Backstepping sliding mode control of stabilized platform for fully rotary steerable drilling system based on nonlinear disturbance observer

To handle the disturbances and uncertainties issue in the control of the stabilized platform toolface angle, a composite control approach based on a backstepping sliding mode control (BSMC) method with a nonlinear disturbance observer (NDO) is proposed. First, a nonlinear mathematical model of the stabilized platform is established, and the NDO is employed to observe and compensate for the external disturbance. Then, the BSMC control approach based on the nonlinear mathematical model is designed to improve the robustness of the stabilized platform control system. The asymptotic stability of the designed controller and NDO convergence are mathematically verified using the Lyapunov theory. Furthermore, to address the dead zone torque caused by mechanical friction on the stabilized platform, which could lead to instability in the toolface angle, a novel dead zone torque estimation algorithm is presented. Finally, in comparison with other advanced control methods, simulation results show the superiority and robustness of the proposed scheme under complex disturbances from the downhole environment. And drilling simulation experiment results are provided to validate the robustness and adaptivity of the proposed method.


| INTRODUCTION
7][8] It is challenging to achieve stable control of the stabilized platform toolface angular position within the intricate downhole environment.Some existing works have proposed various control strategies to address the stability control issue of the stabilized platform toolface angular position.Di 9 conducted initial research work and provided a theoretical exposition of the system design, the control method, and the wellbore trajectory control mechanism for the downhole closed-loop drilling system.Tang et al. 10 designed the cascaded controller for the stabilized platform and conducted an extended experimental study on platform control algorithms and control parameters, which achieved short-term stability of the platform system under specific conditions.Wang 11 established a nonlinear dynamic model for the stabilized platform, laying the foundation for theoretical research in control methods.Wang et al. 12 formulated the nonlinear dynamic equation governing the stabilized platform.Simulation analysis based on the equation had exhibited the regular patterns of rational movement of this platform, which related to the stabilized platform's structure and torque parameters.Wang et al. 13 addressed the issue of wide-ranging variations in drilling fluid flow rates that led to oscillations and control instability in the drilling tool and introduced a turbine motor electromagnetic torque feedforward control method to facilitate stable control of the rotary steerable drilling system with substantial fluctuations in drilling fluid flow.Wang et al. 14 tackled the instability occurrences of the stabilized platform under specific angles attributed to nonlinear eccentric torque and presented an output feedback linearization control technique to eliminate nonlinearity.Zhang et al. 15 used a radial basis function neural network in conjunction with adaptive dynamic programming to devise an online iterative controller to mitigate stick-slip oscillations incorporated with an interval type-2 fuzzy logic control.Zhang et al. 16 proposed an adaptive model predictive control approach to address the instability in the bottom hole assembly during the drilling process, stemming from stratum mutation, and enhanced the stability of the drilling tool control system by alleviating controller conservativeness.Weng et al. 17 developed an integral sliding mode control (SMC) to maintain stable control of the rotary drilling system under varying disturbance conditions.Ke and Song 18 used neural networks to approximate the gradient of the cost function for optimal control input and constructed a controller based on single-neuron adaptive critic dual-heuristic programming method, eliminating vibrations in axial dimensions and torsional dimensions, and improving system stability.Tian and Song. 19roposed an integral barrier Lyapunov function method to mitigate destructive oscillations during the drilling process, guaranteeing the stability of the downhole drilling system.Niu et al. 20 introduced a variational Bayesian-based moving horizon estimation algorithm to handle the toolface estimation problem in dynamic point-the-bit rotary steerable drilling tool systems.Sheng et al. 21proposed a novel polynomial filter based on Carleman approximation to address the estimation of toolface, offering sufficient conditions for input-to-state stability using parameter-dependent linear matrix inequalities and demonstrating effectiveness through experimental validation.Sheng et al. 22 presented a refined particle filtering algorithm and introducing a residual based on the moving average method for accelerometer fault detection in the rotary steerable drilling tool system.The novel approach enhances state estimation accuracy in the presence of strong vibrationinduced measurement noise.Wan et al. 23 proposed observer-based adaptive neural network control law, which compensated for dead zones, employed a dynamic surface control strategy, and demonstrated superior performance in accurate trajectory tracking under challenging downhole conditions and parameter perturbations.Huo et al. 24 established a mathematical and friction model for the stabilized platform in a rotary steerable drilling system and introduced an improved deep deterministic policy gradient attitude control method with the stabilized platform attitude control system.
Most previous control methods of the stabilized platform could only achieve stability of the rotary steerable drilling tool under specific conditions and are not adaptable to complex downhole environments.Moreover, these methods are relatively complicated and difficult be applied to practice.Thus, to improve the system control performance and make the control effect meet various complex practical conditions, it is desired to study a control method that could improve and enhance the robustness and anti-interference capability of the stabilized platform control system.
With dramatic development in nonlinear control strategies, increasing attention has been paid to recent robust control methods.SMC is an effective and popular control algorithm for nonlinear system owing to its strong robustness. 25,26In this article, it has been tried to improve the robustness of the stabilized platform control system by combining two control techniques, that is, SMC and backstepping control method.And an NDO is further employed to estimate the uncertainties and disturbances.The Lyapunov-based approach is utilized to derive the stability condition.
The novelty of this article is to develop the BSMC integrating with the NDO for the stabilized platform and to present the dead zone torque estimation algorithm.The main contributions of this article are summarized as follows: 1.The nonlinear mathematical model of the stabilized platform is established for the fully rotary steerable drilling system, and the relationship expression among the electromagnetic torque of the motor, the flow rate, and control action is constructed to provide a basis for the toolface angle control strategy.2. A robust nonlinear controller, namely, a backstepping sliding mode controller with the NDO, is designed for the stabilized platform control system to achieve stability and improve the robustness of the system.And further its stability is theoretically proven by the Lyapunov approach.3. The dead zone torque is considered for the first time in the stabilized platform of the fully rotary steerable drilling system.A novel method for estimating dead zone torque is proposed to handle the instability issue of the stability platform caused by mechanical friction.
The organization of this article is listed as follows.In Section 2, the nonlinear mathematical model of the stabilized platform is introduced.Section 3 presents the design of the NDO.The BSMC for the stabilized platform toolface angular position is developed in Section 4. Section 5 proves the stability of the BSMC composite control system based on NDO.In Section 6, the performance of our method is illustrated through numerical simulation results comparing with four advanced control methods.Drilling simulation experiments are presented to demonstrate the effectiveness of the proposed control method in Section 7. Conclusions are finally given in the end of the article.

