Control of negative gain nonlinear processes using sliding mode controllers with modified Nelder-Mead tuning equations

Govinda Kumar E.; Arunshankar J.

doi:10.1515/cppm-2020-0092

Article

Control of negative gain nonlinear processes using sliding mode controllers with modified Nelder-Mead tuning equations

Govinda Kumar E. and Arunshankar J.

Published/Copyright: April 12, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Chemical Product and Process Modeling Volume 17 Issue 5

Abstract

This paper proposes a sliding mode controller (SMC) with modified Nelder-Mead tuning, for the control of nonlinear chemical processes, which are represented as first order plus dead time process with negative gain (FOPDT-NG). In the controller design, the SMC controller parameter in continuous part is obtained based on the time constant and dead time of the process, and controller parameters in the discontinuous part is obtained using Nelder-Mead tuning equations. Even though the controller parameters of conventional SMC are tuned using Nelder-Mead tuning, zero dynamics are noticed in the closed loop response of few FOPDT-NG processes and, with few other FOPDT-NG processes tracking of set-point is unachievable. This work proposes modification in the Nelder-Mead tuning equations using Nelder-Mead optimization to overcome the above disadvantages. Four different types of FOPDT-NG processes are considered in this work, and for every type the Nelder-Mead tuning equations are modified, for the design of proposed controllers. The performances of proposed controllers are evaluated for FOPDT-NG processes and also for three different chemical processes taken from literature. A simulation results demonstrate that, the proposed controller prevailed the performance of the conventional SMC in tracking the set-point and the elimination of zero dynamic behavior of FOPDT-NG processes. Hence, the proposed controllers provide improved closed loop performances as compared to the conventional SMC.

Keywords: FOPDT model; negative gain process; Nelder-Mead tuning; sliding mode control; zero dynamics

Corresponding author: Govinda Kumar E., Department of Electronics and Instrumentation Engineering, Karpagam College of Engineering, Coimbatore 641 032, Tamil Nadu, India, E-mail: egovindakumar@gmail.com

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

Nelder-Mead Optimization technique contains of B (Best point), G (Good point), W (Worse point), M (Mid-point), E (Expansion point), R (Reflect point), C (Construction point) and S (Shrink point). The discontinuous part of SMC consists of K_D and δ, which are tuning parameters.

A.1 Algorithm

Generate an initial configuration K randomly, where K₁ = [K_D1 δ₁], K₂ = [K_D2 δ₂] and K₃ = [K_D3 δ₃].
Calculate f(K₁), f(K₂), f(K₃) for finding B, G, W, where B < G < W.
Compute M, E and f(E).
Compare f(E) and f(G), if f(E) < f(G) replace W with E, go to step 8; else Compute R and f(R) go to step 5.
Compare f(R) and f(W), if f(R) < f(W) replace W with R go to step 6.
Compare f(R) and f(G), if f(R) ≥ f(G) Compute C and f(C) go to step 7; else go to step 8.
Compare f(C) and f(W), if f(C) < f(W) replace W with C go to step 8; else compute S, replace G with M and replace W with S go to step 8.
Rearrange the B, G, W, where B < G < W and repeat step (3) until some predefined stopping criteria.

A.2 Symbols used

s ( t )	Sliding surface
	Derivative of sliding surface
r(t)	Reference value (set-point)
x(t)	The measured variable (controlled variable)
e ( t )	Error signal
	Derivative of error signal
K	System gain
τ	System time constant
τ d	Dead time
λ	Tuning parameter of continuous part of SMC
n	Order of the system
u ( t )	Control law of sliding mode control
u c ( t )	Continuous control law of sliding mode control
u d ( t )	Discontinuous control law of sliding mode control
K D	Tuning parameter of reaching mode
δ	Tuning parameter discontinuous part of SMC
G(s)	Process transfer function
x(s)	Laplace transform of controlled variable
u(s)	Laplace transform of manipulated variable
ω n	The natural frequency
ζ	Damping ratio
sgn	The signum function
B	Best point
G	Good point
W	Worse point
M	Mid-point
E	Expansion point
R	Reflect point
C	Construction point
S	Shrink point

Abbreviations

SMC: Sliding Mode Control
PID: Proportional + Integral + Derivative
VSC: Variable Structure Control
FOPDT: First Order Plus Dead Time
FOPDT-NG: First Order Plus Dead Time with Negative Gain
P: Proportional
PI: Proportional + Integral
PD: Proportional + Derivative
VS-PI: Variable Structure based Proportional plus Integral
VS-PID: Variable Structure based Proportional plus Integral plus Derivative
ISE: Integral of the Squared Error
MATLAB: MATrix LABoratory
IAE: Integral of the Amplitude Error
SP: Set-point
CO: Controller Output
PV: Process Variable
ITAE: Integral of the Time Weighted Amplitude Error

References

1. Bequette, BW. Nonlinear control of chemical processes: a review. Ind Eng Chem Res 1991;30:1391–413. https://doi.org/10.1021/ie00055a001.Search in Google Scholar

2. Slotine, JJ, Li, W. Applied nonlinear control. NJ: Englewood Cliffs; 1991.Search in Google Scholar

3. Camacho, O, Smith, CA. Sliding mode control: an approach to regulate nonlinear chemical processes. Instrum Soc Am Trans 2000;39:205–18. https://doi.org/10.1016/S0019-0578(99)00043-9.Search in Google Scholar

4. Utkin, VI. Sliding modes in control optimization. Berlin: Springer-Verlag; 1992.10.1007/978-3-642-84379-2Search in Google Scholar

