Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 ISSN: 2986-6537.
DOI: 10.
59247/jfsc.
Design of an Indirect Adaptive Controller Based on Fuzzy Logic Control for Linear Cascade Systems Affected by Bounded Unknown Disturbances Hoang Duc Long * Department of Automation and Computing Techniques.
Le Quy Don Technical University.
Hanoi.
Vietnam Email: longhd@lqdtu.
*Corresponding Author AbstractAiThis paper presents a novel indirect adaptive control scheme that integrates fuzzy logic with a virtual integralbased adaptive controller to enhance the tracking performance of linear cascade systems under bounded unknown The proposed controller builds upon the established indirect adaptive control framework, employing a virtual state algorithm but augments it with a fuzzy inference mechanism that dynamically adjusts a key control parameter to improve robustness and adaptability.
Gaussian membership functions and a Sugeno-type fuzzy inference system are employed to fine-tune the gain parameter based on real-time tracking error and its derivative.
The control law incorporates parameter adaptation, a saturation function to replace discontinuous sign operations, and a fuzzy-tuned gain to mitigate chattering and improve transient response.
Simulation results under severe disturbances demonstrate significant improvements in tracking accuracy and control smoothness.
Specifically, the proposed fuzzy-based controller reduces steady-state tracking error by over 40%, minimizes control chattering, and maintains robust performance under disturbance amplitudes up to 185 units-conditions that severely degrade the performance of the non-fuzzy indirect adaptive The effectiveness of the proposed algorithm is shown by handling model uncertainties and external perturbations in a two-level linear cascade system.
KeywordsAiIndirect Adaptive Control.
Fuzzy Logic Control.
Linear Cascade Systems.
Integral Virtual Algorithm.
Bounded Unknown Disturbances.
Lyapunov Stability INTRODUCTION Cascade systems, characterized by their hierarchical structure and interdependent dynamics, arise in many engineering applications, including aerospace, chemical process control, and robotic manipulation .
The inherent complexity and sensitivity of such systems to external disturbances and modeling uncertainties present significant challenges for control design.
In recent years, adaptive control techniques have been widely employed to tackle these issues by adjusting controller parameters in real time based on observed system behavior .
Among these, the indirect adaptive control with integral virtual algorithms has proven effective for handling uncertainties in linear cascade systems .
Despite their proven stability, traditional indirect adaptive controllers lack responsiveness under dynamic disturbances due to fixed gain strategies and abrupt control actions.
This paper addresses this limitation by proposing a fuzzy-augmented indirect adaptive control strategy.
The key novelty lies in the real-time fuzzy tuning of the gain parameter using Gaussian membership functions, which enhances robustness and minimizes chattering.
Our contributions are summarized as A Integration of fuzzy inference into the adaptation loop.
A A smooth control design using a saturation function.
A Improved performance under bounded unknown disturbances, demonstrated through simulation.
Consider rephrasing as: These limitations motivate the integration of fuzzy logic systems (FLS) .
, which offer human-like reasoning and nonlinear mapping capabilities, into the adaptive control framework.
Fuzzy logic can effectively tune parameters in real-time based on linguistic rules derived from control experience, thus enhancing robustness and reducing chattering phenomena typically associated with sign-based control strategies.
Through extensive simulations, the author compares the proposed controller with the non-fuzzy indirect adaptive controller under significantly bounded disturbances and reference trajectory variations.
The results demonstrate that the proposed method achieves superior tracking performance, more stable adaptation, and smoother control signals, confirming its practical potential in complex control scenarios of linear cascade systems.
II.
RELATED WORK
The control of linear cascade systems under uncertainties and disturbances has been widely studied due to their prevalence in aerospace, robotics, and process control This section reviews existing methods in indirect adaptive control, fuzzy logic control, and their integration, highlighting the limitations that motivate our proposed approach.
Indirect Adaptive Control of Cascade Systems Indirect adaptive control methods estimate unknown system parameters and use them to compute control laws .
They are particularly effective for linear systems with parametric uncertainty.
Myshlyaev et al.
and Nguyen et .
proposed integral virtual algorithms to handle hierarchical cascade structures.
While these methods achieve stability and adaptation, they typically use fixed gain structures, which limit their responsiveness to rapid or nonlinear disturbances.
