Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 ISSN: 2986-6537.
DOI: 10.
59247/jfsc.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Thanh-Truong Mai 1.
Tuan-Thuong Le 2,*.
Hong-Quang Le 3.
Bao-Duy Than 4.
Hoang-Thien-Hung Trinh 5.
Dinh-Truc Nguyen 6.
Hung-Thinh Tran 7.
Tran-Quang-Huy Le 8.
Tri-Dung Hoang 9.
Sy-Luan Duong 10.
Hoang-Chinh Tran 11.
Ha-Duy Nguyen 12.
Thi-My-Linh Dong 13
1,2,3,4,5,6,7,8,9,10,12,13
Ho Chi Minh City University of Technology and Education (HCMUTE).
Vietnam Cao Thang Technical College.
Vietnam Email: 22151319@student.
vn, 2 22151307@student.
21145644@student.
vn, 4 21145098@student.
vn , 5 21145155@student.
21145308@student.
vn, 7 21145652@student.
vn, 8 21145148@student.
21145100@student.
vn, 10 21145199@student.
vn, 11 tranhoangchinh@caothang.
21145095@hcmute.
vn, 13 21145192@student.
*Corresponding Author AbstractAiThe rotary inverted pendulum (RIP) is a benchmark nonlinear underactuated system commonly used in control research, with various extensions such as multi-link and parallel configurations developed to increase complexity and evaluate advanced controllers.
This paper presents a hybrid control strategy combining Linear Quadratic Regulator (LQR) and a Genetic Algorithm (GA) for stabilizing and tracking control of a rotary double-linked parallel inverted pendulum (RDPIP), a nonlinear under-actuated single input-multi output (SIMO) system.
The LQR controller is designed based on a linearized state-space model at the TOP-TOP equilibrium point.
To enhance performance, the weighting matrices Q and R are optimized using GA with a fitness function minimizing trajectory error.
Simulation results demonstrate that the GAoptimized controller (LQR .
achieves superior performance compared to the trial-based LQR (LQR .
, with a reduced settling time of 0.
5 seconds, lower oscillation amplitudes, and improved tracking of reference signals under sinusoidal and pulse disturbances.
Specifically, the pendulums reached steady state within 2Ae3 seconds, and the arm settled within 6 seconds.
These findings confirm the effectiveness of a hybrid strategy and robustness of the proposed hybrid approach for RDPIP control, laying a foundation for future implementation in real-world KeywordsAiRotary Double Parallel Inverted Pendulum.
Nonlinear System.
LQR Control.
Genetic Algorithm.
SIMO
System
INTRODUCTION
RIP has long served as a standard benchmark for the design and validation of nonlinear control strategies in underactuated systems due to its intrinsic instability and coupled dynamics .
, .
While the basic single-link RIP has been thoroughly explored in both academic and laboratory environments .
, its multi-link variants, such as RDRIP, have attracted increasing attention for testing advanced control methods on higher-order systems .
, .
A further extension.
RDPIP, features two pendulums mounted symmetrically at each end of a horizontal rotating This configuration introduces a higher level of complexity in terms of coupling and control, modeling the system as a nonlinear.
SIMO dynamic model .
, .
Despite its rich dynamics and relevance to real-world applications, the RDPIP has been comparatively less addressed in recent literature, especially with regard to robust trajectory tracking.
Existing studies on RDPIP control have primarily focused on basic stabilization or swing-up techniques, using PID, fuzzy logic, or energy-based control approaches .
More recent efforts have applied model-based strategies such as LQR and H-infinity control to improve stability and reduce energy consumption .
However, these methods often rely on manually tuned parameters, limiting their robustness and adaptability.
To address this, heuristic optimization techniques like GA have been successfully employed to tune control gains and improve performance under nonlinear constraints .
, .
Several recent studies have explored advanced control strategies for underactuated systems.
Mustafa et al.
introduced fractional order control for rotary double pendulums, while Chen and Huang .
addressed adaptive control robustness under time-varying uncertainties.
Sanjeewa and Parnichkun .
proposed HO control for RDPIP but noted its dependence on precise modeling.
overcome such limitations, metaheuristic and soft computing methods have been increasingly adopted.
Notably.
Panjwani et al.
and Karthick et al.
applied NSGA II and SMC, respectively, to optimize LQR gains, improving stability and trajectory tracking.