| NONLINEAR MATHEMATICAL MODEL OF THE STABILIZED PLATFORM
The stabilized platform is a cylindrical body that can freely rotate inside the downhole drilling tool.Successful control of the stabilized platform toolface angular position could be achieved by adjusting the electromagnetic torque of the turbine motor.The torque magnitude can be controlled by modulating the on/off state of the load resistance using a pulse-width modulation circuit.The structural diagram of the control system of the stabilized platform toolface angular position is shown in Figure 1, where θ d denotes the setpoint for the stabilized platform toolface angle, θ is the stabilized platform toolface angle, θ m represents the measured toolface angle, e is the toolface angle error, ω T (t) is the turbine speed, u c (t) represents the control law of the controller, u 2 (t), u 3 (t), u 3 (t), u 4 (t), u 5 (t) are the corresponding outputs of the digital to analog converter (D/A), pulse with modulation (PWM) generator, metal-oxide-semiconductor field-effect transistor (MOS-FET) drive circuit, and load resistor module, u(t) is the electromagnetic torque of the torque motor, Q(t) denotes drilling fluid flow rate, and d(t) is external disturbance torque.
The dynamic equation of the stabilized platform is 12 Jθ where J denotes the moment of inertia of the stabilized platform, K 1 is the viscous friction coefficient of the drilling fluid on the stabilized platform, η represents the equivalent coefficient of the eccentric moment, M 0 is the torque equilibrium point, and θ ̈and θ ̇are the angular acceleration and angular velocity of the stabilized platform, respectively.Neglecting the lag and errors in the process of D/A, analog to digital converter (A/D), measurement sensors, and MOSFET driver circuits, we get The PWM generator can be simply regarded as an amplification section and described as F I G U R E 1 Structural diagram of the control system of the stabilized platform toolface angular position.
LI ET AL.
| 1777 where K s = 1 is the amplification gain. 11Thus, Equation (3) becomes Through the MOSFET driver circuit module, we obtain where R(t) represents the time-varying load resistance, and R 0 is the resistance value of the power load resistor when the PWM pulse width is 100%.
The formula for the electromagnetic torque of an AC motor is where C T is the torque constant, I is the stator current, φ cos is the power factor of the circuit, and ϕ is the magnetic flux density.
In PWM control mode, if inductive and capacitive loads are ignored, the expression for stator current I(t) is where V is the motor phase voltage.When neglecting the voltage drop across the motor stator impedance, V = E.
The induced electromotive force of the motor is where C e is the structural parameter of the motor, and ω r is the relative angular velocity between the stator and rotor of the motor.ω r is expressed as According to the formula derived by Wang et al., 14 the relationship between the turbine speed ω T and the drilling fluid flow rate Q can be expressed as where k T is the flow-speed coefficient of the turbine.
Neglecting the compressibility of drilling fluid, the drilling fluid flow rate can be considered being equal to the displacement of the drilling pump.
Because the drilling pump is reciprocating, the pump displacement is pulsating.An airbag is installed at the pump outlet, and the buffering effect of compressible air achieves a smooth pump displacement.After being buffered by the airbag, according to the principles of hydraulics, the liquid flow rate Q varying with time t is where A, B, C, and ω b represent characteristic parameters related to the pump.Equation ( 11) can be approximately as where Q min represents the minimum displacement of the pump, Q 1 is the pulsation of the displacement, α 0 is the time lag caused by the buffering effect of the airbag, n 0 is the number of pump cylinders, and ω b is the frequency of the pump.Neglecting angular lag, Equation ( 12) is rewritten as where K b is the displacement unevenness coefficient and ω B = n 0 ω b is the pulsation frequency of the pump displacement.
Combining Equations ( 6), ( 7), ( 8), ( 9), (10), and ( 13), the expression relating the electromagnetic torque of the motor to the drilling fluid discharge and the motor load resistance is obtained as where K B is the electromagnetic torque characteristic coefficient of the motor, and its expression is Since the electromagnetic torque of the torque motor is the control driving torque of the stabilized platform, then u t T t ( ) = ( ).Substitute Equations ( 3) and (5) into Equation ( 14) and we obtain , and , and Equation ( 15) can be simplified as where K q and K ω denote the flow-torque and speed-torque characteristic coefficients of the torque motor, respectively.If we ignore the periodic fluctuation of drilling fluid flow and assume drilling fluid flow with constant value, then Equation ( 16) can be further simplified as For controller design, let x 1 and x 2 represent the toolface angle and angular velocity of the stabilized platform, respectively.The dynamic equation of the stabilized platform described by Equation ( 1) can be reformulated as a nonlinear state space model, that is, For simplicity, assume and we get where

