Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 ISSN: 2986-6537.
DOI: 10.
59247/jfsc.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers 1, 2 Hoang Duc Long 1,* .
Vu Xuan Duc 2 Department of Automation and Computing Techniques.
Le Quy Don Technical University.
Vietnam Email: 1 longhd@lqdtu.
vn, 2 ducvx@lqdtu.
*Corresponding Author AbstractAiAccurate estimation of unknown and timevarying disturbances is essential for achieving highperformance control of nonlinear systems.
This paper investigates the design and comparative evaluation of finite-time disturbance observers with different gain adaptation First, a conventional fixed-gain finite-time disturbance observer and a linearly adaptive finite-time disturbance observer are presented.
Then, an adaptive finitetime disturbance observer based on fuzzy logic control is developed to automatically adjust observer gains according to the disturbance estimation error and its rate of change, thereby reducing gain sensitivity and improving transient performance.
Finite-time stability of the closed-loop system is rigorously analyzed using Lyapunov theory, and sufficient conditions for convergence are derived.
Extensive simulation studies on a nonlinear system subject to high-frequency time-varying disturbances demonstrate the effectiveness of the proposed Quantitative results show that the adaptive finitetime disturbance observer based on fuzzy logic control reduces tracking error and disturbance estimation root mean square error by more than 75% compared with the conventional finitetime disturbance observer and by over 50% compared with the linearly adaptive observer, while yielding smoother control These results confirm that the adaptive finite-time disturbance observer based on fuzzy logic control significantly enhances robustness and estimation accuracy, making the proposed observer suitable for practical nonlinear control applications under severe disturbance conditions.
KeywordsAiFiniteAaTime Disturbance Observer.
Fuzzy Logic Control.
Adaptive Observer.
Nonlinear Control.
Disturbance Estimation.
Tracking Error.
Lyapunov Stability INTRODUCTION Disturbances and uncertainties are unavoidable in practical control systems and often degrade tracking accuracy, stability, and robustness, especially for nonlinear External disturbances, unmodeled dynamics, parameter variations, and measurement noise are commonly encountered in engineering applications such as robotic systems, electromechanical actuators, and process control.
Consequently, effective disturbance estimation and compensation have become essential topics in modern nonlinear control theory.
Among existing approaches, disturbance observerAebased control has been widely studied due to its simple structure and strong robustness properties.
Conventional disturbance observers and extended state observers are typically designed to guarantee asymptotic convergence of the estimation error under mild assumptions.
However, asymptotic convergence may be insufficient in applications requiring fast transient response and high precision, particularly in the presence of time-varying or high-frequency disturbances.
To address this limitation, finite-time disturbance observers (FTDOB.
have been proposed to ensure that estimation errors converge to zero within a finite time Finite-time convergence offers faster response and stronger robustness compared with asymptotic schemes and has attracted significant attention in recent years .
Nevertheless, most existing FTDOB designs rely on fixed observer gains, which must be chosen conservatively to ensure stability.
Large fixed gains may lead to chattering, noise amplification, or excessive control effort, while small gains may degrade convergence speed and disturbance rejection performance.
To reduce gain sensitivity, several studies have introduced adaptive finite-time disturbance observers, where observer gains are adjusted online according to estimation errors .
Although linear or parameter-based adaptive laws can improve robustness to some extent, their adaptation capability is often limited when dealing with rapidly changing or high-frequency disturbances.
Moreover, the tuning process may still be nontrivial, and performance degradation can occur under severe disturbance conditions.
Fuzzy logic control has been widely recognized as an effective tool for handling nonlinearities, uncertainties, and imprecise information without requiring an accurate mathematical model .
By incorporating heuristic knowledge through linguistic rules, fuzzy systems can adapt controller or observer parameters in a flexible and intuitive Despite these advantages, the systematic integration of fuzzy logic-based gain adaptation into finite-time disturbance observers, together with rigorous finite-time stability analysis and quantitative performance comparison, remains insufficiently explored in the existing literature.
