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In this study, a new systematic approach is proposed to design the fuzzy controller for a class of Takagi–Sugeno fuzzy-partial differential equation (TS fuzzy-PDE) systems which describe the non-linear distributed parameter system formulated by first-order semi-linear hyperbolic PDEs. In this study, non-quadratic Lyapunov function is utilised and some slack matrices are introduced to derive stability conditions in terms of linear matrix inequalities (LMIs). The proposed approach has three main features. First, stability conditions are not derived in the form of spatial differential LMI. Second, conservativeness of LMI conditions is reduced. Third, there is no restriction on the form of semi-linear hyperbolic PDE systems and therefore more semi-linear systems classes can be stabilised. Also, the proposed approach is more suitable for practical implementation compared with the recently published papers.
References
-
-
1)
-
6. Ray, W.H.: ‘Advanced process control’ (McGraw-Hill, New York, 1981).
-
2)
-
25. Aksikas, I., Winkin, J.J., Dochain, D.: ‘Asymptotic stability of infinite dimensional semi-linear systems: application to a nonisothermal reactor’, Syst. Control Lett., 2007, 56, (2), pp. 122–132 (doi: 10.1016/j.sysconle.2006.08.012).
-
3)
-
14. Karafyllis, I., Daoutidis, P.: ‘Control of hot spots in plug flow reactors’, Comput. Chem. Eng., 2002, 26, (7–8), pp. 1087–1094 (doi: 10.1016/S0098-1354(02)00027-3).
-
4)
-
7. Sira-Ramirez, H.: ‘Distributed sliding mode control in systems described by quasilinear partial differential equations’, Syst. Control Lett., 1989, 13, (2), pp. 177–181 (doi: 10.1016/0167-6911(89)90036-4).
-
5)
-
20. Patwardhan, A.A., Wright, G.T., Edgar, T.F.: ‘Nonlinear model predictive control of distributed parameter systems’, Chem. Eng. Sci., 1992, 47, (4), pp. 721–735 (doi: 10.1016/0009-2509(92)80264-D).
-
6)
-
19. Dubljevic, S., Mhaskar, P., El-Farra, N.H., Christofides, P.D.: ‘Predictive control of transport reaction processes’, Comput. Chem. Eng., 2005, 29, (11–12), pp. 2335–2345 (doi: 10.1016/j.compchemeng.2005.05.008).
-
7)
-
2. Aggelogiannaki, E., Sarimveis, H.: ‘Robust nonlinear H∞ control of hyperbolic distributed parameter systems’, Control Eng. Pract., 2009, 17, (6), pp. 723–732 (doi: 10.1016/j.conengprac.2008.11.005).
-
8)
-
4. Montaseri, G., Yazdanpanah, M.J.: ‘Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model’, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, (1), pp. 388–404 (doi: 10.1016/j.cnsns.2011.05.019).
-
9)
-
37. Tanaka, K., Wang, H.O.: ‘Fuzzy control systems design and analysis: a linear matrix inequality approach’ (John Wiley & Sons, Inc., 2001).
-
10)
-
17. Christofides, P.D., Daoutidis, P.: ‘Robust control of hyperbolic PDE systems’, Chem. Eng. Sci., 1998, 53, (1), pp. 85–105 (doi: 10.1016/S0009-2509(97)87571-9).
-
11)
-
22. Aksikas, I., Winkin, J.J., Dochain, D.: ‘Optimal LQ-feedback for a class of first-order hyperbolic distributed parameter systems’, ESAIM, Control, Optimisation Calc. Var. (France), 2008, 14, (4), pp. 897–908 (doi: 10.1051/cocv:2008015).
-
12)
-
32. Wang, J.W., Wu, H.N., Li, H.X.: ‘Distributed proportional–spatial derivative control of nonlinear parabolic systems via fuzzy PDE modeling approach’, IEEE Syst. Man Cybern. Soc., 2012, 42, (3), pp. 927–938 (doi: 10.1109/TSMCB.2012.2185046).
-
13)
-
38. Slotine, E., Li, W.: ‘Applied nonlinear control’ (Prentice-Hall, Englewood Cliffs, NJ, 1991).
-
14)
-
18. Shang, H., Forbes, J.F., Guay, M.: ‘Feedback control of hyperbolic distributed parameter systems’, Chem. Eng. Sci., 2005, 60, (4), pp. 969–980 (doi: 10.1016/j.ces.2004.09.067).
-
15)
-
11. Callier, F., Winkin, J.J.: ‘LQ-optimal control of infinite-dimensional systems by spectral factorization’, Automatica, 1992, 28, (4), pp. 757–770 (doi: 10.1016/0005-1098(92)90035-E).
-
16)
-
28. Manai, Y., Benrejeb, M.: ‘New condition of stabilisation for continuous Takagi-Sugeno fuzzy system based on fuzzy lyapunov function’, Int. J. Control Autom., 2011, 4, (3), pp. 51–63.
-
17)
-
24. Aksikas, I., Winkin, J.J., Dochain, D.: ‘Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization’, IEEE Trans. Autom. Control, 2007, 52, (7), pp. 1179–1193 (doi: 10.1109/TAC.2007.900823).
-
18)
-
29. Wu, H.N., Li, H.X.: ‘Finite-dimensional constrained fuzzy control of a class of nonlinear distributed process systems’, IEEE Trans. Syst. Man Cybern., 2007, 37, (5), pp. 1422–1430 (doi: 10.1109/TSMCB.2007.904026).