| DESIGN OF THE NONLINEAR DISTURBANCE OBSERVER
To reduce the impact of the disturbances and enhance the control precision and robustness of the stabilized platform control system, an NDO is designed.The structural diagram illustrating the NDO with the BSMC for the stabilized platform is given in Figure 2, where u k represents the BSMC law, and u d denotes the control effort needed to counteract disturbances.
Following the NDO form from Zhang et al., 27 Dutta and Kumar, 28 we have where d ˆis the observer estimate value of d, z is the internal state of the observer, p(x) is the nonlinear function to be designed, and L(x) is the gain of the nonlinear observer satisfying F I G U R E 2 Structural diagram nonlinear disturbance observer (NDO) with the backstepping sliding mode control (BSMC) of the stabilized platform control system.
LI ET AL.
| 1779 The observation error of the NDO is defined as Generally, there is no prior knowledge of the derivative of the disturbance d.Assuming that the disturbance varies slowly relative to the observer's dynamic characteristics, we have By combining Equations ( 22) and ( 25), the dynamic equation of the observer error system can be derived as Therefore, the solution for Equation ( 26) is If L(x) > 0, the observer error exponentially converges, and the convergence speed of the observer relies on L(x).
Letting L(x) = q, where q is a constant, from Equation ( 23), we obtain Equation ( 18) can be expressed as where is the equivalent force of the disturbance on the stabilized platform.The output d ˆof the NDO is feedbacked to the gain adjustment module, and then . Thus, we get Thus, u is compensated using the estimation u d of external disturbances.

