Robust Adaptive Super-Twisting Sliding Mode Stability Control of Underactuated Rotational Inverted Pendulum With Experimental Validation

In this study, an adaptive proportional-integral-derivative (PID) sliding mode control technique combined with the super-twisting algorithm is planned for the stabilization of rotational inverted pendulum in the appearance of exterior perturbation. The state-space model of rotational inverted pendulum in the existence of exterior disturbance is attained. Then, the super-twisting PID sliding mode controller is designed for finite time stability control of the considered underactuated control system. The upper bounds of perturbation are presumed to be unknown; consequently, the adaptive control procedure is taken into account to approximate the uncertain bounds of external disturbances. The stability control of rotational inverted pendulum system is verified by means of the Lyapunov stability theory. In order to validate the accuracy and efficiency of the recommended control technique, some simulation outcomes are prepared and compared with other existing scheme. Finally, the experimental results are implemented to show the success of the designed method.

of position and attitude of aircrafts, and robot system have 23 been originated from the model of RIP system [14], [15], 24 [16]. For this reason, stability and control of RIP system is 25 still in consideration [17], [18], [19], [20]. Hence, the con- 26 The associate editor coordinating the review of this manuscript and approving it for publication was Jason Gu . trol problem of RIP system is divided into two subsystems. 27 In the first subsystem, the stability of position and angular 28 velocity relevant to the arm of RIP system is investigated. 29 In addition, in the second subsystem, the main goal of con- 30 trol is the balancing of pendulum to be stand up-right [21], 31 [22], [23], [24]. Therefore, some control methods including 32 proportional-integral-derivative (PID), linear quadratic regu-33 lator (LQR), linear quadratic Gaussian (LQG), linear matrix 34 inequality (LMI) [25], sliding mode control (SMC), adaptive- 35 control, fuzzy logic and neural network [26] techniques have 36 been applied for both stability and balancing control of RIP 37 systems [18], [27], [28], [29], [30], [31]. 38 In [32], LOR and LQG methods based on the fuzzy logic 39 controltechnique has been proposed aimed at stability con- 40 trol of double-RIP system under perturbation. Besides, these 41 methods are compared with the classical LOR and LQG 42 VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ techniques which confirm better performance of the pro- 43 posed methods. In [33], an LQR control scheme decoupled 44 PID control technique is designed in order to stability and 45 balancing control of RIP systems. In [34], a robust LQR 152ÿ 154ẏ 155ÿ 4 (t) = F 4 y 2 (t) + F 5 y 4 (t) + F 6 156 + M 21 τ 1 (t) + 2 (t).

195
The objective of the following theorem is the finite time 196 stability of the rotational inverted pendulum in the existence 197 of external disturbance with known bounds. Taking derivative of (24) with respect to time and using 207 (20), it can obtain 208 where substituting the control laws (21)-(23) into (25), one

218
After some simplification, we have Considering Assumption 1 and doing some mathematical 222 operations, it yields

225
whereas σ 1 |s (t)| 3 2 is a positive expression; so, it can be 226 removed, therefore we have

243
The adaptive laws can be offered as 245β

246
where a 1 and a 2 are the positive constants and b m1 and b m2 247 are achieved by the following equations:

250
while k m1 and k m2 signify the positive constants. Thus, the 251 control input is designed as Theorem 2: For the rotary inverted pendulum system 260 (11)-IV under known external disturbance which holds 261 Assumption 1, the control inputs (37)- (39) are designed based 262 on the sliding surface (17) and adaptive laws (33)- (36). Thus, 263 the sliding surface is converged to the origin as well as the 264 stability control of the underactuated system is fulfilled.
where respect to the time-derivative of Eq. (40) and consid-269 Now, the equations (20) and (33) Using the control laws (37)-(39), it gets According to Assumption 1, we have From Eqs. (31) and (32) and removing the similar terms,

292
we can obtain

308
where by removing the same expressions, it leads to Hence, we obtainV (t) ≤ 0. Therefore, it is demonstrated 311 that the proposed switching surface converges to origin. The 312 proof is finished.

