Abstract
Real industrial processes are almost all of multi-input and multi-output (MIMO) nature, and any change or disturbance occurring in one loop will affect the other loops through the cross-couplings between loops, which is the most important features of a MIMO system, and which implies that the control engineer cannot design each loop independently as they do for SISO systems. Therefore, the requirements for high performance in MIMO control are known to be much more difficult than in the SISO control. Despite great advances in modern control theory, the PID controller is still the most popular controller type used in process industries due to its simplicity and reliability. There are rich theories and designs for the SISO PID control, but little has been done for MIMO PID control while much is demanded for the latter to reach the same maturity and popularity as the single-loop PID case. In this chapter, we first introduce some fundamentals for MIMO systems such as transfer function matrices, poles, zeros, and feedback system stability. Then, we present a graphical method for the design of a multiloop PI controller to achieve the desired gain and phase margins for each loop. Finally, we define the loop gain margins and compute them for multivariable feedback systems. In this way, the stability and robustness of an multivariable feedback system can be really achieved and guaranteed.
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References
Rosenbrock, H.H.: State-Space and Multivariable Theory. Nelson, London (1970)
Desoer, C.A., Shulman, J.D.: Zeros and poles of matrix transfer-functions and their dynamical interpretation. IEEE Trans. Circuits Syst. CAS-21, 3–8 (1974)
Desoer, C.A., Chan, W.S.: The feedback interconnection of linear time-invariant systems. J. Franklin Inst. 300, 335–351 (1975)
Desoer, C., Wang, Y.T.: On the generalized Nyquist stability criterion. IEEE Trans. Autom. Control 25, 187–196 (1980)
Jeng, J.C., Huang, H.P., Lin, F.Y.: Modified relay feedback approach for controller tuning based on assessment of gain and phase margins. Ind. Eng. Chem. Res. 45, 4043–4051 (2006)
Wang, Y.G., Shao, H.H.: PID autotuner based on gain and phase margin specifications. Ind. Eng. Chem. Res. 45, 3007–3012 (1999)
Wang, Q.G., Fung, H.W., Zhang, Y.: PID tuning with exact gain and phase margins. ISA Trans. 38, 243–249 (1999)
Keel, L.H., Bhattacharyya, S.P.: Robust fragile, or optimal? IEEE Trans. Autom. Control 42, 1098–1105 (1997)
Astrom, K.J., Hagglund, T.: Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica 20, 645–651 (1984)
Ho, W.K., Hang, C.C., Cao, L.S.: Tuning of PID controllers based on gain and phase margin specifications. Automatica 31, 497–502 (1995)
Fung, H.W., Wang, Q.G., Lee, T.H.: PI tuning in terms of gain and phase margins. Automatica 34, 1145–1149 (1998)
Wang, Q.G., Lin, C., Ye, Z., Wen, G.L., He, Y. Hang C.C.: A quasi-LMI approach to computing stabilizing parameter ranges of multi-loop PID controllers. J. Process Control 17, 59–72 (2007)
Palmor, Z.J., Halevi, Y., Krasney, N.: Automatic tuning of decentralized PID controllers for TITO processes. Automatica 31, 1001–1010 (1995)
Wang, Q.G., Zou, B., Lee, T.H., Bi, Q.: Auto-tuning of multivariable PID controllers from decentralized relay feedback. Automatica 33, 319–330 (1997)
Wang, Q.G., Ye, Z., Cai, W.J., Hang, C.C.: PID Control for Multivariable Processes. Springer, Berlin (2008)
Ho, W.K., Lee, T.H., Gan, O.P.: Tuning of multiloop proportional-integral-derivative controllers based on gain and phase margin specifications. Ind. Eng. Chem. Res. 36, 2231–2238 (1997)
Kookos, I.K.: Comments on “Tuning of multiloop proportional-integral-derivative controllers based on gain and phase margin specifications”. Ind. Eng. Chem. Res. 37, 1574 (1998)