Therefore, the following gaps can be identified:
a Most FTDOBs employ fixed or weakly adaptive gains, resulting in a trade-off between convergence speed and a Existing adaptive schemes lack flexibility in responding to fast and high-frequency disturbances.
a Comparative studies that quantitatively evaluate different gain adaptation mechanisms under the same disturbance conditions are limited.
This work is licensed under a Creative Commons Attribution 4.
0 License.
For more information, see https://creativecommons.
org/licenses/by/4.
Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 These gaps motivate the development of a more flexible and robust disturbance observer that combines finite-time convergence with intelligent gain adaptation.
In this paper, we investigate and compare finite-time disturbance observers with different gain adaptation strategies.
Unlike existing adaptive FTDOBs that rely on linear gain adaptation, the proposed fuzzy mechanism adapts gains using both error magnitude and error rate, enabling improved performance under high-frequency disturbances.
The main contributions are summarized as follows:
a A conventional fixed-gain finite-time disturbance observer and a linearly adaptive finite-time disturbance observer are formulated for nonlinear systems subject to unknown disturbances.
a A fuzzy logic-based adaptive finite-time disturbance observer (AFTDOB-Fuzz.
is proposed, in which observer gains are adjusted online according to the disturbance estimation error and its derivative, reducing gain sensitivity and improving transient performance.
a Rigorous finite-time stability analysis is provided using Lyapunov theory, and sufficient conditions for convergence are derived.
a Extensive simulation studies under high-frequency timevarying disturbances are conducted, including quantitative comparisons based on root mean square error (RMSE), demonstrating the superior performance of the proposed method.
The paper is organized as follows: Section i describes the system model and revisits the fixedAagain finiteAatime disturbance observer presented in previous studies.
The limitations of FTDOB are discussed, motivating the development of adaptive algorithms for observer gain Section IV introduces the adaptive finiteAatime disturbance observer integrated with a backstepping controller for the nonlinear system.
Stability analysis is conducted based on a suitably chosen Lyapunov function.
Section V, an enhanced adaptive finiteAatime disturbance observer is developed using an adaptive finiteAatime disturbance observer based on fuzzy logic control.
Section VI presents simulation results and quantitative comparisons among the three observers: FTDOB.
AFTDOB, and AFTDOBAaFuzzy.
Finally.
Section VII concludes the study.
II.
RELATED WORK
Disturbance observer-based control has been extensively studied as an effective approach to compensate for uncertainties and external disturbances in control systems.
Existing works related to the present study can be broadly classified into three categories: conventional disturbance observers, finite-time disturbance observers with fixed or adaptive gains, and fuzzy logicAebased observer and control Conventional Disturbance Observers Traditional disturbance observers and extended state observers have been widely applied to linear and nonlinear systems to estimate lumped disturbances and model uncertainties .
These methods typically guarantee asymptotic convergence of the estimation error and are relatively simple to implement.
However, their convergence speed is often insufficient in applications requiring fast transient response.
Moreover, performance may degrade significantly in the presence of rapidly time-varying or highfrequency disturbances, which motivates the development of finite-time estimation techniques.
Finite-Time Disturbance Observers To enhance convergence speed and robustness, finitetime disturbance observers (FTDOB.
have been proposed in recent years .
, .
, .
By employing nonlinear correction terms and finite-time stability theory, these observers ensure that estimation errors converge to zero within a finite time interval.
Despite these advantages, most existing FTDOBs rely on fixed observer gains, which must be conservatively selected to guarantee stability.
Large gains may cause chattering and noise amplification, whereas small gains may slow convergence and reduce disturbance rejection To address this issue, several adaptive finite-time disturbance observers have been developed .
, where observer gains are adjusted online based on estimation errors or predefined adaptation laws.
While such approaches improve robustness and reduce gain sensitivity, the adaptation mechanisms are often linear or parameterdependent, which limits their flexibility when dealing with severe or high-frequency disturbances.