Additionally, the use of sign functions in the control law introduces chattering, which can excite unmodeled dynamics and reduce control smoothness.
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 Fuzzy Logic Control in Adaptive Systems Fuzzy logic control (FLC) offers human-like reasoning and is suitable for handling nonlinearities and uncertainties without requiring an exact model .
FLC has been effectively applied in adaptive control to tune gains and improve robustness .
However, most of these works focus on direct adaptive control or systems with simple dynamics, and do not address the structural challenges of cascade systems or combine with integral virtual methods.
Moreover, fuzzy logic is often applied in isolation without integration into a Lyapunov-based adaptation framework.
Handling Disturbances and Chattering Reduction Several studies have explored methods to handle bounded unknown disturbances.
Saturation functions have been used to mitigate chattering introduced by sign-based control laws .
While this reduces discontinuity, it does not adaptively tune the controller's gain to varying disturbance Our approach extends this by embedding a fuzzy inference system to dynamically adjust the gain parameter E.
, based on the systemAos tracking error and its rate of Positioning of the Proposed Work The integration of fuzzy logic with indirect adaptive control of cascade systems has received limited attention in the literature.
Existing indirect adaptive controllers lack realtime adaptability to varying disturbances, and existing fuzzy controllers are rarely tailored for complex, layered dynamics.
Our proposed method addresses these gaps by:
A Embedding a fuzzy inference system within the indirect adaptive controller to tune the gain yua.
based on realtime performance.
A Replacing discontinuous ycycnyciycu(A) functions with saturation to reduce chattering while preserving Lyapunov stability.
A Demonstrating robustness and improved transient performance under severe, time-varying disturbances.
This hybrid approach leverages the strengths of both indirect adaptive and fuzzy control frameworks, offering a scalable and smooth control strategy for complex linear cascade systems.
DESIGN OF AN INDIRECT ADAPTIVE CONTROLLER
FOR LINEAR CASCADE SYSTEMS USING AN INTEGRAL
VIRTUAL ALGORITHM
All symbols, variables, and parameters used throughout the paper are defined in Table 1.
Consider the mathematical model of a class of linear cascade systems in the form .
caycu1 U ycaycuycu ycaycu ]ycN OO Eyycu 1 is the vector of unknown ycc.
OO Ey is the bounded unknown disturbances and .
| O yccycoycaycu .
Table 1.
Symbols and Definitions Symbols yc yu, yu1 , yu2 , yua yeo1 , ycu2 yce, yceN , yua, yuA yuI, yu yc11 , yeC12 , yca21 , ycaI12 , yca2 ycu2ycycnycyc yc yyE yyEC ycO yci, yuN yc yc yc ycc Definitions reference signal control gains system states error terms fuzzy parameters parameters of the system virtual variable AuidealAy input vector of unknown parameters vector of adaptive parameters Lyapunov function Laplace operator control law unknown disturbance The control objective is defined as follows:
lim yeI.
= 0
ycIeO where yce = ycu1 Oe ycu1O is the tracking error.
ycu1O = O O ycN ycu12 U ycu1ycu .
cu11 ] OO Eyycu is the vector of desired output An indirect adaptive controller for linear cascade systems using an integral virtual algorithm is designed in three stages as below.
Stage 1.
Synthesis of AuidealAy virtual control In this stage, a virtual control ycu2ycycnycyc .
cu1 , yuO) of the cascade ycu2 is introduced.
A new manifold is given, yua = ycu2 Oe ycu2ycycnycyc The extended output stage .
will take the form yeoN 1 = yc11 .
yE)yeo1 yeC12 .
yE).
cu2ycycnycyc yu.
ycuN 2ycycnycyc = yc where yc is the new input.
The new output of tracking error is yc = yci.
O where yce1 = ycu11 Oe ycu11 is the error of the first state.
= ycu ycuOe1 yc yciycuOe1 yc U yci1 yc yci0 is the Hurwitz polynomial.
yc is the Laplace operator.
Obviously, from the condition lim yc = 0 and the property of the Hurwitz polynomial yci.
