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Received September 26, 2014
Accepted December 22, 2014
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A novel and computationally efficient algorithm for stability analysis of multi input-multi output process control systems
Hooman Fatoorehchi1 2
Hossein Abolghasemi1 2†
Reza Zarghami2 3
Randolph Rach4
Sebastian von Freeden5
1Center for Separation Processes Modeling and Nano-Computations, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran 11155-4563, Iran 2Advanced Process Control Research Laboratory, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran 11155-4563, Iran 3Multiphase Systems Research Laboratory, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran 11155-4563, Iran 4The George Adomian Center for Applied Mathematics, 316 South Maple Street, Hartford, Michigan 49057-1225, U.S.A., USA 5Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), University of Stuttgart, Stuttgart 70174, Germany
hoab@ut.ac.ir
Korean Journal of Chemical Engineering, September 2015, 32(9), 1733-1743(11), 10.1007/s11814-014-0385-8
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Abstract
An efficient method based on the Faddeev-Leverrier algorithm combined with the Adomian decomposition method is devised to facilitate the stability analysis of multi-input multi-output control systems. In contrast to prior eigenvalue algorithms, our method affords all eigenvalues of the state matrix, either real or complex. Specifically, the calculation of the complex eigenvalues is made possible through special canonical forms, mainly involving square root operators, of the characteristic equation of the state matrix. Moreover, the proposed method does not require an initial guess, which is often a matter of concern since an inappropriate guess can cause failure in such available schemes. For the sake of illustration, a number of numerical examples, including chemical reaction processes, are also provided that demonstrate the efficiency of our new technique.
Keywords
References
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Rach R, Duan JS, Wazwaz AM, J. Math. Chem., 52, 255 (2014)
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Rach R, Duan JS, Wazwaz AM, Chem. Eng. Commun., DOI:10.1080/00986445.2014.900054.
Hasseine A, Barhoum Z, Attarakih M, Bart HJ, Adv. Polym. Technol., (In press), DOI:10.1016/j.apt.2014.08.011.
Kundu B, Int. J. Refrig., 32, 1657 (2009)
Kundu B, Wongwises S, J. Franklin Inst., 349, 966 (2011)
Rach R, Kybernetes, 41, 1087 (2012)
Saravanan A, Magesh N, J. Egyptian Math. Soc., 21, 259 (2013)
Siddiqui AM, Hameed M, Siddiqui BM, Ghori QK, Comm. Nonlinear Sci. Numer. Simulat., 15, 2388 (2010)
Wazwaz AM, Rach R, Duan JS, Appl. Math. Comput., 219, 5004 (2013)
Hoffman JD, Frankel S, Numerical Methods for Engineers and Scientists, McGraw-Hill, New York (2001).
Shanks D, J. Math. Phys. Sci., 34, 1 (1955)
Bakosova M, Puna D, Dostal P, Zavacka J, Chem. Paper Chem. Zvesti, 63, 527 (2009)
Pottmann M, Seborg DE, J. Proc. Cont., 2, 189 (1992)
Watkins DS, Fundamentals of Matrix Computations, Wiley, New York (1991).
Bronson R, Costa GB, Linear algebra: An introduction, Academic Press, New York (2007).
Arpornwichanop A, Kittisupakorn P, Hussain MA, Korean J. Chem. Eng., 19(2), 221 (2002)
Coughanowr DR, Koppel LB, Process Systems Analysis and Control, McGraw-Hill, New York (1965).
Dorf RC, Bishop RH, Modern Control Systems, Prentice-Hall, New Jersey (2001).
Gopal M, Digital Control Engineering, John Wiley and Sons, New York (1988).
Han M, Clough DE, Korean J. Chem. Eng., 23(4), 540 (2006)
Ogata K, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ (1999).
Stephanopoulos G, Chemical Process Control: An Introduction to Theory and Practice, Prentice Hall, Englewood Cliffs, NJ (1984).
Allaire GA, Kaber SM, Numerical Linear Algebra, Springer, New York (2002).
Pantazis RD, Szyld DB, Numer. Lin. Algebra Appl., 2, 251 (1995)
Saad Y, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester (1992).
Wilkinson JH, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford (1988).
Fatoorehchi H, Abolghasemi H, J. Egyptian Math. Soc., 22, 6 (2014)
Gopal M, Modern Control System Theory, Wiley, New York (1993).
Helmberg G, Wagner P, Lin. Algebra Appl., 185, 219 (1993)
Householder AS, The Theory of Matrices in Numerical Analysis, Blaisdell, New York (1964).
Adomian G, Rach R, J. Math. Anal. Appl., 91, 39 (1983)
Adomian G, Solving frontier problems of physics: The decomposition method, Kluwer Academic, Dordrecht (1994).