Yildiran .
employed reinforcement learning to adapt LQR controllers, and Le et al.
analyzed how GA variant choices affect tuning outcomes.
Nguyen et .
demonstrated the effectiveness of PSO-LQR for rotary pendulum stabilization.
These efforts support the growing trend toward hybrid intelligent control and reinforce the relevance of GA-based LQR optimization for complex systems such as RDPIP.
In this paper, we propose a hybrid LQR-GA approach to optimize the state-feedback controller of RDPIP.
The novelty of our study lies in combining optimal control design with evolutionary search techniques for automatic tuning of weighting matrices.
This not only enhances system response but also strengthens disturbance rejection capability.
Furthermore, we evaluate our method on the TOP-TOP equilibrium point, incorporating a reference tracking scenario with dynamic setpoints - a feature often neglected in earlier RDPIP works .
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 2, 2025 Compared to prior papers such as study .
Fahmizal .
, which used ANFIS for control synthesis, and Dang et .
, which applied PID to simpler inverted pendulum models, our approach demonstrates superior transient performance and better generalizability.
With detailed simulation results and quantitative metrics, this paper contributes a practical and extendable solution for high-order under-actuated control systems.
II.
MATHEMATICAL EQUATIONS AND SIMULATION
Mathematical Equations Mathematical model of PRIP is illustrated in Fig.
A DC
motor is mounted vertically on a fixed base, and its output shaft is connected to center of a rigid horizontal link, referred to as the arm.
At both ends of arm, rotary encoders are Each encoder serves as a pivot point for an attached link, referred to as a pendulum.
These pendulums are allowed to rotate freely about axis of their respective encoders.
The angular displacement of each pendulum is continuously measured by the corresponding encoder.
Parameters of PRIP are listed in Table 1 and Table 2.
The kinetic energy K (N) of the RDPIP is then given by:
ya = ya0 yuoN 2 ya1 yuEN2 yco1 yc1 2 yco2 yc2 2 In this context, yc1 and yc2 represent the velocities of pendulum 1 and pendulum 2, respectively.
The total kinetic energy of the system is then rewritten as follows:
ya0 yuoN 2 ya1 yuEN1 ya2 yuEN2 yco1 .
coyci1 ycycnycu yuE1 yuoN)2 yco1 .
ayuoN)2 yco1 .
coyci1 yuE1 ) Oe yco1 ycoyci1 ya ycaycuyc yuE1 yuEN1 yuoN yco2 .
coyci2 ycycnycu yuE2 yuoN )2 yco2 .
ayuoN)2 yco2 .
coyci2 yuE2 ) Oe yco2 ycoyci2 ya ycaycuyc yuE2 yuEN2 yuoN ya= .
The potential energy U (N) of RDPIP is expressed as:
ycO = yco1 yciycoyci1 ycaycuyc yuE1 yco2 yci ycoyci 2 ycaycuyc yuE2 The dissipated energy W (N) of the RDPIP is given by:
ycO = yca0 yuoN 2 yca1 yuEN1 yca2 yuEN22 From .
, the Lagrangian function is computed as ya=yaOeycO = ya0 yuoN 2 ya1 yuEN1 ya2 yuEN22 yco1 .
coyci1 ycycnycu yuE1 yuoN) yco1 .
ayuoN)2 yco1 .
coyci1 yuE1 ) Oe yco1 ycoyci1 ya ycaycuyc yuE1 yuEN1 yuoN yco2 .
coyci2 ycycnycu yuE2 yuoN)2 yco2 .
ayuoN )2 yco2 .
coyci2 yuEN2 ) Oe yco2 ycoyci2 ya ycaycuyc yuE2 yuEN2 yuoN Oeyco1 yciycoyci1 ycaycuyc yuE1 Oe yco2 yciycoyci2 ycaycuyc yuE2 Fig.
Mathematical model of RDPIP Table 1.
System parameters of RDPIP .
=1, .
Parameters ycoycn Unit ycoyciycn yaycn ya ya0 yci yaycn ya0 ycoyci.
yco 2 yco ycoyci.
yco 2 yco/yc 2 ycAyco.
yc ycAyco.
yc Table 2.