315
In this part, the simulation results for RIP system are per-316 formed based on the adaptive super-twisting PID-SMC tech-317 nique as exposed in Fig.1. The constant parameters of RIP 318 system and the design values are given in Table 1 and Table 2, 319 correspondingly.

320
The simulation outcomes based on the planned scheme are         presented in Fig. 5. The applied torque of the system which 332 is gained by the super-twisting PID-SMC is shown in Fig.6. 333 From these figures, it can be seen that not only the suggested 334 method has quick response respect to the method of [1], but 335 also the transient performnace of the recomende method is 336 much better than method of [1].

337
Now, it is persumed that the upper bound of exterior 338 disturbance is unknown. So, the simulation results are reim-339 plemented using the adaptive control technique. The stability 340 control of RIP system based on the adaptive super-twisitng 341 PID sliding mode controller is exposed in Fig.7 and Fig.8. 342 Also, time trajectory of reachability of sliding surface to 343 origin is represented in Fig.9. Time response of the applied 344 torque based on the adaptive super-twisting PID-SMC is 345    with the method of [1]. At last, the adapation laws related 348 to the approximation of upper bound of exterior disturbances 349 are illustrated in Fig.11. Acording to these figures and com-350 parisions, it can be seen that the recommended method based 351 on the adptive super-twisting PID-SMC presents the fast and 352 better transient response in comparison with technique of [1]. 353

354
In this part, some experimental outcomes are implemented on 355 a real electro-mechanical engineering control system which 356 is developed by the TERASOFT company in Future Tech-357 nology Research Center (FTRC) in National Yunlin Uni-358 versity of Science and Technology. The components of this 359 system are shown in Fig.12. Moreover, this control system 360 has support package in MATLAB as the embedded coder 361 toolbox that supports Texas instruments C2000 Processors. 362 In addition, block diagram of the platform is depicted in 363 Fig.13. The laboratory environment for implementation of the 364 suggested method on real RIP system is exposed in Fig.14. 365 The applied voltage for motor in the control of RIP system 366 is calculated as the following equation:    where R m and K t are the motor armature resistance and motor 369 torque constants. After implementing the suggested method on the RIP sys-371 tem, the subsequent outcomes are found. Time responses of 372 the position and angular velocities of the arm and pendulum 373 are shown in Fig.15 and Fig.16, individually. It can be seen 374 that the position of arm is stabilized near 0.7 degree which 375 is equal to 0.012 radian. Additionally, the pendulum position 376 is converged to zero (around 3.11 degree). So, the positions 377 of the arm and pendulum are stabilized to a region near the 378 origin. In VI, time trajectory of the applied voltage in DC 379 motor is displayed. Hence, the validation of the suggested 380 method is proved.

382
In this paper, the dynamical model of rotational inverted 383 pendulum system was studied in the form of state-space 384 model. The finite time stability of the rotary inverted pen-385 dulum system under known bounded exterior disturbance 386 was accomplished according to the super-twisting PID slid-387 ing mode control. Whereas the upper bound of perturbation 388 was assumed to be unknown and the adaptive-tuning con-389 trol scheme was designed to estimate the unknown bounds. 390 In addition, the Lyapunov stability theory was used to attest 391 the stability control of underactuated rotary inverted pendu-392 lum based on the adaptive super-twisting PID sliding mode 393 control technique. As well, simulation results were provided 394 based on the recommended method. The simulation outcomes 395 were compared with another method which was confirmed 396 the proficiency and efficacy of the suggested procedure in 397 comparison with the other method. Furthermore, experimen-398 tal results on real RIP were provided to demonstrate the 399 efficiency of the planned method. He has published several papers in the national and international 631 journals. His research interests include control theory, sliding mode control, 632 robust tracking, non-holonomic robots, and chaotic systems. He is a member 633 of the IEEE Control Systems Society and serves as a member for program 634 committee of several international conferences. He is an associate editor of 635 several international scientific journals and has acted as a symposium/the 636 track co-chair in numerous IEEE flagship conferences. He has been a world's 637 top 2% scientist from Stanford University, since 2019, and has been ranked 638 among 1% top scientists in the world in the broad field of electronics and 639 electrical engineering. He is also recognized in the list of Top Electronics 640 and Electrical Engineering Scientists in Iran.