Furthermore, quantitative comparisons among different adaptive strategies are rarely reported, making it difficult to assess their relative Fuzzy LogicAeBased Observer and Control Approaches Fuzzy logic control has been widely recognized as an effective tool for handling nonlinearities, uncertainties, and imprecise information without relying on accurate system models .
Fuzzy systems have been successfully applied to adaptive control, observer design, and disturbance estimation by incorporating heuristic rules derived from expert knowledge.
In particular, fuzzy-based gain tuning has been shown to enhance adaptability and robustness in nonlinear control systems.
However, most fuzzy logic-based observer designs focus on asymptotic convergence or lack rigorous finite-time stability analysis .
, .
, .
In addition, the integration of fuzzy logic-based adaptive gain mechanisms into finitetime disturbance observers has not been systematically investigated, and comparative studies with conventional fixed-gain and linearly adaptive FTDOBs remain limited.
Relationship to the Present Study The present work differs from the above studies in several important aspects.
First, unlike conventional FTDOBs with fixed gains, the proposed method employs a fuzzy logicAe based adaptive mechanism to adjust observer gains online according to both the disturbance estimation error and its rate of change.
Second, in contrast to existing adaptive FTDOBs with linear adaptation laws, the fuzzy-based strategy provides greater flexibility in responding to time-varying and highfrequency disturbances.
Third, rigorous finite-time stability analysis is provided using Lyapunov theory, and explicit convergence conditions are derived.
Finally, extensive simulation studies are conducted to quantitatively compare the proposed approach with fixed-gain and linearly adaptive observers using root mean square error (RMSE) metrics.
Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 thereby clearly demonstrating the advantages of the proposed To further clarify the similarities and differences between the proposed approach and existing works, a comparative summary is provided in Table I.
Table 1.
Comparison Between Existing Disturbance Observer Methods and the Proposed Approach Conventional DOB .
, .
FTDOB .
Adaptive FTDOB .
Fuzzy-based .
Proposed i.
Gain
Strategy Convergence Handles HighFrequency Disturbance Constant
Asymptotic Limited Constant
Finite-time Moderate Adaptive Linear
Finite-time Improved Adaptive Fuzzy Asymptotic/ finite-time Improved FTDOB
Fuzzy Finite-time High
Observer
Type
Method
Fixedgain
Fixedgain
DESCRIPTION AND FIXEDAaGAIN FINITEAaTIME
DISTURBANCE OBSERVER
Consider a general class of nonlinear systems of order with the following dynamics .
ycuN ycn = ycuycn 1 , ycn = i 1, ycu Oe 1 cuN ycu = yce.
eo, y.
yc = ycu1 where ycu = .
cu1 ycu2 U ycuycu ]ycN OO Eyycu is the vector of state yce.
OO Ey and yci.
OO Ey are smooth nonlinear ycc.
cu, y.
OO Ey is a bounded unknown disturbance.
OO Ey is the control input.
yc = ycu1 is the measurable output.
Assumptions:
a A1: The reference trajectory ycu1ycc .
is ycuAatimes continuously differentiable.
a A2: The disturbance ycc.
cu, y.
is bounded by a known constant: .
cu, y.
| O ya.
a A3: yci.
is smooth and bounded away from zero: yci.
O 0 with OAycu.
The fixedAagain finiteAatime disturbance observer of the nonlinear system .
is given as below .
ycuCN = yce.
yc { ycu yc = yc1 ycuEycu yc2 tanh.
caycuEycu ) .
where ycuCycu is the estimate of ycuycu .
yc is the estimate of ycc.
ycuEycu = ycuycu Oe ycuCycu denotes the estimation error.
yc1 and yc2 are positive observer gains.
yca is a positive constant.
) is the hyperbolic tangent function.
Despite the effective disturbance estimation and the reduced chattering offered by using a hyperbolic tangent function in the finiteAatime disturbance observer, several important limitations persist:
a Manual Gain Tuning Required: The observer gains yc1 , yc2 and the saturation constant yca must be carefully selected by trialandAaerror.
