**3.** ð Þ� *α***,** *β* **pythagorean fuzzy descriptor systems**

As T-S fuzzy descriptor systems are very familiar to us, and pythagorean fuzzy sets are a new tool to deal with vagueness. So we decide to study the new (*α*, *β*)-

pythagorean fuzzy descriptor systems in order to solve practical control problems more easily and feasible. Next, the related definitions of ð Þ� *α*, *β* pythagorean fuzzy descriptor systems are gradually given.

**Definition 2.1** ð Þ� *α*, *β* pythagorean fuzzy descriptor systems are as follows: Rule *<sup>i</sup>*: if *<sup>x</sup>*1ð Þ*<sup>t</sup>* is *Pi* <sup>1</sup> and...and *xn*ð Þ*<sup>t</sup>* is *<sup>P</sup><sup>i</sup> <sup>n</sup>*, then.

$$
\dot{E}\dot{\mathbf{x}}(t) = A\_i \mathbf{x}(t) + B\_i \mu(t) \tag{1}
$$

$$\mathbf{y}(t) = \mathbf{C}\_i \mathbf{x}(t) + D\_i \boldsymbol{\mu}(t) \tag{2}$$

where *x t*ðÞ¼ *<sup>x</sup>*1ð Þ*<sup>t</sup>* � �, *<sup>x</sup>*2ð Þ*<sup>t</sup>* , <sup>⋯</sup>, *xn*ð Þ� *<sup>t</sup> <sup>T</sup>* <sup>∈</sup> *<sup>R</sup>n*and *<sup>μ</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup>*Rm*are the state vector and the control input vector, respectively;*y t*ð Þis the measurable output vector; *Ai*, *Bi*, *Ci* and *Di*are known real constant matrices with appropriate dimension;*E*is a singular matrix; *P<sup>i</sup>* 1,*Pi* 2,...,*Pi <sup>n</sup>*(*i* ¼ 1, 2, … ,*r*) are all pythagorean fuzzy sets.

By fuzzy blending, the overall fuzzy model is inferred as follows.

$$E\dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t) + B(t)\mu(t)$$

$$\mathbf{y}(t) = \mathbf{C}(t)\mathbf{x}(t) + D(t)\mu(t)$$

where

$$A(t) = \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) A\_i, B(t) = \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) B\_i,$$

$$\mathbf{C}(t) = \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) \mathbf{C}\_i, D(t) = \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) D\_i,$$

and *hi*ð Þ *x t*ð Þ is the normalized grade of membership, given as.

$$h\_i(\mathbf{x}(t)) = \frac{h\_{i(a,\boldsymbol{\beta})}(\mathbf{x}(t))}{\sum\_{i=1}^r h\_{i(a,\boldsymbol{\beta})}(\mathbf{x}(t))}, i = 1, 2, 3, \cdots, r;$$

where

$$h\_{i(a,\emptyset)}(\mathbf{x}(t)) = \begin{cases} h\_i^1(\mathbf{x}(t)) & \text{when } h\_i^1(\mathbf{x}(t)) \ge a \text{ or } h\_i^2(\mathbf{x}(t)) \le \emptyset \\ 0 & \text{else} \end{cases}, a + \beta \le 1, i = 1, 2, 3, \cdots, r;$$

$$h\_i^1(\mathbf{x}(t)) = \frac{\mu\_{p^i}(\mathbf{x}(t))}{\sum\_{i=1}^r \mu\_{p^i}(\mathbf{x}(t))}, h\_i^2(\mathbf{x}(t)) = \frac{\nu\_{p^i}(\mathbf{x}(t))}{\sum\_{i=1}^r \nu\_{p^i}(\mathbf{x}(t))},$$

where *h*<sup>1</sup> *<sup>i</sup>*ð Þ *x t*ð Þ and *<sup>h</sup>*<sup>2</sup> *<sup>i</sup>*ð Þ *x t*ð Þ are respectively positive and negative membership functions.

$$\sum\_{i=1}^{r} h\_{i1}(\mathbf{x}(t)) = \mathbf{1}, \sum\_{i=1}^{r} h\_{i2}(\mathbf{x}(t)) = \mathbf{1};$$

$$\mu\_{p^i}(\mathbf{x}\_j(t)) = \prod\_{j=1}^{r} \mu\_{p^i\_j}(\mathbf{x}\_j(t)), \nu\_{p^i}(\mathbf{x}\_j(t)) = \prod\_{j=1}^{r} \nu\_{p^i\_j}(\mathbf{x}\_j(t));$$

*μP j <sup>i</sup> <sup>x</sup> <sup>j</sup>*ð Þ*<sup>t</sup>* � �and *<sup>ν</sup>P<sup>i</sup> j <sup>x</sup> <sup>j</sup>*ð Þ*<sup>t</sup>* � �is the membership function value of *<sup>x</sup> <sup>j</sup>*ð Þ*<sup>t</sup>* that belongs and does not belong to the intuitionistic fuzzy numbers set *P<sup>i</sup> j :*

#### **Remark 2.1:**


Firstly, the relation of T-S fuzzy descriptor systems and (*α*,*β*)-pythagorean fuzzy descriptor systems is studied through an example.

When *α* ¼ 0, *β* ¼ 1, then

$$\begin{split} h\_i(\mathbf{x}(t)) &= h\_{i(a,\boldsymbol{\theta})}(\mathbf{x}(t)) = h\_{i1}(\mathbf{x}(t)) = \frac{\mu\_i^M(\mathbf{x}(t))}{\sum\_{i=1}^r \mu\_i^M(\mathbf{x}(t))}, \\ &= \prod\_{j=1}^n \mu\_{ij}^M(\mathbf{x}\_j(t)). \end{split} \quad h\_{i2}(\mathbf{x}(t)) = \mathbf{0}, \mu\_i^M(\mathbf{x}(t)) = \mathbf{0}, \quad \forall i \in \{1, \ldots, r\}.$$

Then the special (0,1)-pythagorean fuzzy descriptor systems are T-S fuzzy descriptor systems. In other words, T-S fuzzy descriptor systems are all the special (0,1)-pythagorean fuzzy descriptor systems. Therefore, it is easy to get the following Theorem 3.1.

Theorem 3.1 T-S fuzzy descriptor systems are all the (*α*,*β*)-pythagorean fuzzy descriptor systems.

Proof:It is so easy, so omit.
