Abstract
The present paper solves the approximation problem of T–S fuzzy linear singular system for a class of nonlinear singular system with impulses. Consider a special nonlinear singular bio-economic system with impulses; the T–S 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 T–S 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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Appendix
Appendix
Specific process of Sect. 4.
-
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.
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.
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
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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 \({M_{52}}\left( {{z_{5}}(t)} \right)\)
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THEN\(E\dot{x}\left( t \right) = {A_{28}}x\left( t \right) + {B_{28}}u\left( t \right)\)
Model Rule 29
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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 \({M_{51}}\left( {{z_{5}}(t)} \right)\)
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THEN\(E\dot{x}\left( t \right) = {A_{29}}x\left( t \right) + {B_{29}}u\left( t \right)\)
Model Rule 30
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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 \({M_{52}}\left( {{z_{5}}(t)} \right)\)
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THEN\(E\dot{x}\left( t \right) = {A_{30}}x\left( t \right) + {B_{30}}u\left( t \right)\)
Model Rule 31
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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 \({M_{51}}\left( {{z_{5}}(t)} \right)\)
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THEN\(E\dot{x}\left( t \right) = {A_{31}}x\left( t \right) + {B_{31}}u\left( t \right)\)
Model Rule 32
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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 \({M_{52}}\left( {{z_{5}}(t)} \right)\)
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THEN\(E\dot{x}\left( t \right) = {A_{32}}x\left( t \right) + {B_{32}}u\left( t \right)\)
where
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4.
The T–S 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 T–S 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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DOI: https://doi.org/10.1007/s00521-019-04576-0