TELKOMNIKA Telecommunication Computing Electronics and Control Vol.
No.
April 2026, pp.
ISSN: 1693-6930.
DOI: 10.
12928/TELKOMNIKA.
Adaptive fuzzy sliding mode control with exponential reaching law and MPL method for the coupled-tank system Thanh Tung Pham1.
Le Minh Thien Huynh2 Faculty of Electrical and Electronics Engineering.
Vinh Long University of Technology Education.
Vinh Long Province.
Vietnam Faculty of Engineering and Technology.
Saigon University.
Ho Chi Minh City.
Vietnam
Article Info
ABSTRACT
Article history:
This study develops an adaptive fuzzy sliding mode control (ASMC) scheme incorporating an exponential reaching law (ERL) and a minimum parameter learning (MPL) strategy to achieve liquid-level regulation in a coupled-tank Such systems are widely used in industrial applications, including chemical and petrochemical processing, water treatment, power generation, and the manufacturing of construction materials, as well as in boilers, evaporators, reactors, and distillation columns.
The ERL-based sliding mode controller is formulated to guarantee finite-time tracking of the desired liquid level while effectively suppressing chattering near the sliding surface.
The MPL approach is embedded within the fuzzy system (FS), resulting in a single online adaptive parameter, which significantly reduces computational complexity and enhances real-time performance.
The stability of the closedloop system is rigorously established using Lyapunov theory.
Simulation studies conducted in MATLAB/Simulink validate the effectiveness of the proposed controller, demonstrating a rise time of 6.
1918 s, a settling time of 2553 s, zero overshoot, convergence of the steady-state error to zero, and a noticeable reduction in chattering.
Received Jul 27, 2025 Revised Jan 7, 2026 Accepted Jan 30, 2026 Keywords:
Coupled-tank Exponential reaching law Fuzzy system Minimum parameter learning Sliding mode control This is an open access article under the CC BY-SA license.
Corresponding Author:
Le Minh Thien Huynh Faculty of Engineering and Technology.
Saigon University Ho Chi Minh City.
Vietnam Email: leminhthien.
huynh@sgu.
INTRODUCTION
Liquid level control is a fundamental problem in industrial process systems, as it directly influences product quality, operational safety, and energy efficiency.
This issue commonly arises in chemical and petrochemical industries, water treatment plants, power generation units, and various production processes such as boilers, reactors, evaporators, and distillation columns.
Consequently, regulating fluid levels in storage and process tanks has long been regarded as a core challenge in process control engineering .
Ae.
In recent years, coupled-tank systems have received considerable attention due to their nonlinear behavior and strong inter-tank coupling.
Classical control approaches combining proportionalAeintegralAe derivative (PID) control and fuzzy logic (FL) have been widely studied and shown to provide satisfactory tracking performance with zero steady-state error under nominal conditions .
Ae.
Model reference adaptive control (MRAC) schemes have also been reported to outperform conventional proportionalAeintegral (PI) and fuzzy controllers in specific operating scenarios, particularly in terms of robustness and steady-state accuracy .
In addition, comparative studies indicate that FL controllers can achieve faster transient responses than traditional proportional (P).
PI, and PID controllers for tank-level regulation problems .
To further improve performance, adaptive and optimal control strategies have been developed for multi-tank systems.
Adaptive fuzzy/proportionalAederivative (PD) controllers have demonstrated enhanced Journal homepage: http://journal.
id/index.
php/TELKOMNIKA A ISSN: 1693-6930 tracking capability with reduced steady-state error .
, while optimal control techniques have been employed to mitigate modeling uncertainties and modelAeplant mismatches, leading to improved transient behavior .
Nevertheless, classical PI controllers still suffer from relatively long settling times and high sensitivity to parameter tuning .
Driven by these limitations, robust nonlinear control methods have gained increasing interest.
Sliding mode control (SMC) is well known for its robustness against uncertainties and external disturbances .
, but conventional SMC is affected by chattering, which degrades tracking accuracy, induces mechanical vibrations, and increases thermal losses in power electronic components .
To overcome these issues, this paper proposes an adaptive fuzzy sliding mode control (AFSMC) with an exponential reaching law (ERL) and minimum parameter learning (MPL) for liquid level control of a coupled-tank system, aiming to reduce transient times, steady-state error, chattering effects, and online computational burden.
The remainder of this paper is organized as follows.
Section 2 presents the mathematical model of the coupled-tank system.
Section 3 describes the design of the proposed AFSMC-ERL-MPL controller.
Section 4 discusses the simulation results and performance evaluation, and section 5 concludes the paper.