5. Biswas, PP, Srivastava, R, Ray, S, Samanta, AN. Sliding mode control of quadruple tank process. Mechatronics 2009;19:548–61. https://doi.org/10.1016/j.mechatronics.2009.01.001.Search in Google Scholar

6. Mirakhorli, E, Farrokhi, M. Sliding-mode state-feedback control of non-minimum phase quadruple tank system using fuzzy logic. IFAC Proc Vol 2011;44:13546–51. https://doi.org/10.3182/20110828-6-IT-1002.01235.Search in Google Scholar

7. Khan, MK, Spurgeon, SK. Robust MIMO water level control in inter-connected twin-tanks using second order sliding mode control. Contr Eng Pract 2006;14:375–86. https://doi.org/10.1016/j.conengprac.2005.02.001.Search in Google Scholar

8. Boiko, I, Sun, X, Tamayo, E. Variable-structure PI controller for tank level process. In: IEEE proceedings of American control conference, Seattle, WA; 2008.10.1109/ACC.2008.4587236Search in Google Scholar

9. Boiko, I. Variable-structure PID controller for level process. Contr Eng Pract 2013;21:700–7. https://doi.org/10.1016/j.conengprac.2012.04.004.Search in Google Scholar

10. Camacho, O, Rojas, R. A general sliding mode controller for nonlinear chemical Processes. J Dyn Syst Meas Contr 2000;122:650–5. https://doi.org/10.1115/1.1318351.Search in Google Scholar

11. Mingzhong, L, Wang, F, Gao, F. PID-based sliding mode controller for nonlinear processes. Ind Eng Chem Res 2001;40:2660–7. https://doi.org/10.1021/ie990715e.Search in Google Scholar

12. Shahraz, A, Bozorgmehry, RB. A fuzzy sliding mode control approach for nonlinear chemical processes. Contr Eng Pract 2009;17:541–50. https://doi.org/10.1016/j.conengprac.2008.10.011.Search in Google Scholar

13. Galluzzo, M, Carmelo, C. Sliding mode fuzzy logic control of an unstable bioreactor. Chem Eng Trans 2013;32:1213–8. https://doi.org/10.3303/CET1332203.Search in Google Scholar

14. Tham, HJ, Ramachandran, KB, Hussain, MA. Sliding mode control for a continuous bioreactor. Chem Biochem Eng Q 2003;17:267–75.Search in Google Scholar

15. Sutha, S, Lakshmi, P, Sankaranarayanan, S. Fractional order sliding mode controller design for a modified quadruple tank process via multi-level switching. Comput Electr Eng 2015;45:10–21. https://doi.org/10.1016/j.compeleceng.2015.04.012.Search in Google Scholar

16. Govinda Kumar, E, Arunshankar, J. Control of nonlinear two-tank hybrid system using sliding mode controller with fractional-order pi-d sliding surface. Comput Electr Eng 2018;71:953–65. https://doi.org/10.1016/j.compeleceng.2017.10.005.Search in Google Scholar

17. Giriraj Kumar, SM, Jain, R, Anantharaman, N, Dharmalingam, V, Sheriffa Begum, KMM. Genetic algorithm based PID controller tuning for a model bioreactor. Indian Chem Eng 2008;50:214–26.Search in Google Scholar

18. Arbogasta, JE, Beauregard, BM, Cooper, DJ. Intuitive robust stability metric for PID control of self- regulating processes. Instrum Soc Am Trans 2008;47:420–8. https://doi.org/10.1016/j.isatra.2008.06.001.Search in Google Scholar PubMed

19. Utkin, VI. Variable structure systems with sliding modes. IEEE Trans Automat Contr 1977;22:212–22. https://doi.org/10.1109/TAC.1977.1101446.Search in Google Scholar

20. Sivaramakrishnan, S, Tangirala, AK, Chidambaram, M. Sliding mode controller for unstable systems. Chem Biochem Eng Q 2008;22:41–7.Search in Google Scholar

21. Musmade, BB, Munje, RK, Patre, BM. Design of sliding mode controller to chemical processes for improved performance. Int J Contr Autom 2011;4:15–32.Search in Google Scholar

22. Samantaray, AK, Bouamama, B. Model-based process supervision: a bond graph approach. London: Springer-Verlag; 2008.Search in Google Scholar

23. Rojas, R, Camacho, O, Caceres, R, Castellano, A. A sliding mode control proposal for nonlinear electrical systems. In: 15th Triennial World Congress. Spain: Barcelona; 2002.Search in Google Scholar

24. Camacho, O, Rojas, R, Garcia-Gabin, W. Some long time delay sliding mode approaches. Instrum Soc Am Trans 2007;46:95–101. https://doi.org/10.1016/j.isatra.2006.06.002.Search in Google Scholar PubMed

25. Rojas, R, Camacho, O, Caceres, R, Castellano, A. On sliding mode control for nonlinear electrical systems. In: 3rd WSEAS International Conference on Applications of Electrical Engineering. Mexico: Cancun; 2004.Search in Google Scholar

26. Sinlapakun, V, Assawinchaichote, W. PID controller based Nelder-Mead algorithm for electric furnace system with disturbance. ECTI Trans Comp Inf Technol 2016;10:71–9.10.37936/ecti-cit.2016101.58160Search in Google Scholar

27. Mathews, JH, Fink, KD. Numerical methods using MATLAB. NJ: Prentice-Hall; 1999.Search in Google Scholar

Received: 2020-09-25

Accepted: 2021-03-26

Published Online: 2021-04-12

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/cppm-2020-0092

Keywords for this article

FOPDT model; negative gain process; Nelder-Mead tuning; sliding mode control; zero dynamics