation…

DOI: http://dx.doi.org/10.5772/intechopen.83479

<sup>K</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup>

<sup>K</sup><sup>8</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup>

The fuzzy observer validation is performed using real data from a distillation

Parameter Magnitude Units EtOH volume in the boiler 1000 ml H2O volume in the boiler 1000 ml Process total pressure 662 mmHg

Parameter Ethanol Water Units Density (pi) 0.789 1 kg/m<sup>3</sup> Molecular weight (Wi) 46.069 18.0528 g/mol Boiling temperature (Tbi) 78.4 100 °C

The process operates during 50 min, taking the initial compositions xð Þ¼ 0 [0.8555, 0.8525, 0.8480, 0.8412, 0.8309, 0.8148, 0.7896, 0.7483, 0.6767,

process; the main characteristics are shown in Tables 1 and 2.

4. Results and discussion

Figure 3. Observer scheme.

Table 1. Process parameters.

Table 2.

21

Mixture parameters.

0.5369, 0.2300] in steady state.

⋮ <sup>K</sup><sup>7</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup>

N<sup>1</sup>

N<sup>2</sup>

N<sup>7</sup>

(27)

N<sup>8</sup>

The LMIs that represent the membership functions overlaps are expressed in Eq. (26):

$$\begin{aligned} A'\_1P - C'\_2N'\_1 + PA\_1 - N\_1C\_2 + PA'\_2 - C'\_1N'\_2 + PA\_2 - N\_2C\_1 &< 0\\ A'\_1P - C'\_3N'\_1 + PA\_1 - N\_1C\_3 + PA'\_3 - C'\_1N'\_3 + PA\_3 - N\_3C\_1 &< 0\\ \vdots\\ A'\_1P - C'\_7N'\_1 + PA\_1 - N\_1C\_7 + PA'\_7 - C'\_1N'\_7 + PA\_7 - N\_7C\_1 &< 0\\ A'\_2P - C'\_3N'\_2 + PA\_2 - N\_2C\_3 + PA'\_3 - C'\_2N'\_3 + PA\_3 - N\_3C\_2 &< 0\\ A'\_2P - C'\_4N'\_2 + PA\_2 - N\_2C\_4 + PA'\_4 - C'\_2N'\_4 + PA\_4 - N\_4C\_2 &< 0\\ \vdots \end{aligned}$$

$$A\_2'P - C\_7'N\_2' + PA\_2 - N\_2C\_7 + PA\_7' - C\_2'N\_7' + PA\_7 - N\_7C\_2 < 0 \tag{26}$$

$$A\_3'P - C\_4'N\_3' + PA\_3 - N\_3C\_4 + PA\_4' - C\_3'N\_4' + PA\_4 - N\_4C\_3 < 0$$

$$A\_3'P - C\_5'N\_3' + PA\_3 - N\_3C\_5 + PA\_5' - C\_3'N\_5' + PA\_5 - N\_5C\_3 < 0$$

$$\begin{aligned} &A\_6'P - \mathbf{C}\_7'\mathbf{N}\_6' + PA\_6 - N\_6\mathbf{C}\_7 + PA\_7' - \mathbf{C}\_6'\mathbf{N}\_7' + PA\_7 - N\_7\mathbf{C}\_6 \le \mathbf{0} \\\\ &A\_7'P - \mathbf{C}\_8'\mathbf{N}\_7' + PA\_7 - N\_7\mathbf{C}\_8 + PA\_8' - \mathbf{C}\_7'\mathbf{N}\_8' + PA\_8 - N\_8\mathbf{C}\_7 \le \mathbf{0} \end{aligned}$$

⋮

where P is positive definite diagonal matrix P . 0 of 11 � 11 dimension, and N is an auxiliary matrix dependent on the number of states (11) and the measured outputs (7), resulting in a 7 � 11 dimension.

The resulting system of 36 LMIs is solved using the MATLAB® lmiedit tool. Once the LMI system is solved, with P . 0 to guarantee the closed-loop stability of each subsystem, the Ki gains are calculated in Eq. (27):

Fuzzy Logic Modeling and Observers Applied to Estimate Compositions in Batch Distillation… DOI: http://dx.doi.org/10.5772/intechopen.83479

Figure 3. Observer scheme.

x t ^\_ðÞ¼ <sup>∑</sup> 8 i¼1

Distillation - Modelling, Simulation and Optimization

hið Þ z tð Þ ð Þ Aix tð Þþ Biu tðÞþ Kið Þ e tð Þ

<sup>1</sup> þ PA<sup>1</sup> � N1C<sup>1</sup> , 0

<sup>2</sup> þ PA<sup>2</sup> � N2C<sup>2</sup> , 0

<sup>7</sup> þ PA<sup>7</sup> � N7C<sup>7</sup> , 0

<sup>8</sup> þ PA<sup>8</sup> � N8C<sup>8</sup> , 0

<sup>2</sup> þ PA<sup>2</sup> � N2C<sup>1</sup> , 0

<sup>3</sup> þ PA<sup>3</sup> � N3C<sup>1</sup> , 0

<sup>7</sup> þ PA<sup>7</sup> � N7C<sup>1</sup> , 0

<sup>3</sup> þ PA<sup>3</sup> � N3C<sup>2</sup> , 0

<sup>4</sup> þ PA<sup>4</sup> � N4C<sup>2</sup> , 0

<sup>7</sup> þ PA<sup>7</sup> � N7C<sup>2</sup> , 0

<sup>4</sup> þ PA<sup>4</sup> � N4C<sup>3</sup> , 0

<sup>5</sup> þ PA<sup>5</sup> � N5C<sup>3</sup> , 0

<sup>7</sup> þ PA<sup>7</sup> � N7C<sup>6</sup> , 0

<sup>8</sup> þ PA<sup>8</sup> � N8C<sup>7</sup> , 0

(24)

