Frontiers in Computing and Intelligent Systems ISSN: 2832-6024 | Vol. 3, No. 3, 2023 30 Stick slip vibration control of drill string based on torque feedforward and sliding mode controller Yinling Jiang 1,2, Ke Zhu 1, Qiang Zhang 3, Shuai Zhao 3 1 School of Electrical Engineering and Information, Northeast Petroleum University, Daqing 163318, China 2 College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China 3 School of Mechanical Science and Engineering, Northeast Petroleum University, Daqing 163318, China Abstract: The drill string system in drilling engineering has significant geometric nonlinearity. Due to the frictional torque generated by the interaction between the drill bit and the rock and soil, it is easy to cause the drill string to fall into a sticky sliding vibration state. Based on the two lumped mass torsion pendulum model of the drill string, this paper establishes an observer of the bit speed, and designs a sliding mode controller (TF-SMC) considering the bit torque to suppress the stick slip vibration of the drill string. The simulation results show that compared with PID controller and sliding mode controller, torque feed-forward sliding mode controller can achieve better control effect on drill string at low speed. When torque feedforward sliding mode controller and PID controller are used in parallel, the control effect is better than the first three controllers. Keywords: Stick slip vibration control of drill string; Torque feedforward; Sliding mode controller. 1. Introduction The drill string system has a large aspect ratio, resulting in significant geometric nonlinearity of the drill string. During rotary drilling, the contact and collision between the drill string and the wellbore, as well as the rock breaking of the drill bit and the flow of drilling fluid, can cause vibration in the drill string system[1-3]. Among them, the vibration form that causes significant damage to the drill bit assembly (BHA) is the stick-slip vibration phenomenon[2]. When the drill string is in a stick-slip vibration state, the drill bit will enter a stationary high-speed sliding motion cycle. The speed of a drill bit in a general high-speed sliding state can be Several times the normal working speed, which can lead to high-speed wear, tooth skipping, and even drill string fracture accidents. Therefore, it is necessary to control the stick-slip vibration behavior of the drill string. In order to suppress the stick-slip vibration of the drill string, researchers have studied methods such as increasing the rigidity of the drill string system and reducing impact vibration [4-5], but these methods require modification of the original control system. Without changing the basic structure of the drill string system, reference [6] studied a stick-slip vibration control system that directly controls the drilling speed of the drill bit. However, due to factors such as high- frequency vibration of the drill string itself, drilling fluid flow induced vibration, and the need for frequent single connections in the drill string system, there are great difficulties in measuring and transmitting the drilling speed at the bottom of the drill bit. Therefore, there are significant difficulties in the application of control strategies that directly use real drill bit drilling speed as signal