áp dụng LQR nghiên cứu xe hai bánh

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Regulation of the two-wheeled inverted pendulum robot based on the settling time and overshoot assignment method and LQR controller Nam H Nguyen1,∗, Xuan D Tran1 , Long T Pham1 , Trung D Trinh1 , Khanh G Tran1 , Phuoc D Nguyen1 No 1, Dai Co Viet Street, Hanoi, Vietnam Abstract In this paper, the novel overshoot and settling time assignment (OSA) technique in combination with a linear quadratic regulator (LQR) is proposed to control a two-wheeled inverted pendulum robot (TWIPR) such that it is kept balanced while following a path The proposed TWIPR control system consists of two control loops The inner loop has two PI controllers for two DC motors’ currents, which are separately designed based on the OSA method The outer loop contains a LQR controller for the tilt angle, heading angle and position of the TWIPR The OSA method is compared to the existing methods such as the magnitude optimum (MO) and genetic algorithm (GA) methods The proposed control scheme is verified through simulations and practical tests with a TWIPR, and it is also compared to the MO-LQR and GA-LQR strategies The results show that the control strategy based on the OSA - LQR provides better performance in terms of the TWIPR’s tilt angle and position Keywords: PID tuning, dc motor control, settling time, overshoot, two-wheeled inverted pendulum Introduction 10 15 20 25 TWIPRs are widely studied and applied in practice and literature [1, 2, 3] The input to the TWIPR is usually torques produced by two DC motors These torques are 30 directly proportional to currents caused by the DC motors Thus, the current controllers are crucial to the TWIPR However, most of the works [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] about TWIPR have not addressed the problem of current or torque control In addition, the 35 voltages, applied to the DC motors, are considered as the input to the TWIPR in stead of torques In [4], an adaptive backstepping controller combined with two PD controllers is proposed for an electric scooter The input to the vehicle are torques produced by two DC 40 motors, but there is neither torque nor current controller designed The proposed control scheme is tested through both simulation and experiment Sliding mode controllers and model based friction compensation are proposed in [5] for a TWIPR This approach is also verified via simula- 45 tion and real test However, the torque/current control problem is also not dealt with In [6], a nonlinear state transformation based controller is proposed Then, the regulator is tested through simulation A combination of two PD controllers and a time-delayed controller [7] for 50 fast movement of TWMR is proposed, in which the first ∗ Corresponding author Email address: nam.nguyenhoai@hust.edu.vn (Nam H Nguyen) Department of Automatic Control, School of Electrical Engineering, Hanoi University of Science and Technology Preprint submitted to Elsevier 55 PD controller is designed for the pitch angle, the other PD controller is applied for the orientation, and the timedelayed controller is synthesised for the position Practical test is carried out to prove the proposed method In [8], the proposed control scheme consists of local controller and a global planner for the TWIPR based personal transportation vehicle, in which the local controller contains three PID controllers for the pitch angle, the yaw angle and the position The vehicle is tested with different passengers Three fuzzy logic inference system based controllers are designed for the TWIPR in [9] The control system is verified through simulation and experimental test In [10], an indirect adaptive fuzzy controller based on trajectory planner for the TWIPR is given The control strategy is verified through simulation In [11], adaptive sliding mode control in combination with direct fuzzy control is applied for balancing and trajectory tracking of the TWIPR A statedependent nonlinear model based