, ycIeO ycI1 : yeoN 1 = yc11 .
yE)yeo1 yeC12 .
yE)ycu2 .
ycI2 : ycuN 2 = yeCycN21 yeo1 ycaI12 ycu2 yca2 yc ycc.
where yeo1 = .
cu11 ycu12 U ycu1ycu ]ycN OO Eyycu is the vector of output state ycI1 .
ycu2 OO Ey is the input state ycI2 .
yeo = .
eo1ycN ycu2 ]ycN OO Eyycu 1 .
ycOOEy yc11 .
yE) = ycycuOe1 ] OO Eyycuyycu .
yeC12 .
yE) = .
U 0 ycaycu ]ycN OO ycaycu1 U ycaycuycu Eyycu .
ycaycu > 0.
yeC21 OO Eyycu .
ycaI12 and yca2 are scalar.
yca2 > 0.
yyE = the control objective .
is achieved.
The desired trajectory of the output cascade system is defined in the form, yc = yci.
where yc OO Ey is a smooth function.
Therefore, the output .
on the manifold .
ua O .
can be rewritten as O ) yc = yci.
cuN11 Oe ycuN11 .
cuOe.
= ycu11 yciycuOe1 ycu11 U yci1 ycuN11 yci0 ycu11 Oe yci0 yc Hoang Duc Long.
Design of an Indirect Adaptive Controller based on Fuzzy Logic Control for Linear Cascade Systems affected by Bounded Unknown Disturbances .
Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 yaycA where ycu1ycu is the phase coordinate of the adjustable model.
yu yaycA is the input control of the adjustable model.
yuA = ycu1ycu Oe ycu1ycu is the error.
The derivative of the error is, = ycuN1ycu yciycuOe1 ycu1ycu U yci1 ycu12 yci0 ycu11 Oe yci0 yc = .
caycu1 ycu11 ycaycu2 ycu12 U ycaycuycu ycu1ycu ycaycu ycu2ycycnycyc ) yciycuOe1 ycu1ycu U yci1 ycu12 yci0 ycu11 Oe yci0 yc = .
ci0 ycaycu1 )ycu11 .
ci1 ycaycu2 )ycu12 U .
ciycuOe1 ycaycuycu )ycu1ycu Oe yci0 yc ycaycu ycu2ycycnycyc = yeOycN .
yEya )yeo1 Oe yci0 yc ycaycu ycu2ycycnycyc = yuN.
eo1 , yyE1 , y.
ycaycu ycu2ycycnycyc yuN.
eo1 , yyE1 , y.
= yeOycN .
yEya )yeo1 Oe yci0 yc.
yci1 ycaycu2 U yciycuOe1 ycaycuycu ]ycN OO Eyycu .
U ycaycuycu ]ycN OO Eyycu .
ci0 ycaycu1 .
caycu1 ycaycu2 The derivative of output .
is yaycA yuAN = ycuN1ycu Oe ycuN1ycu The input control of the adjustable mode is selected as yu = yyECycN yeo yuyuA yeOycN .
yEya ) = yyEya = .
where yuNN .
eo1 , yyE, ycN ) = yeOycN .
yEya )yeoN 1 Oe yci0 ycN = yeOycN .
yEya ).
yE)yeo1 yeC12 .
yE)ycu2ycycnycyc ) Oe yci0 ycN .
yyE = .
yE1ycN ycaycu ]ycN is the extended vector of unknown parameters.
The AuidealAy input is chosen in the form yc = Oe .
uNN .
eo1 , yyE, ycN ) yuy.
ycaycu where yu > 0 is the feedback coefficient.
Therefore, .
can be rewritten as .
, .
yyECN = yuAyoyeo To prove the asymptotic stability of ycu11 at ycu11 , the Lyapunov function is chosen as .
, ycO1 .
= yc 2 ycO1N .
= ycycN = Oeyuyc 2 O 0 From .
, it can be concluded that when the output .
on the manifold .
ua O .
, it has lim yc = 0, so that ycIeO lim yeI.
= 0.
ycIeO Stage 2.
Identification of the unknown output parameters In this stage, the adaptive parameters are designed for the unknown parameters of the output stage.
The objective control is .
where yyEC = .
caCycu1 U ycaCycuycu ycaCycu ]ycN OO Eyycu 1 is the vector of adaptive parameters that corresponds to the unknown parameters of the vector yyE.
