Observer-based adaptive neural network control for stabilized platform in rotary steerable system with unknown input dead-zone

Abstract

The core task of the stabilized platform in the rotary steerable system is to control the toolface angle, so that the trajectory of the whole bit can move forward to the set direction. Due to many downhole interference factors, the uncertainties of parameters related to internal friction and interference torque of stabilized platform always exist, which poses great challenges to model establishment and controller design. Moreover, the actuator dead-zone nonlinearity makes the control more complicated. Hence, a dynamic model of stabilized platform considering a variety of nonlinear factors is established, and an observer-based adaptive neural network (NN)-control law is proposed. An NN state observer is developed to cope with the uncertain states composed of unknown friction parametric uncertainties and unmodeled disturbances, the dead-zone inverse is constructed to compensate for a dead-zone, and the dynamic surface control (DSC) strategy is used to solve the “differential explosion,” all signals in the system are proved to be semi-global uniformly ultimately bounded (SGUUB) by the Lyapunov function. Finally, the MATLAB simulation is set up, and the accurate tracking effect of the system is proved by the simulation in the presence of friction, modeling error, and dead-zone nonlinearity. The validity and the superior performance of the proposed control method under the downhole harsh environment and parameter perturbation is verified by the comparison simulation experiments.

Keywords

Rotary steerable system stabilized platform dynamic surface control radial basis function neural network state observer dead-zone inverse

Introduction

Currently, the industry classifies directional drilling technology to two common groups, steerable motor drilling and rotary steerable drilling system. Several works (de Wardt and Rogers, 2011; Zhong et al., 2019) have found that steerable motor drilling is a relatively inefficient process, due to the large sliding friction resistance of drill string, the adjustment of toolface angle is rather slow, and most drilling operations rely on multiple groups of personnel to perform various tasks required for measurement manually. Compared with this, rotary steerable system offers the potential to drill longer horizontal sections faster and more safely. Its penetration rate is 50% faster, delivering substantially higher overall Rate of Penetration (ROP) especially in extended reach and horizontal applications where orienting is particularly difficult. It enhances accuracy and precision in True Vertical Depth (TVD) control and improves production in comparison with traditionally drilled horizontal well. There are fewer stuck pipe incidents, fewer washout conditions, and a lower, more constant Equivalent Circulating Density (ECD) (Molayee and Teymoori, 2006), with its value, rotary steerable system has gradually become a research hotspot.

In steerable drilling, toolface angle is an important parameter in well trajectory control (Sheng et al., 2021), Liu (2017) established directional deflection equations for steerable drilling tools and presented a process control technique for wellbore trajectory by establishing the relation curves between the build-up rate and toolface angel and the relationship between built-up rate and orientation azimuth angle of steerable drilling tool. He has confirmed build-up rate of the drilling tool determines the well trajectory curvature (WTC) and the toolface determines the distribution relationship between tool build-up rate and azimuth angle. The stabilized platform located within the drill collar is responsible for adjusting the toolface angle and the Electronic Control Unit in the platform will set the direction and magnitude of the force vector applied to the bit by the Bias Unit, adjust the toolface angle to make it track the given toolface angle command quickly and smoothly, and remain stable under external interference.