-
19)
-
16. Christofides, P.D.: ‘Nonlinear and robust control of PDE systems: methods and applications to transport reaction processes’ (Birkhauser, Boston, 2001).
-
20)
-
72. Lam, H.K., Lauber, J.: ‘Membership-function-dependent stability analysis of fuzzy-model-based control systems using fuzzy Lyapunov functions’, Inf. Sci., 2013, (232), pp. 253–266, (doi: 10.1016/j.ins.2012.12.027).
-
21)
-
30. Wu, H.N., Li, H.X.: ‘H∞ fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints’, IEEE Trans. Fuzzy Syst., 2008, 16, (2), pp. 502–516 (doi: 10.1109/TFUZZ.2007.896351).
-
22)
-
79. Guerra, T.M., Bernal, M., Guelton, K., Labiod, S.: ‘Non-quadratic local stabilization for continuous-time Takagi–Sugeno models’, IEEE Fuzzy Sets Syst., 2012, 201, pp. 40–54 (doi: 10.1016/j.fss.2011.12.003).
-
23)
-
3. Aksikas, I., Fuxman, A., Forbes, J.F., Winkin, J.J.: ‘LQ control design of a class of hyperbolic PDE systems: application to fixed-bed reactor’, Automatica, 2009, 45, (6), pp. 1542–1548 (doi: 10.1016/j.automatica.2009.02.017).
-
24)
-
5. Mahadev, N., Hoo, K.A.: ‘Wavelet-based model reduction of distributed parameter systems’, Chem. Eng. Sci., 2000, 55, (19), pp. 4271–90 (doi: 10.1016/S0009-2509(00)00062-2).
-
25)
-
L.A. Mozelli ,
R.M. Palhares ,
G.S. Avellar
.
A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems.
Inf. Sci.
,
8 ,
1149 -
1162
-
26)
-
35. Wu, H.N., Wang, J.W., Li, H.X.: ‘Design of distributed H∞ fuzzy controllers with constraint for nonlinear hyperbolic PDE systems’, Automatica, 2012, 48, (10), pp. 2535–2543 (doi: 10.1016/j.automatica.2012.06.043).
-
27)
-
9. Hanczyc, E.M., Palazoglu, A.: ‘Nonlinear control of a distributed parameter process: the case of multiple characteristics’, Ind. Eng. Chem. Res., 1995, 34, (12), pp. 4406–4412 (doi: 10.1021/ie00039a032).
-
28)
-
21. Shang, H., Forbes, J.F., Guay, M.: ‘Model predictive control for quasi-linear hyperbolic distributed parameter systems’, Ind. Eng. Chem. Res., 2004, 43, (9), pp. 2140–2149 (doi: 10.1021/ie030653z).
-
29)
-
34. Wang, J.W., Wu, H.N., Li, H.X.: ‘Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters’, Automatica, 2012, 48, (3), pp. 569–576 (doi: 10.1016/j.automatica.2012.01.006).
-
30)
-
1. Wang, J.W., Wu, H.N., Li, H.X.: ‘Distributed fuzzy control design of nonlinear hyperbolic PDE systems with application to nonisothermal plug-flow reactor’, IEEE Trans. Fuzzy Syst., 2011, 19, (3), pp. 514–526 (doi: 10.1109/TFUZZ.2011.2116028).
-
31)
-
8. Hanczyc, E.M., Palazoglu, A.: ‘Sliding mode control of nonlinear distributed parameter chemical processes’, Ind. Eng. Chem. Res., 1995, 34, (2), pp. 557–566 (doi: 10.1021/ie00041a016).
-
32)
-
12. Curtain, R.F., Zwart, H.J.: ‘An introduction to infinite-dimensional linear systems theory’ (Springer-Verlag, New York, 1995).
-
33)
-
15. Alotaibi, S., Sen, M., Goodwine, B., Yang, K.T.: ‘Flow based control of temperature in long ducts’, Int. J. Heat Mass Transf., 2004, 47, (23), pp. 4995–5009 (doi: 10.1016/j.ijheatmasstransfer.2004.06.017).
-
34)
-
13. Christofides, P.D., Daoutidis, P.: ‘Feedback control of hyperbolic PDE systems’, AIChE J., 1996, 42, (11), pp. 3063–3086 (doi: 10.1002/aic.690421108).
-
35)
-
31. Chen, B.S., Chang, Y.T.: ‘Fuzzy state-space modeling and robust observer-based control design for nonlinear partial differential systems’, IEEE Trans. Fuzzy Syst., 2009, 17, (5), pp. 1025–1043 (doi: 10.1109/TFUZZ.2009.2020506).
-
36)
-
10. Callier, F., Winkin, J.J.: ‘Spectral factorization and LQ-optimal regulation for multivariable distributed systems’, Int. J. Control, 1990, 52, (1), pp. 55–75 (doi: 10.1080/00207179008953524).
-
37)
-
33. Wu, H.N., Wang, J.W., Li, H.X.: ‘Exponential stabilization for a class of nonlinear parabolic PDE systems via fuzzy control approach’, IEEE Trans. Fuzzy Syst., 2012, 20, (2), pp. 318–329 (doi: 10.1109/TFUZZ.2011.2173694).
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