Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 19, 2017

SSGM Based Multivariable Control of Unstable Non-Square Systems

Manickam Chidambaram and Dhanya Ram V ORCID logo

Abstract

Davison (1976) proposed a method to design controllers for multivariable systems using the knowledge of only the Steady-State Gain Matrix (SSGM) of the system. In the present work, Davison's method is suitably modified to design controllers for unstable non-square multivariable systems. A single-stage multivariable PI controller is designed using the SSGM. Simulation results show that the overshoots of the closed loop responses are larger, hence a two-stage P-PI control system is proposed. For the proposed two- stage design (i) the system is first stabilized by a simple proportional controller matrix designed based on SSGM using modified Davison's method (1976). (ii) Diagonal PI controllers are designed for this stabilized system based on gain matrix. Simulation studies are carried out to compare the closed loop performance of the single stage multivariable PI control system with that of the two-stage control system (inner loop centralized P controllers and outer loop diagonal PI controllers). A method to identify the steady state gain matrix of a non-square multivariable (SSGM) unstable system under closed loop control is presented. The effect of disturbances and measurement noise on the identification of SSGM is also discussed.

Appendix

Let A be a m × n matrix. For a non-square matrix, the inverse of ATA does not exist if m < n. The pseudoinverse can be defined as A=(ATA)1AT

References

[1] Sree R, Chidambaram M. Control of Unstable Systems. New Delhi: Narosa, 2006.Search in Google Scholar

[2] Jacob EF, Chidambaram M. Design of controllers for unstable first-order plus time delay systems. Comput Chem Eng. 1996;20:579–584.10.1016/0098-1354(95)00210-3Search in Google Scholar

[3] Luyben M, Luyben W. Essentials of process control. McGraw-Hill, 1997.Search in Google Scholar

[4] Maciejowski J. Multivariable feedback design. Workingham England: Addison, 1989.Search in Google Scholar

[5] Tanttu JT, Lieslehto J. A comparative study of some multivariable PI controller tuning methods. In: Devanathan R, editors. Intelligent tuning and adaptive control. IFAC Symposia Series. Oxford: Pergamon Press, 1991:357–362.10.1016/B978-0-08-040935-1.50062-4Search in Google Scholar

[6] Katebi R. Robust multivariable tuning methods. In: Vilanova R, Visioli A, editors. PID Control in the Third Millennium. London: Springer Verlag, 2012:225–280.Search in Google Scholar

7 Reddy PDS, Pandit M, Chidambaram M. Comparision of multivariable controllers for non-minimum phase systems. Int J Model Simul. 2006;26:237–243.10.1080/02286203.2006.11442373Search in Google Scholar

8 Wang Q, Ye Z, Cai C, Hang WJ. PID Control for Multivariable Processes. Berlin: Springer, 2008.Search in Google Scholar

9 Wang Q, Nie Z. PID control for MIMO systems. In: Vilanova R, Visioli A, editors. PID Control in the Third Millennium. London: Springer Verlag, 2012:177–204.10.1007/978-1-4471-2425-2_6Search in Google Scholar

10 Georgiou A, Georgakis C, Luyben WL. Control of a multivariable open-loop unstable process. Ind. Eng. Chem. Res. 1989;28:1481–1489.10.1021/ie00094a008Search in Google Scholar

11 Agamennoni OE, Desages AC, Romagnoli JA. A multivariable delay compensator scheme. Chem Eng Sci. 1992;47:1173–1185.10.1016/0009-2509(92)80239-9Search in Google Scholar

12 Govindakannan G, Chidambaram M. Multivariable PI control of unstable systems. Process Control Qual. 2000;10:319–329.Search in Google Scholar

13 Govindakannan G, Chidambaram M. Two stage multivariable controllers for unstable plus time dealy systems. Indian Chem Eng. 2000;42:34–38.Search in Google Scholar

14 Ram VD, Rajapandiyan C, Chidambaram M. Steady-state gain identification and control of multivariable unstable systems. Chem Eng Commun. 2015;202:151–162.10.1080/00986445.2013.832226Search in Google Scholar

15 Treiber S. Multivariable control of non-square systems. Ind Eng Chem Process Des Dev. 1984;23:854–857.10.1021/i200027a040Search in Google Scholar

16 Reeves DE, Arkun Y. Interaction measures for nonsquare decentralized control structures. AIChE J. 1989;35:603–613.10.1002/aic.690350410Search in Google Scholar

17 Loh EJ, Chiu M. Robust decentralized control of non-square systems. Chem Eng Commun. 1997;158:157–180.10.1080/00986449708936586Search in Google Scholar

18 Ganesh P, Chidambaram M. Multivariable controller tuning for non-square systems with RHP zeros by genetic algorithm. Chem Biochem Eng Q. 2010;24:17–22.Search in Google Scholar

19 Rao AS, Chidambaram M. Smith delay compensator for multivariable non-square systems with multiple time delays. Comput Chem Eng. 2006;30:1243–1255.10.1016/j.compchemeng.2006.02.017Search in Google Scholar

20 Chen L, Li J, Ding R. Identification for the second-order systems based on the step response. Mathematical and Computer Modelling. 2011;53:1074–1083.10.1016/j.mcm.2010.11.070Search in Google Scholar

21 Xu C, Shin YC. A self-tuning fuzzy controller for a class of multi-input multi-output nonlinear systems. Eng Appl Artif Intell. 2011;24:238–250.10.1016/j.engappai.2010.10.021Search in Google Scholar

22 Sarma KLN, Centralized Chidambaram M. PI/PID controllers for nonsquare systems with RHP zeros. J. Indian Inst Sci. 2005;85:201–214.Search in Google Scholar

23 Davison E. Multivariable tuning regulators: The feed-forward and robust control of general servo- mechanism problem 1976;AC-21:35–41. IEEE Trans Autom Control.10.1109/TAC.1976.1101126Search in Google Scholar

24 Papastathopoulou HS, Luyben WL. A new method for the derivation of steady-state gains for multivariable processes. Ind Eng Chem Res. 1990;29:366–369.10.1021/ie00099a011Search in Google Scholar

25 Chidambaram M. Applied Process Control. New Delhi: Allied Publishers, 1998.Search in Google Scholar

26 Venkatashankar V, Chidambaram M. Design of P and PI controllers for unstable first order plus time delay systems. Int J. Control. 1994;60:137–144.10.1080/00207179408921455Search in Google Scholar

27 Valentine CC, Chidambaram M. PID control of unstable time delay systems. Chem Eng Commun. 1997;162:63–74.10.1080/00986449708936632Search in Google Scholar

28 Jevtovic BT, Matauek MR. PID controller design of TITO system based on ideal decoupler. J Process Control. 2010;20:869–876.10.1016/j.jprocont.2010.05.006Search in Google Scholar

Received: 2017-5-16
Accepted: 2017-6-3
Published Online: 2017-7-19

© 2018 Walter de Gruyter GmbH, Berlin/Boston