The estimation performance is highly sensitive to these parameters, and inappropriate tuning may lead to degraded accuracy, instability, or overly aggressive responses.
a Lack of Adaptive Capability: FTDOB uses fixed gains, which do not adjust in response to changing system dynamics or disturbance characteristics.
As a result, it may underperform in environments with timeAavarying or unknown disturbances, especially when disturbance amplitudes or frequencies change rapidly.
a Trade-off Between Convergence Speed and Smoothness:
A large value of the saturation parameter yca increases convergence speed but may introduce peaking or residual chattering, especially near zero crossings of the estimation error.
A small yca, while providing smoother estimates, significantly slows down the observerAos response time.
a Limited Robustness to Noise and Nonlinearities: The design assumes ideal measurements and does not explicitly account for measurement noise, actuator saturation, or unmodeled nonlinearities.
This limits its applicability in realAaworld systems with sensor imperfections or dynamic uncertainties.
a Only Bounded Estimation Performance: Due to the use of the ycycaycuEa(.
) function, the disturbance estimation error is guaranteed to remain uniformly ultimately bounded (UUB), but not asymptotically convergent to zero.
This means the observer cannot fully eliminate the estimation error, especially under persistent disturbances.
These limitations motivate the development of enhanced observers, such as AFTDOB or AFTDOBAaFuzzy, which incorporate adaptive gain mechanisms to improve robustness, estimation accuracy, and adaptability to varying operating IV.
ADAPTIVE FINITEAaTIME DISTURBANCE OBSERVER
AND CONTROLLER DESIGN
Adaptive FiniteAaTime Disturbance Observer To overcome the limitations of the FiniteAaTime Disturbance Observer (FTDOB), the fixed observer gains yc1 and yc2 are replaced with adaptive gains yu1 .
and yu2 .
, which evolve in real time based on the systemAos estimation error dynamics.
Define the estimation error for the highestAaorder state:
ycuEycu = ycuycu Oe ycuCycu The adaptive observer is designed as:
ycuCN = yce.
yc yco0 ycuEycu { ycu yc = yu1 .
ycuEycu yu2 .
caycuEycu ) .
where the positive constant yco0 > 0, the adaptive gains yu1 .
and yu2 .
are defined as below:
= yu1,0 yuC1 .
cuEycu | yu2 .
= yu2,0 yuC2 ycuEycu2 where yu1,0 , yu2,0 , yuC1 , yuC2 , yca are positive constants.
Backstepping Control Backstepping control is a recursive, systematic design methodology used to develop stabilizing controllers for nonlinear systems, particularly those with hierarchical or Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 cascaded structures .
, .
, .
The authors outline the main design of backstepping control as below .
Define the error coordinate:
ycuOe1 ycn=1 yc1 = ycu1 Oe ycu1ycc .
c = ycu Oe yu .
yc = i 2, ycu ycOe1 ycn=1 ycn=2 .
= yc12 Oc ycycn2 ycycu2 ycn=2 Replace into .
yu1 = ycuN1ycc Oe yco1 yc1 yuycn = yuN ycnOe1 Oe ycoycn ycycn .
ycn = i 2, ycu Oe 1 where ycoycn > 0 are control gains.
The backstepping control law is:
ycON O Oe Ocycuycn=1 ycoycn ycycn2 ( yc12 OcycuOe1 ycn=2 ycycn 2 ycycu ) .
cycu ycuEycu )yccE Oe yco0 ycuEycu2 = Oe .
co1 Oe ) yc12 Oe OcycuOe1 ycn=2 .
coycn Oe .
ycycn Oe .
coycu Oe (Oeyce.
Oe yc yuN ycuOe1 Oe ycoycu ycycu ) .
Stability Analysis a Theorem.
Under Assumptions A1AeA3, the closedAaloop system with the proposed controller and AFTDOB ensures that yc1 , yc2 .
A , ycycu , ycuEycu Ie 0 in finite time.
a Proof.