Han et al. (2004) studied the control principle of the stabilized platform of a modulated rotary steerable drilling tool. Based on the analysis of the internal motor torque control principle, a three-loop control scheme was proposed, the mathematical model of the stabilized platform was established, and the transfer function was obtained. It is proved that the stability control of the platform is a typical closed-loop system, which can be controlled by mature control theory. In order to ensure the state variables required in the process of rotary steerable drilling, Tang et al. (2004) established a state-space model of the stabilized platform, and they carried out the pole assignment of the state feedback to verify that the system can meet the control requirements of the toolface angle, which also provided the basis for the advanced control theory. In practical engineering, traditional proportional–integral–derivative (PID) control is generally used. Based on the requirements of system performance, Duan and Peng (2009) compared the traditional PID and Fuzzy Adaptive proportional–integral (PI)-variable damping control schemes of modulated rotary steering drilling tool (MRST) stabilized platform and it was proved that the fuzzy adaptive PI-variable damping schemes have better control effect than the traditional PID control, having ability to adapt to more than one diverse drilling condition. But this modeling is too simplistic, it is necessary to explain the system further. For nonlinear moment problem in stabilized platform, Wang et al. (2014) proposed an output feedback linearizing control method to eliminate the effect of nonlinear eccentric torque and change the closed-loop system into a linear one. Besides, an online estimation method is also proposed by for the unmeasurable nonlinear factors caused by eccentric torque and friction torque which will make instability of the platform. In this method, an external force will be applied to the system and it needs to record the angular velocity and acceleration at each moment to change the applied force, and use the least squares method to get a best estimate. The platform needs to interrupt the closed-loop angle control, and change to open-loop control, and the estimated time is 10–20 seconds, which is too complex. Recently, Wang et al. (2019) proposed a three-loop compound toolface control method using a Model-based Active Disturbance Rejection Control algorithm, which can obtain external disturbances and traces the directional command well. But it depends on exact model excessively, and the dead-zone nonlinear was not considered in it. It must be pointed out that due to the harsh downhole conditions and limitations of measuring equipment, nonlinear factors and modeling uncertainties exist in stabilized platform system widely, and an accurate model is impossible to get.

According to the structure of the stabilized platform, Huo et al. (2009) analyzed the causes producing friction torque. The theoretical derivation and numerical analysis of bearing friction torques, viscous friction torques, and pulsating friction torques are carried out, which provide a theoretical basis for control and decision-making of the stabilized platform. It is also pointed out that there will be some phenomena of creeping and the dead-zone in the platform due to the existence of these frictions. But there have not been any dead-zone considered in the control research of the stabilized platform of the rotary steerable system. Tang et al. (2009) found in hydraulic experiments on the dynamic characteristics of the stabilized platform that because of the factors such as the connection clearance and the sensitivity limit of the measurement system, the dead-zone nonlinear factors existing in the stabilized platform object have a great influence on the platform control results, and necessary control measures should be taken.

In the above papers, the modeling of the controlled object is simply based on the structure of the stabilized platform, without considering the model error, friction forces and the actuator dead-zone, causing these proposed controllers have no adaptive ability and cannot overcome the nonlinearities, which will quickly fail when they encounter unknown friction downhole, so their references to engineering practice are very limited. In order to improve the system control performance and make the control effect of stabilized platform can meet various complex practical working conditions, it is necessary to study the structure and control mechanism, establish the control object model including frictions, the dead-zone, and other disturbances, and discuss the control strategy and control algorithm to meet the requirements of downhole microprocessor and adapt to downhole working conditions, so as to further improve the intelligent level of stabilized platform control procedures and provide the basis for the control strategy of guided drilling tools.

Actually, nonlinear characteristics such as frictions, interferences, crawling, and the dead-zone are very common in practical mechanical systems. Their existence seriously restricts the performance of the system, and has been widely studied in the control field. When the nonlinearity is known, it can be parameterized first, but many mechanical systems are unknown or unmeasurable. In this case, many adaptive fuzzy and neural network (NN) backstepping control methods have been developed. Lu et al. (2020) presented an adaptive NN dynamic surface control (DSC) approach for the post-capture tethered spacecraft, where model uncertainties, input saturation, and state constraints exist. The NN is adopted to compensate the model uncertainties and the effects of input saturation. Wan and Huang (2020) investigated a novel adaptive output feedback control scheme for non-strict feedback nonlinear systems with uncertainties, disturbances, and asymmetric time-varying output constraints and used an adaptive fuzzy state observer to obtain the estimated values of the states. An adaptive fuzzy backstepping controller was developed for a two continuous stirred tank reactors process based on DSC approach (Xin et al., 2020). Xu (2015) proposed an adaptive NN DSC method for ultrasonic vehicle with unknown dynamics and input dead-zone. Radial basis function neural network (RBFNN) is used to approximate unknown dynamic systems, no precise model is required, which is useful in many systems with unmodeled parts (Dai et al., 2016; Van Cuong and Nan, 2016). Wu et al. (2019) took full account of the dead-zone of the actuator, proposing a robust adaptive backstepping controller for the photoelectric gyro stabilized platform widely used in rockets and missiles, and the controller can effectively compensate for the uncertainties caused by the parameter uncertainties and the dead-zone of the actuator. As a stabilized platform similar to rotary steering system, the research is very valuable.