Huang, H.P., Jeng, J.C., Chiang, C.H., Pan, W.: A direct method for multi-loop PI/PID controller design. J. Process Control 13, 760–786 (2003)
Tavakoli, S., Griffin, I., Fleming, J.: Tuning of decentralized PI (PID) controllers for TITO processes. Control Eng. Pract. 14, 1069–1080 (2006)
Zhang, Y., Wang, Q.G., Astrom, K.J.: Dominant pole placement for multi-loop control systems. Automatica 38, 1213–1220 (2002)
Nie, Z.Y., Wang, Q.G., Wu, M., He, Y.: Tuning of multi-loop PI controllers based on gain and phase margin specifications. J. Process Control 21, 1287–1295 (2011)
Majhi, S.: On-line PI control of stable processes. J. Process Control 15, 859–867 (2005)
Astrom, K.J., Panagopoulos, H., Hagglund, T.: Design of PI controllers based on non-convex optimization. Automatica 34, 585–601 (1998)
Bissell, C.: Control Engineering. Chapman and Hall, New York (1994)
Wood, R.K., Berry, M.W.: Terminal composition control of a binary distillation column. Chem. Eng. Sci. 28, 1707–1717 (1973)
Luyben, W.L.: Simple method for tuning SISO controllers in multivariable systems. Ind. Eng. Chem. Process Des. Dev. 25, 654–660 (1986)
Lee, T.T., Wang, F.Y., Newell, R.B.: Robust multivariable control of complex biological processes. J. Process Control 14, 193–209 (2004)
Kristiansson, B., Lennartson, B.: Evaluation and simple tuning of PID controllers with high frequency robustness. J. Process Control 16, 91–102 (2006)
Cvejn, J.: Sub-optimal PID controller settings for FOPDT systems with long dead time. J. Process Control 19, 1486–1495 (2009)
Wang, Q.G.: Decoupling Control. Springer, New York (2003)
He, M.J., Cai, W.J., Wu, B.F., He, M.: Simple decentralized PID controller design method based on dynamic relative interaction analysis. Ind. Eng. Chem. Res. 44, 8334–8344 (2005)
Doyle, J.: Analysis of feedback systems with structured uncertainty. IEE Proc. Part D. Control Theory Appl. 129, 242–250 (1982)
Fan, M.K.H., Tits, A., Doyle, J.C.: Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Trans. Autom. Control 36, 25–38 (1991)
Braatz, R.P., Young, P.M., Doyle, J.C., Morari, M.: Computation complexity of calculation. IEEE Trans. Autom. Control 39, 1000–1002 (1994)
Safonov, M.G.: Stability margins of diagonally perturbed multivariable feedback systems. IEE Proc. Part D. Control Theory Appl. 129, 251–256 (1982)
Safonov, M.G., Athans, M.: A multiloop generalization of the circle criterion for stability margin analysis. IEEE Trans. Autom. Control 26, 415–422 (1981)
De Gastion, R.R.E., Safonov, M.G.: Exact calculation of the multiloop stability margin. IEEE Trans. Autom. Control 33, 156–171 (1988)
Bar-on, J.R., Jonckheere, E.A.: Multivariable gain margin. Int. J. Control 54, 337–365 (1991)
Li, Y., Lee, E.B.: Stability robustness characterization and related issues for control systems design. Automatica 29, 479–484 (1993)
Hong, Y., Yang, O.W.W.: Self-tuning multiloop PI rate controller for an MIMO AQM router with interval gain margin assignment. High Perform. Switch. Routing 12, 401–405 (2005)
Ye, Z., Wang, Q.G., Hang, C.C.: Frequency domain approach to computing loop phase margins of multivariable systems. Ind. Eng. Chem. Res. 47, C4418–C4424 (2008)
Galin, D.M.: Real matrices depending on parameters. Usp. Mat. Nauk 27, 241–242 (1972)
Ballantine, C.S.: Numerical range of a matrix: some effective criteria. Linear Algebra Appl. 19, 117–188 (1987)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Morari, M., Zafiriou, E.: Robust Process Control. Prentice Hall, Englewood Cliffs (1989)
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Wang, QG., Nie, ZY. (2012). PID Control for MIMO Processes. In: Vilanova, R., Visioli, A. (eds) PID Control in the Third Millennium. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-2425-2_6
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DOI: https://doi.org/10.1007/978-1-4471-2425-2_6
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