Define the Lyapunov candidate:
) ycycu2 .
cycu ycuEycu )yccE Oe yco0 ycuEycu2 If we choose gains ycoycn :
yco1 > , ycoycn > 1 .
cn = 2.
A , ycu Oe .
, ycoycu > yco1 > , ycn = 1 Compute its time derivative:
Set:
yuIycn : = .
coycn Oe 1, 2 O ycn O ycu Oe 1 ycO = Ocycuycn=1 ycycn2 ycuEycu2 ycoycu Oe , ycn = ycu So that, ycON = Ocycuycn=1 ycycn ycNycn ycuEycu ycuENycu Backstepping structure yields:
ycON O Oe Ocycuycn=1 yuIycn ycycn2 .
cycu ycuEycu )yccE Oe yco0 ycuEycu2 Apply YoungAos inequality ycN = ycycn 1 Oe ycoycn ycycn yceycuyc ycn = 1.
A , ycu Oe 1 { ycn ycNycu = yce.
cu, y.
Oe yuN ycuOe1 cycu ycuEycu )yccE O ycycu2 ycuEycu2 yccE 2 Then:
Substitute yc.
into ycNycu :
ycuOe1 ycN = Oeycoycu ycycu yccE { ycu yccE = ycc.
cu, y.
Oe yc ycON O Oe Oc yuIycn ycycn2 Oe .
uIycu Oe ) ycycu2 Oe .
co0 Oe ) ycuEycu2 yccE 2 ycn=1 Also, the observer error dynamics:
ycuENycu = ycuN ycu Oe ycuCNycu = yccE Oe yco0 ycuEycu So, the derivative of ycO becomes:
ycON = OcycuOe1 ycn=1 ycycn .
cycn 1 Oe ycoycn ycycn ) ycycu (Oeycoycu ycycu ycc ) ycuEycu .
cc Oe yco0 ycuEycu ) With yc = yu1 .
ycuEycu yu2 .
caycuEycu ) in .
From Assumption A2, ycc.
cu, y.
is bounded: .
cu, y.
| O yccI .
ccI is boundar.
, and the function tanh(.
) is also bounded, so there exist constants ya1 , ya2 Ou 0 such that .
ccE | O yccI yu1 .
cuEycu | yu2 yccE 2 O ya1 ycuEycu2 ya2 Apply YoungAos inequality ycayca O yca2 yca 2 for each element ycycn ycycn 1 :
ycycn ycycn 1 O ycycn2 ycycn 1 , ycn = 1.
A , ycu Oe 1 From this we have .
simple square-form boun.
ycu = OcycuOe1 Eycu )yccE Oe yco0 ycuEycu2 ycn=1 ycycn ycycn 1 Oe Ocycn=1 ycoycn ycycn .
cycu ycu Therefore, ycu ycuOe1 where the virtual control laws yuycn .
cn = i 1, ycu Oe .
are defined:
= ycuOe1 Oc ycycn ycycn 1 O Oc ycycn2 Oc ycycn2 Therefore, ycON O Oe OcycuOe1 ycn=1 yuIycn ycycn Oe .
uIycu Oe ) ycycu Oe .
co0 Oe Oe ya1 ) ycuEycu2 ya2 Set positive constants .
f gain choice is satisfie.
Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 uyc Oi ycn=1,A ,ycuOe1 ycoycn > 0, uycu Oi yuIycu Oe > 0, .
uycu : = yco0 Oe Oe ya1 > 0 From ycON O Oeuyc OcycuOe1 Eycu2 ya2 ycn=1 ycycn Oe u ycu ycycu Oe u ycu ycu O Oemin.
yc , uycu , uycu )(Ocycuycn=1 ycycn2 ycuEycu2 ) ya2 = Oe2uycO ya2 with u O= min.
yc , uycu , uycu ) > 0.
We consider two cases:
a ya2 = 0.
Then ycON O Oe2uycO yields exponential decay ycO.
O ycO.
yce Oe2ut .
exponential decay alone does not imply finite-time convergence.
