2.2 Distillation column Takagi-Sugeno model

The Takagi-Sugeno model is the fuzzy representation of a nonlinear model obtained from the linear subsystems interpolation according to fuzzy rules having the form (Eq. 9):

$$\begin{aligned} \textbf{Model } i \text{ rule} \\ \textbf{If } z\_i \ (t) \text{ is } M\_{i1} \text{ and } \dots \text{ and } z\_p(t) \text{ is } M\_{p1} \\ \textbf{Then} \\ \dot{x}(t) = A\_i x(t) + B\_i u(t) \\ \dot{\phantom{x}} = \mathbf{1}, \mathbf{2}, \dots, r \end{aligned} \tag{9}$$

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 subsystem and represents an operating point of the nonlinear system.

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 defuser, as described in Eq. (10):

$$\begin{aligned} x(t) &= \frac{\sum\_{i=1}^{r} w\_i(z(t)) (A\_i \ge (t) + B\_i \, u(t))}{\sum\_{i=1}^{r} w\_i(z(t))} \\ &= \sum\_{i=1}^{r} h\_i(z(t)) (A\_i \, \mathbf{x}(t) + B\_i \, u(t)) \\ y(t) &= \frac{\sum\_{i=1}^{r} w\_i(z(t)) (C\_i \, \mathbf{x}(t))}{\sum\_{i=1}^{r} w\_i(z(t))} = \sum\_{i=1}^{r} h\_i(z(t)) (C\_i \, \mathbf{x}(t)) \end{aligned} \tag{10}$$

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

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

$$z(t) = \begin{bmatrix} z\_1(t), z\_2(t), \dots, z\_p(t) \end{bmatrix} \tag{11}$$

In addition, the calculated weight wið Þ z tð Þ for each i rule from the membership functions is defined by Eq. (12):

$$w\_i(\mathbf{z}(t)) = \prod\_{j=1}^p M\_{ij}\mathbf{z}\_j(t) \tag{12}$$

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

$$h\_i(z(t)) = \frac{w\_i(z(t))}{\sum\_{i=1}^r w\_i(z(t))}\tag{13}$$