(25)

(26)

hið Þ z tð Þ ð Þ Cix t ^ð Þ

The fuzzy observer scheme proposed for the distillation column of the case study is

shown in Figure 3, taking as inputs the heating power and the reflux action; the estimated states are the molar compositions in all the plates, and the measured outputs are the temperatures in the condenser; in plates 2, 4, 6, 8, and 10; and in the boiler. Using the LMI system for eight rules, the LMI system for the fuzzy observer is obtained. The LMI's characteristics of each function are expressed in Eq. (25):

1N<sup>0</sup>

2N<sup>0</sup>

7N<sup>0</sup>

8N<sup>0</sup>

⋮

The LMIs that represent the membership functions overlaps are expressed in

<sup>2</sup> � C<sup>0</sup> 1N<sup>0</sup>

<sup>3</sup> � C<sup>0</sup> 1N<sup>0</sup>

<sup>7</sup> � C<sup>0</sup> 1N<sup>0</sup>

<sup>3</sup> � C<sup>0</sup> 2N<sup>0</sup>

<sup>4</sup> � C<sup>0</sup> 2N<sup>0</sup>

<sup>7</sup> � C<sup>0</sup> 2N<sup>0</sup>

<sup>4</sup> � C<sup>0</sup> 3N<sup>0</sup>

<sup>5</sup> � C<sup>0</sup> 3N<sup>0</sup>

<sup>7</sup> � C<sup>0</sup> 6N<sup>0</sup>

<sup>8</sup> � C<sup>0</sup> 7N<sup>0</sup>

where P is positive definite diagonal matrix P . 0 of 11 � 11 dimension, and N is an auxiliary matrix dependent on the number of states (11) and the measured out-

The resulting system of 36 LMIs is solved using the MATLAB® lmiedit tool. Once the LMI system is solved, with P . 0 to guarantee the closed-loop stability of each

⋮

⋮

⋮

^y ¼ ∑ 8 i¼1

A0 <sup>1</sup>P � C<sup>0</sup>

A0 <sup>2</sup>P � C<sup>0</sup>

A0 <sup>7</sup>P � C<sup>0</sup>

A0 <sup>8</sup>P � C<sup>0</sup>

<sup>1</sup> þ PA<sup>1</sup> � N1C<sup>2</sup> þ PA<sup>0</sup>

<sup>1</sup> þ PA<sup>1</sup> � N1C<sup>3</sup> þ PA<sup>0</sup>

<sup>1</sup> þ PA<sup>1</sup> � N1C<sup>7</sup> þ PA<sup>0</sup>

<sup>2</sup> þ PA<sup>2</sup> � N2C<sup>3</sup> þ PA<sup>0</sup>

<sup>2</sup> þ PA<sup>2</sup> � N2C<sup>4</sup> þ PA<sup>0</sup>

<sup>2</sup> þ PA<sup>2</sup> � N2C<sup>7</sup> þ PA<sup>0</sup>

<sup>3</sup> þ PA<sup>3</sup> � N3C<sup>4</sup> þ PA<sup>0</sup>

<sup>3</sup> þ PA<sup>3</sup> � N3C<sup>5</sup> þ PA<sup>0</sup>

<sup>6</sup> þ PA<sup>6</sup> � N6C<sup>7</sup> þ PA<sup>0</sup>

<sup>7</sup> þ PA<sup>7</sup> � N7C<sup>8</sup> þ PA<sup>0</sup>

Eq. (26):

A0 <sup>1</sup>P � C<sup>0</sup>

A0 <sup>1</sup>P � C<sup>0</sup>

A0 <sup>1</sup>P � C<sup>0</sup>

A0 <sup>2</sup>P � C<sup>0</sup>

A0 <sup>2</sup>P � C<sup>0</sup>

A0 <sup>2</sup>P � C<sup>0</sup>

A0 <sup>3</sup>P � C<sup>0</sup>

A0 <sup>3</sup>P � C<sup>0</sup>

A0 <sup>6</sup>P � C<sup>0</sup>

A0 <sup>7</sup>P � C<sup>0</sup>

20

2N<sup>0</sup>

3N<sup>0</sup>

7N<sup>0</sup>

3N<sup>0</sup>

4N<sup>0</sup>

7N<sup>0</sup>

4N<sup>0</sup>

5N<sup>0</sup>

7N<sup>0</sup>

8N<sup>0</sup>

puts (7), resulting in a 7 � 11 dimension.

subsystem, the Ki gains are calculated in Eq. (27):

$$\begin{aligned} K\_1 &= P^{-1} N\_1 \\ K\_2 &= P^{-1} N\_2 \\ &\vdots \\ K\_{\mathcal{T}} &= P^{-1} N\_{\mathcal{T}} \end{aligned} \tag{27}$$
 
$$K\_8 = P^{-1} N\_8$$
