State Estimation and Optimal Control of Four-Tank System with Stochastic Approximation Approach


  • Xian Wen Sim Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Johor, Malaysia
  • Sie Long Kek Department of Mathematics and Statistics, Universiti Tun Hussein Onn Malaysia, Johor, Malaysia
  • Sy Yi Sim Department of Electrical Engineering, Universiti Tun Hussein Onn Malaysia, Johor, Malaysia
  • Jiao Li School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan, China



four-tank system, nonlinear optimal control, stochastic approximation, state estimation


This study aims to optimally control the level of a four-tank system at the steady state in the random disturbance environment using the stochastic approximation (SA) approach. Firstly, the stochastic optimal control problem of an equivalent discrete-time is introduced, where the voltages to the pumps are the control inputs. By minimizing the sum of squared errors, the liquid levels are estimated. Then, first-order necessary conditions are derived by defining the Hamiltonian function. Thus, the optimal voltages are calculated based on the estimated liquid levels to update the gradient of the cost function. Finally, for illustration, parameters in the system are considered and a simulation is conducted. The simulation results show that the state estimation and control law design can perform well, and the liquid levels are addressed along the steady state. In conclusion, the applicability of the SA approach for handling a four-tank system with random disturbances is demonstrated.


J. K. Pradhan, A. Ghosh, and C. N. Bhende, “Two-Degree-of-Freedom Multi-Input Multi-Output Proportional–Integral–Derivative Control Design: Application to Quadruple-Tank System,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 233, no. 3, pp. 303-319, March 2019.

B. A. Kumar, R. Jeyabharathi, S. Surendhar, S. Senthilrani, and S. Gayathri, “Control of Four Tank System Using Model Predictive Controller,” IEEE International Conference on System, Computation, Automation and Networking, pp. 1-5, March 2019.

G. Rigatos, G. Cuccurullo, P. Siano, K. Busawon, and M. Abbaszadeh, “Nonlinear Optimal Control for the Quadruple Water Tank System,” AIP Conference Proceedings, vol. 2293, no. 1, article no. 310005, November 2020.

L. Zhang, H. Xie, Q. Duan, C. Lu, J. Li, and Z. Lv, “Power Level Control of Nuclear Power Plant Based on Asymptotical State Observer Under Neutron Sensor Fault,” Science and Technology of Nuclear Installations, vol. 2021, article no. 8833729, 2021.

N. A. Al-Awad, “Optimal Control of Quadruple Tank System Using Genetic Algorithm,” International Journal of Computing and Digital Systems, vol. 8, no. 1, pp. 51-59, January 2019.

R. Kashyap, N. Jaggi, and B. Pratap, “Robust Controller Design and Performance Analysis of Four-Tank Coupled System,” 1st International Conference on Signal Processing, VLSI and Communication Engineering, pp. 1-6, March 2019.

S. A. Panahi and S. B. M. Noor, “Model Predictive Control Design for a Nonlinear Four-Tank System,” IFAC Proceedings Volumes, vol. 43, no. 8, pp. 382-387, 2010.

R. Hedjar and M. Bounkhel, “Wireless Model Predictive Control: Application to Water-Level System,” Advances in Mechanical Engineering, vol. 8, no. 4, pp. 1-13, 2016.

P. Prasad, “The Performance Analysis of Four Tank System for Conventional Controller and Hybrid Controller,” International Research Journal of Engineering and Technology, vol. 3, no. 6, pp. 2182-2187, June 2016.

K. H. Johansson, “The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero,” IEEE Transactions on Control Systems Technology, vol. 8, no. 3, pp. 456-465, May 2000.

S. B. Prusty, U. C. Pati, and K. K. Mahapatra, “Model Based Predictive Control of the Four Tank System,” Advances in System Optimization and Control, vol. 509, pp. 249-260, 2019.

W. A. Altabey, “Model Optimal Control of the Four Tank System,” International Journal of Systems Science and Applied Mathematics, vol. 1, no. 4, pp. 30-41, November 2016.

A. K. Ali and M. S. Mahmoud, “Distributed Estimation and Control of a Four-Tank Process,” Global Journal of Engineering and Technology Advances, vol. 9, no. 3, pp. 133-142, December 2021.

X. Meng, H. Yu, J. Zhang, T. Xu, and H. Wu, “Liquid Level Control of Four-Tank System based on Active Disturbance Rejection Technology,” Measurement, vol. 175, article no. 109146, April 2021.

X. Meng, H. Yu, T. Xu, and H. Wu, “Disturbance Observer and L2-Gain-Based State Error Feedback Linearization Control for the Quadruple-Tank Liquid-Level System,” Energies, vol. 13, no. 20, article no. 5500, October 2020.

C. Renotte, A. V. Wouwer, and M. Remy, “Neural Modeling and Control of a Heat Exchanger Based on SPSA Techniques,” Proceedings of the 2000 American Control Conference, vol. 5, pp. 3299-3303, June 2000.

X. Kong, X. Wang, and Z. Ma, “A Novel Method for Controller Parameters Optimization of Steam Generator Level Control,” 21st International Conference on Nuclear Engineering, article no. 16294, July-August 2013.

I. Kong, I. Qian, and I. Wang, “SPSA-Based PID Parameters Optimization for a Dual-Tank Liquid-Level Control System,” IEEE International Conference on Industrial Engineering and Engineering Management, pp. 1463-1467, December 2016.

M. Jesmani, B. Jafarpour, M. C. Bellout, and B. Foss, “A Reduced Random Sampling Strategy for Fast Robust Well Placement Optimization,” Journal of Petroleum Science and Engineering, vol. 184, article no. 106414, January 2020.

J. C. Spall, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control, 1st ed., Hoboken: Wiley-Interscience, 2003.

A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, 1st ed., New York: Taylor and Francis, 1975.

F. L. Lewis, D. L. Vrabie, and V. L. Syrmos, Optimal Control, 3rd ed., Hoboken: John Wiley & Sons, 2012.

D. E. Kirk, Optimal Control Theory: An Introduction, New York: Dover Publications Mineola, 2004.

K. L. Teo, B. Li, C. Yu, and V. Rehbock, Applied and Computational Optimal Control: A Control Parametrization Approach, 1st ed., Switzerland: Springer Cham, 2021.




How to Cite

Xian Wen Sim, Sie Long Kek, Sy Yi Sim, and Jiao Li, “State Estimation and Optimal Control of Four-Tank System with Stochastic Approximation Approach”, Adv. technol. innov., vol. 8, no. 2, pp. 150–161, Apr. 2023.