MATHEMATICAL MODEL OF THE COUPLED-TANK SYSTEM
The coupled-tank system comprises two tanks arranged in series, as illustrated in Figure 1 .
, .
Let ya1 and ya2 denote the liquid levels in Tank 1 and Tank 2, respectively, while ya1 and ya2 represent their corresponding cross-sectional areas.
The interconnecting flow between the two tanks is denoted by ycEycu3 .
The variables ycEycn1 and ycEycn2 correspond to the pump inflow rates to Tank 1 and Tank 2, respectively, whereas ycEycu1 and ycEycu2 indicate the outflow rates from each tank.
Figure 1.
System configuration and structure of the coupled-tank process Assuming that the inflow rates ycEycn1 and ycEycn2 are fixed, the system reaches steady-state liquid levels ya1 and ya2 .
Small variations in the inflows, represented by yc1 and yc2 , induce corresponding perturbations in the liquid levels, denoted as Ea1 and Ea2 .
, .
The dynamic behavior of the system is represented by the transfer function in .
Ea2 .
ya1 ya2 yua1 yua2 yc 2 .
ua1 yua2 )yc .
Oeya12 ya21 ) yua1 = ya1 yu1 yu3 2Ooya1 2Ooya1 Oeya2 yu3 2Ooya1 Oeya2 yu2 yu3 2Ooya1 2Ooya1Oeya2 ya21 = yua2 = ya2 yu2 yu3 2Ooya2 2Ooya1 Oeya2 ya1 = yu1 yu3 2Ooya1 2Ooya1 Oeya2 ya2 = yu2 yu3 2Ooya1 2Ooya1 Oeya2 ya12 = yu3 2Ooya1 Oeya2 yu1 yu3 2Ooya1 2Ooya1Oeya2 , yu1 , yu2 , yu3 are proportionality constants.
Within the plant model, the valve .
ump actuato.
is approximated as a gain element, whose dynamics are characterized by the differential equation presented in .
ycNyca yccycycn .
yccyc ycycn .
= ycEyca .
where, ycNyca is the time constant of the value/pump actuator, ycycn .
is the time-varying input flow rate, and ycEyca .
is the computed or commanded flow rate.
TELKOMNIKA Telecommun Comput El Control.
Vol.
No.
April 2026: 707-716 TELKOMNIKA Telecommun Comput El Control Define the state variables as .
Ea2 .
= ycu1 .
ycuN1 .
= ycu2 .
Substituting .
, we have .
ycuN 2 .
= Oe 1Oeya12 ya21 yua1 yua2 yua yua2 ycu1 .
Oe 1 yua1 yua2 ya ya ycu2 .
1 2 yc1 .
yua1 yua2 Define:
= Oe 1Oeya12 ya21 yua1 yua2 yua yua2 ycu1 .
Oe 1 yua1 yua2 The space state of the coupled-tank system as .
ycuN1 .
= ycu2 .
ycuN 2 .
= yce.
1 2 yc.
Ea2 .
= ycu1 .
ya ya yua1 yua2 where, ycu.
= .
]ycN is the state vector of the system, yc1 .
= yc.
is the control input, ycc.
is the disturbance, .
| O ya.
DESIGN THE AFSMC-ERL-MPL METHOD
The overall structure of the proposed AFSMC-ERL-MPL control strategy is illustrated in Figure 2.
This block diagram shows the closed-loop configuration, in which the reference signal is compared with the measured output to produce the tracking error used by the controller.
Based on this error signal, the AFSMCERL-MPL algorithm computes the control input to regulate the coupled-tank system dynamics, ensuring accurate liquid-level tracking and robust disturbance rejection.
The block diagram of the AFSMC-ERL-MPL method as Figure 2.
Figure 2.
Block diagram of the AFSMC-ERL-MPL Design the SMC based on the ERL The sliding surface is formulated as in .
, .
yc = ycayce yceN where the coefficient yca is selected to satisfy the Hurwitz stability condition with yca > 0.
The tracking error is defined in .
yce = Ea2 Oe Ea2ycc Ea2ycc denotes the desired liquid level, and Ea2 represents the actual liquid level of the coupled-tank Taking the derivative of .
, we have .
yceN = EaN 2 Oe EaN 2ycc yceO = EaO 2 Oe EaO 2ycc Adaptive fuzzy sliding mode control with exponential reaching law and MPL method A (Thanh Tung Pha.
A ISSN: 1693-6930 Taking the derivative of .
, we have .
ycN = ycayceN yceO Substituting .
