2.3 Fuzzy observer

B ¼

where Hvap

the form (Eq. 9):

defuser, as described in Eq. (10):

14

y tðÞ¼ <sup>∑</sup><sup>r</sup>

Lx<sup>1</sup> M<sup>1</sup>

0

Distillation - Modelling, Simulation and Optimization

BBBBBBBBBBBB@

0 ⋮ 0

0

the relative volatility ð Þ <sup>α</sup><sup>i</sup> , the saturation pressure Psat

2.2 Distillation column Takagi-Sugeno model

Hvap

<sup>V</sup> <sup>¼</sup> Qb Hvap

<sup>i</sup> is the light component vapor enthalpy, <sup>H</sup>vap

vapor enthalpy, and xn is the light component composition in the boiler.

<sup>i</sup> xn <sup>þ</sup> <sup>H</sup>vap

<sup>i</sup> xn <sup>þ</sup> <sup>H</sup>vap

L ¼ 1 � Rf

The G xi ð Þ ; α<sup>i</sup> function is determined by the light component composition ð Þ xi ,

VPsat <sup>i</sup> γ<sup>i</sup> PT

PT, the vapor molar ð Þ V , and the activity coefficient γ<sup>i</sup> ð Þ, as expressed in Eq. (8):

The Takagi-Sugeno model is the fuzzy representation of a nonlinear model obtained from the linear subsystems interpolation according to fuzzy rules having

> Model i rule If zi ð Þt is Mi<sup>1</sup> and … and zpð Þt is Mp<sup>1</sup> Then x t \_ðÞ¼ Aix tð Þþ Biu tð Þ

> > i ¼ 1, 2, …, r

Given the pair ð Þ x tð Þ; u tð Þ , the complete fuzzy model is obtained by using a singleton fuzifier, product-type inference mechanism, and center of gravity as a

\_

∑r

<sup>i</sup>¼<sup>1</sup>wið Þ z tð Þ <sup>¼</sup> <sup>∑</sup><sup>r</sup>

<sup>i</sup>¼<sup>1</sup>wið Þ z tð Þ ð Þ Ai x tðÞþ Bi u tð Þ

<sup>i</sup>¼<sup>1</sup>wið Þ z tð Þ

<sup>i</sup>¼<sup>1</sup>hið Þ z tð Þ ð Þ Ci x tð Þ

<sup>i</sup>¼<sup>1</sup>hið Þ z tð Þ ð Þ Ai x tðÞþ Bi u tð Þ

subsystem and represents an operating point of the nonlinear system.

<sup>i</sup>¼<sup>1</sup>wið Þ z tð Þ ð Þ Ci x tð Þ

x tðÞ¼ <sup>∑</sup><sup>r</sup>

<sup>¼</sup> <sup>∑</sup><sup>r</sup>

∑r

where ziðÞ� t zpð Þt is the premise variables, Mi<sup>1</sup> � Mp<sup>1</sup> is the fuzzy sets, r is the number of linear subsystems, x tð Þ is the state vector, u tð Þ is the input vector, Aiis the linear submodel state matrices, and Bi is the input vector for each subsystem. Each consecutive linear equation represented by Aix tð Þþ Biu tð Þ is called a

G xi ð Þ¼ ; α<sup>i</sup>

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

0

0 ⋮ 0 xnð Þ 1 � G xð Þ <sup>n</sup> α<sup>n</sup>

� �

<sup>j</sup> ð Þ 1 � xn Mn

<sup>j</sup> ð Þ 1 � xn

i

1

CCCCCCCCCCCCA

� �V (7)

<sup>j</sup> is the heavy component

� �, the pressure in the column

(6)

(8)

(9)

(10)

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

$$\begin{aligned} \hat{\boldsymbol{x}}(t) &= \sum\_{i=1}^{r} h\_i(\mathbf{z}(t)) (A\_i \mathbf{x}(t) + B\_i \boldsymbol{u}(t) + K\_i(\mathbf{e}(t))) \\ \hat{\boldsymbol{y}} &= \sum\_{i=1}^{r} h\_i(\mathbf{z}(t)) (\mathbf{C}\_i \hat{\boldsymbol{x}}(t)) \end{aligned} \tag{14}$$

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

$$
\dot{e}(t) = \dot{\jmath}(t) - \dot{\jmath}(t) \tag{15}
$$

The fuzzy observer stability is demonstrated if each Ai, Ci pair is observable and P complies with the Lyapunov equation expressed in Eq. (16):

$$P\_i \overline{A}\_i + \overline{A}\_i' P\_i < 0 \tag{16}$$

where

$$\overline{\mathcal{A}}\_i = A\_i - K\_i C\_i$$

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

$$\begin{aligned} P &> 0\\ N\_i &> 0\\ A'\_i P - C'\_i N'\_i + PA\_i - N\_i C\_i &< 0\\ A'\_i P - C'\_j N'\_i + PA\_i - N\_i C\_j + PA'\_j - C'\_i N'\_j + PA\_j - N\_j C\_i &< 0\\ i &< j \end{aligned} \tag{17}$$

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

$$K\_i = P\_i^{-1} N\_i \tag{18}$$