The innovation of this paper lies in the fact that we aim to make the output toolface angle track the input command quickly and smoothly under uncertain conditions. To accomplish this mission, a three-loop control method is designed and a backstepping control method is proposed. To avoid “integral explosion,” speed up the operation, a DSC method is introduced. The adaptive law of NN is derived by proper transformation to approximate the friction force, unmodeled term is handled by the state observer, and the dead-zone will be compensated in the process. Finally, the stability of the system is analyzed, and the effectiveness of the control law is verified by some simulation experiments.

The main contributions of this paper are summarized as follows:

Motivated by the above studies, a three loop-control method of stabilized platform is proposed; unknown friction, unmodeled disturbances, and dead-zone nonlinear are catered for the system.

Different from Wang et al. (2014), the RBFNN is used to overcome the influence of unknown friction torque instead of using a constant external force or the least square method for estimation calculation, the redundant calculation is reduced and it is unnecessary to record the speed and acceleration of the platform all the time. For the unmodeled part, a state observer is introduced to compensate it. Then, an observer-based NN controller is designed to achieve asymptotic tracking performance, mean while ensuring the stability of the closed-loop system. This method appears for the first time in the study of stabilized platform control, in this way, not only in the case of friction and external interferences system can maintain stability, and even in the case of error caused by parameter changes can have adaptability.

The dead-zone is considered for the first time in the stabilized platform of the rotary steerable system. It is difficult to obtain the dead-zone in the stabilized platform, and it is impossible to measure the specific structure of the dead-zone so far, so parameterization of dead-zone is avoided, for the unknown slope and width, using the dead-zone smooth inverse can obtain perfect tracking performance and well compensate the influence of the nonlinearity of the dead-zone on the system.

This paper is arranged as follows. In section “System nonlinear model,” the structure and nonlinear model of the stabilized platform will be presented and influences of the dead-zone and friction torques will be discussed. Section “Adaptive NN controller design” gives the adaptive NN dynamic surface controller; the unmodeled part and the dead-zone are solved by the state observer and smooth inverse, respectively, and then the stability of the system will be proved. The simulation results will be given in section “Simulation results” and section “Conclusion” draws a conclusion.

System nonlinear model

The structure of stabilized platform

As it is shown in Figure 1, the stabilized platform is mounted on bearings so that it is able to rotate independently relative to the drill collar, which spins with the bottom hole assembly (BHA) and drill string about a common longitudinal axis (Alturbeh et al., 2019; Wang et al., 2018).

It is composed of upper turbine motor, electronic control unit, turbine torque motor, and position sensors. The two motors here are permanent magnet synchronous motor (PMSM). The control unit provides a regulated voltage power supply using energy extracted from the inner annular mud flow (between the collar and the control unit outside diameter) pumped from the surface using the torquer rotor-mounted impellers. The upper one supplies constant power for all the platforms because of its constant load resistance, the output torque will not change. As a torque generator, the lower turbine torque motor adjusts the output torque by changing the current, so as to change the toolface and speed. Position sensors are in the electronic control unit, which include gyroscope, linear accelerometer, amplifier, and so on (Li et al., 2015; Noureldin et al., 2000).

Figure 2 shows the three-loop control method, including position loop, current loop, and speed loop. The position loop controls the tool rotation by checking the feedback signal of the accelerometer to make the steering tool reach the ideal toolface position. The current loop can ensure the load current generated by the lower generator meets the control requirements, so that the appropriate torque can be reached. In the speed loop, the gyroscope is used for detection and feedback to ensure the stable working state of the stabilized platform. When the toolface angle changes, motor speed loop outputs a voltage signal different from the original one because of the toolface angle deviation, and the difference between this voltage through pulse-width modulation (PWM) circuit of PWM effects on MOS tube and exports a new current. The electromagnetic torque emitted by the lower motor varies with the armature current, drive disk valve rotating shaft to control the direction of the changes, until the toolface angle error goes to zero, then a control action is completed. In this way, the stabilized platform can tack the command toolface angle quickly, and steady the control axis of the stabilized platform at the required toolface angle. So broadly, we regard a bias voltage signal as the input signal, the toolface angle as output signal and the system model can be obtained.