feedback in engineering. When the physical quantities to be measured in the general control system cannot be measured accurately, the engineering personnel will use the state observer to estimate the physical quantities to be measured. Therefore, reference [7] designed a bit speed observer. Then, researchers designed observer-based SMC [8], H∞ [9] controller, etc. to control the bit speed and suppress the stick slip vibration behavior of the drill string. However, these controllers generally consider the frictional torque generated by the interaction between the drill bit and rock soil as interference, and there is not much research on incorporating the frictional torque of the drill string into the structural design of the controller. The frictional torque generated by the interaction between the drill bit and the rock is the main cause of the stick-slip vibration of the drill string. Due to the fact that the input torque of the top drive in actual drilling engineering may lose its original control performance due to changes in the bottom layer or low top drive speed. In the study of control systems for drill strings, the friction torque at the drill bit is usually described using Stribeck and Karnopp models that consider the maximum static friction and viscous resistance. This paper will establish a dual mass torsion pendulum model of the drill string based on the Karnopp model, design the torque feedforward link of the drill string system, and combine SMC, torque feedforward controller, and PID controller to design a new stick-slip vibration controller for the drill string. 2. Dynamic model of drill string According to the concentrated mass torsion pendulum model of the drill string used in reference [7], the dynamic model of the drill string established in this paper is shown in Figure 1. Figure 1. Drill string two concentrated mass torsion pendulum model. π‘˜ 𝑐 π‘‡π‘š 𝑇𝑓 π½π‘Ÿ 𝐽𝑏 π‘π‘Ÿ 𝑐𝑏 31 For the system in Figure 1, the dynamic equation of the system can be established based on the theory of rigid body dynamics. The dynamic equation at the top drive turntable is: π½π‘ŸοΏ½Μ‡οΏ½π‘Ÿ = π‘‡π‘š βˆ’ π‘‡π‘Žπ‘Ÿ βˆ’ 𝑇𝑑 (1) π‘‡π‘Žπ‘Ÿ = πœ”π‘Ÿπ‘π‘Ÿ (2) 𝑇𝑑 = 𝑐(πœ”π‘Ÿ βˆ’ πœ”π‘) + π‘˜(πœ‘π‘Ÿ βˆ’ πœ‘π‘) (3) The dynamic equation of the drill bit part: 𝐽𝑏�̇�𝑏 = 𝑇𝑑 βˆ’ π‘‡π‘Žπ‘ βˆ’ 𝑇𝑓 (4) π‘‡π‘Žπ‘ = π‘π‘πœ”π‘ (5) where, π½π‘Ÿ and Jb is the rotational inertia of the top drive rotary table and the equivalent drill bit; πœ‘π‘Ÿ, πœ”π‘Ÿ, πœ‘π‘, πœ”π‘ is the torsional angle and rotational speed of the top drive rotary table and drill bit, respectively; π‘‡π‘š is the input speed of the top drive, π‘‡π‘Žπ‘Ÿ is the viscous torque of the top drive rotary table; 𝑇𝑑 is the coupling torque of the drill string; π‘‡π‘Žπ‘ is the viscous torque of the drill bit; 𝑇𝑓 is the friction torque of the drill bit. π‘π‘Ÿ, 𝑐𝑏, 𝑐 is the damping coefficient of the top drive rotary table, drill bit, and equivalent drill string, respectively; π‘˜ is the torsional stiffness of the drill string. Take the state variable π‘₯ = (πœ”π‘Ÿ , πœ”π‘ , πœ‘π‘Ÿ βˆ’ πœ‘π‘) 𝑇 , The friction torque 𝑇𝑓 of the drill bit is represented by the Karnopp model in Equation 6. 𝑇𝑓(π‘₯) = { 𝑇𝑒𝑏(π‘₯) , |πœ”π‘| < 𝐷𝑣, |𝑇𝑒𝑏| < 𝑇𝑠𝑏 𝑇𝑠𝑏𝑠𝑔𝑛(𝑇𝑒𝑏(π‘₯)), |πœ”π‘| < 𝐷𝑣, |𝑇𝑒𝑏| β‰₯ 𝑇𝑠𝑏 𝑓𝑏(πœ”π‘)𝑠𝑔𝑛(πœ”π‘), |πœ”π‘| β‰₯ 𝐷𝑣 (6) where sgn(…) is the sign function, 𝐷𝑣 is the Karnopp velocity boundary layer. 