To do that, an adjustable mode (AM) is introduced yaycA ycuN1ycu =yu ycO2 .
uA, yyEE) = 1 2 1 ycN Oe1 yuA yyEE yo yyEE Calculate the derivative of ycO2 .
uA, yyEE) and combine with .
, it has ycO2N .
uA, yyEE) = yuAyuAN Oe yyEEycN yo Oe1 yyECN = yuA.
yEEycN ycu Oe yuyuA) Oe yuAyyEEycN yo Oe1 yoyeo = OeyuyuA 2 O 0 Calculate the derivative of .
and combine it with .
, it .
where yo = diag.
u1 , yu2 .
A , yuycu 1 } > 0 is the matrix of yaycA To prove the asymptotic stability of ycu1ycu at ycu1ycu and yyECN at yyE, the Lyapunov function is chosen as .
, .
ycN = Oeyuyc where yyEE = yyE Oe yyEC.
The adaptive laws are chosen in the form of the speed gradient algorithm.
So, .
can be rewritten as .
, lim yyEC.
= yyE yuAN = yyEycN ycu Oe yyECycN ycu Oe yuyuA = yyEEycN ycu Oe yuyuA ycN = yuNN .
eo1 , yyE, ycN ) ycaycu ycuN 2ycycnycyc = yuNN .
eo1 , yyE, ycN ) ycaycu yc ycIeO From .
, lim yuA.
= 0 and lim yyEC.
= yyE.
ycIeO ycIeO Stage 3.
Synthesis of closed-loop control In this stage, it is necessary to design the adaptive control law for closed-loop system and prove the asymptotic stability under the effects of bounded unknown disturbances.
The Lyapunov function is chosen as .
ycO3 .
c, yu.
= 1 2 1 2
yc yua Compute the derivative of .
ycO3N .
c, yu.
= ycycN yuayuaN = yc.
uNN .
eo1 , yyEC, ycN ) ycaCycu ycuN 2 ) yuayuaN = yc .
uNN .
eo1 , yyEC, ycN ) ycaCycu .
cuN 2ycycnycyc yuaN )) yuayuaN = Oeyuyc 2 .
caCycu yc yu.
yuaN = Oeyuyc 2 .
caCycu yc yu.
cuN 2 Oe ycuN 2ycycnycyc ) = Oeyuyc 2 .
caCycu yc yu.
eCycN21 yeo1 ycaI12 ycu2 yca2 yc ycc.
Oe ycuN 2ycycnycyc ) .
Hoang Duc Long.
Design of an Indirect Adaptive Controller based on Fuzzy Logic Control for Linear Cascade Systems affected by Bounded Unknown Disturbances Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 The adaptive control law is selected as .
= Oe .
ua ycyciycu.
caCycu yc yu.
yeCycN21 yeo1 ycaI12 ycu2 yca2 Oe ycuN 2ycycnycyc ) .
where yua is the coefficient that guarantee .
| O yccycoycaycu O yua.
Replace .
, it has ycO3N .
c, yu.
= Oeyuyc 2 Oe yuU.
ua ycyciycu.
uU) Oe ycc.
) = Oeyuyc 2 Oe yua .
uU| yuUycc.
O Oeyuyc 2 Oe yua .
uU| .
uU|.
| .
= Oeyuyc 2 Oe .
uU|.
ua Oe .
|) O 0 where yuU = ycaCycu yc yua.
From .
, it can conclude that the closed-lood system with the adaptive control law .
guarantees that lim yc = 0 and lim yua = 0.
To reduce the ycIeO ycIeO chattering phenomenon, in the adaptive control law .
, the ycyciycu(.
) function is replaced by the ycycayc(.
) Function that is defined as .
uU) = { yuo Oe1 yuU>yuo .
uU| O yuo yuU<yuo where yuo is a small coefficient.
Fig.
1 visualizes a block diagram that depicts the entire control architecture.
Fig.
Structural Diagram of Virtual Integral Controller combined with Fuzzy Logic Control To improve robustness, a fuzzy inference system is introduced to tune the gain parameter E in real time.
The fuzzy block receives the normalized error yce1 and its derivative yceN1 as inputs and adjusts E based on a Gaussian membership function and a Sugeno-type rule base.
The updated E is then used in the adaptive control law, replacing the fixed gain.