Abbaoui K, Cherruault Y, Math. Comput. Model., 20, 69 (1994)
Abbaoui K, Cherruault Y, Comput. Math. Appl., 28, 103 (1994)
Abdelrazec A, Pelinovsky D, Numer. Meth. Part. Differ. Equat., 27, 749 (2011)
Babolian E, Biazar J, Appl. Math. Comput., 130, 383 (2002)
Cherruault Y, Adomian G, Math. Comput. Model., 18, 103 (1993)
Fatoorehchi H, Abolghasemi H, J. Appl. Comput. Sci. Math., 5, 85 (2011)
Babolian E, Javadi S, Appl. Math. Comput., 153, 253 (2004)
Duan JS, Appl. Math. Comput., 216, 1235 (2010)
Duan JS, Appl. Math. Comput., 217, 6337 (2011)
Rach R, J. Math. Anal. Appl., 102, 415 (1984)
Rach R, Kybernetes, 37, 910 (2008)
Abbasbandy S, Appl. Math. Comput., 145, 887 (2003)
Abbasbandy S, Darvishi MT, Appl. Math. Comput., 163, 1265 (2005)
Adomian G, J. Math. Anal. Appl., 102, 402 (1984)
Adomian G, Nonlinear Stochastic Operator Equations, Academic Press, New York (1986).
Adomian G, Math. Comput. Model., 13, 17 (1990)
Adomian G, Rach R, J. Math. Anal. Appl., 105, 141 (1985)
Adomian G, Rach R, J. Math. Anal. Appl., 112, 136 (1985)
Adomian G, Rach R, J. Math. Anal. Appl., 113, 504 (1986)
Adomian G, Rach R, Kybernetes, 15, 33 (1986)
Adomian G, Rach R, Sarafyan D, J. Math. Anal. Appl., 111, 423 (1985)
Bougoffa L, Rach RC, Kybernetes, 42, 641 (2013)
Bougoffa L, Rach RC, Mennouni A, Appl. Math. Comput., 218, 1785 (2011)
Bougoffa L, Mennouni A, Rach RC, Appl. Math. Comput., 219, 4423 (2013)
Duan JS, Rach R, Baleanu D, Wazwaz AM, Commun. Fract. Calc., 3, 73 (2012)
El-Sayed AMA, Gaber M, Phys. Lett., 359, 175 (2006)
Fatoorehchi H, Abolghasemi H, Adv. Nat. Appl. Sci., 5, 26 (2011)
Fatoorehchi H, Abolghasemi H, Appl. Appl. Math., 7, 717 (2012)
Fatoorehchi H, Abolghasemi H, Intermetallics, 32, 35 (2013)
Fatoorehchi H, Abolghasemi H, Appl. Math. Model., 37, 6008 (2013)
Fatoorehchi H, Abolghasemi H, J. Taiwan Inst. Chem. Eng., 45, 880 (2014)
Fatoorehchi H, Rach R, Tavakoli O, Abolghasemi H, Chem. Eng. Commun., 202(3), 402 (2015)
Fatoorehchi H, Abolghasemi H, J. Egyptian Math. Soc., 22, 524 (2014)
Fatoorehchi H, Abolghasemi H, Rach R, Assar M, Can. J. Chem. Eng., 92(12), 2211 (2014)
Fatoorehchi H, Gutman I, Abolghasemi H, Kragujevac J. Sci., 36, 69 (2014)
Fatoorehchi H, Abolghasemi H, Rach R, J. Petrol. Sci. Eng., 117, 46 (2014)
Rach R, Duan JS, Wazwaz AM, J. Math. Chem., 52, 255 (2014)
Dib A, Haiahem A, Bou-Said B, Powder Technol., 269, 193 (2015)
Rach R, Duan JS, Wazwaz AM, Chem. Eng. Commun., DOI:10.1080/00986445.2014.900054.
Hasseine A, Barhoum Z, Attarakih M, Bart HJ, Adv. Polym. Technol., (In press), DOI:10.1016/j.apt.2014.08.011.
Kundu B, Int. J. Refrig., 32, 1657 (2009)
Kundu B, Wongwises S, J. Franklin Inst., 349, 966 (2011)
Rach R, Kybernetes, 41, 1087 (2012)
Saravanan A, Magesh N, J. Egyptian Math. Soc., 21, 259 (2013)
Siddiqui AM, Hameed M, Siddiqui BM, Ghori QK, Comm. Nonlinear Sci. Numer. Simulat., 15, 2388 (2010)
Wazwaz AM, Rach R, Duan JS, Appl. Math. Comput., 219, 5004 (2013)
Hoffman JD, Frankel S, Numerical Methods for Engineers and Scientists, McGraw-Hill, New York (2001).
Shanks D, J. Math. Phys. Sci., 34, 1 (1955)
Bakosova M, Puna D, Dostal P, Zavacka J, Chem. Paper Chem. Zvesti, 63, 527 (2009)
Pottmann M, Seborg DE, J. Proc. Cont., 2, 189 (1992)
Watkins DS, Fundamentals of Matrix Computations, Wiley, New York (1991).
Bronson R, Costa GB, Linear algebra: An introduction, Academic Press, New York (2007).