, we have .
ya ya ycN = ycayceN yce.
1 2 yc ycc.
Oe EaO 2ycc yua1 yua2 The SMC based on the ERL is .
, .
yc= yua1 yua2 ya1 ya2 [OeycayceN Oe yce.
EaO 2ycc .
Oe yuCycycnyciycu.
Oe yuNy.
where, yuC > 0, yuN > 0.
Now, ycN = OeyuCycycnyciycu.
Oe yuNyc ycc.
, so if we design yuC Ou ya, yuN > 0, we have .
ycycN = yc(OeyuCycycnyciycu.
Oe yuNyc ycc.
) = OeyuC.
Oe yuNyc 2 ycycc.
O 0
The component yce.
is very difficult to measure in actual control.
Thus, the fuzzy system (FS) with the MPL method is used in this study to approximate this component .
, .
Uncertainty approximation using an FS In this section, the unknown nonlinear function yce.
is approximated by an FS yceC.
to implement feedback control.
Based on the universal approximation theorem, the fuzzy modeling procedure is developed through the following steps .
yco yco First, for the input variables ycu1 and ycu2 , the corresponding fuzzy sets ya11 and ya22 are defined, where ycu ycoycn = 1,2.
A ,5.
Second, a total of Oaycn=1 ycyycn = ycy1 y ycy2 = 25 fuzzy rules are constructed to form the FS yceC.
cu O yuEyce ), expressed by rules of the form:
ycI .
: if ycu1 is ya11 and A and ycu2 is ya12 then yceC is yaA1 ycI .
: if ycu1 is ya15 and A and ycu2 is ya52 then yceC is yaA25 where, ycoycn = 1,2, .
,5, ycn = 1,2, ycy1 = ycy2 = 5.
Finally, the output of the FS is obtained using standard fuzzy inference mechanisms:
yco yco yuEyce ) = 1 2 (Oa2 yuN Oc5 cuycn )) yco =1 Ocyco =1 ycE yce ycn=1 yaycn Oc5 yco =1 Ocyco =1(Oaycn=1 yuN yco .
cuycn )) yaycn ycn where, yuN ycoycn .
cuycn ) is the membership function of ycuycn .
yaycn yco yco Let ycEyce1 2 be a free parameter belonging to the admissible set yuECyce OO ycI .
A column vector yuO.
is then introduced, enabling .
to be rewritten in the compact form given by .
yuEyce ) = yuECyceycN yuO.
where, ycu = .
cu1 ycu2 ]ycN , yuO.
is the Oaycuycn=1 ycyycn = ycy1 y ycy2 = 25 the dimensional column vector, and yco1 , yco2 denote the corresponding elements .
Oa2 ycn=1 yuN yco .
cuycn ) yuOyco1 yco2 .
= yaycn ycn ycy ycy .
Ocyco 1=1 Ocyco 2=1(Oa2 ycn=1 yuN yco .
cuycn )) yaycn ycn TELKOMNIKA Telecommun Comput El Control.
Vol.
No.
April 2026: 707-716 TELKOMNIKA Telecommun Comput El Control The membership functions need to be selected according to experience.
Moreover, all the states must be Design of the AFSMC-ERL-MPL Assume that the optimal parameter vector is defined as in .
yuEyceO = ycaycyci ycoycnycu .
ceC .
yuEyce ) Oe yce.
|] ycuOOycI ycu yuEyce OOyuyce .
where, yce denotes the admissible set of yuEyce , i.
, yuEyce OO yce .
Accordingly, the nonlinear function yce can be expressed as .
yce = yuEyceOycN yuO.
where ycu represents the input vector of the FS, yuO.
is the fuzzy basis vector, and yuA denotes the approximation error, which is bounded by yuA O yuAycA .
The FS is employed to approximate the unknown function yce.
The input of the FS is selected as ycu = .
cu1 ycu2 ]ycN , and the corresponding FS output is given in .
, .
yuEyce ) = yuECyceycN yuO.
Using MPL, define yuo = AnyuEyceO An , where yuo is a positive constant, and let yuoC be an estimation of yuo.
The FSMC-ERL-MPL controller is designed as .
yc= yua1 yua2 [Oe ycyuoC yuO ycN yuO EaO 2ycc Oe ycayceN Oe yuCycycnyciycu.
Oe yuNy.
ya1 ya2 where, yuC Ou yuAycA ya, yuN > 0.
Then we have .
ya ya ycN = yce 1 2 yc ycc Oe EaO 2ycc ycayceN = yuECyceycN yuO.
yuA Oe ycyuoCyuO ycN yuO Oe yuC ycyciycu.