Mathematics model of the stabilized platform

The lower turbine uses field-oriented control (FOC) method to adjust the output torque, in the standard version of FOC schemes, the current component is used as the torque control quantity. Under constant rotor flux amplitude, it adjusts the torque directly. Ignoring the coupling of two motors, according to Newton’s second law, the dynamic model of the motor can be summarized as

{\begin{matrix} L \frac{di}{dt} + iR + C_{e} ω = u \\ J \frac{d ω}{dt} = C_{m} i - T_{f} - e \\ \frac{d θ}{dt} = ω \end{matrix}

(1)

where u denotes motor voltage, i denotes the motor current, R denotes the motor resistance, L denotes the motor inductance, J denotes the motor inertia moment, $C_{e}$ denotes the electromotive force coefficient, and $C_{m}$ denotes the motor torque coefficient. $T_{f}$ represents the frictional disturbance torque in the stabilized platform, and e represents the modeling error, which is a bounded unknown term, including the error caused by modeling uncertainty and parameter deviation. From this, the relationship among the toolface angle, angular velocity, acceleration and torque, voltage, friction, and error can be obtained as

{\begin{matrix} \frac{d θ}{dt} = ω \\ \frac{LJ}{C_{m}} \frac{d^{2} ω}{d t^{2}} = U - \frac{RJ}{C_{m}} \frac{d ω}{dt} - \frac{T_{f} R}{C_{m}} - \frac{eR}{C_{m}} - \frac{L {\overset{\cdot}{T}}_{f}}{C_{m}} - \frac{L \overset{\cdot}{e}}{C_{m}} - C_{e} ω \end{matrix}

(2)

On the basis of the principle of PWM power converter (Liang and Nwankpa, 1999), when the control voltage changes, the output voltage will not change until the next cycle, so we regard it as a time-delay part and its transfer function is $G_{PWM} (s) = K_{PWM} e^{- Ts}$ , where T is the switching period of PWM square wave, and $K_{PWM}$ is the ratio of power supply voltage and triangle amplitude voltage. Due to the switching frequency of the PWM square wave is very high, much higher than the cut-off frequency of the motor, the conversion circuit is simplified to an amplification link $K_{PWM}$ . The gyroscope can be attributed to a proportional link. Besides that, the proportional coefficient of torque and current of generator is expressed by $K_{E}$ . Thus, the block diagram of the controlled object can be obtained which is shown in Figure 3.

Define state variables as $x_{1} = θ, x_{2} = ω$ , where $θ$ is the toolface angle and $ω$ is the angular velocity. In general, the inductance of motor is very small, which is ignored.

Combined with the above, the stabilized platform state equation is obtained as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - \frac{(K_{PWM} K_{W} K_{E} C_{m} + C_{e} C_{m})}{RJ} x_{2} - \frac{C_{m}}{RJ} T_{f} \\ y = x_{1} \end{matrix} - \frac{C_{m}}{RJ} e + \frac{K_{PWM} K_{E} C_{m}}{RJ} u

(3)

In addition, in theoretical analysis, the phenomenon of dead-zone in PMSM-based drive system is often ignored, but it is proved by Huo et al. (2009) and Tang et al. (2009) that the nonlinearity of dead-zone is inevitable because of frictions, actuator clearances, and other factors (Wang and Wang, 2016). We describe the dead-zone term as follows, although it is a simplified model, it can describe the servomotor well

u = D (v) = {\begin{matrix} m_{r} (v - b_{r}), v \geq b_{r} \\ 0, - b_{l} < v < b_{r} \\ m_{l} (v + b_{l}), u \leq - b_{l} \end{matrix}

(4)

where v is actual input signal, $D (v)$ represents the dead-zone function, and $u = D (v)$ is the output of dead-zone, which is unavailable for measurement.

Friction torque analysis

Generally, the existence of nonlinear characteristic of friction torques to the control axis cannot be ignored in rotary steerable drilling stabilized platform control. In this paper, the friction model analyzed by Huo et al. (2009) is introduced to describe the nonlinear friction characteristics in rotary steerable drilling stabilized platform. Considering the friction torque of three parts on the stabilized platform.