Variables yuo yuoN yuE1 yuEN1 yuE2 yuEN2 Unit ycycaycc ycycaycc/yc ycycaycc ycycaycc/yc ycycaycc ycycaycc/yc Description Mass of pendulum i Distance to center of mass of Moment of inertia of pendulum i Length of arm Moment of inertia of the arm Gravitational acceleration Friction coefficient of pendulum i Friction coefficient of the arm The RDPIP is actuated by a DC motor.
The relationship between torque yua (N.
and input voltage ycOycnycu (V) is given by the following equation:
yua = .
ayc ycOycnycu Oe yayc yayca yuoN)/ycIyco .
According to .
, the linearized dynamic equations of the RDPIP are derived from .
as follows:
System variables of PRIP ya1 .
a2 ya3 Description Angular displacement of the arm Angular velocity of the arm Angular displacement of pendulum 1 Angular velocity of pendulum 1 Angular displacement of pendulum 2 Angular velocity of pendulum 2 Dynamic equations of system based on Euler-Lagrange formulation are as follows .
ycc yuiya yuiya yuiycO = yaycn yccyc yuiycN ycn yuiycycn yuiycN ycn ya1 yuoO ya ycOycnycu ya2 ] .
uEO1 ] .
cN2 ] = yc [ 0 ] ycIyco ya3 yuEO Where:
ya1 = ya0 yco1 ycoyci1 ycycnycu2 yuE1 yco1 ya2 yco2 ycoyci2 ycycnycu2 yuE2 yco2 ya2 .
ya1 = Oeyco1 yaycoyci1 ycaycuyc yuE1 .
ya1 = Oeyco2 yaycoyci2 ycaycuyc yuE2 .
ya2 = 0.
2 N N
ycN1 = yco1 ycoyci1 yuE1 yuo ycycnycu( 2yuE1 ) yco1 ycoyci1 yayuEN 2 1 ycycnycu yuE1 yca0 yuoN yayc yayca yco2 ycoyci2 yuE2 yuoN ycycnycu( 2yuE2 ) .
ca0 )yuoN.
ycIyco ya2 = Oeyco1 yaycoyci1 ycaycuyc yuE1 .
ya2 = ya1 yco1 ycoyci1 2 .
ya2 = 0.
Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 ycN2 = Oeyco1 ycoyci1 2 yuoN 2 ycycnycu yuE1 ycaycuyc yuE1 Oe yco1 yciycoyci1 ycycnycu yuE1 yca1 yuEN1 .
ya3 = Oeyco2 yaycoyci2 ycaycuyc yuE1 .
ya3 = 0.
ya3 = ya2 yco2 ycoyci2 2 .
ycN3 = Oeyco2 ycoyci2 2 yuoN 2 ycycnycu yuE2 ycaycuyc yuE2 Oe yco2 yciycoyci2 ycycnycu yuE2 yca2 yuEN2 .
Control Algorithm yuo OO yuoN OO yuE1 OO yuEN1 OO yuE2 OO yuEN2 OO 0.
ycOycnycu OO 0.
ycu = .
uo yuoN yuE1 yuEN1 yuE2 yuEN2 ]ycN .
The linearized state-space equation of the RDPIP is given Linear Quadratic Regulator (LQR) The Linear Quadratic Regulator (LQR) is utilized to stabilize the RDPIP system based on its linearized state-space model Fig.
The controller computes an optimal statefeedback gain matrix that minimizes a quadratic cost function involving both state deviations and control effort.
The resulting control law yc = Oeyaycu enables the system to maintain stability around the equilibrium point while reducing oscillations.
Appropriate selection of weighting matrices Q and R ensures a balance between performance and energy consumption.
This approach is well-suited for RDPIP due to the systemAos high sensitivity and strong coupling .
ycuN = yaycu yaAyc Where:
ce1 ) yuiyuo yui.
ce2 ) yuiyuo yui.
ce3 ) yuiyuo ya= yui.
ce4 ) yuiyuo yui.
ce5 ) yuiyuo yui.
ce6 ) [ yuiyuo yui.
ce1 ) yuiyuoN yui.
ce2 ) yuiyuoN yui.
ce3 ) yuiyuoN yui.
ce4 ) yuiyuoN yui.
ce5 ) yuiyuoN yui.
ce6 ) yuiyuoN yui.
ce1 ) yuiyuE1 yui.
ce2 ) yuiyuE1 yui.
ce3 ) yuiyuE1 yui.
ce4 ) yuiyuE1 yui.
ce5 ) yuiyuE1 yui.