Oe yuNyc ycc yua1 yua2 The Lyapunov function is defined as .
, .
ycO = yc2 yuoE 2 where, yu > 0, yuoE = yuoC Oe yuo.
Substituting .
into the derivative of .
, we have .
yuECyceycN yuO.
yuA Oe ycyuoCyuO ycN yuO ycO = ycycN yuoyuo = yc ( ) yuoEyuoCN OeyuC ycyciycu.
Oe yuNyc ycc O yc 2 yuoyuO ycN yuO yuAyc Oe yc 2 yuoCyuO ycN yuO Oe yuC.
Oe yuNyc 2 yccyc yuoEyuoCN = Oe yc 2 yuoEyuO ycN yuO yuAyc Oe yuC.
Oe yuNyc 2 yccyc yuoEyuoCN = yuoE (Oe yc 2 yuO ycN yuO yuoCN) Oe yuNyc 2 yuAyc Oe yuC.
yccyc O yuoE (Oe yc 2 yuO ycN yuO yuoCN) Oe yuNyc 2 The adaptive law is designed as .
yuoCN = yc 2 yuO ycN yuO Oe yuIyuyuoC where yuI > 0 Then we have .
ycON O OeyuIyuoEyuoC Oe yuNyc 2 O Oe .
uoE 2 Oe yuo 2 ) Oe yuNyc 2 = Oe yuoE 2 Oe yuNyc 2 ( yuo 2 ) Define yuI = 2yuN , then we have .
Adaptive fuzzy sliding mode control with exponential reaching law and MPL method A (Thanh Tung Pha.
A ISSN: 1693-6930
ycON O Oe yuoE 2 Oe yuNyc 2 ( yuo 2 ) = Oe2yuN ( yuoE 2 yc 2 ) ( yuo 2 ) = Oe2yuNycO ycE where ycE = yuo 2 .
By applying Lemma 1.
3, the solution of the differential inequality ycON O Oe2yuNycO ycE is obtained as .
ycON O .
Oe ) yce Oe2yuNyc Then, ycoycnycoycO = ycIeO yuI 2 1 yuIyuo2 1 2yuN 2 = yu From the above-mentioned, we can discern that the overlap precision depends on yu and yuN.
RESULTS AND DISCUSSION
Figure 3 presents the MATLAB/Simulink schematic of the proposed AFSMC-ERL-MPL control The parameters of the coupled-tank system are presented as .
: ya1 = ya2 = 32 .
, yu1 = yu2 = 14.
m3/2/.
, yu3 = 20 .
m3/2/.
, ycNyca = 1.
, ycEycnycoycaycu = 300 .
, yua1 = 7.
445, yua2 = 6.
ya1 = 0.
23267, ya2 = 0.
1939, ya12 = 0.
6453, and ya21 = 0.
yca = 0.
35, yuC = 12, yuN = 2, yu = 25, and ycc.
= ycycnycu.
These are the parameters of the proposed controller.
Figure 4 depicts the system response and tracking error obtained using the AFSMC-ERL-MPL controller for a reference level of 9 cm.
As observed, the actual liquid level accurately follows the desired trajectory, achieving a rise time of 6.
1918 s, a settling time of 11.
2553 s, zero percent overshoot, and convergence of the steady-state error to zero.
These performance indices are summarized in Table 1 and compared with those of the PID FL controller .
and the conventional PI method .
, both evaluated under identical parameter settings and operating conditions.
The comparative results in Table 1 clearly demonstrate the superior control performance of the proposed AFSMC-ERL-MPL approach relative to the reference Figure 3.
Simulation schematic of the AFSMC-ERL-MPL controller in MATLAB/Simulink Figure 4.
Response and error of the AFSMC-ERL-MPL with Ea2ycc = 9 .
TELKOMNIKA Telecommun Comput El Control.
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April 2026: 707-716 TELKOMNIKA Telecommun Comput El Control Table 1.
The achieved quality criteria of the AFSMC-ERL-MPL controller Quality criteria Rising time .
Settling time .
Overshoot (%) Steady state error .
AFSMC-ERL-MPL
PID FL .
PI controller .
The control signal of the proposed controller is shown in Figure 5.
The chattering phenomenon was significantly reduced when using the ERL.
This result demonstrates the effectiveness of the AFSMC-ERLMPL algorithm in controlling the coupled-tank system.
Figure 6 presents the estimation result of yuo.
Figure 5.
Control input of the proposed controller with Ea2ycc = 9 .
Figure 6.