LQR is investigated in [12] It is compared to the classical LQR controller via simulation and real test, and its performance is better in term of high yaw rate In [13], a backstepping controller is designed for trajectory tracking and sliding mode controllers are applied for the TWIPR’s velocities and pitch angle Four interval type fuzzy logic inference system based controllers [14] are designed with usage of TakagiSugeno model for the TWIPR such that the vehicle is able to reach the desired point and orientation while its balancing is kept The proposed control strategy is verified through experiments In [15], a passivity based controller with usage of two rule Takagi-Sugeno fuzzy model is proposed for the TWIPR This control scheme is tested via February 27, 2019 60 65 70 ☎ ✖ ✘✓ ✗✘✔ simulation A novel TWIPR with four-bar parallel mechanism is developed in [16], in which the horizontal posture is guaranteed Simulation and practical test are carried out In [17], a model predictive controller for the trajectory tracking of TWIPR is proposed, and verified through numerical simulations A trajectory planning and tracking control for obstacle avoidance of the TWIPR is given in [18] A control of cooperative double-wheel inverted pendulum robots based on Udwadia-control approach is recently proposed in [19] for the first time In this work, the novel overshoot and settling time assigment method [20] is applied to design PI controllers for the TWIPR for the first time, and it is compared with the other methods Then, these controllers are combined with an LQR controller to keep the TWIPR balanced and able to move straight forwad, make an U-turn and even take a load, since the LQR controller is the simplest control strategy among the other advanced control methods ☎ ✖ ✕✓ ✗✕✔ ☎✆ ✝ ✁ 85 ✄ ☎✎ ✝ ☛ ☎✟ ✝ ☞✡ ☎✌ ✞ ☎✌ ✞ ✒ ✄ ☎✌ ✝ ✑ [I3 + 2K + md2 /2 + Jd2 /(2r2 ) (I3 I1 M l2 )sin2 ()]ă + [M lx˙ − 2(I3 − I1 − M l2 )]ψsin(θ) ˙ /(2r2 ) = (iR − iL )Km d/(2r) + C ψd (3) Let define state variables and inputs as x = [x1 = [x 2.1 Mathematical Model 95 ✂ ✡ ✠ Figure 1: Schematic diagram of a TWIPR The next section presents the mathematical model and the proposed control system for the TWIPR In section 3, the OSA based PI controllers are designed for two DC motors’ currents of the TWIPR and they are compared to the MO and GA methods through simulations and experiments In section 4, a LQR controller is designed to maintain that the TWIPR is vertically stable and it able to reach desired position and orientation Some simulations and practical tests for the TWIPR are given in section The final section provides a summary and future works Two-Wheeled Inverted Pendulum Robot 90 ☎✎ ✞ ✍✎✏   75 80 ☎✆ ✞ ☎ ✖ ✙✓ ✗✙✔ x2 θ x4 x5 x6 ]T ˙T x˙ θ˙ ψ] x3 ψ (4) and In this work, the mathematical model of TWIPR in [21] is used for simulation and controller design The schematic diagram of a TWIPR is shown in Fig The notations and parameters of the TWIPR are shown in Tab 1,where J = mr2 /2, K = mr2 /4, I1 = I3 = M d2 /2, I2 = M l2 /2 The inputs of the model [21] are torques produced by motors, but in this work, the two motors’ currents are used instead of torques as the inputs to the TWIPR Thus, the motion equations of the TWIPR are represented as in Eq 1, and u = [u1 u2 ]T = [iL iR ]T (5) respectively Then, x˙ = f (x, u) (6) in which x˙ = x4 , x˙ = x5 , x˙ = x6 , x˙ = f4 (x, u), x˙ = f5 (x, u) and x˙ = f6 (x, u) where f4 (x, u) = Ω1 +Ω , Ω Ω3 +Ω4 Ω5 f5 (x, u) = Ω , f4 (x, u) = Ω6 with Ω1 =r2 (M l2 + I2 ) Km (u1 + u2 )/r + 2C(x5 − x4 /r)/r + M lsin(x2 )(x25 + x26 ) , 2J )ă x M l( + )sin() + M lcos()ă r2 (1) x˙ Km ˙ + ( − θ) = (iL + iR ) C r r (M + 2m + Ω2 =M lr2 cos(x2 ) − cos(x2 )sin(x2 )(M l2 + I1 − I3 )x26 + Km (u1 + u2 ) + 2C(x5 − x4 /r) − M glsin(x2 ) , M lcos()ă x (M l2 + I2 )ă + (I3 I1 M l2 ) sin(θ)cos(θ) x˙ ˙ = −Km (iL + iR ) − M lgsin(θ) − 2C( − θ) r (2) Ω3 = 2J + (M + 2m)r2 cos(x2 )sin(x2 )(M l2 + I1 − I3 )x26 − Km (u1 + u2 ) − 2C(x5 − x4 /r) + M glsin(x2 ) , Ω4 = − M lr2 cos(x2 ) Km (u1 + u2 )/r + 2C(x5 − x4 /r)/r + M lsin(x2 )(x25 + x26 ), Table 1: TWIP’s notations and parameters Notation x θ ψ d l r M m J K Km iL , iR τL , τR C I1 , I1 , I1 Definition Position of the TWIPR Pitch angle of the pendulum body Yaw angle of the TWIPR Distance from the left to right wheels Length from the center of the pendulum to the wheel axis Radius of wheels Mass of the pendulum (without wheels) Mass of left (right) wheel Mass moment of inertia (MOI) of the wheel with respect to (w r t.) the wheel axis MOI of the wheel w r t the vertical axis the motor torque constant the left, right armature current motor’s left (right) torque coefficient of viscous friction on wheel axis MOI of the pendulumn w r t the frame at the center of the pendulumn Figure 2: Block diagram of the TWIPR control system Ω5 = − 2r2 Km d(u1 − u2 )/(2r) + x6 sin(x2 ) M lx4 + 2x5 cos(x2 )(M l2 + I1 − I3 ) + Cd2 x6 /(2r2 ) , Figure 3: Front view of a TWIPR under control in laboratory Ω6 = 2I3 + 4K + md2 + 2(I1 − I3 + M l2 )sin2 (x2 ) r2 + Jd2 , and Ω =(M lr)2 (1 − cos2 (x2 )) + 2I2 J + 2JM l2 + (I2 M + 2I2 m + 2M ml2 )r2 100 The motion equations 1, and are used to design the LQR controller in Section In the next section, PID controllers are designed to stabilize two motors’ currents, which create desired inputs to the TWIPR 2.2 TWIPR Control System 105 110 The block diagram of the proposed control strategy is shown in Fig It contains three control loops, in which the first two inner loops consist of two PI controllers and the third loop has a LQR controller Pictures of the controlled TWIPR in the laboratory is shown in Fig and Fig Figure 4: Side view of a TWIPR under control in laboratory The TWIPR control system mainly consists of two DC motors, two current sensors based on the INA219, two incremental encoders, two wheels, pendulum body, the MPU115 2.2.1 DC Motor Each DC motor has a gear train with a gear ratio of n = 6050 based tilt angle sensor, three batteries and the Dis1/34 This means that the wheel, attached to the output covery kit with STM32F411VE MCU (powered by a 9V shaft of the gear train, rotates n times for each revolution and 200mAh battery) 125 130 of the motor shaft These two motors are powered by three 3.7V and 4000mAh batteries (UltraFire) 2.2.2 Incremental Encoder 135 An encoder is also attached to each DC motor’s rotor shaft The encoder has two channels and each channel has N = 13 pulses per revolution of the motor This incremental encoder are used with the highest resolution, so there are 4N/n = 1768 pulses per revolution of the wheel It is applied to measure both the angle and velocity of the wheel, then from these measurements, the position, head-140 ing angle, velocity and vertical speed of the TWMR are estimated 2.2.3 Kalman Filter In this subsection, the Kalman filter [22] is applied to estimate the tilt angle of the pendulum Let ζk be the145 gyroscope signal from the MPU 6050 at discrete point k in time, ζkbias be the deviation of the ζk , and ∆T be the sampling time Then, the tilt angle of the pendulum can be approximately calculated as θk = θk−1 + (ζk − ζkbias )∆T Define state variables as zk = [θk space model is • Step 1: Calculate the Kalman gain using Eq 10 • Step 2: Update estimate with measurement using Eq • Step 3: Determine error covariance for updated estimate using Eq 11 • Step 4: Project ahead using Eq 12, then go back the first step This algorithm will be used to estimate the tilt angle of the pendulum based on the MPU 6050 sensor In Fig 5, an example of the measured tilt angle is shown, where the black curve is the angle with Kalman filter and the blue curve is the angle without filter (7) yk = Cθ zk + vk Without Kalman With Kalman ζkbias ]T Then, the state zk = Aθ zk−1 + Bθ uk + wk θ Thus, the Kalman filter can be implemented as the following steps: Start with a prior estimate z0 and its error covariance matrix P0− -5 -10 Degrees 120 (8) -15 -20 where Aθ = Bθ = −∆T , -25 -30 ∆T , -35 Cθ = [1 0], uk = ζk , wk is the input white noise, and vk is the measurement