zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fractional order control of a coupled tank. (English) Zbl 1204.93086
Summary: A hybrid system that combines the advantages in terms of robustness of the fractional control and the Sliding Mode Control (SMC) will be proposed. The proposed fractional order SMC is applied to a level control in a nonlinear coupled tank, as a case study. To investigate the capability of the method, a Sliding Mode Controller is alternatively designed. Primarily a sliding surface based on linear compensation networks PD or PID is designed. The work is followed by designation of a fractional form of these networks, $PD \mu $ or $PI \lambda D \mu $ . Finally, the performance of the proposed technique is also investigated under disturbance and variation in parameters of system. The simulation results indicate the significance of the fractional order sliding mode controllers.

93C95Applications of control theory
34A08Fractional differential equations
93B12Variable structure systems
76B75Flow control and optimization
Full Text: DOI
[1] Ahn, H.S., Chen, Y., Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, 27--34 (2007) · Zbl 1123.93074 · doi:10.1016/j.amc.2006.08.099
[2] Valério, D., Costa, J.: Tuning of fractional PID controllers with Ziegler--Nichols-type rules. Signal Process. 86, 2771--2784 (2006) · Zbl 1172.94496 · doi:10.1016/j.sigpro.2006.02.020
[3] Feliu-Batlle, V., Rivas Pérez, R., Sánchez Rodríguez, L.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng. Pract. 15, 673--686 (2007) · doi:10.1016/j.conengprac.2006.11.018
[4] Calderón, A.J., Vinagre, B.M., Feliu, V.: Fractional order control strategies for power electronic buck converters. Signal Process. 86, 2803--2819 (2006) · Zbl 1172.94377 · doi:10.1016/j.sigpro.2006.02.022
[5] Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136 (1999) · doi:10.1103/PhysRevLett.82.1136
[6] Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Automat. Control 29(5), 441--444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[7] Laskin, N.: Fractional market dynamics. Physica A 287, 482 (2000) · Zbl 0948.81595 · doi:10.1016/S0378-4371(00)00387-3
[8] El-Sayed, A.M.A.: Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35(2), 311 (1996) · Zbl 0846.35001 · doi:10.1007/BF02083817
[9] Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) · Zbl 30.0801.03
[10] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[11] Yu, C., Gao, G.: Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26--29 (2005) · Zbl 1088.34501 · doi:10.1016/j.jmaa.2004.12.015
[12] Ladaci, S., Loiseau, J.J., Charef, A.: Fractional order adaptive high-gain controllers for a class of linear systems. Commun. Nonlinear Sci. Numer. Simul. 13, 707--714 (2008) · Zbl 1221.93128 · doi:10.1016/j.cnsns.2006.06.009
[13] Oustaloup, A., Bluteau, B., Nouillant, M.: First generation scalar CRONE control: application to a two DOF manipulator and comparison with nonlinear decoupling control. Int. Conf. Syst., Man Cybern. 4, 453--458 (1993)
[14] Oustaloup, A.: Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation. IEEE Trans. Circ. Syst. 28(10), 1007--1009 (1981) · doi:10.1109/TCS.1981.1084917
[15] Serrier, P., Moreau, X., Sabatier, J., Oustaloup, A.: Taking into account the non-linearities in the CRONE approach: application to vibration isolation. In: 32nd Annual Conference on Industrial Electronics, IECON 2006, pp. 5360--5365 (2006)
[16] Lanusse, P., Benlaoukli, H., Nelson-Gruel, D., Oustaloup, A.: Fractional-order control and interval analysis of SISO systems with time-delayed state. Control Theory Appl., IET 2(1), 16--23 (2008) · doi:10.1049/iet-cta:20060491
[17] Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991) · Zbl 0753.93036
[18] Khan, M.Kh., Spurgeon, S.K.: Robust MIMO water level control in interconnected twin-tanks using second order sliding mode control. Control Eng. Pract. 14, 375--386 (2006) · doi:10.1016/j.conengprac.2005.02.001
[19] Almutairi, N.B., Zribi, M.: Sliding mode control of coupled tanks. Mechatronics 16, 427--441 (2006) · doi:10.1016/j.mechatronics.2006.03.001
[20] Yau, H.T., Chen, C.L.: Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos Solitons Fractals 30, 709--718 (2006) · doi:10.1016/j.chaos.2006.03.077
[21] Shahnazi, R., Shanechi, H., Pariz, N.: Position control of induction and DC servomotors: a novel adaptive fuzzy PI sliding mode control. In: Power Engineering Society General Meeting, pp. 1--9 (2006)
[22] Wang, T., Tong, Sh.Ch.: Fuzzy sliding mode control for nonlinear systems. Int. Conf. Mach. Learn. Cybern. 2, 839--844 (2004)
[23] Moshiri, B., Jalili-Kharaajoo, M., Besharati, F.: Application of fuzzy sliding mode based on genetic algorithms to control of robotic manipulators. In: Emerging Technologies and Factory Automation, vol. 2, pp. 169--172 (2003)
[24] Khoei, A., Hadidi, Kh., Khorasani, M.R., Amirkhanzadeh, R.: Fuzzy level control of a tank with optimum valve movement. Fuzzy Sets Syst. 150, 507--523 (2005) · Zbl 1060.93551 · doi:10.1016/j.fss.2004.09.009
[25] Poulsen, N.K., Kouvaritakis, B., Cannon, M.: Nonlinear constrained predictive control applied to a coupled-tanks apparatus. IEE Proc. Control Theory Appl. 148, 17--24 (2001) · Zbl 1010.93043 · doi:10.1049/ip-cta:20010231
[26] Delavari, H., Ranjbar, A.: Robust intelligent control of coupled tanks. In: WSEAS International Conferences, pp. 1--6, Istanbul (2007)
[27] Delavari, H., Ranjbar, A.: Genetic-based fuzzy sliding mode control of an interconnected twin-tanks. In: IEEE Region 8 EUROCON 2007 Conference, pp. 714--719, Poland (2007)
[28] Alli, H., Yakut, O.: Fuzzy sliding-mode control of structures. Eng. Struct. 27, 277--284 (2005) · doi:10.1016/j.engstruct.2004.10.007
[29] Liang, C.Y., Su, J.P.: A new approach to the design of a fuzzy sliding mode controller. Fuzzy Sets Syst. 139, 111--124 (2003) · Zbl 1045.93027 · doi:10.1016/S0165-0114(02)00480-3
[30] Hung, L.Ch., Lin, H.P., Chung, H.Y.: Design of self-tuning fuzzy sliding mode control for TORA system. Expert Syst. Appl. 32, 201--212 (2007) · doi:10.1016/j.eswa.2005.11.008
[31] Yau, H.T., Chen, C.L.: Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos Solitons Fractals 30, 709--718 (2006) · doi:10.1016/j.chaos.2006.03.077
[32] Hosein Nia, S.H., Ranjbar, A.N., Ganji, D.D., Soltani, H., Ghasemi, J.: Maintaining the stability of nonlinear differential equations by the enhancement of HPM. Phys. Lett. A 372, 2855--2861 (2008) · Zbl 1220.70018 · doi:10.1016/j.physleta.2007.12.054
[33] Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional order predator--prey and rabies models. J. Math. Anal. Appl. 325(1), 542--553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087