3.2 Case of study: Takagi-Sugeno fuzzy model

The state-space model of the case of study expressed in Eq. (10) contains nonlinearities 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) when it is activated.

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 evaporate the mixture.

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

$$M1 = \begin{cases} 1, z < a \\ \frac{b-z}{b-a}, a \le z \le b \\ 0, z > b \end{cases} \tag{21}$$

Model 4 rule If z1ð Þt is M1, z2ð Þt is and M4 z2ð Þt is M6 Then x t \_ðÞ¼ A4x tð Þþ B4u tð Þ y ¼ C4x tð Þ Model 5 rule If z1ð Þt is M2, z2ð Þt is M3 and z3ð Þt is M5 Then x t \_ðÞ¼ A5x tð Þþ B5u tð Þ y ¼ C5x tð Þ Model 6 rule If z1ð Þt is M2, z2ð Þt is and M3 z2ð Þt is M6 Then x t \_ðÞ¼ A6x tð Þþ B6u tð Þ y ¼ C6x tð Þ Model 7 rule If z1ð Þt is M2, z2ð Þt is M4 and z3ð Þt is M5 Then x t \_ðÞ¼ A7x tð Þþ B7u tð Þ y ¼ C7x tð Þ Model 8 rule If z1ð Þt is M2, z2ð Þt is and M4 z2ð Þt is M6 Then x t \_ðÞ¼ A8x tð Þþ B8u tð Þ y ¼ C8x tð Þ

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

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

where M1 and M2 are fuzzy sets for z1ð Þt , M3 and M4 are fuzzy sets for z2ð Þt ,

According to the fuzzy rules, the Takagi-Sugeno fuzzy model of the EDF-100

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

hið Þ z tð Þ ð Þ Aix tðÞþ Biu tð Þ

(23)

hið Þ z tð Þ ð Þ Cix tð Þ

and M5 and M6 are fuzzy sets for z3ð Þt .

3.3 Case of study: fuzzy observer

19

distillation pilot plant is expressed in Eq. (23):

x t \_ðÞ¼ ∑ 8 i¼1

11-plate distillation column is expressed in Eq. (24):

y tðÞ¼ ∑ 8 i¼1

$$M2 = \left\{ \frac{\text{l}, z < a}{\text{l} - a}, a \le z \le b \right\} \tag{22}$$
  $M2 = \begin{cases} a - a \\ b - a \\ 1, z > b \end{cases} \tag{23}$ 

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