𝑇𝑒𝑏 = 𝑇𝑑 βˆ’ π‘‡π‘Žπ‘ (7) 𝑇𝑠𝑏 = πœ‡π‘ π‘π‘Šπ‘œπ‘π‘…π‘ (8) 𝑓𝑏 = πœ‡π‘(πœ”π‘)π‘Šπ‘œπ‘π‘…π‘ (9) πœ‡π‘(πœ”π‘) = πœ‡π‘π‘ + (πœ‡π‘ π‘ βˆ’ πœ‡π‘π‘)𝑒 βˆ’π›Ύπ‘|πœ”π‘|/𝑣𝑓 ) where, 𝑇𝑒𝑏 is the static equilibrium torque of the drill string; 𝑇𝑠𝑏 is the maximum static friction torque; 𝑓𝑏 is the dynamic friction torque; πœ‡π‘ (...) is the dynamic friction coefficient function; πœ‡π‘π‘ and πœ‡π‘ π‘ is the coefficient of viscous friction and the coefficient of dynamic friction; 𝛾𝑏 and 𝑣𝑓 are karnopp empirical constants. In summary, the dynamic equation of the drill string system is: { π½π‘ŸοΏ½ΜˆοΏ½π‘Ÿ + 𝑐(οΏ½Μ‡οΏ½π‘Ÿ βˆ’ �̇�𝑏) + 𝑐(πœ‘π‘Ÿ βˆ’ πœ‘π‘) + π‘π‘ŸοΏ½Μ‡οΏ½π‘Ÿ = π‘‡π‘š π½π‘οΏ½ΜˆοΏ½π‘ + 𝑐(οΏ½Μ‡οΏ½π‘Ÿ βˆ’ �̇�𝑏) + 𝑐(πœ‘π‘Ÿ βˆ’ πœ‘π‘) + 𝑐𝑏�̇�𝑏 = βˆ’π‘‡π‘“(π‘₯) (11) ⟹ [ οΏ½Μ‡οΏ½π‘Ÿ οΏ½Μ‡οΏ½π‘Ÿ βˆ†οΏ½Μ‡οΏ½ ] = [ βˆ’ 𝑐+π‘π‘Ÿ π½π‘Ÿ πœ”π‘Ÿ + 𝑐 π½π‘Ÿ πœ”π‘ βˆ’ π‘˜ π½π‘Ÿ βˆ†πœ‘ + 1 π½π‘Ÿ π‘‡π‘š 𝑐 𝐽𝑏 πœ”π‘Ÿ βˆ’ 𝑐+𝑐𝑏 𝐽𝑏 πœ”π‘ + π‘˜ 𝐽𝑏 βˆ†πœ‘ βˆ’ 1 𝐽𝑏 𝑇𝑓𝑏 πœ”π‘Ÿ βˆ’πœ”π‘ ] (12) 3. Design of speed observer and controller Based on the proportional integral observer design method, this paper considers 𝑇𝑓 as an observable interference. The observer structure of the drill string rotary speed is as follows [7]: { οΏ½Μ‡Μ‚οΏ½π‘Ÿ = βˆ’ 𝑐+π‘π‘Ÿ π½π‘Ÿ οΏ½Μ‚οΏ½π‘Ÿ + 𝑐 π½π‘Ÿ �̂�𝑏 βˆ’ π‘˜ π½π‘Ÿ βˆ†οΏ½Μ‚οΏ½ + 1 π½π‘Ÿ π‘‡π‘š +𝐾1(πœ”π‘Ÿ βˆ’ οΏ½Μ‚οΏ½π‘Ÿ) �̇̂�𝑏 = 𝑐 𝐽𝑏 οΏ½Μ‚οΏ½π‘Ÿ βˆ’ 𝑐+𝑐𝑏 𝐽𝑏 �̂�𝑏 + π‘˜ 𝐽𝑏 βˆ†οΏ½Μ‚οΏ½ βˆ’ 1 𝐽𝑏 𝑇𝑓 + 𝐾2(πœ”π‘Ÿ βˆ’ οΏ½Μ‚οΏ½π‘Ÿ) βˆ†οΏ½Μ‡Μ‚οΏ½ = οΏ½Μ‚οΏ½π‘Ÿ βˆ’ �̂�𝑏 +𝐾3(πœ”π‘Ÿ βˆ’ οΏ½Μ‚οΏ½π‘Ÿ) �̇̂�𝑓 = 𝐾4(πœ”π‘Ÿ βˆ’ οΏ½Μ‚οΏ½π‘Ÿ) (13) Due to the observation of four physical quantities by the observer, according to the optimal damping principle, the optimal damping polynomial corresponding to the observer is of order 4. The gain coefficients of the observer are: 𝐾1 = 1 𝐷2𝐷3𝐷4𝑇𝑒 βˆ’ 𝑐+π‘π‘Ÿ π½π‘Ÿ βˆ’ 𝑐+π‘π‘Ÿ 𝐽𝑏 (14) 𝐾2 = π‘˜(βˆ’π½π‘Ÿπ‘π‘+𝐾1π½π‘Ÿπ½π‘βˆ’π‘π‘π·2𝑇𝑒 2𝐾4+π‘π‘Ÿπ½π‘+𝐽𝑏𝑇𝑒𝐾4) 𝑐𝑏(π‘π‘π‘βˆ’π‘˜π½π‘) βˆ’π‘π‘ 2(π‘π‘Ÿ+𝑐+𝐾1π½π‘Ÿ)+𝑐(π‘π‘π‘π‘Ÿ+𝑐𝑏𝐾1π½π‘Ÿ+𝐽𝑏𝐾4) 𝑐𝑏(π‘π‘π‘βˆ’π‘˜π½π‘) (15) 𝐾3 = π‘˜2(π½π‘Ÿ+𝐽𝑏+𝐷2𝑇𝑒 2𝐾4)+π‘˜(π‘π‘π‘π‘Ÿ+𝐾1π½π‘Ÿπ‘π‘βˆ’π‘πΎ4𝑇𝑒)+𝐾4𝑐 2 π‘˜(βˆ’π‘π‘π‘+π‘˜π½π‘) (16) 𝐾4 = π½π‘Ÿπ½π‘ π‘˜π·4𝐷3 2𝐷2 3𝑇𝑒 4 (17) where 𝐷2, 𝐷3, 𝐷4 Take as 0.5; the time coefficient 𝑇𝑒 is taken as 0.2. In order to suppress stick slip vibration of drill string, an observer-based torque feedforward sliding mode controller (TF-SMC-PID) is designed in this paper. The controller structure is shown in Figure 2. Figure 2. Controller Structure According to equation 12, when the drill string system does not experience stick-slip vibration, οΏ½Μ‡οΏ½=0. The top drive torque π‘‡π‘šπ‘  of the system is: π‘‡π‘šπ‘  = πœ”π‘Ÿπ‘’π‘“(π‘π‘Ÿ + 𝑐𝑏) + �̂�𝑓 (18) Therefore，this paper uses π‘‡π‘šπ‘  as the output of the torque feedforward controller and refers to it as π‘‡π‘šπ‘“. This paper takes πœ”π‘ β†’ πœ”π‘Ÿ , πœ”π‘Ÿ β†’ πœ”ref is the control target of the sliding mode controller, and an integral sliding mode surface is selected for the sliding mode controller. The structure of the sliding mode surface is: 𝑠 = (πœ”π‘Ÿ βˆ’ πœ”π‘Ÿπ‘’π‘“) + πœ† ∫ (πœ”π‘Ÿ βˆ’ πœ”π‘Ÿπ‘’π‘“ 𝑑 0 ) + (πœ”π‘Ÿ βˆ’πœ”π‘)π‘‘πœ (19) where πœ† is the sliding mode area partition coefficient? The control law of sliding mode controller is π‘’π‘ π‘šπ‘ = π½π‘Ÿ ( 𝑐+π‘π‘Ÿ π½π‘Ÿ πœ”π‘Ÿ βˆ’ 𝑐 π½π‘Ÿ πœ”π‘ + π‘˜ π½π‘Ÿ βˆ†πœ‘ + οΏ½Μ‡οΏ½π‘Ÿπ‘’π‘“ βˆ’ πœ†(2πœ”π‘Ÿ βˆ’ πœ”π‘ βˆ’ πœ”π‘Ÿπ‘’π‘“) βˆ’ πœ€ π‘ π‘Žπ‘‘(𝑠) βˆ’ πœ‚π‘ ) (20) where, sat(… ) is the saturation function; πœ€, πœ‚ is a sliding mode controller parameter. οΏ½Μ‚οΏ½f State Observer Plant TF SMC πœ”ref π‘‡π‘šπ‘“ π‘’π‘ π‘šπ‘ π‘‡π‘š πœ”π‘Ÿ , π‘‡π‘š �̂�𝑏 PID - πœ”π‘Ÿ 32 The PID controller in Figure 2 indirectly controls the bit speed only with the top drive speed as the feedback signal. 