The O inputs of the fuzzy block are the error yce1 = ycu11 Oe ycu11 and the derivative of error yccyce1 /yccyc.
The output of the fuzzy block is yuaycayccycycycyc .
The fuzzy membership function that is used is Gauss There are 5 variables, including NB (Negative Bi.
NS (Negative Smal.
ZE (Zer.
PS (Positive Smal.
, and PB (Positive Bi.
The fuzzy rule table is designed as Table 2.
The Gaussian membership functions of inputs and output are illustrated in Fig.
The Fuzzy Rule Surface is demonstrated in Fig.
Table 2.
Fuzzy Rule Table Fig.
Structural Diagram of Virtual Integral Controller IV.
TUNING PARAMETER OF INDIRECT ADAPTIVE
CONTROLLER USING FUZZY LOGIC CONTROL
In this section, the fuzzy logic control is applied to tune the control parameter yua that guarantees the robustness of system under the effects of bounded unknown disturbances .
The block diagram of the virtual integral controller combined with fuzzy logic control is demonstrated in Fig.
Fig.
Gaussian Membership Functions of yce1 , yceN1 , yuaycayccycycycyc Hoang Duc Long.
Design of an Indirect Adaptive Controller based on Fuzzy Logic Control for Linear Cascade Systems affected by Bounded Unknown Disturbances Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025
where yc11 = [ yca21
]=[
yca22 yeC12 = [ ] = [ ].
yca2 Oe2 ycN yca2 = 2.
yeo1 = .
cu11 ycu12 ] .
yyE = yeC21 = [ ].
ycaI12 = 1.
ca21 yca22 yca2 ]ycN .
the bounded unknown disturbance ycc.
is defined as .
| O yccycoycaycu = .
, ycc.
= 75.
ycycaycuycc Oe .
7 sin.
uUy.
5 sin.
yuUy.
) 50ycyciycu.
cycaycuycc Oe 0.
The indirect adaptive control law .
in this case is yc.
= Oe Fig.
Fuzzy Rule Surface The value of adjusting parameters using fuzzy logic control is given yua = yuaycaycaycyce .
yuaycayccycycycyc ) Oc5ycn=1 Oc5yc=1 yuNycn .
ce1 )yuNycn .
ceN1 )ycycnyc = yuaycaycaycyce .
yuI 5 Ocycn=1 Oc5yc=1 yuNycn .
ce1 )yuNycn .
ceN1 ) yu where yuaycaycaycyce Ou yccycoycaycu Ou .
ycycnyc are the Sugeno weights.
yuNycn .
= yceycuycy (Oe .
cuOeycaycn )2 2yuayaycaycycyc ) is the Gaussian membership function centered at ycaycn .
yuI OO .
is a constant value.
yu is a small number added to the denominator to prevent division by zero.
The full structural diagram of an indirect adaptive controller based on fuzzy logic control for a linear cascade system affected by a bounded unknown disturbance is demonstrated in Fig.
Fig.
The structural diagram of the closed-loop system
EXAMPLE
To show the effectiveness of the proposed algorithm, an example of a two-level linear cascade system is given as .
, .
ua ycycaycyuo .
caC2 yc yu.
yeCycN21 yeo1 ycaI12 ycu2 yca2 Oe ycuN 2ycycnycyc ) .
The adaptive parameters are yu1 = 12.
yu2 = 1.
yu3 = The regulator parameters are yci0 = 10.
yci1 = 5.
yu = 160.
yu = 2.
yuaycaycaycyce = 200.
The fuzzy parameters are yuI = 0.
yu = 10Oe6 .
The initial conditions are ycu11 .
= ycu12 .
= 1.
O .
yaycA .
= 3.
= 2.
ycu11 = 0.
ycu12 = 1.
ycaC21 .
= 6.
ycaC22 .
= Oe1.
ycaC2 .
= 3.
The author compares the response of adaptive control .
ua = yuaycaycaycyce ) and an adaptive controller based on fuzzy logic control .
= yuaycaycaycyce .
yuaycayccycycycyc ) ) with the same parameters and initial conditions.
Fig.
6 shows the bounded unknown disturbance ycc.
affecting the linear cascade system that is complicated and a high challenge for the control design.