| BACKSTEPPING SLIDING MODE CONTROL DESIGN
Substituting Equation (30), the second subsystem of the system in Equation ( 18) can be rewritten as The above equation illustrates that with the NDO implementation, the system disturbance is d ˜, and the total disturbance has been diminished.
The original system in Equation ( 18) can be redescribed as The details of designing the BSMC method for system in Equation ( 32) are as follows.The tracking error is defined as follows: Substituting differentiated Equation (36) into Equation ( 18) yields The sliding mode surface is designed as where c > 0 to satisfy the Hurwitz condition.Consequently, the control output signal of the BSMC control law is expressed as where h > 0 and λ > 0 are both controller design parameters.Normally, the SMC method has chattering phenomena, which affects the accuracy of toolface angle control.To eliminate chattering, the continuous function θ s ( ) with relay characteristics is used to replace the sign s ( ) function to restrict the trajectory in a boundary layer of ideal sliding mode.Then Equation (40) can be rewritten as where

| STABILITY ANALYSIS
The minimum control requirement ensures the stability for dynamic systems.This section mainly analyzes the stability of the stabilized platform control system combining the BSMC with the NDO which is theoretically proven using Barbalat lemma 29,30 : Lemma 1. Barbalat Lemma.If a differentiable function f t ( ) has a limit as t  , and if f t ̇( ) is uniformly continuous, the f t ( ) 0  as t  .
Theorem 1.Consider the stabilized platform modeled by Equation (18), if the controller is designed as Equation (41), the tracking error and the NDO error go asymptotically to zero.
Proof: A Lyapunov function is defined for the entire closed-loop system as Taking the derivative of Equation ( 42) and substituting Equations (33), (36), (38), and (39) yield Substituting the BSMC law from Equation (41) into Equation (43) yields Since G, c, h, and q are all positive constants, V ̇0  .Equations ( 42) and (44) prove that V is positive semidefinite with a lower bound and V ̇is negative semidefinite.Furthermore, the uniform continuity of V ṡhould be checked.The derivative of V ̇is It can be demonstrated that V ̈is bounded through Equations ( 33), (34), and (26).So V ̇is uniform continuity.Now the application of Barbalat's lemma indicates V ̇0  as t  .Then, it can be concluded that e 0 1  , e 0 2  , and d ˜0  as t  .Thus, the proposed control method could guarantee the convergence of the trajectory tracking error and NDO error to be both zero.

| SIMULATION ANALYSIS
This section demonstrates the performance of the BSMC method of the stabilized platform based on NDO.Therefore, we established the nonlinear dynamic model of the stabilized platform based on Equation (18) using MATLAB.The system parameters of the stabilized platform are designed as J = 0.0285 kg•m 2 , K 1 = 0.0008 kg•m 2 /s, η = 0.5 Nm, K q = 1/12, K ω = 0.08/π, and Q = 36 L/s. 12 The primary control objective is to track the desired toolface angle.
The numerical simulation experiments are categorized into two groups.In the first group, we impose complex disturbances on the stabilized platform control system to evaluate the NDO's estimation performance.In the second group, we compare the conventional SMC, conventional SMC with NDO, backstepping control, and backstepping control with NDO considering significant variations of parameters, such as drilling fluid flow rate, toolface angle, dead zone torque, and measurement delay, to further demonstrate the superior performance of the proposed control approach.
Each parameter of the proposed control method is significantly crucial to enhance the control performance of the stabilized platform control system.The flowchart in Figure 3 illustrates the procedure for determining the controller parameters.We use a trial-and-error method to fine-tune these parameters to meet the system's performance requirements.As shown in Figure 3, the controller parameters are adjusted in three steps, that is, system modeling, controller design, and control system performance verification.The controller parameters of the BSMC method and NDO parameters are finally tuned as c = 55, G = 40, h = 70, λ = 60, and q = 30.

| Effectiveness verification
Assume that the stabilized platform system is subject to the following two sets of external disturbances: Disturbance signal 1: πt 50 cos(0.3) + 15.Disturbance signal 2: πt πt 10 cos(0.15 ) + 20 sin(0.4 ).These signals are generated using the simulation on Simulink conducting 40 s.To validate the effectiveness of the designed NDO, observations are implemented for the above two sets of external disturbances separately.
Figures 4 and 5 display the disturbance signals 1 and 2, respectively being observed using the proposed NDO.These disturbance estimation plots demonstrate that the NDO rapidly converges to the actual disturbance values with no oscillations.Furthermore, Figures 6 and 7 give the estimation error of the disturbance signals 1 and 2, respectively.In Figure 6, the maximum estimation error of the first disturbance signal is approximately 2 Nm.In Figure 7, the maximum estimation error of the second disturbance signal is approximately 1.5 Nm.These results indicate a small NDO's observation error, which demonstrates the effectiveness of the proposed observer to estimate external disturbances.