4. Controller performance analysis The parameters of the system are shown in Table 1. Set Expected Torque π‘‡π‘šπ‘Ÿπ‘’π‘“ =12000Nmm. According to equation 12, when the system is stable, οΏ½Μ‡οΏ½=0, the relationship between the expected speed and the expected torque is πœ”ref = πœ”π‘Ÿ = πœ”π‘ = 𝑇mβˆ’π‘‡π‘“π‘  π‘π‘Ÿ+π‘π‘Ÿ (21) The expected speed of the system is obtained from equation 21. πœ”ref=9.32rad/s. The performance of the system is shown in Figure 3. Table 1. Parameters Parameter Value Parameter Value Parameter Value π½π‘Ÿ 2122 kgΒ·m2 𝑐𝑏 50 NΒ·mΒ·sΒ·rad-1 π‘Šπ‘œπ‘ 97533 N 𝐽𝑏 374 kgΒ·m2 πœ‡π‘π‘ 0.5 𝑅𝑏 0.156 m 𝑐 23.2 NΒ·mΒ·sΒ·rad-1 πœ‡π‘ π‘ 0.8 π‘π‘Ÿ 425 NΒ·mΒ·sΒ·rad-1 π‘˜ 473 NΒ·mΒ·rad-1 𝛾𝑏 0.9 𝑣𝑓 1 𝐷𝑣 0.001 mΒ·s-1 0 10 20 30 40 50 4 10 16 22 w / r ad -s t / s wref wr wb Figure 3. Raw system performance From Figure 3, it can be seen that there is a periodic oscillation in the top drive rotary table speed, and the drill bit speed is in a stagnant, high-speed sliding periodic vibration state, that is, a stick-slip vibration state. At the same desired speed, this paper studies the influence of PID, SMC, torque feedforward sliding mode controller (TF-SMC), torque feedforward sliding mode controller and PID parallel controller (TF-SMC-PID) on the performance of the drill string system. The system performance is shown in Figure 4. From Figure 4, it can be seen that the time when the top drive speed of the system reaches stability is basically the same, and the speed vibration peak value of the PID controller scheme is relatively low. In terms of drill speed control, compared to PID controllers, the other three controllers have significantly better control effects. The drill bit speed can basically reach a steady state within 20 seconds. Among them, the TF-SMC-PID controller has significantly better overshoot indicators for drill bit speed, and the peak to peak oscillation is significantly lower compared to other controllers. 0 10 20 30 0 3 6 9 12 15 w / r ad -s t / s wref wr TF-SMC-PID wr TF-SMC wr SMC wr PID (a) 0 10 20 30 0 4 8 12 16 20 w / r ad -s t / s wref wr TF-SMC-PID wr TF-SMC wr SMC wr PID (b) Figure 4. The Influence of several controllers on system performance 0 10 20 30 0 4 8 12 16 20 w / r ad -s t / s wref wb TF-SMC-PID wb TF-SMC wb SMC wb PID (a) 0 10 20 30 0 6 12 18 24 30 w / r ad -s t / s wref wb TF-SMC-PID wb TF-SMC wb SMC wb PID (b) Figure 5. System performance: (a) low speed (b) high bit pressure Set the expected speed πœ”ref =5.5rad/s, and the performance of each controller is shown in Figure 5(a). After setting the expected speed πœ”ref =9.32rad/s and increasing the π‘Šπ‘œπ‘ to 150000N, the performance of each controller is 33 shown in Figure 5.2(b). From Figure 5.1(a), it can be seen that TF-SMC-PID and TF-SMC can still achieve effective control of the drill string at low rotational speeds. SMC and PID controllers cannot effectively control the