ce6 ) yuiyuE1 yui.
ce1 ) yuiyuEN1 yui.
ce2 ) yuiyuEN1 yui.
ce3 ) yuiyuEN1 yui.
ce4 ) yuiyuEN1 yui.
ce5 ) yuiyuEN1 yui.
ce6 ) yuiyuEN1 yui.
ce1 ) yuiyuE2 yui.
ce2 ) yuiyuE2 yui.
ce3 ) yuiyuE2 yui.
ce4 ) yuiyuE2 yui.
ce5 ) yuiyuE2 yui.
ce6 ) yuiyuE2 yuiyce2 ycOycnycu yuiyce3 ycOycnycu yuiyce4 ycOycnycu yuiyce5 ycOycnycu yuiyce1 yaA=[ ycOycnycu Fig.
Block diagram of the principle of the system controlled by the LQR controller The state variables of the system are defined as follows:
Equation .
is converted into the following system of ycu3 = yuoO = Ea1 .
uo, yuoN , yuE1 , yuEN1 , yuE2 , yuEN2 ) .
cu6 = yuEO1 = Ea2 .
uo, yuoN, yuE1 , yuEN1 , yuE2 , yuEN2 ) ycu9 = yuEO2 = Ea3 .
uo, yuoN, yuE1 , yuEN1 , yuE2 , yuEN2 ) yuiyce6 ycN ycOycnycu Where:
yce1 = ycuN 1 = ycu2 .
yce2 = ycuN 2 = ycu3 .
yce3 = ycuN 4 = ycu5 yce4 = ycuN 5 = ycu6 .
yce5 = ycuN 7 = ycu8 .
yce6 = ycuN 8 = ycu9 Weighting matrix Q:
ycu1 = yuoycu4 = yuE1 ycu7 = yuE2 ycu2 = yuoNycu5 = yuEN1 ycu8 = yuEN2 ycu3 = yuoOycu6 = yuEO1 ycu9 = yuEO2 yui.
ce1 ) yuiyuEN2 yui.
ce2 ) yuiyuEN2 yui.
ce3 ) yuiyuEN2 yui.
ce4 ) yuiyuEN2 yui.
ce5 ) yuiyuEN2 yui.
ce6 ) yuiyuEN2 ] .
From Equation .
, combined with the AusolveAy function in MATLAB referred in Fig.
3, the result for Equation .
is Fig.
Illustrates the computation performed using MATLAB The LQR controller is linear, whereas the RDPIP system is inherently nonlinear.
Therefore, the system must be linearized around an operating point, specifically the TOPTOP equilibrium point.
ycE1 ycE= ycE2 ycE3 ycE4 ycE5 ycE6 ] .
Energy coefficient R (R > .
Solving the Riccati equation, the solution is matrix P:
ycEya yaycN ycE ycE Oe ycEyaAycI Oe1 yaA ycN ycE = 0 Control matrix K:
ya = ycI Oe1 yaAycN ycE Caculate matrix K in Matlab:
ya = ycoycyc.
a, yaA, ycE, ycI) .
Genetic Algorithms (GA) for LQR To enhance the performance of the LQR controller.
GA is applied to optimize the weighting matrices Q and R in the energy coefficient.
This approach enables automatic tuning of control parameters without relying on manual trial-anderror procedures.
The GA iteratively evolves a population of candidate solutions by employing selection, crossover, and mutation operations, with the fitness function defined based on the system's tracking accuracy and control effort.
optimizing the LQR gains in this manner, the controller achieves improved stabilization and reduced oscillation for the nonlinear and highly coupled RDPIP system.
This method demonstrates robustness and adaptability, particularly suitable for systems with complex dynamics such as RDPIP.
Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 In this study.
GA is implemented in an offline manner.
The GA configuration includes a population size of N=100, with selection based on linear ranking using a selection pressure coefficient =0.
Decimal coding is adopted for chromosome representation, and two-point crossover is used to promote genetic diversity.
The crossover and mutation probabilities are set to 0.
9 and 0.
1, respectively, ensuring a balance between exploration and exploitation during the optimization process.
Choose fitness function:
ycu ya = Oc.
ce1 2 .
yce2 2 .
yce3 2 .