Friction torque of the upper and lower main support bearing

Structurally, stabilized platform is fixed inside the drill collar through the upper and lower support bearing, during drilling, the bearing rotates with the drill collar and the stabilized platform remains stationary; thus, the stabilized platform and the support bearings are in a state of relative rotation with relatively low rotational speed, so the mechanical friction torque between these two bearings and the stabilized platform are in the same direction, and the control torque in opposite direction. The lubrication of bearings is constant ignoring the change of temperature, the rolling bearing friction torque is given by

M_{1} = sign (n_{0} - n) \cdot r_{1} \cdot G \cdot (f_{u 1} \sin α + f_{u 2} \cos α)

(5)

Friction torque of the upper and lower turbine motors bearing

When mud is driving two turbine motors, there will be dynamic frictions, and because the upper turbine generator is the same as the rotation direction, the torque motor is opposite to the rotation direction, they will provide the opposite two friction torque $M_{2}$ and $M_{3}$

M_{2} = r_{2} \cdot G_{2} \cdot (f_{u 1} \sin α + f_{u 2} \cos α)

(6)

M_{3} = r_{3} \cdot G_{3} \cdot (f_{u 1} \sin α + f_{u 2} \cos α)

(7)

Viscous friction torque caused by drilling collar annular mud

As rotating of drill collar, drill collar drives rotating annulus mud and also drives the rotary lower valve plate during drilling. So, the rotation of the slurry produces a viscous friction torque $M_{4}$ for the steady platform

M_{4} = 4 π μ L \frac{{r_{0}}^{2} r^{2}}{{r_{0}}^{2} - r^{2}} (ω_{0} - ω)

(8)

where G is the total weight of stabilized platform, $G_{2}$ is the weight of upper torquer, and $G_{3}$ is the weight of lower torquer. $r_{0}$ and r are the radius within the drill collar and the radius outside the platform, respectively, and $r_{1}$ is the average radius of bearing friction. Because the sizes of the upper and lower turbine motors are the same, the average friction radius of the turbine motor bearings is the same and $r_{2} = r_{3}$ , $μ$ is the slurry viscosity coefficient, and L is the length of mud. $f_{u 1}$ and $f_{u 2}$ are the rolling friction coefficient of radial bearings and axial bearings separately, and $α$ is the inclination. $n_{0}$ is the rotating speed of the drill collar and stabilized platform.

Assuming that there is no coupling among all parts, they act simultaneously at runtime, and there is a total friction torque

T_{f} = M_{1} + M_{2} - M_{3} + M_{4}

(9)

The state variables $x_{1} = θ, x_{2} = ω$ , angular velocity, and speed satisfy the equation $2 π n = ω$ , so the friction torque can be expressed as

\begin{matrix} T_{f} = sign (n_{0} - \frac{x_{2}}{2 π}) \cdot r_{1} \cdot G \cdot (f_{u 1} \sin α + f_{u 2} \cos α) + \\ (r_{2} \cdot G_{2} - r_{3} \cdot G_{3}) \cdot (f_{u 1} \sin α + f_{u 2} \cos α) \\ + 4 π μ L \frac{{r_{0}}^{2} r^{2}}{{r_{0}}^{2} - r^{2}} (2 π n_{0} - x_{2}) \end{matrix}

(10)

Adaptive NN controller design

Based on the stabilized platform’s model (3), a dead-zone model (4), and the friction torque (9), we rewrite the system model in a state-space form as

{\begin{matrix} \overset{\cdot}{X} = AX + B [D (v) + f (X) + d_{e}] \\ y = C^{T} X \end{matrix}

(11)

where $X = [x_{1}, x_{2}]^{T}$ is the state vector

\begin{matrix} A = [\begin{matrix} 0 \begin{matrix} 1 \end{matrix} \\ 0 \begin{matrix} 0 \end{matrix} \end{matrix}], B = [\begin{matrix} 0 \\ \frac{K_{PWM} K_{E} C_{m}}{RJ} \end{matrix}], C = {[1 \begin{matrix} 0 \end{matrix}]}^{T} \\ f (X) = (K_{W} + \frac{C_{e}}{K_{PWM} K_{E}}) x_{2} - \frac{1}{K_{PWM} K_{E}} T_{f} \\ d_{e} = - \frac{1}{K_{PWM} K_{E}} e \end{matrix}

and $d_{M}$ is an unknown positive value, $| d_{e} | \leq d_{M}$ , and $D (v)$ is the input signal with an unknown dead-zone.

