Control of two - wheeled inverted pendulum robot using robust pi and lqr controllers

 𝑏62], (24) 
with 𝑎42 =
−(𝑀𝑔𝑙)2𝑔
𝛬
, 𝑎44 =
−2𝐶((𝑀𝑙2+𝐼2)+𝑀𝑙𝑟)
𝛬
, 𝑎45 =
2𝐶𝑟(𝑀𝑙2+𝐼2)+2𝐶𝑀𝑙
2
𝛬
, 𝑎52 =
𝑀𝑔𝑙(2𝐽+𝑀𝑟2+2𝑚𝑟2)
𝛬
, 𝑎54 =
2𝐶𝑀𝑙+2𝐶(2𝐽+𝑀𝑟2+2𝑚𝑟2)/𝑟
𝛬
, 𝑎55 =
2𝐶(2𝐽+𝑀𝑟2+2𝑚𝑟3)+2𝐶𝑀𝑙𝑟
−𝛬
, 
𝑎66 =
−𝐶𝑑2
2𝐽𝑑2+(2𝐼3+4𝐾+𝑚𝑑2)𝑟2
, 𝑏41 = 𝑏42 =
𝐾𝑚𝑟[(𝑀𝑙
2+𝐼2)+𝑀𝑙𝑟]
𝛬
, 𝑏51 = 𝑏52 =
−𝐾𝑚(2𝐽+𝑀𝑟
2+2𝑚𝑟2+𝑀𝑙𝑟)
𝛬
, 𝑏61 = −𝑏62 =
−𝐾𝑚𝑑𝑟
𝐽𝑑2+(2𝐼3+4𝐾+𝑑2𝑚)𝑟2
, and 𝛬 = 2𝐼2𝐽 +
2𝐽𝑀𝑙2 + 𝐼2𝑀𝑟
2 + 2𝐼2𝑚𝑟
2 + 2𝑀𝑙2𝑚𝑟2. 
4.2. LQR Controller 
From the linearized model (22), a LQR controller [26] is designed such that the 
following cost function is minimized. 
𝐹 = ∫ (𝑥𝑇𝑄𝑐𝑥 + 𝑢
𝑇𝑅𝑐𝑢)𝑑𝑡
∞
0
→ 𝑚𝑖𝑛, (25) 
in which 𝑄𝑐 is a symmetric positive definite matrix, and 𝑅𝑐 is a symmetric 
nonnegative definite matrix. The controller, which satisfies the cost function (25), 
is 𝑢 = −𝐾𝐿𝑄𝑅𝑥, where 
𝐾𝐿𝑄𝑅 = 𝑅𝑐
−1𝐵𝑇𝑃𝑐, (26) 
and 𝑃𝑐 is the solution to the Riccati equation 
𝑃𝑐𝐵𝑅𝑐
−1𝐵𝑇𝑃𝑐 − 𝑃𝑐𝐴 − 𝐴
𝑇𝑃𝑐 = 𝑄𝑐. (27) 
In this paper, the values of the TWIPR’s parameters are determined from a real 
TWIPR as follows M = 0.5kg, m = 0.04kg, l = 0.08m, d = 0.16m, r = 0.033m, g = 
9.81m/s2, C = 0.005, Km = 0.41202Nm/A. Thus, the matrix B is 
 𝐵 = [0 0; 0 0; 0 0; 55.5 55.5; −548.6 − 548.6; −138.9 138.9 ], (28) 
and the matrix A has the form as in Eq. (29) with 𝑎42 = −11.4, 𝑎44 = −40.8, 
𝑎45 = 1.35, 𝑎52 = 176.8, 𝑎54 = 403.4, 𝑎55 = −13.31, 𝑎66 = −8.1. 
𝐴 =
[
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 𝑎42 0 𝑎44 𝑎45 0
0 𝑎52 0 𝑎54 𝑎55 0
0 0 0 0 0 𝑎66 ]
 (29) 
Electronics & Automation 
T. G. Khanh, , N. D. Phuoc, “Control of two-wheeled  robust PI and LQR controllers.” 10 
The matrices of the cost function are chosen as 
𝑄𝑐 = 𝑑𝑖𝑎𝑔([4, 3, 2, 3.5, 0.001, 0.9 ]) (30) 
and 
𝑅𝑐 = [1 0; 0 1 ]. (31) 
From Eq. (26) and Eq. (27), the gain matrix of the LQR controller is obtained as 
 𝐾𝐿𝑄𝑅 = [
−1.41 −3.09 −3.16 −2.14 −0.33 −0.65
−1.41 −3.09 3.16 −2.14 −0.33 0.65
] . (32) 
The output of the LQR controller will be setpoint for the current controllers. 
Remark 3. Since the input to the LQR controller consists of state variable pairs 
(x;�̇�), (θ;�̇�) and (ψ;�̇�), the LQR output is similar to the sum of the three PD 
controllers’ output as in works [4, 7, 8], in which each state variable pair is the 
input to one PD controller. 
Remark 4. The LQR controller in [12] is also compared to the state dependent 
LQR controller when the high yaw rate is considered. 
Figure 10. The tilt angle of the TWIPR 
through simulation. 
Figure 11. The heading angle of the 
TWIPR through simulation. 
5. SIMULATION AND PRACTICAL TEST 
Figure 12. The position of the TWIPR 
through simulation. 
Figure 13. The tilt angle of the TWIPR 
through practical test. 