drill string system at low speeds. From this, it can be seen that the torque feedforward link can significantly increase the lower limit of the system's operating speed. From Figure 5.1(b), it can be seen that as the drilling pressure increases, the PID controller is no longer able to effectively control the drill string system. TF-SMC-PID, TF- SMC, and SMC can all suppress the stick-slip vibration behavior of the drill string. Among them, TF-SMC-PID and TF-SMC have faster response speed and lower peak to peak vibration compared to SMC. 5. Conclusion This paper establishes a dual mass torsion pendulum model for the drill string and a drill bit drilling speed observer. Based on the drill speed observer, this paper studies the control effect of torque feedforward sliding mode controller (TF- SMC), PID controller and torque feedforward sliding mode controller and PID parallel controller (TF-SMC-PID) on the drill string. The research shows that both PID controller and torque feedforward sliding mode controller can effectively control the stick slip vibration of the system. After the torque feedforward is added, the system can guarantee the normal operation of the system at a lower speed or a higher WOB. Compared to PID controllers, TF-SMC-PID controllers significantly increase the response speed of drill bit speed, shorten the adjustment time, significantly reduce overshoot, and significantly lower the peak value of vibration when controlling the drill string system. References [1] Lei Chen,Jianjun Xu,Shuang Chen,Hui Yang,Linhu Liu. Power Quality Disturbance Signal Denoising Based on Overcomplete Representation. IEEJ TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERING.2022; 17: 544–555. [2] Xu Jianjun, Huang Lida, Yan Limei, Yi Na. Insulator Self- Explosion Defect Detection Based on Hierarchical Multi-Task Deep Learning. Transactions of China Electrotechnical Society, 2021,36(07):1407-1415. [3] Yan Limei,LIU Yongqiang,XU Jianjun,et al. Broken string diagnosis of composite insulator based on Grabcut segmentation and filler area discrimination. Power System Protection and Control,2021,49(22):114-119. [4] Yi, Q. Wang, L. Yan, et al., A multi-stage game model for the false data injection attack from attacker’s perspective. Sustainable Energy Grids & Networks 28 (2021). [5] Na Yi,Jianjun Xu,Limei Yan,Lin Huang. Task Optimization and Scheduling of Distributed Cyber-physical System Based on Improved Ant Colony Algorithm. Future Generation Computer Systems, 109(Aug. 2020),134-148. [6] Yang Zhao, Jianjun Xu, Jingchun Wu. A New Method for Bad Data Identification of Oilfield Power System Based on Enhanced Gravitational Search-Fuzzy C-Means Algorithm. IEEE Transactions on Industrial Informatics. VOL. 15, NO. 11, NOVEMBER 2019 5963-5970. [7] VAHID VAZIRI,MARCIN KAPITANIAK & MARIAN WIERCIGROCH.(2018).Suppression of drill-string stick–slip vibration by sliding mode control: Numerical and experimental studies. (5),805-825.G.R. Mettam, How to prepare an electronic version of your article, in: B.S. Jones, R.Z. Smith (Eds.), Introduction to the Electronic Age, E-Publishing Inc., New York, 1999, pp. 281-304. [8] Riane Rami et al. (2022). Observer-Based H∞ Controller Design for High Frequency Stick-Slip Vibrations Mitigation in Drill-String of Rotary Drilling Systems. Vibration, 5(2), pp. 264-289.