) .
ycn=0 With yce1 = yuo.
yce2 = yuE1 .
yce3 = yuE2 , n as the number of samples during the simulation, value of function J depends on the errors of the arm angle and the two pendulum angles.
this case, with a simulation time of 100 seconds and a sampling interval of 0.
01 seconds, we have n = 10001 As illustrated in Fig.
GA is employed to optimize the weighting matrices Q and R of the LQR controller.
The process begins with the initialization of a population of candidate solutions, where each individual represents a pair of Q-R matrices encoded as chromosomes.
For each individual in population, fitness function J is evaluated based on the cumulative tracking error of the RDPIP, considering deviations of both pendulums and rotating arm from their reference trajectories.
If the current fitness J is lower than the best-known fitness value J min, the algorithm updates J min and stores the corresponding individual as the current best solution.
, two-poin.
, and mutation, to generate new individuals for the next generation.
This process continues iteratively, refining the solution space until a predefined number of generations or convergence criterion is met.
Once the termination condition is satisfied, the individual with the lowest fitness value J min is selected as the optimal This individual defines the final optimized values of Q and R matrices, which are then used in the LQR controller to improve system performance in terms of settling time, stability, and trajectory tracking.
Simulation Program The selection of control parameters plays a critical role in determining the overall performance and stability of the control system.
In practical applications, several metaheuristic optimization techniques have been employed to identify optimal parameters, with PSO .
and GA .
being among the most commonly used.
In this study, the GA is employed to optimize the controller parameters for all three control strategies under consideration.
Specifically, for LQR controller, optimal feedback gain matrix K must be Since K is computed based on weighting matrices Q and R, which directly influence the trade-off between system performance and control effort.
GA is utilized to identify the most suitable values for Q and R that minimize cost function and enhance overall system behavior.
Optimization process is performed offline with fixed simulation duration and sampling time, allowing the evaluation of control performance through a well-defined fitness function.
In addition to the main fitness function J, metrics such as RMSE and settling time are used to assess the quality of control performance.
These indicators help quantify oscillation, response speed, and steady-state This approach not only improves response quality but also ensures robustness against modeling uncertainties and nonlinearities inherent in RDPIP.
Matrices A and B are calculated based on the parameters presented in Table 3 and Table 4 as follows:
0 Oe13.
ya= 0 Oe45.
Oe33.
yaA = .
Table 3.
Parameters yayca yayc ycIyco Table 4.
Fig.
Flowchart of GA used to optimize LQR weighting matrices Q and The fitness function minimizes trajectory error across the arm and two pendulums .
The population then undergoes a series of genetic operations, including selection .
ypically via linear rankin.
Parameters ycoycn ycoyciycn yaycn ya ya0 yci yaycn ya0 Oe0.
Oe0.
Oe0.
ycN Oe0.
Oe0.
Oe0.
Parameters of DC Motor Unit ycO/.
cycaycc/ ycyceyc.
ycO/.
cycaycc/ ycyceyc.
yu Value Parameters of RDPIP Pendulum 1 Pendulum 1 Arm The controllability of the system is analyzed as follows:
Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 yayc = .
aA yayaA ya2 yaA ya3 yaA ya4 yaA ya5 yaA] .
Substituting matrices.
A and B into .
, we obtain:
ayc ) = 6.
yccyceyc( yayc ) = Oe0.
Since the rank of the matrix equals the order of the system .
th orde.
and the determinant is non-zero, the system is Table 5.
Initial state variables Parameters x1_init x2_init x4_init x5_init x7_init x8_init Value The MATLAB/Simulink R2022a.
The sampling interval was set 01 seconds, with a total simulation time of 10 seconds.
The solver used was the variable-step ODE45, and all simulations were executed under identical conditions to ensure consistency in controller performance evaluation.
Fig.
Fig.
Fig.
7 represent the simulation of the control algorithms using Matlab/Simulink software version 2022a.
the Rotary Inverted Pendulum block, the equations are those outlined in section A of II.
Similarly, the blocks for LQR Controller contain the equations presented in section B of II.
The output response of the system for each algorithm is observed through the Scope block.
All results are presented in section i.
A trial-and-error approach is employed to determine suitable values for the weighting matrices Q and R.
ycE= 0
[ 0
ycI = 0.
The stability of the LQR controller is ensured by solving the Riccati equation at the steady-state operating condition.
In this case, the LQR 1 controller stabilizes the RDPIP system when operating around the selected equilibrium point presented in Table 5.
ya = 106 .