Since there are model errors and disturbances in the system, making some available states, causing poor control effect and even instability. The state observer is a dynamic system, which estimates the state variables according to feedback of the input and output errors of the controlled system and a state observer is incorporated to estimate the unmeasured states here, which is always an effective method to deal with unmodeled dynamics (Lin et al., 2020; Zhu et al., 2020).

The basic state observer is designed as follows

{\begin{matrix} \overset{\cdot}{\hat{X}} = A \hat{X} + B [\hat{f} (\hat{X}) + D (v)] + K (y - C^{T} \hat{X}) \\ \hat{y} = C^{T} \hat{X} \end{matrix}

(12)

where $\hat{X} = [{\hat{x}}_{1}, {\hat{x}}_{2}]^{T}$ is the state observation

K = [\begin{matrix} k_{1} \\ k_{2} \end{matrix}]

is the observer gain vector, and

A = [\begin{matrix} - k_{1} \begin{matrix} 1 \end{matrix} \\ - k_{2} \begin{matrix} 0 \end{matrix} \end{matrix}]

$\hat{f} (\hat{X})$ is the state estimate of $f (X)$ , and $\tilde{y} = y - C^{T} \hat{X}$ is the estimate error of y. By choosing appropriate parameter K to ensure A is a Hurwitz matrix, such that

A^{T} P + PA = - 2 Q

(13)

and there P and Q are any positive definite matrices.

Dead-zone equivalence inverse

As it is discussed, there is a dead-zone nonlinear part in the system, which is described as

u = D (v) = {\begin{matrix} m_{r} (v - b_{r}), v \geq b_{r} \\ 0, \begin{matrix} b_{l} < v \end{matrix} < b_{r} \\ m_{l} (v + b_{l}), v \leq - b_{l} \end{matrix}

(14)

before designing the controller, some assumptions (Tong et al., 2016) should be given as follows.

Assumption 1

Unknown constants satisfy $b_{r} \geq 0$ , $b_{l} \leq 0$ , which represent the right break-point and left break-point of the dead-zone.

Assumption 2

Unknown slopes of the dead-zone $m_{r}$ and $m_{l}$ satisfy $m_{r} > 0, m_{l} > 0$ , which represent the right slope and the left slope of the dead-zone, respectively. And there exist some known positive constants $m_{r min}$ , $m_{r max}$ , $m_{l min}$ , and ${m_{l}}_{max}$ , and the ranges of them are $m_{r} \in [m_{r min}, m_{r max}]$ and $m_{l} \in [m_{l min}, m_{l max}]$ .

Define $D^{- 1} (v)$ as the inverse function of $D (v)$ , $u_{d}$ is the input of the plant without a dead-zone, then we can obtain the control signal v from the dead-zone inverse (Shojaei et al., 2019)

v = D^{- 1} (u_{d}) = \frac{u_{d} + {\hat{b}}_{mr}}{{\hat{m}}_{r}} δ + \frac{u_{d} - {\hat{b}}_{ml}}{{\hat{m}}_{l}} (1 - δ)

(15)

where ${\hat{b}}_{mr}, {\hat{b}}_{ml}, {\hat{m}}_{r}, {\hat{m}}_{l}$ are the estimates of $b_{mr}, b_{ml}, m_{r}, m_{l}$ , and

δ = {\begin{matrix} 1, u_{d} \geq 0 \\ 0, u_{d} < 0 \end{matrix}

Based on that, one resulting error between them can be obtained as

\begin{matrix} u - u_{d} \\ = D (v) - u_{d} \\ = ({\tilde{b}}_{mr} - \frac{u_{d} + {\hat{b}}_{mr}}{{\hat{m}}_{r}} {\tilde{m}}_{r}) δ + ({\tilde{b}}_{ml} - \frac{u_{d} - {\hat{b}}_{ml}}{{\hat{m}}_{l}} {\tilde{m}}_{l}) (1 - δ) + ε_{d} \end{matrix}