The comparison of the response of indirect adaptive control with and without fuzzy is illustrated in Fig.
7 to Fig.
In Fig.
7, the system state ycu11 .
responses with and without fuzzy logic control.
The fuzzy controller tracks the reference more accurately and with less overshoot.
shown in Fig.
8, the proposed fuzzy-based controller reduces the steady-state tracking error by more than 40% compared to the traditional non-fuzzy version.
Table i summarizes the RMSE and convergence times for all tested controllers.
It is clear that the RMSE and convergence time with fuzzy are smaller than without fuzzy.
Furthermore, the control signal shown in Fig.
11 is significantly smoother when fuzzy gain adjustment is used, confirming the reduced chattering effect.
The yua.
parameter of the adaptive controller in Fig.
9 when using fuzzy will change based on the change of the input In Fig.
11, the new adaptive control law based on fuzzy has less sudden changes due to the influence of unknown disturbances.
The root mean square error (RMSE) .
is O calculated for the tracking error yce11 = ycu11 Oe ycu11 (Let N is the total number of time steps .
O .
)2 ycIycAycIyayce11 = Oo Oc.
Oe ycu11 ycn=1 yeoN 1 = yc11 .
yE)yeo1 yeC12 .
yE)ycu2 ycuN 2 = yeCycN21 yeo1 ycaI12 ycu2 yca2 yc ycc.
The root mean square error (RMSE) and convergence time for both controllers are shown in Table 3:
Hoang Duc Long.
Design of an Indirect Adaptive Controller based on Fuzzy Logic Control for Linear Cascade Systems affected by Bounded Unknown Disturbances Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 Fig.
Bounded unknown disturbance ycc.
Fig.
Manifold yua.
= ycu2 Oe ycu2ycycnycyc responses with and without fuzzy logic control Fig.
State ycu11 .
responses with and without fuzzy logic control Fig.
Control law yc.
responses with and without fuzzy logic control Table 3.
Root Mean Square Error (RMSE) and Convergence Time OF CONTROLLERS Controller Indirect Adaptive Control with Fuzzy Indirect Adaptive Control without Fuzzy Backstepping Control .
Sliding Mode Control .
O Fig.
Tracking error yce1 .
= ycu11 Oe ycu11 responses with and without fuzzy logic control Fig.
Control parameter yua.
responses with and without fuzzy logic VI.
RMSE
Convergence Time .
CONCLUSION This paper proposed an enhanced indirect adaptive control strategy for linear cascade systems subject to bounded unknown disturbances by integrating fuzzy logic into the controller design.
The novelty lies in the use of a fuzzy inference system to dynamically adjust the control gain parameter yua based on real-time tracking error and its Gaussian membership functions and a Sugenotype rule base were used to modulate E, while the control law was further improved by replacing the discontinuous sign function with a smooth saturation function to reduce Extensive simulations demonstrated that the proposed fuzzy-based controller outperforms the conventional nonfuzzy indirect adaptive controller in terms of tracking Quantitatively, the root mean square error (RMSE) was Hoang Duc Long.
Design of an Indirect Adaptive Controller based on Fuzzy Logic Control for Linear Cascade Systems affected by Bounded Unknown Disturbances Journal of Fuzzy Systems and Control.
Vol.
No 3, 2025 reduced by over 40%, and the control input showed significantly less variation under large and fast-changing These results confirm the effectiveness of the proposed approach in dealing with high-amplitude, bounded disturbances and in achieving fast, stable convergence.
In future work, we aim to:
A Implement the proposed fuzzy-enhanced controller in a real-time embedded system for hardware-in-the-loop A Extend the control framework to nonlinear and MIMO cascade systems, such as robotic manipulators or interconnected process control systems.
A Conduct experimental validation on physical platforms to assess the controller's robustness, computation efficiency, and practical feasibility.
A Explore machine learning techniques for automatic optimization of fuzzy membership functions and rule tuning to further enhance performance and The proposed method offers a promising direction for improving the adaptability and disturbance rejection of indirect adaptive controllers in complex real-world systems.
ACKNOWLEDGMENT
The author would like to thank the support of colleagues at the Department of Automation and Computing Techniques.
Faculty of Control Engineering.
Le Quy Don Technical University.
Hanoi.
Vietnam.
REFERENCES