white noise Denote covariance matrices for wk and vk as Qk and Rk , respectively Let zˆk− be the prior estimate, e− ˆk− be the estik = zk − z mation error, and the respective error covariance matrix − T θ Pk− = E[e− k (ek ) ] Then, the measurement yk will be used to improve the prior estimate as follows zˆk = zˆk− + Kk (θ yk − Cθ zˆk− ) − Pk+1 = Aθ Pk ATθ + Qk 10 12 14 (9)150 3.1 Current Model Identification In this section, the transfer function for the current of DC motors will be built based on the unit step response The electrical equation [23] for a DC motor is described as (10) di (13) dt where u is the motor armature voltage, R is armature coil resistance, L is armature coil inductance, i is the armature current, e = Ke ω is the back electromotive-force voltage, ω is the motor angular speed, τ = Km i is the motor torque, Ke is an electrical constant, and Km is the motor torque constant u = e + Ri + L (11) At time k + 1, − zˆk+1 = Aθ zˆk + Bθ uk Seconds 2.2.4 Stability Analysis Current Controller Design The covariance matrix with respective to the optimal estimate can be calculated as Pk = (I − Kk Cθ )Pk− Figure 5: Measured pitch angles with and without Kalman filter where Kk is a Kalman gain or blending factor and zˆk is an updated estimate The Kalman gain is computed as Kk = Pk− CθT (Cθ Pk− CθT + Rk )−1 (12) Thus, the transfer function for the current of the motor is Gi (s) = I(s) = U (s) R + Ls (14) k Ts + (15) or Gi (s) = 155 are determined as kP = − 0.13919 1.23 ∗ 10−4 s + (19) TI = T1 + T2 , (20) T1 T2 T1 + T2 (21) and where k = 1/R and T = L/R To determine these parameters, a unit-step response based approach is applied An voltage of 6V is supplied to the motor, then the motor’s current is measured as shown in Fig Based on this measured current, the current transfer function is obtained as follows Gi (s) = a (T1 + T2 )ln 100 , kTa% TD = Remark With this OSA method, one can design a PID controller such that the settling time is smaller than Ta% , where < a ≤ 100 This proposed method also guarantees that the overshoot is zero (16)170 Remark For the system 15, T = so the parameters a T ln 100 , TI = T , and TD = of the PID controller is kP = − kTa% For the plant 16, the desired settling time is chosen as Ta% = ms Thus, the parameters of the PI controller are computed as kP = 3.4563 and TI = 1.23 ∗ 10−4 from Eq 19 and Eq 21 with T2 = As the magnitude optimum method [24] is applied for the current object, the parameter of the I controller 0.9 0.8 Model's output Motor's current Current (A) 0.7 0.6 0.5 0.4 Rmo (s) = 0.3 Ti s (22) 0.2 is calculated as 0.1 Ti = 2kT = 3.4232 ∗ 10−5 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 (23) Time (Seconds) The GA method [25] is also utilized to determine the current controller for the plant 16 with the usage of following objective function Figure 6: Motor’s current and model’s current From the identified model (16), the current controllers are designed with usage of different methods in the next section 160 3.2 Current Controller Design based on the OSA Method 165 In this section, the novel assignment method (OSA) in [20] will be applied to design a PI controller for the DC motor’s current and compared to the other methods such as GA and MO methods via simulations and practical experiments tf inal k (T1 s + 1)(T2 s + 1) The GA based PI controller is obtained as follows Rga (s) = 0.064 + 175 (17) 180 and a desired settling time Ta% , then the parameters of the PID controller with transfer function Rp (s) = kP (1 + s + TD s) TI (24) Lemma [20] Given a plant with transfer function of second-order system G(s) = t|e(t)|dt → J= (18)185 9936 s (25) These PID controllers are utilized to control the motor’s current Both simulation and practical test are carried out to compare the OSA method with the other methods as shown in Fig 7, and In Fig 7, the controlled current is plotted via simulation It can be seen that the OSA method provides shorter settling time than the GA and MO methods It also produces a zero