139 Oe1.
3632 Oe0.
Fig.
Simulation for LQR controller for RDPIP Fig.
Simulation using GA to optimize LQR controller for RDPIP
RESULTS AND DISCUSSION
Simulation Results of the LQR 1 Controller The simulation results of the LQR 1 controller using the Q and R matrices selected by the trial-and-error method, as described in .
, are presented in Fig.
Fig.
Fig.
In this figure, the arm initially oscillates to regulate the two pendulums around the designated operating point.
Pendulum 1 reaches its equilibrium state after approximately 2 seconds, while pendulum 2 stabilizes after around 3 This represents a significant achievement in controlling both pendulums with fast settling times.
Once the pendulums are stabilized, the arm converges to 0 rad, with a settling time of approximately 6 seconds.
Fig.
LQR controller test simulation tracking set signal Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 system response.
Specifically, the integral of absolute error (IAE) is reduced from 0.
189227 to 0.
027819, and the rootmean-square error (RMSE) decreases from 0.
034370 to 010544 radians.
In addition, the oscillation amplitude of the arm is attenuated, narrowing from a range of Oe0.
09 to 0.
(LQ .
to Oe0.
058 to 0.
039 (LQ .
The settling time is shortened by approximately 80%, from 5.
1679 seconds to 0444 seconds, and the rise time is reduced from 0.
0291 to 0128 seconds.
These findings demonstrate the superior performance of the GA-tuned controller in terms of speed, accuracy, and stability.
Fig.
Angular displacement of the arm Table 6.
Angular displacement of the arm
IAE
RMSE
Oscillation amplitude Settling time Rise time Fig.
Angular displacement of the pendulum 1 Simulation Results of LQR 2 Controller After Using GA With LQR 1 controller, controller parameters have not been fully optimized.
The result of running the GA program converged after approximately 10 generations.
The value of the cost function J is illustrated in Fig.
11, and the optimized LQR 2 controller parameters obtained are as follows:
Oe4.
Oe0.
LQR 1
LQR 2
Fig.
Angular displacement of the arm Fig.
Angular displacement of the pendulum 2 ya = 105 .
ya = 0.
Performance metrics of the rotary arm under LQR 1 and LQR 2 Table 7 and Fig.
13 present the performance metrics for the angular displacement of pendulum 1 under both LQR 1 and LQR 2 control strategies.
In contrast to the improvement observed in the rotary arm, the GA-optimized controller (LQR .
demonstrates a mixed effect on pendulum 1.
Although the settling time is reduced from 1.
5184 seconds to 2225 seconds and the rise time remains nearly unchanged, the error metrics - namely IAE and RMSE - increase slightly under LQR 2, from 0.
024246 to 0.
033698 and from 0.
013985, respectively.
Moreover, the oscillation amplitude expands from [Oe0.
017, 0.
(LQ .
to [Oe0.
(LQR .
, indicating stronger overshoot or more aggressive control action.
Table 7.
Performance metrics of angular displacement of the pendulum 1
Angular displacement of the arm
IAE
RMSE
Oscillation amplitude Settling time
Rise time
LQR 1
LQR 2
Fig.
The change of J over generations A quantitative comparison of control performance between the conventional LQR (LQR .
and the GAoptimized LQR (LQR .
is presented in Table 6 and Fig.
The results clearly indicate that LQR 2 significantly improves Fig.
Angular displacement of the pendulum 1 Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 Table 8 and Fig.
14 present the performance metrics for the angular displacement of pendulum 2 under both LQR The GA-optimized LQR (LQR .
shows a clear improvement in most performance indicators compared to the manually tuned version (LQR .
Notably, the integral of absolute error (IAE) is significantly reduced from 022783 to 0.
007281, and the RMSE drops from 0.
011777 radians.
These reductions reflect a more accurate tracking performance of LQR 2.
Additionally, the rise time improves substantially, decreasing from 0.
0393 seconds (LQR .
0077 seconds (LQR .
, which indicates a much faster initial response.
Table 8.
Performance metrics of angular displacement of the pendulum 2
Angular displacement of the arm
IAE
RMSE
Oscillation amplitude Settling time
Rise time
LQR 1
LQR 2
and negligible steady-state error.
The pendulums remain stable around the upright position during the entire trajectory, as seen in Fig.
16 and Fig.