| Comparison study
To verify the superior performance of the proposed control method in terms of fast convergence and tracking accuracy under various simulated drilling conditions, our method is compared with other advanced control methods including the conventional SMC, SMC with NDO, backstepping control method, and backstepping control method with NDO.The traditional SMC law is expressed as where ε, β, and τ are controller design parameters with positive constants.These parameters are selected using a trial-and-error approach with the values ε = 15, β = 55, and τ = 70.
In actual drilling engineering scenarios, the stabilized platform is subject to various disturbance effects, such as hydraulic shocks, alternative frictions, and near-bit vibrations. 12These torque effects can be described using pulse disturbance d 1 (t), cosine-form disturbance d 2 (t), and random disturbance d 3 (t).
Here we assume that the stabilized platform experienced strong disturbance effects, with the pulse disturbance d 1 (t) amplitude 0.46 Nm, pulse width 17 s, alternating frequency 6 Hz, d 2 (t) = 0.41 cos(πt) Nm, and d 3 (t) = 0.1 rand(t) Nm. 12 The system tracking curve and tracking error curve of toolface angle are given in Figures 8 and 9 for our proposed method and four advanced control methods, respectively.
Figure 8 shows that five plots all have rapid tracking of the desired trajectory and almost overlap with each other.Oscillations in SMC with NDO are higher than other control methods.A closer inspection of the tracking curve between time interval [19.8, 21.4]  indicates that the SMC and the Backstepping controller both lack an environmental disturbance compensation mechanism and exhibit greater sensitivity to disturbances, which leads to slow oscillations in the stabilized platform toolface angle response and a relatively delayed response speed.The proposed control method has a shorter transient time.Meanwhile, the steady plot with smoother oscillation of our control method is closer to reference than other four plots after time point 20.4,which demonstrates better robustness and higher tracking accuracy of the proposed approach.
The tracking error of the stabilized platform toolface angle in Figure 9 indicates favorable performance, with the tracking error consistently around zero. Figure 9 exhibits three pairs of peak values of the tracking error, which is reasonable because the desired stabilized platform toolface angle suddenly changes from 5 rad to 6 rad, then from 6 rad to 9 rad, and finally from 9 rad to 5 rad.From closer examination between time [1,4] and [24.5, 26.5], the tracking error of the designed BSMC with NDO controller is smaller than that of the other control methods.The plot using BSMC with NDO method is the smoothest one with minimum fluctuation.
In drilling engineering, the drilling fluid flow rate must satisfy diverse geological conditions and drilling process requirements.Such variations in drilling fluid flow rate induce changes in the turbine drilling speed, causing fluctuations in the electromagnetic torque of the turbine motor and control actions, and resulting in control instability of the stabilized platform.Therefore, we evaluate the robustness of the proposed control method by setting drilling fluid flow rates Q(t) as 16, 36, 50, and 70 L/s, maintaining the stabilized platform toolface angular position setpoint θ d = 0.5π.The dynamic curves are shown in Figure 10.
Figure 10 gives the dynamic curve responses of the five control methods in different flow rates, respectively.A comparative analysis of these figures reveals that all controllers can meet the control requirements for drilling fluid flow rates ranging from 16 to 70 L/s.However, from Figure 10B when Q = 16 L/s, the system exhibits overshoot of about 11.1%, a steady-state error of approximately 2°, and fluctuations in the stabilized platform toolface angle with an amplitude of around 1.5°during steadystate operation.And the SMC with NDO control method in Figure 10C exhibits oscillations and increasing steadystate error as the flow rate ranges from 16 to 70 L/s.As shown in Figure 10A, compared with the other four control methods, the proposed control method has a fast response speed, guarantee the control error in an extremely small range, and rapidly track the trajectory in a short time under a wide range of drilling fluid flow rate conditions.The stability of the stabilized platform system is correlated with the assigned toolface angle.In practice, the assigned toolface angle is stable within certain ranges with the same control method and same parameters and potentially becomes unstable in other ranges. 