In this section, the proposed control scheme based on the robust PI design 
method and LQR controller is verified for the TWIPR through simulation and 
experiments. Then, it is compared to the other methods. The TWIPR is controlled 
Research 
Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 11 
to move from the initial point to the end point along a line, where the distance 
between these two points is 1 meter. 
5.1. The proposed control strategy 
The simulation results are shown in fig. 10, 11 and 12 whereas the experimental 
results are shown in fig. 13, 14 and 15. The left motor’s measured current and right 
motor’s measured current are shown in fig. 16 and 17. For simulation, the origin 
model as Eq. (6) and its parameters given in section 4.2 are applied for the TWIPR, 
in which the current control loops are modeled as saturation blocks since their 
settling time is very short 1 (ms). 
Figure 14. The real heading angle of 
the TWIPR. 
Figure 15. The real position of 
the TWIPR. 
For experimental test, the sampling time for PID and LQR controllers is chosen 
as 0.25 (ms) and 2.5 (ms), respectively. From fig. 10 to fig. 15, it can be implies 
that the real-time results are similar to the simulation results. This implies that a 
good model of the TWIPR was built and the controllers work well. The proposed 
control strategy produces good performance such as the pitch angle lies within the 
interval [−3 3] (degrees), the TWIPR reaches the destination at 1 (m) after 3.5 
(seconds) following a straight line because the yaw angle is approximately within 
the range [−0.3 0.3] (degrees). 
5.2. Comparisons 
The proposed control scheme based on the robust PI-LQR method is compared 
to the MO-LQR and GA-LQR methods through practical tests. The desired 
trajectory is a straight line with a length of 1 meter. Fig. 18 shows the positions of 
the TWIPR with different methods, where the blue curve, the black curve and the 
red curve are positions of the TWIPR provided by the proposed method, the MO-
LQR method and the GA-LQR method, respectively. 
The comparison of the pitch and yaw angles of the TWIPR with different 
methods are shown in fig. 19 and fig. 20, respectively. Since the yaw angles are 
very small for all methods, the TWIPR moves nearly along the straight line with 
distance of 1 meter. The robust PI-LQR based TWIPR moves more smoothly from 
the start point 0 to the end point with smaller pitch angle and less oscillation than 
the other methods. Hence, the proposed controller provides better performance 
than the others. 
Electronics & Automation 
T. G. Khanh, , N. D. Phuoc, “Control of two-wheeled  robust PI and LQR controllers.” 12 
Figure 16. Left motor’s real current. 
Figure 17. Right motor’s real current. 
Figure 18. The position of the TWIPR with different methods. 
Figure 19. The pitch angle of the 
TWIPR with different methods. 
Figure 20. The yaw angle of the TWIPR 
with different methods. 
Remark 5: The proposed control scheme is also tested with different loads and 
inclined surfaces. The practical results prove that the TWIPR is kept balanced and 
movable. However, they are skipped to show here. 
6. CONCLUSIONS 
In this work, the TWIPR control system based on the robust PI controller design 
method is proposed. First, the proposed method is compared to the MO and GA 
Research 
Journal of Military Science and Technology, Special Issue, No.66A, 5 - 2020 13 
methods through the DC motor’s current. It produces better performance than the 
others. Then, the proposed control system is verified and compared to the other 
methods through simulations and practical tests. The obtained results show that the 
TWIPR with the proposed method is kept balanced and able to reach the desired 
position and direction, and it produces better performance than the others. Future 
works will focus on the combination of the proposed method with other advanced 
control methods for the TWIPR. 
ACKNOWLEDGEMENT 
This research is funded by the Hanoi University of Science and Technology 
(HUST) under project number T2018-PC-052. 
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TÓM TẮT 
ĐIỀU KHIỂN XE HAI BÁNH CÂN BẰNG 
SỬ DỤNG BỘ ĐIỀU KHIỂN PI BỀN VỮNG VÀ LQR 
Trong bài báo này, một bộ điều khiển PI bền vững kết hợp với bộ điều 
khiển LQR được đề xuất được đề xuất để điều khiển xe hai bánh sao cho xe 
thằng bằng khi di chuyển. Hệ thống điều khiển gồm hai vòng. Vòng trong có 
hai bộ điều khiển dòng PI để điều khiển dòng động cơ một chiều, được thiết kế 
riêng sử dụng cấu trúc PI bền vững. Vòng ngoài có bộ điều khiển LQR cho 
góc nghiêng, góc hướng và vị trí xe. Phương pháp thiết kế bộ điều khiển PI đề 
xuất được so sánh với phương pháp tối ưu độ lớn và giải thuật di truyền. Cấu 
trúc điều khiển đề xuất được kiểm chứng thông qua mô phỏng và thực nghiệm, 
và nó được so sánh với các phương pháp MO-LQR và GC-LQR. 
Từ khóa: Chỉnh định PID; Xe hai bánh; Điều khiển động cơ; Thời gian xác lập. 
Received date, 02nd March 2020 
Revised date, 14th April 2020 
Published 06th May, 2020 
Author affiliations: 
1Hanoi University of Science and Technology, No. 1, Dai Co Viet Street, Hanoi, Vietnam; 
2Viện Hàng không Vũ trụ Viettel; 
3Công ty Ô tô Toyota Việt Nam. 
*Corresponding author: nam.nguyenhoai@hust.edu.vn.