17, where their angular displacements stay within A0.
01 rad with quick damping.
Fig.
Angular displacement of the arm and set sine signal Fig.
Angular displacement of the pendulum 1 Fig.
Angular displacement of the pendulum 2 The results in Table 6.
Table 7.
Table 8 confirm that the GA-optimized LQR controller significantly improves tracking accuracy and response speed for the RDPIP system.
Notable reductions in IAE.
RMSE, and settling time were achieved for both the rotary arm and pendulum 2, while pendulum 1 showed improved dynamic response with minor trade-offs in error metrics.
These outcomes demonstrate the effectiveness of GA in tuning LQR parameters, offering a robust and adaptive control strategy for nonlinear SIMO The proposed method provides a practical and scalable solution for advanced pendulum control LQR Setpoint Tracking Control To evaluate whether LQR controller 2 is capable of maintaining the balance of both pendulums at designated operating point, a reference tracking scenario is implemented by varying the desired arm angle as illustrated in Fig.
this study, the reference signal is defined as a sine wave with an amplitude of A/2 and a frequency of 0.
005A, time stop is 1000 seconds representing a smooth and continuous disturbance to test the robustness of controller.
To evaluate the tracking capability of the proposed GAoptimized LQR controller, the RDPIP system was subjected to reference trajectories with different profiles: a continuous sinusoidal signal and a discrete pulse signal.
Fig.
Fig.
Fig.
17 illustrate the system response under the sinusoidal reference, where the rotary arm tracks a sine wave of amplitude A A/2 rad.
As shown in Fig.
15, the arm output closely follows the reference curve with minimal phase delay Fig.
Angular displacement of the pendulum 2 The reference signal is a square wave with an amplitude 1 rad, a period of 50 seconds, and a duty cycle of 50%, which is passed through a filter to smooth out the sharp Under the pulse signal reference shown in Fig.
Fig.
Fig.
20, the controller again demonstrates robust In Fig.
18, the arm successfully tracks the square wave reference with sharp transitions and minimal Despite the abrupt changes in desired angle, the system maintains good precision, and the response remains consistent across cycles.
Fig.
19 and Fig.
20 reveal that both pendulums exhibit small, bounded oscillations with magnitudes below A0.
01 rad.
The low amplitude and consistent damping after each transition highlight the controllerAos ability to maintain pendulum stability during dynamic reference shifts.
Fig.
Angular displacement of the arm and set pulse signal The results shown in Fig.
Fig.
Fig.
17 and Fig.
Fig.
Fig.
20 demonstrate that the LQR-GA controller not Thanh-Truong Mai.
Trajectory Control for Double-Linked Parallel Rotary Inverted Pendulum Journal of Fuzzy Systems and Control.
Vol.
No 2, 2025 only ensures stabilization at a fixed point but also provides reliable tracking for time-varying references.
Its robustness against abrupt disturbances and continuous variation validates the method's suitability for real-time control of nonlinear and under-actuated systems.
Fig.
Angular displacement of the pendulum 1 Fig.
Angular displacement of the pendulum 1 IV.
CONCLUSION
This paper has presented a hybrid LQR-GA control strategy for the stabilization and trajectory tracking of RDPIP.
The system was linearized around the TOP-TOP equilibrium, and a state-space model was used for LQR controller design.
To enhance performance.
GA was applied to optimize the weighting matrices Q and R.
The novelty of this work lies in integrating GA-based optimization into LQR tuning for a nonlinear SIMO system, which has been rarely addressed in the context of RDPIP control.
Simulation results confirmed that the optimized controller significantly improved system behavior.
For the rotary arm, the settling time was reduced from 5.
17 s (LQ .
04 s (LQ .
, and MSE decreased from 0.
0344 to 0.
0105 rad.
Similar improvements were observed for the pendulums, particularly in steady-state error and response time.
The LQR-GA controller also demonstrated robust tracking of both sinusoidal and pulse reference signals, maintaining pendulum stability under time-varying conditions.
These findings demonstrate the effectiveness of GA-based tuning for complex nonlinear SIMO systems and provide a solid foundation for future experimental implementation and extension to higher-order inverted pendulum models.
ACKNOWLEDGEMENT
We want to give thanks to PhD.
Van-Dong-Hai Nguyen .
ecturer of HCMUTE) due to his supervision for us to complete this research.
REFERENCES