11Simulations are conducted with setpoints of 0.5π, 0.6π, and π to demonstrate the adaptability of the proposed control method.The results are shown in Figure 11.
In Figure 11B, the dynamic curve of the conventional SMC has system overshoot with increasing magnitude as toolface angle setpoint gets larger.Without the NDO, the controlled variable exhibits fluctuations during steadystate operation.The control system using SMC with NDO, backstepping control method, and backstepping control with NDO all exhibit oscillations at the specified tool face angle from Figure 11C-E.And SMC with NDO has the largest oscillation amplitude with approximately 20°.However, when implementing the proposed control method, the system achieves stability across all stabilized platform toolface angle setpoints and exhibits no overshoot, no fluctuations during steady-state, smoothest curves, and accurate predictions of external disturbances, which improves the system control performance as shown in Figure 11A.
During actual drilling operations, downhole measurement sensors measure the real-time spatial dynamical attitude of the drilling tool.However, due to the data processing time delay, the torque motor could not promptly control the actuation mechanism.Consequently, this measurement delay could lead to the accumulation of control inputs and instability of the stabilized platform.We conduct a simulation by incorporating a measurement delay component and setting the toolface angle setpoint and measurement delay time at 0.5π and 30 ms, respectively.The simulation results are given in Figure 12.
In Figure 12, the dynamic processes of all controllers oscillatorily converge with measurement delay.Using backstepping control method and backstepping control with NDO, their adjustment time are approximately 7 s, and the oscillation amplitude is approximately ±20°.Moreover, SMC with NDO exhibits poor control performance with a steady-state error of approximately 30°.However, the proposed control method has an adjustment time of approximately 1.5 s and the oscillation amplitude of approximately ±2°, which illustrates that our control method outperforms other four advanced control methods.The mechanical motion of the stabilized platform relies on torque difference effect.However, there is a significant dead zone between the forward and reverse rotations of the stabilized platform.The dead zone could result in instability of the stabilized platform toolface angle.N and E are the critical points for rotational speed transitions in Figure 13 which is the conventional Coulomb friction model.When the control torque of the stabilized platform exceeds the critical point, the stabilized platform enters a rapid rotation state.Therefore, achieving stable control of the stabilized platform at the desired toolface angle primarily involves operating near zero rotational speed to make slight forward and reverse rotations to constantly adjust the angle difference.We simulate real drilling conditions by incorporating a dead zone component, and the dynamic curves of the five control methods are shown in Figure 14.
It can be seen from Figure 14 that although all control strategies successfully achieve stable control of the stabilized platform system except for SMC with NDO, the proposed control method facilitates rapid convergence with an adjustment time of approximately 2 s, no steady-state error, and an oscillation amplitude of approximately ±5°.The results shown in Figures 8-12 and 14 verify the robustness of our control method to complex disturbances and uncertainties of various levels.