If the dissipation is strengthened to ycON O yuI1 ycO yuI2 ycO yu Fuzzy Logic Inputs and Outputs The proposed fuzzy system accepts two realAatime input * Error = ycuEycu = ycuycu Oe ycuCycu : estimation error * dError = ycuENycu = ycuN ycu Oe ycuCNycu : time derivative of estimation error The fuzzy controller outputs:
* gamma1 = yu1 .
cuEycu , ycuENycu ): adaptive proportional gain.
* gamma2 = yu2 .
cuEycu , ycuENycu ): adaptive nonlinear saturation gain.
Fuzzy Rules and Membership Functions Each input is assigned three Gaussian membership functions: Error (Negative (N).
Zero (Z), and Positive (P)) and dError (Negative Derivative (DN).
Zero Derivative (DZ), and Positive Derivative (DP)) that are illustrated in Fig.
Each output is assigned three trimf membership functions:
gamma1 (Low (L).
Medium (M), and High (H)) and gamma2 (Low (L).
Medium (M), and High (H)) that demonstrated in Fig.
The fuzzy rule table is given as Table II.
where yuI1 , yuI2 > 0, 0 < yu < 1.
Then, the finite-time upper bound is given as below:
ycNyce O .
Oey.
yuI ln .
1 ycO 1Oeyu .
) yuI2 where ycO.
is the value of ycO at yc = 0.
a ya2 > 0.
Inequality .
only guarantees ultimate boundedness and not convergence to zero.
To recover finitetime decrease one may pick yuI2 > 0 and define the threshold yu ya ycOO = ( 2 ) .
While ycO.
Ou ycOO we have ya2 O yuI2 ycO yu so ycON O yuI2 yuI1 ycO yuI2 ycO yu and the time to reach ycOO is bounded by yuI ycNyce,O O .
Oey.
yuI ln .
1 ycO 1Oeyu .
Oe ycOO1Oeyu ) yuI2 Fig.
Membership Functions of Inputs (Error and dErro.
Below ycOO further finite-time convergence to zero requires either eliminating the residual ya2 by design or enforcing a global nonlinear dissipation term OeyuI2 ycO yu .
To avoid instability with sudden change of disturbances, the adaptive gains with saturation are used:
= yu1,0 yuC1 min.
cuEycu , yuU1 ) yu2 .
= yu2,0 yuC2 min.
cuEycu2 , yuU2 ) .
where yuU1 , yuU2 > 0 are gain saturation bounds.
ADAPTIVE FINITEAaTIME DISTURBANCE OBSERVER
BASED ON FUZZY LOGIC CONTROL
While AFTDOB enhances performance by adjusting observer gains in response to the estimation error magnitude, it still relies on heuristic tuning of base gains yu1,0 , yu2,0 , yuC1 , and yuC2 .
In dynamic or highly nonlinear systems, static adaptation rules may underperform or cause slow To address this, the authors propose AFTDOBAa Fuzzy that adaptively modifies the observer gains online.
Fig.
Membership Functions of outputs .
amma1 and gamma.
The Gaussian function is defined .
, .
= exp (Oe .
cuOeyc.
2 2yua 2 where yca is the center and yua is the standard deviation.
This choice ensures smooth transitions between fuzzy regions.
Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 Table 2.
Fuzzy Rule Table Error dError The backstepping control law is designed yc.
= ycn Oc9 yc=1 ycyc yuNyc .
cuEycu ,ycuEycu ) Oc9 yc=1 yuNyc .
cuEycu ,ycuEycu ) i , for ycn = 1,2 yc = yc1 ycuEycu yc2 tanh.
caycuEycu ) .
with yuNyayc and yuNyaAyc being the membership functions of the ruleAos input terms.