(16)

where ${\tilde{b}}_{mr} = b_{mr} - {\hat{b}}_{mr}$ , ${\tilde{b}}_{ml} = b_{ml} - {\hat{b}}_{ml}$ , ${\tilde{m}}_{r} = m_{r} - {\hat{m}}_{r}$ , and ${\tilde{m}}_{l} = m_{l} - {\hat{m}}_{l}$ are dead-zone’s parameter errors, and $ε_{d} = - m_{r} χ_{r} (v - d_{r}) - m_{l} χ_{l} (v - d_{l})$ is bounded, where

χ_{r} = {\begin{matrix} 1, 0 \leq v < b_{r} \\ 0, otherwise \end{matrix}, χ_{l} = {\begin{matrix} 1, b_{l} \leq v < 0 \\ 0, otherwise \end{matrix}

(17)

RBF NNs approximator system

RBFNN has been proved by Hartman et al. (1990) to be universal approximators of nonlinear functions; the control architecture consists of a linear input layer, a linear output layer, and a nonlinear hidden layer with neurons, each having a radial basis activation function. In this paper, the Gaussian activation function is used to design the algorithm, which is given by

σ_{j} (x) = \exp (- \frac{‖ {X - c_{j} ‖}^{2}}{2 {b_{j}}^{2}}), j = 1, \dots, m

(18)

the parameter $c_{j}$ is the center of the hidden unit and $b_{j}$ is the width of it. W denotes the connection weight of RBFNN and $σ_{j} (X)$ is the activation function, so the output node can be expressed as

y = W^{T} σ_{j} (X)

(19)

Lemma 1

Suppose $f (X)$ is a continuous function, which is defined on a compact set $Ω \in R^{n}$ , there exist an ideal optimal weight $W^{*}$ letting the approximation error $ε$ be satisfied for any given small positive number $κ$ , such that

| ε | = | f (X) - \hat{f} (\hat{X}, W^{*}) | < κ

(20)

Define optimal weight as $W^{*} = \arg \min_{W \in R} {\sup_{σ \in Ω} | f (X) - W_{j}^{T} σ (X) |}$ .

Besides, $W^{*} - \hat{W} = \tilde{W}$ , where $\hat{W}$ is the observe value of W, and $\tilde{W}$ is the error of W.

According to Lemma 1, the NN estimation value of $f (X)$ can be written as

\hat{f} (\hat{X}) = {\hat{W}}^{T} σ_{j} (\hat{X})

(21)

$σ_{j} (\hat{X})$ is the Gaussian function for approximating $\hat{f} (\hat{X})$ , so the errors of $f (X)$ can be defined as

ε = f (X) - {\hat{W}}^{T} σ_{j} (\hat{X})

(22)

RBF NNs state observer design

The RBFNNs state observer is designed as

{\begin{matrix} \dot{\hat{X}} = A \hat{X} + B [{\hat{W}}^{T} σ_{j} (\hat{X}) + D (v)] + K (y - C^{T} \hat{X}) \\ \hat{y} = C^{T} \hat{X} \end{matrix}

(23)

Define observation error as $\tilde{X} = X - \hat{X}$ , from equations (10) and (21), we have observer error dynamic equation

\begin{array}{l} \dot{\tilde{X}} = \dot{X} - \dot{\hat{X}} \\ = A \tilde{X} + B [f (X) - \hat{f} (\hat{X}) + d_{e}] \\ = A \tilde{X} + B [ε + d_{e}] \\ = A \tilde{X} + δ \end{array}

(24)

where $δ = ε + d_{e}$ .

