The approximation of the TS fuzzy model for a class of nonlinear singular systems with impulses

Abstract

The present paper solves the approximation problem of TS fuzzy linear singular system for a class of nonlinear singular system with impulses. Consider a special nonlinear singular bio-economic system with impulses; the TS fuzzy linear singular system of the nonlinear singular system has been calculated. The relationship between the impulse of the singular system and the singular induced bifurcation is proved for the first time. For this particular case, it is extended to more generally nonlinear singular system. For a class of nonlinear singular system that is bounded impulse-free item and separable impulse item with singularity-induced bifurcation, we proved that it can be approximated by TS fuzzy singular system with arbitrary accuracy. Finally, a numerical simulation is carried out to show the consistency with theoretical analysis and illustrate the effectiveness of approximation.

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Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No. 61673099, National Natural Science Foundation of China under Grant Nos. 11661050, National Natural Science Foundation of China under Grant No. 61673100 and Fundamental Research Funds for Central Universities under Grant No. 150504011.

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Correspondence to Zhenghong Jin.

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Appendix

Appendix

Specific process of Sect. 4.

  1. 1.

    Define \({z_1}(t) = 2\left( {1 - {x_1}(t)} \right)\), \({z_2}(t) = - {x_1}(t)\), \({z_3}(t) = 0.9{x_1}(t)\), \({z_4}(t) = - {x_3}(t)\) and \({z_5}(t) = 2{x_3}(t) - 1\). Then, it has

    $$\begin{aligned} \begin{array}{l} E\dot{x}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{z_1}(t)}&{}\quad 0&{}\quad {{z_2}(t)}&{}\quad 0\\ 0&{}\quad { -\,1.1}&{}\quad {{z_3}(t)}&{}\quad 0\\ 0&{}\quad {0.8}&{}\quad { -\,0.3}&{}\quad {{z_4}(t)}\\ 0&{}\quad 0&{}\quad 0&{}\quad {{z_5}(t)} \end{array}} \right] x\left( t \right) + \left[ {\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ 1 \end{array}} \right] \sin \left( t \right) \end{array} \end{aligned}$$
    (28)
  2. 2.

    Since \(x_1(t) \in \left[ {0,2} \right]\), \(x_2(t) \in \left[ {0,2.5} \right]\) and \(x_3(t) \in \left[ {0,4} \right]\), \(z_1(t) \in \left[ {-\,2,2} \right]\), \(z_2(t) \in \left[ {-\,2,0} \right]\), \(z_3(t) \in \left[ {0,1.8} \right]\), \(z_4(t) \in \left[ {-\,4,0} \right]\) and \(z_5(t) \in \left[ {-\,1,7} \right]\).

    According the maximum and minimum values of \(z_i(t),i=1,2,3,4,5\), the \(z_i(t)\) can be represented by

    $$\begin{aligned}\begin{array}{l} {z_1}(t) = {M_{11}}\left( {z_1(t)} \right) \cdot 2 + {M_{12}}\left( {z_1(t)} \right) \cdot \left( { - 2} \right) \\ {z_2}(t) = {M_{21}}\left( {z_2(t)} \right) \cdot 0 + {M_{22}}\left( {z_2(t)} \right) \cdot \left( { - 2} \right) \\ {z_3}(t) = {M_{31}}\left( {z_3(t)} \right) \cdot 1.8 + {M_{32}}\left( {z_3(t)} \right) \cdot 0 \\ {z_4}(t) = {M_{41}}\left( {z_4(t)} \right) \cdot 0 + {M_{42}}\left( {z_4(t)} \right) \cdot \left( { - 4} \right) \\ {z_5}(t) = {M_{51}}\left( {z_5(t)} \right) \cdot 7 + {M_{52}}\left( {z_5(t)} \right) \cdot \left( { - 1} \right) \\ \end{array}\end{aligned}$$

    where

    $$\begin{aligned} {M_{i1}} + {M_{i2}} = 1,\quad i = 1,2,3,4,5 \end{aligned}$$

    So,

    $$\begin{aligned}\begin{array}{l} {M_{11}}\left( {z_1(t)} \right) = \frac{{{z_1}(t) + 2}}{4},\quad {M_{12}}\left( {z_1(t)} \right) = \frac{{2 - {z_1}(t)}}{4} \\ {M_{21}}\left( {z_2(t)} \right) = \frac{{{z_2}(t) + 2}}{2},\quad {M_{22}}\left( {z_2(t)} \right) = - \frac{{{z_2}(t)}}{2} \\ {M_{31}}\left( {z_3(t)} \right) = \frac{{{z_3}(t)}}{{1.8}},\quad {M_{32}}\left( {z_3(t)} \right) = \frac{{1.8 - {z_3}(t)}}{{1.8}} \\ {M_{41}}\left( {z_4(t)} \right) = \frac{{{z_4}(t) + 4}}{4},\quad {M_{42}}\left( {z_4(t)} \right) = - \frac{{{z_4}(t)}}{4} \\ {M_{51}}\left( {z_5(t)} \right) = \frac{{{z_5}(t) + 1}}{8},\quad {M_{52}}\left( {z_5(t)} \right) = \frac{{7 - {z_5}(t)}}{8} \\ \end{array} \end{aligned}$$
  3. 3.