Using the fuzzy logic controller, the observer is rewritten as:
ycuCNycu = yce.
yc yco0 ycuEycu yc = yu1 .
cuEycu , ycuENycu )ycuEycu yu2 .
cuEycu , ycuENycu )tanh.
caycuEycu ) .
where yc1 = 125, yc2 = 100, yca = 5.
a AFTDOB:
where yuycn OO .
u1 , yu2 } are the fuzzy outputs .
bserver gain.
ycycycn is the singleton output value.
yuNyc is the firing strength of rule yc, computed as:
yuNyc = yuNyayc .
cuEycu )yuNyaAyc ( ycuENycu ) (Oeyc yuN 1 yuE2 ycu2 Oe yco2 yc2 Oe yc1 ) where yco1 = 50, yco2 = 25.
yc is the output of the disturbance observer that is defined by three algorithms below:
a FTDOB .
Adaptive Observer based on Fuzzy Logic Control The generic equation for yu1 and yu2 .
, .
cuEycu , ycuENycu ) = yuE1 yc = yu1 .
ycuEycu yu2 .
caycuEycu ) .
where yu1,0 = 100, yu2,0 = 40, yuC1 = 50, yuC2 = 30, yca = 5, yuU1 = 10, yuU1 = 50.
a AFTDOBAaFuzzy:
yc = yu1 .
cuEycu , ycuENycu )ycuEycu yu2 .
cuEycu , ycuENycu )tanh.
caycuEycu ) .
where yca = 5.
Fig.
3 to Fig.
6 illustrate the system output ycu1 , the tracking error yc1 = ycu1 Oe ycu1ycc , the disturbance estimation error yccE = ycc Oe yc , and control input yc.
for all three algorithms.
where yu1 .
cuEycu , ycuENycu ) and yu2 .
cuEycu , ycuENycu ) are the adaptive observer gains, which serve as the outputs of the fuzzy logic block.
VI.
SIMULATION RESULTS
To demonstrate the effectiveness of the proposed algorithm, an example in the previous study is used and the comparison of AFTDOB.
AFTDOBAaFuzzy and FTDOB is given.
The dynamics of the mechanical system are considered .
ycuN1 = ycu2 ycuN 2 = yuE1 yc Oe yuE2 ycu2 Oe yuE3 sgn.
cu2 ) ycc.
where ycu1 and ycu2 are system states.
yuE1 = 10, yuE2 = 15, and yuE3 = 5 are actual system parameters.
is a bounded unknown disturbance that is given as below:
= 8sin10yc 4cos25yc Fig.
The output ycu1 and the desired trajectory ycu1ycc The tracking errors:
yc1 = ycu1 Oe ycu1ycc { 1 = ycuN1ycc Oe yco1 yc1 yc2 = ycu2 Oe yu1 The virtual control derivative:
yuN 1 = ycuO1ycc Oe yco1 .
cuN1 Oe ycuN1ycc ) = ycuO1ycc Oe yco1 .
cu2 Oe ycuN1ycc ) = ycuO1ycc Oe yco1 yc2 yco12 yc1 Fig.
The tracking error yc1 = ycu1 Oe ycu1ycc Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 a For disturbance estimation error yccE = ycc Oe yc :
RMSEyccE = Oo OcycA 1 .
Oe ycc .
) .
ycA Quantitatively, the proposed AFTDOB-Fuzzy reduces the tracking error RMSE by approximately 78% and 53% compared with the fixed-gain FTDOB and the linearly adaptive FTDOB, respectively, while the disturbance estimation RMSE is reduced by over 80% and 57%, confirming the superior performance of the fuzzy-based adaptive scheme.
The above results clearly highlight the influence of different gain adaptation mechanisms on disturbance estimation and tracking performance.
The fixed-gain FTDOB provides finite-time convergence but exhibits a trade-off between convergence speed and robustness, leading to either slower response or increased oscillations when highfrequency disturbances are present.
The linearly adaptive FTDOB improves robustness by adjusting gains online.
however, its adaptation capability remains limited when the disturbance varies rapidly.
In contrast, the proposed AFTDOB-Fuzzy consistently achieves faster convergence, lower steady-state error, and smoother control input.