overshoot as the GA method The practical result is shown in Fig 8, where the OSA scheme produces the best performance in comparison with the GA and MO methods This will improve the control performance of the TWIPR Fig shows the actual control signal (the applied voltage to the DC motor) produced by the different controllers The overshoot and settling time given by the controllers through both simulation and real test are compared in Tab For the overall, 0.7 0.6 Set Point Proposed Method GA Magnitude Optimum 0.4 Voltage (V) 0.5 Current(A) Proposed Method Magnitude Optimum GA 0.3 0.2 0.1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.005 0.01 0.015 Time(s) Figure 7: Simulation of the controlled current using different controllers 0.7 Current (A) 0.6 Set Point Proposed Method GA Magnitude Optimum 0.3 where 0.2 0.1 A= -0.1 0.005 0.01 0.015 0.03 [∗ ∗ 0 0]T , where ∗ are any values of x1 and x3 , which are the desired position and heading angle of the TWIPR Without loss of generality, it is assumed that ∗ is zero, this means xe = Thus, a linear model of the system around the equillibrium point xe can be obtained as follows x˙ = Ax + Bu (26) 0.8 0.4 0.025 Figure 9: Practical control signal using different controllers 0.9 0.5 0.02 Time (s) 0.02 0.025 0.03 ∂f ∂x x=xe Time (s)  0  0 = 0  0 0 0 a42 a52 0 0 0 a44 a54 0 a45 a55  0        a66 (27) Figure 8: Practical regulated current using different controllers  190 the OSA method is better than the MO and GA methods in terms of both overshoot and settling time The OSA based PI controller will be applied to control the two motors’ currents of the TWIPR In the next section, a LQR controller combined with the proposed PI controller is designed to keep the TWIPR balanced and movable with desired trajectories B= OSA MO GA T2% (ms) Simulation Test 1.1 1.04 2.42 2.5 Overshoot Simulation 4.32 (%) Test 1.36 29.92 x=xe  0  0  b42   b52  b62 (28) −(M gl)2 g , a44 = −2C((M l Λ+I2 )+M lr) , a45 = Λ 2 2Cr(M l2 +I2 )+2CM lr , a52 = M gl(2J+MΛr +2mr ) , a54 = Λ 2 2CM l+2C(2J+M r +2mr )/r )+2CM lr , a55 = 2C(2J+M r +2mr , Λ −Λ 2 Km r[(M l +I2 )+M lr] −Cd a66 = 2Jd2 +(2I3 +4K+d2 m)r2 , b41 = b42 = , Λ −Km (2J+M r +2mr +M lr) b51 = b52 = , b61 = −b62 = Λ −Km dr 2 Jd2 +(2I3 +4K+d2 m)r , and Λ = 2I2 J + 2JM l + I2 M r + 2 2I2 mr + 2M l mr with a42 = Table 2: Comparison of methods through simulation and real test Method ∂f ∂u 0  0 = b41  b51 b61 200 4.2 LQR Controller 195 From the linear model 26, a LQR controller [26] is designed such that the following cost function is minimum LQR Controller Design 4.1 Linearized Model of the TWIPR Let xe be an equilibrium point of the system 6, then xe is the solution to the equation f (x, 0) = Thus, xe = F = ∞ (xT Qc x + uT Rc u)dt → (29) Simulation and Practical Test KLQR = Rc−1 B T Pc (30)220 and Pc is the solution to the Riccati equation Pc BRc−1 B T Pc − Pc A − AT Pc = Qc (31) 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 =225 9.81m/s2 , C = 0.005, Km = 0.41202N m/A Thus, the system matrices   0 0 0 0    0 0 0    (32) A= 1.35  0 −11.41 −40.85  0 176.81 403.48 −13.31  0 0 −8.17 In this section, the propsed control scheme based on the OSA 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 to move from the initial point to the end point along a line, where the distance between these two points is meter 5.1 The proposed control strategy The simulation results are shown in Fig 10, 11 and 12 whereas the real-time run 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 0.4 0.3 (rad) where Qc is a symmetric positive definite matrix and Rc is a symmetric nonnegative definite matrix The controller, which satisfies the cost function 29, is u = −KLQR x, where 0.2 0.1   0   0     0  B=  55.53 55.53    −548.60 −548.60 −138.92 138.92 The matrices of the cost function are chosen as   0 0 0 0 0   0 20 0 0   Qc =  0  0 0 3.5 0 0 0.001  0 0 0.9 -0.1 (33) 10 12 14 16 18 20 Time(s) Figure 10: The tilt angle of the pendulumn through simulation 10-15 (34) (rad) and and -2 (35) From Eq 30 and Eq 