    The model rules are given. For \(i=1,2, \ldots ,32\), it has

Model Rule 1

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{1}}x\left( t \right) + {B_{1}}u\left( t \right)\)

Model Rule 2

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{2}}x\left( t \right) + {B_{2}}u\left( t \right)\)

Model Rule 3

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{3}}x\left( t \right) + {B_{3}}u\left( t \right)\)

Model Rule 4

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{4}}x\left( t \right) + {B_{4}}u\left( t \right)\)

Model Rule 5

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{5}}x\left( t \right) + {B_{5}}u\left( t \right)\)

Model Rule 6

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{6}}x\left( t \right) + {B_{6}}u\left( t \right)\)

Model Rule 7

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{7}}x\left( t \right) + {B_{7}}u\left( t \right)\)

Model Rule 8

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{8}}x\left( t \right) + {B_{8}}u\left( t \right)\)

Model Rule 9

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{9}}x\left( t \right) + {B_{9}}u\left( t \right)\)

Model Rule 10

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{10}}x\left( t \right) + {B_{10}}u\left( t \right)\)

Model Rule 11

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{11}}x\left( t \right) + {B_{11}}u\left( t \right)\)

Model Rule 12

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{12}}x\left( t \right) + {B_{12}}u\left( t \right)\)

Model Rule 13

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{13}}x\left( t \right) + {B_{13}}u\left( t \right)\)

Model Rule 14

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{14}}x\left( t \right) + {B_{14}}u\left( t \right)\)

Model Rule 15

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{15}}x\left( t \right) + {B_{15}}u\left( t \right)\)

Model Rule 16

  • IF\(z_1(t)\) is \({M_{11}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{16}}x\left( t \right) + {B_{16}}u\left( t \right)\)

Model Rule 17

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{17}}x\left( t \right) + {B_{17}}u\left( t \right)\)

Model Rule 18

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{18}}x\left( t \right) + {B_{18}}u\left( t \right)\)

Model Rule 19

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{19}}x\left( t \right) + {B_{19}}u\left( t \right)\)

Model Rule 20

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{20}}x\left( t \right) + {B_{20}}u\left( t \right)\)

Model Rule 21

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{21}}x\left( t \right) + {B_{21}}u\left( t \right)\)

Model Rule 22

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{22}}x\left( t \right) + {B_{22}}u\left( t \right)\)

Model Rule 23

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{23}}x\left( t \right) + {B_{23}}u\left( t \right)\)

Model Rule 24

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{21}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{24}}x\left( t \right) + {B_{24}}u\left( t \right)\)

Model Rule 25

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{25}}x\left( t \right) + {B_{25}}u\left( t \right)\)

Model Rule 26

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{26}}x\left( t \right) + {B_{26}}u\left( t \right)\)

Model Rule 27

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{27}}x\left( t \right) + {B_{27}}u\left( t \right)\)

Model Rule 28

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{31}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{28}}x\left( t \right) + {B_{28}}u\left( t \right)\)

Model Rule 29

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{29}}x\left( t \right) + {B_{29}}u\left( t \right)\)

Model Rule 30

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{41}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{30}}x\left( t \right) + {B_{30}}u\left( t \right)\)

Model Rule 31

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{51}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{31}}x\left( t \right) + {B_{31}}u\left( t \right)\)

Model Rule 32

  • IF\(z_1(t)\) is \({M_{12}}\left( {{z_1}(t)} \right)\) and \(z_2(t)\) is \({M_{22}}\left( {{z_2}(t)} \right)\) and \(z_3(t)\) is \({M_{32}}\left( {{z_3}(t)} \right)\) and \(z_{4}(t)\) is \({M_{42}}\left( {{z_{4}}(t)} \right)\) and \(z_{5}(t)\) is \({M_{52}}\left( {{z_{5}}(t)} \right)\)

  • THEN\(E\dot{x}\left( t \right) = {A_{32}}x\left( t \right) + {B_{32}}u\left( t \right)\)

where

$$\begin{aligned} {A_1}& = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_2} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_3} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] , \quad {A_4} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_5}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_6} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] \\ {A_7}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_8} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_9}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{10}} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{11}}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{12}} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] \\ {A_{13}}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{14}} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{15}}= & {} \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] , \quad {A_{16}} = \left[ {\begin{array}{*{20}{r}} 2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{17}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{18}} =\left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] \\ {A_{19}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{20}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{21}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] , \quad {A_{22}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{23}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{24}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] \\ {A_{25}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{26}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{27}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{28}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad {1.8} &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] ,\\ {A_{29}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{30}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] \\ {A_{31}}= & {} \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 7 \\ \end{array}} \right] ,\quad {A_{32}} = \left[ {\begin{array}{*{20}{r}} -\,2 &{}\quad 0 &{}\quad { -\,2} &{}\quad 0 \\ 0 &{}\quad { -\,1.1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {0.8} &{}\quad { -\,0.3} &{}\quad { -\,4} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad { -\,1} \\ \end{array}} \right] .\end{aligned}$$
  1. 4.

    The TS fuzzy model is given as:

    $$\begin{aligned} E\dot{x}\left( t \right) = \sum \limits _{i = 1}^{32} {{h_i}\left( {z(t)} \right) \left[ {{A_i}x(t) + {B_i}u(t)} \right] } \end{aligned}$$
    (29)

    where \({h_i} = \prod \nolimits _{j = 1}^5 {{M_{jk}}} ,k = 1,2,i=1,2, \ldots ,32\).

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Jin, Z., Zhang, Q. & Ren, J. The approximation of the TS fuzzy model for a class of nonlinear singular systems with impulses. Neural Comput & Applic 32, 10387–10401 (2020). https://doi.org/10.1007/s00521-019-04576-0

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Keywords

  • TS fuzzy singular systems
  • Impulses
  • Approximation
  • Singularity-induced bifurcations