This improvement is mainly attributed to the fuzzy logicAebased gain adaptation, which flexibly adjusts observer gains according to both the magnitude and rate of change of the estimation error.
As a result, the proposed method demonstrates superior disturbance rejection capability and reduced sensitivity to gain tuning, as confirmed by the quantitative RMSE comparisons, as shown in Table i.
Fig.
The control law yc.
Fig.
The disturbance estimation error yccE = ycc Oe yc Table 3.
RMSE Table a FTDOB demonstrates the slowest convergence rate and produces the largest steadyAastate disturbance estimation While the use of a hyperbolic tangent function helps reduce chattering, the fixed gains limit adaptability to varying disturbance profiles.
a AFTDOB achieves faster convergence and improved disturbance rejection by adaptively adjusting the observer However, under highAafrequency disturbances, its performance can fluctuate, and the gain adaptation may introduce control input oscillations.
a AFTDOBAaFuzzy provides the best overall performance, combining rapid convergence, minimal steadyAastate error, and smooth control input transitions.
By using a fuzzy logicAabased gain tuning mechanism, the observer dynamically adapts to varying disturbance conditions without the need for manual gain scheduling.
Quantitative comparisons confirm the advantages of AFTDOBAaFuzzy, which outperforms both baseline methods in terms of tracking accuracy and disturbance estimation The Root Mean Square Error (RMSE) .
, .
is calculated for the tracking error and the disturbance estimation (Let N be the total number of time steps .
a For the tracking error yc1 = ycu1 Oe ycu1ycc :
RMSEyc1 = Oo OcycA Oe ycu1ycc .
) ycA
Observer
FTDOB
AFTDOB
AFTDOB-Fuzzy yacyaUyaeyaEyeuya yacyaUyaeyaEyeIE VII.
CONCLUSION
This paper investigated finite-time disturbance observerAe based control for nonlinear systems and presented a systematic comparison between three observer designs: a fixed-gain finite-time disturbance observer (FTDOB), a linearly adaptive finite-time disturbance observer (AFTDOB), and a fuzzy logicAebased adaptive finite-time disturbance observer (AFTDOB-Fuzz.
The objective was to evaluate how different gain adaptation mechanisms influence disturbance estimation accuracy, convergence speed, and robustness under time-varying disturbances.
Finite-time stability of the observer error dynamics was established using Lyapunov analysis, and sufficient conditions for convergence were derived.
Simulation results high-frequency time-varying demonstrated clear performance differences among the three The fixed-gain FTDOB achieved finite-time convergence but exhibited sensitivity to gain selection and degraded performance under rapidly varying disturbances.
The linearly adaptive FTDOB improved robustness and reduced steady-state error.
however, its adaptation capability remained limited for high-frequency disturbances.
contrast, the proposed AFTDOB-Fuzzy consistently Hoang Duc Long.
Enhanced Disturbance Estimation for Tracking Control of Nonlinear Systems Using Adaptive Fuzzy Finite-Time Observers Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 achieved faster convergence, smoother control input, and significantly lower tracking and disturbance estimation Quantitative evaluation using root mean square error (RMSE) metrics showed that the proposed method reduced tracking and disturbance estimation errors by more than 50% compared with the linearly adaptive observer and over 75% compared with the fixed-gain FTDOB.
Despite these advantages, the present study has several The performance of the fuzzy-based observer depends on the selection of membership functions and rule bases, which were designed empirically.
In addition, the computational complexity is higher than that of fixed-gain observers, and the validation was limited to numerical Experimental implementation and robustness analysis in the presence of measurement noise were not Future research will focus on experimental verification on real-time platforms, systematic tuning methods for fuzzy parameters, and robustness enhancement under sensor noise and actuator constraints.
Extensions to networked, multiagent, and large-scale nonlinear systems will also be ACKNOWLEDGMENT The authors would like to thank the support of colleagues at the Department of Automation and Computing Techniques.
Institute of Control Engineering.
Le Quy Don Technical University.
Hanoi.
Vietnam.
REFERENCES