| DRILLING SIMULATION EXPERIMENT
We conduct drilling simulation experiments to further validate the effectiveness of the proposed control strategy.The equipments for the drilling simulation experiment are shown in Figure 15, where the power distribution cabinet supplies power to the entire system, and offer flexibility to adjust the water flow rate and drilling tool's rotation speed. 31The variable frequency motor and the centrifugal pump are used for water circulation of the test rack which is for the functional testing of the stabilized platform.The water tank simulates a drilling fluid mud pit.
The steerable drilling tool adopts an independent dual-central processing unit structure which is responsible for controlling the stabilized platform toolface angle, and the other handles data recording.The control frequency is set to above 300 Hz, and the recording period is 1 s.
Engineering practices indicate that dead zone torque could deteriorate the stabilized platform stability.The dead zone torque is sensitive to various factors, such as the processing, assembly, and operational conditions of the stabilized platform system.To deal with this issue, we specially design a program considering these factors and | 1787 incorporate it into the main control algorithm to facilitate real-time estimation of dead zone torque.
Apply control to the stabilized platform based on the measured stabilized platform rotational rate, and make the stabilized platform rotate slowly forward by gradually increases the control effort force to u c1 (t).Subsequently, gradually decreases control force to u c2 (t), and make the stabilized platform to rotate slowly in reverse.The estimation of the torque equilibrium point of the stabilized platform is expressed as where M ˆ0 denotes the torque equilibrium point of the stabilized platform, and u c1 (t) and u c2 (t) represent the magnitudes of the dead zone torque near the stabilized platform's torque equilibrium point.
Drilling simulation experiments are conducted on the aforementioned steerable drilling tool using the proposed control method.Experimental scenario is with drill collar rotation speed 86 rpm, drilling fluid flow rate 56 L/s, and well inclination angle 2°.The experimental control program has three key steps.First, execute the selfchecking program of the steerable drilling tool to primarily evaluate the operational status of the system's sensors and other mechanical components.Next, implement the online estimation and open-loop control programs for the torque equilibrium point of the stabilized platform and the performance examination of various pivotal functional system modules.Finally, execute the stabilized platform toolface angle control program with a setpoint of 250.3°.The stabilized platform toolface angle signals are collected using the data recording device.The data are read to computer and illustrated in Figure 16, where rational speed is plotted in Figure 16A, toolface angle in Figure 16B, and toolface angle error in Figure 16C.
As shown in Figure 16A, the stabilized platform system operate self-checking mode on time interval [0, 90], search for the torque equilibrium point on [91, 413], stay in transient process on [414,445], and finally enter the toolface angle control mode after time point t = 446.According to Figure 16B, the stabilized platform system is in self-checking and torque equilibrium point search mode from 0 to 446 s.During this period, the platform is in an open-loop control state, so the toolface angle is uncontrollable.Then the stabilized platform system entries the toolface angle control mode and reaches a stable state, which is consistent with Figure 16A with the rotational speed approximately being zero.Since the drill collar is always in a rotating state and the torque acting on the stabilized platform is dynamically altering, the toolface angle curve of the stabilized platform could not be straight.In addition, long-term drilling simulation experiments indicate that the toolface angle of the fully rotary steerable drilling tool can only exhibit stable oscillations.It can be seen from Figure 16C that the maximum toolface angle error of the stabilized platform is approximately 20°, which could meet the control requirements.
Figure 17 illustrates the dynamic curve of the stabilized platform toolface angle for various angle setpoints.The stable platform toolface angle is successfully stabilized at about 245.7°, 312.6°, and 21.2°.After adjustment time of approximately 20 s, the toolface angle transits from 245.7°to 312.6°, and stabilize at 312.6°with an overshoot of around 25°.The third dynamic process gradually converges to 21.2°after time point t = 1750 with approximately 40 s adjustment time.Multiple conducted drilling simulation experiments have validated that the proposed control method can achieve stable control at all angles.
Over 200 drilling simulation experiments were conducted for the proposed control method with well inclinations from 0.5°to 5°, drilling fluid flow rates from 36 to 74 L/s, and drill collar rotation speeds from 60 to 110 rpm.The experimental results demonstrate that the rotary steerable drilling tool could maintain continuous stability at the specified angle using the proposed control method.The duration part during which the toolface angle control error is within ±10°accounts for 98% of the total time of angle control, which meets the requirements of drilling engineering.All simulation and experimental results verify robustness and adaptivity of our proposed method.In this article, the stabilized platform for the fully rotary steerable drilling system was investigated, and a nonlinear mathematical model for the stabilized platform was established.Then to enhance the robustness of the stabilized platform control system, a toolface angle control method based on BSMC with NDO was proposed.Furthermore, a novel dead-zone torque estimation algorithm was presented to eliminate instability of the stabilized platform.From the results obtained in this study, the following conclusion could been drawn: 1.According to numerical simulation, BSMC based on NDO had within ±5°steady-state oscillation in complex drilling environments, which outperformed SMC, SMC with NDO, backstepping control method, and backstepping control with NDO.Control accuracy and antidisturbance capability were successfully verified.2. Drilling simulation experiments showed that our proposed method could achieve stable control for various angles.These results confirmed the effectiveness of the control method and dead-zone torque estimation algorithm.3. Multiple conducted drilling simulation experiments demonstrated that the proposed control method in this article could meet the requirements of drilling engineering well.

F I U R E 3
Flowchart of the proposed methodology.BSMC, backstepping sliding mode control; NDO, nonlinear disturbance F I G U R E 4 Disturbance signal 1 estimation curve.

F
I G U R E 5 Disturbance signal 2 estimation curve.F I G U R E 6 Disturbance signal 1 estimation error curve.F I G U R E 7 Disturbance signal 2 estimation error curve.LI ET AL. | 1783 F I G U R E 8 Trajectory tracking of angle position.F I G U R E 9 Tracking errors of angle position.

F
I G U R E 12 Dynamic curve considering measurement delay.F I G U R E 13 Coulomb friction model.LI ET AL.

F
I G U R E 14 Dynamic curve considering dead zone.F I G U R E 15 Drilling simulation experiment apparatus.

F
I G U R E 16 Drilling simulation experiment of the stabilized platform.(A) Rotational speed of the stabilized platform.(B) Stabilized platform toolface angle.(C) Toolface angle error of the stabilized platform.