2.1.1 Distillation column state-space model

The state-space dynamics of the distillation column, considering as states the light component composition in all the plates and as control inputs to the heating power, Qf , and the reflux, Rf , is expressed in Eq. (5):

$$
\dot{\boldsymbol{\omega}} = \mathbf{A} \begin{pmatrix} \boldsymbol{\omega}\_1 \\ \boldsymbol{\omega}\_2 \\ \vdots \\ \boldsymbol{\omega}\_{n-1} \\ \boldsymbol{\omega}\_n \end{pmatrix} + B \begin{pmatrix} R\mathbf{f} \\ Q\boldsymbol{b} \end{pmatrix} \tag{5}
$$

where A and B matrixes are defined by

$$A = \begin{pmatrix} \begin{array}{cccc} -\frac{(V+D)}{M\_i} & \frac{V\cdot G(\mathbf{x}\_i a\_i)}{M\_i} & \cdots & 0 & 0\\ & \frac{L}{M\_i} & -\frac{V\cdot G(\mathbf{x}\_{i+1} a\_{i+1}) - L}{M\_i} & \cdots & 0 & 0\\ & & & \vdots & & \vdots\\ \vdots & \vdots & \cdots & & \frac{V\cdot G(\mathbf{x}\_{n-1} a\_{n-1}) - L}{M\_{n-1}} & \frac{V\cdot G(\mathbf{x}\_n a\_n)}{M\_{n-1}}\\ 0 & 0 & \cdots & & \frac{L}{M\_n} & -\frac{L}{M\_n} \end{array} \end{pmatrix}$$

$$B = \begin{pmatrix} \underline{L\boldsymbol{\omega}\_1} & & \mathbf{0} \\ \mathbf{0} & & & \\ \mathbf{0} & & & \\ \vdots & & \vdots & \\ \mathbf{0} & & & \mathbf{0} \\ \mathbf{0} & & \overline{\boldsymbol{\omega}\_n(\mathbf{1} - G(\boldsymbol{\omega}\_n \boldsymbol{\alpha}\_n))} \\ \mathbf{0} & & \overline{\left(H\_i^{vap}\mathbf{x}\_n + H\_j^{vap}(\mathbf{1} - \boldsymbol{\omega}\_n)M\_n\right)} \end{pmatrix}$$

where the molar fluids of liquid, L, and vapor,V, are determined by Eqs. (6) and (7):

$$V = \frac{Q\_b}{H\_i^{up} \varkappa\_n + H\_j^{up} (1 - \varkappa\_n)}\tag{6}$$

$$L = \left(\mathbf{1} - \mathbf{R}\_f\right) \mathbf{V} \tag{7}$$

where the vector for p premise variables z tð Þ is defined by Eq. (11):

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

wið Þ¼ z tð Þ <sup>Y</sup>

and the normalized weight hi for each rule is defined by Eq. (13):

^y ¼ ∑ r i¼1

P complies with the Lyapunov equation expressed in Eq. (16):

A0 i P � C<sup>0</sup> i N<sup>0</sup>

3. Case of study: EDF-1000 distillation pilot plant

<sup>i</sup> þ PAi � NiCj þ PA<sup>0</sup>

x t ^\_ðÞ¼ <sup>∑</sup> r i¼1

The estimation error is determined by Eq. (15):

(LMIs) is found, as shown in Eq. (17):

A0 i P � C<sup>0</sup> j N<sup>0</sup>

functions is defined by Eq. (12):

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

2.3 Fuzzy observer

where

15

In addition, the calculated weight wið Þ z tð Þ for each i rule from the membership

p

j¼1

According to the structure of the fuzzy observer [24–26] expressed in Eq. (14),

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

hið Þ z tð Þ ð Þ Ci x t ^ð Þ

The fuzzy observer stability is demonstrated if each Ai, Ci pair is observable and

0

Ai ¼ Ai � KiCi In [25], the demonstration that the observer is stable is presented as long as a positive definite matrix P that satisfies the system of linear matrix inequalities

> P . 0 Ni . 0

> > i , j

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

The observer gains are defined by the LMI systems solution defined in Eq. (18):

As a case of study, an EDF-1000 distillation pilot plant is used (Figure 2), consisting of 11 perforated plates, having 7 RTD (PT100) temperature sensors

<sup>i</sup> þ PAi � NiCi , 0

<sup>j</sup> � C<sup>0</sup> i N<sup>0</sup>

PiAi þ Ai

hið Þ¼ z tð Þ wið Þ z tð Þ ∑r

z tðÞ¼ <sup>z</sup>1ð Þ<sup>t</sup> ; <sup>z</sup>2ð Þ<sup>t</sup> ; <sup>⋯</sup> ; zpð Þ<sup>t</sup> � � (11)

Mijzjð Þt (12)

<sup>i</sup>¼<sup>1</sup>wið Þ z tð Þ (13)

e tðÞ¼ y tðÞ� ^y tð Þ (15)

Pi , 0 (16)

<sup>j</sup> þ PAj � NjCi , 0

<sup>i</sup> Ni (18)

(14)

(17)

where Hvap <sup>i</sup> is the light component vapor enthalpy, <sup>H</sup>vap <sup>j</sup> is the heavy component vapor enthalpy, and xn is the light component composition in the boiler.

The G xi ð Þ ; α<sup>i</sup> function is determined by the light component composition ð Þ xi , the relative volatility ð Þ <sup>α</sup><sup>i</sup> , the saturation pressure Psat i � �, the pressure in the column PT, the vapor molar ð Þ V , and the activity coefficient γ<sup>i</sup> ð Þ, as expressed in Eq. (8):

$$G(\mathbf{x}\_i, a\_i) = \frac{VP\_i^{\text{at}} \mathbf{y}\_i}{P\_T} \tag{8}$$