31, the gain matrix of the LQR controller is obtained as Rc = -4 210 10 12 14 16 18 20 Figure 11: The heading angle of the TWIPR through simulation −2.147 −2.147 −0.33 −0.659 −0.33 0.659 (36) The output of the LQR controller will be the setpoints for the current controllers, which are belong to the inner control loop 0.8 0.6 x (m) 205 Time(s) KLQR = −1.414 −3.098 −3.162 −1.414 −3.098 3.162 Remark Since the input to the LQR controller consists ˙ and (ψ; ψ), ˙ the LQR of state variable pairs (x; x), ˙ (θ; θ) 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 0.4 0.2 -0.2 10 12 14 16 18 20 Time(s) 215 Remark The LQR controller in [12] is also compared to the state-dependent LQR controller when the high yaw rate is considered Figure 12: The position of the TWIPR through simulation 0.6 1.5 0.5 IR (A) 0.4 (rad) 0.3 0.5 0.2 -0.5 0.1 -1 -1.5 10 12 14 16 18 20 Time(s) -0.1 10 12 14 16 18 20 Time(s) Figure 17: Right motor’s real current Figure 13: The real tilt angle of the pendulumn 10-3 230 (rad) -5 -10 235 -15 From these figures, one can see that the real-time results look like the simulation results, this implies that a good model of the TWIPR is built and the controllers works well The proposed control strategy produces good performance such as the pitch angle lies within the range [−3 3] (deg), the TWIPR reaches the destination at (m) after 3.5 (seconds) following a straight line because the yaw angle is approximately within the range [−0.3 0.3] (deg) -20 10 12 14 16 18 20 5.2 Comparisons Time(s) Figure 14: The real heading angle of the TWIPR 240 1.2 x (m) 0.8 0.6 245 0.4 The proposed control scheme based on the OSA-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 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 OSA-LQR method, the MO-LQR method and the GA-LQR method, respectively The comparison of the pitch and yaw angles of the 0.2 1.2 -0.2 10 12 14 16 18 20 Time(s) 0.8 OSA -LQR MO - LQR GA - LQR x (m) Figure 15: The real position of the TWIPR 0.6 0.4 0.5 IL (A) 0.2 0 -0.5 -0.2 -1 10 Time(s) -1.5 10 12 14 16 18 20 Figure 18: The position of the TWIPR with different methods Time(s) Figure 16: Left motor’s real current 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 265 0.8 OSA - LQR MO - LQR GA - LQR 0.6 (rad) 0.4 270 0.2 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 is kept balanced and able to reach the desired position and direction, and the proposed controller produces better performance than the others Future works will focus on the combination of the OSA method with other advanced control methods for the TWIPR and other robots Acknowledgments -0.2 275 -0.4 10 Time (s) Figure 19: The pitch angle of the TWIPR with different methods 250 References straght line with a distance of meter The OSA-LQR280 based TWIPR moves more smoothly from the start point to the end point with smaller pitch angle and less oscillation than the other methods Hence, the proposed con285 troller provides better performance than the others 10-3 290 OSA - LQR MO - LQR GA - LQR (rad) 295 -5 -10 300 -15 305 -20 10 Time (s) Figure 20: The yaw angle of the TWIPR with different methods 310 255 Remark The proposed control scheme is also tested with different loads and inclined surfaces The practical315 results prove that the TWIPR is kept balanced and movable However, they are skipped to show here 260 Conclusions This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2017-TT-004 320 In this work, the TWIPR control system based on the OSA-LQR controller is proposed First, the OSA method is compared to the MO and GA methods through the325 DC motor’s current It produces better performance than [1] Zhijun L., Chenguang Y., and Liping F (2013) Advanced Control of Wheeled Inverted Pendulum Systems, Springer [2] Ronald P M C., Karl A S., and Roger H (2013) Review of modelling and control of two-wheeled robots Annual Reviews in Control, 37 (1), pp 89-103 [3] Khanh G T., Phuoc D N., and Nam H N (2017) Advanced control 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