Change of yuoC Figure 7 illustrates the system response and tracking error of the proposed controller under stepwise varying reference inputs.
The liquid level continues to converge to the desired trajectory within a finite time, while the steady-state error tends toward zero.
These results confirm the effectiveness and suitability of the proposed control approach for liquid-level regulation in the coupled-tank system.
Table 2 presents the quantitative error performance measures derived from the test sample data.
Figure 7.
Response and error of the AFSMC-ERL-MPL controller with the variable input Table 2.
Various error performance measures of the AFSMC-ERL-MPL controller Different measures of Average absolute deviation (AAD) Mean squared error (MSE) Root mean
squared error
(RMSE)
Mean
percentage error
(MPE)
Mean absolute percentage error
(MAPE)
Mean relative error (MRE) At yc = 25 s, the second motor pump connected to the second tank is activated, injecting an additional flow rate of 17 cm3 /s into the system.
This external disturbance is intentionally introduced to evaluate the robustness of the proposed control strategy.
The corresponding output response of the coupledtank process under this disturbance, when regulated by the AFSMC-ERL-MPL scheme, is illustrated in Figure 8.
Adaptive fuzzy sliding mode control with exponential reaching law and MPL method A (Thanh Tung Pha.
A ISSN: 1693-6930 Figure 8.
Dynamic response of the coupled-tank system using the AFSMC-ERL-MPL controller in the presence of an external disturbance, with Ea2ycc = 9 cm As shown in Figure 8, the system output rapidly converges back to the desired reference level, demonstrating that the applied disturbance is effectively attenuated and rejected by the proposed controller.
Furthermore.
Figure 9 presents the controller performance in the presence of white noise, which represents sensor noise with an amplitude of 0.
01 W and a sampling period of 0.
1 s acting on the system output.
Despite the presence of this stochastic perturbation, the controlled system maintains stable operation, continues to track the reference signal accurately, and achieves convergence within a finite time.
Figure 9.
Dynamic responses of the coupled-tank system using the AFSMC-ERL-MPL controller when subjected to white noise disturbances From the obtained results, the actual liquid level is consistently shown to converge to the reference value within a finite time under all investigated operating scenarios, including nominal conditions, external disturbances, and measurement noise.
These results clearly demonstrate the robustness, stability, and effectiveness of the proposed AFSMC-ERL-MPL control strategy in regulating the liquid level of the coupled-tank system.
CONCLUSION
This study develops an AFSMC scheme incorporating an ERL and a MPL mechanism for liquidlevel regulation in a coupled-tank system.
The proposed controller guarantees finite-time convergence of the actual liquid level to the desired reference while effectively suppressing chattering near the sliding surface.
MATLAB/Simulink simulation results demonstrate that the controller achieves a rise time of 6.
1918 s, a settling time of 11.
2553 s, zero overshoot, and vanishing steady-state error, along with a noticeable reduction in chattering effects.
Comparative evaluations further confirm that the proposed approach outperforms both TELKOMNIKA Telecommun Comput El Control.
Vol.
No.
April 2026: 707-716 TELKOMNIKA Telecommun Comput El Control the PID FL controller and the conventional PI controller under identical operating conditions.
Overall, these results verify the suitability, robustness, and effectiveness of the AFSMC-ERL-MPL strategy for coupledtank level control applications.
Future work will focus on integrating advanced optimization algorithms for parameter tuning and validating the proposed controller through experimental implementation on a real coupled-tank system.
FUNDING INFORMATION
This research was funded by the joint collaborative research program between Sai Gon University (SGU) and Vinh Long University of Technology Education in Vietnam.
AUTHOR CONTRIBUTIONS STATEMENT
This journal uses the Contributor Roles Taxonomy (CRediT) to recognize individual author contributions, reduce authorship disputes, and facilitate collaboration.
Name of Author Thanh Tung Pham Le Minh Thien Huynh C : Conceptualization M : Methodology So : Software Va : Validation Fo : Formal analysis ue ue ue ue ue ue ue ue ue ue ue ue ue I : Investigation R : Resources D : Data Curation O : Writing - Original Draft E : Writing - Review & Editing ue ue ue ue ue ue ue ue ue ue ue ue ue Vi : Visualization Su : Supervision P : Project administration Fu : Funding acquisition CONFLICT OF INTEREST STATEMENT The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The research was conducted independently and objectively, without any financial, professional, or personal conflict of interest.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author.
Le Minh Thien Huynh, upon reasonable request.
All simulation models, figures, and analysis scripts were developed by the authors as part of this study and can be shared for academic and non-commercial purposes.
REFERENCES