To evaluate the property of the state observer, the following Lyapunov function candidate $V_{0}$ is considered as

V_{0} = \frac{1}{2} {\tilde{X}}^{T} P \tilde{X}

(25)

where $P = P^{T} > 0$ is a positive definite matrix. Now, we have its time derivative

\begin{matrix} {\overset{\cdot}{V}}_{0} = \frac{1}{2} {\overset{\cdot}{\tilde{X}}}^{T} P \tilde{X} + \frac{1}{2} {\tilde{X}}^{T} P \overset{\cdot}{\tilde{X}} \\ = \frac{1}{2} {\tilde{X}}^{T} [A^{T} P + PA]] X + {\tilde{X}}^{T} P δ \\ = - {\tilde{X}}^{T} Q \tilde{X} + {\tilde{X}}^{T} P δ \end{matrix}

(26)

Using inequality $2 ab \leq a^{2} + b^{2}$ , we can have

{\tilde{X}}^{T} P δ \leq \frac{1}{2} ({‖ \tilde{X} ‖}^{2} + {‖ P δ ‖}^{2})

(27)

In addition, the following inequality is true for a positive definite matrix Q (Shojaei et al., 2019)

λ_{min} (Q) ‖ \tilde{X} ‖^{2} \leq {\tilde{X}}^{T} Q \tilde{X} \leq λ_{max} (Q) ‖ \tilde{X} ‖^{2}

(28)

Then, we have

\begin{array}{l} {\dot{V}}_{0} = - {\tilde{X}}^{T} Q \tilde{X} + {\tilde{X}}^{T} P δ \\ \leq - λ_{\min} (Q) {‖ \tilde{X} ‖}^{2} + \frac{1}{2} ({‖ \tilde{X} ‖}^{2} + {‖ P δ ‖}^{2}) \\ \leq - (λ_{\min} (Q) - \frac{1}{2}) {‖ \tilde{X} ‖}^{2} + \frac{1}{2} {‖ P δ ‖}^{2} \end{array}

(29)

It is noticeable that if $λ_{min} (Q) - (1 / 2) > 0$ , in the sense of the Lyapunov function we can know the state observer equation (29) is asymptotically stable, and the inequality (29) will be used for stability proof of the designed controller later.

DSC controller design

In order to solve the “differential explosion” problem of traditional backstepping control algorithm, DSC technology will be introduced (Han et al., 2019). Regard $β_{1}$ as output of the low-pass filter $1 / τ s + 1$ , and $α_{1}$ is the virtual input, satisfying

{\begin{matrix} τ {\overset{\cdot}{β}}_{1} + β_{1} = α_{1} \\ β_{1} (0) = α_{1} (0) \end{matrix}

so we can obtain

{\overset{\cdot}{β}}_{1} = \frac{α_{1} - β_{1}}{τ}

(30)

The filter error generated by the definition is

e_{1} = β_{1} - α_{1}

(31)

Step 1. Define toolface angle tracking error as $z_{1} = y - y_{d}$ , the second error is defined as

z_{2} = {\hat{x}}_{2} - β_{1}

(32)

and

\begin{matrix} {\overset{\cdot}{z}}_{1} = \overset{\cdot}{y} - {\overset{\cdot}{y}}_{d} \\ = x_{2} - {\overset{\cdot}{y}}_{d} \\ = {\hat{x}}_{2} + {\tilde{x}}_{2} - {\overset{\cdot}{y}}_{d} \end{matrix}

(33)

By substituting equations (31) and (32) into equation (33), we have

{\overset{\cdot}{z}}_{1} = z_{2} + α_{1} + e_{1} + {\tilde{x}}_{2} - {\overset{\cdot}{y}}_{d}

(34)

Consider the Lyapunov function candidate as

V_{1} = V_{0} + \frac{1}{2} {e_{1}}^{2} + \frac{1}{2} {z_{1}}^{2}

(35)

where $P = P^{T} > 0$ is a positive definite matrix. By utilizing equations (29) and (34), we can calculate its time derivative

\begin{matrix} {\overset{\cdot}{V}}_{1} = {\overset{\cdot}{V}}_{0} + e_{1} {\overset{\cdot}{e}}_{1} + z_{1} {\overset{\cdot}{z}}_{1} \\ = {\overset{\cdot}{V}}_{0} + e_{1} {\overset{\cdot}{e}}_{1} + z_{1} (z_{2} + α_{1} + e_{1} + {\tilde{x}}_{2} - {\overset{\cdot}{y}}_{d}) \\ \begin{matrix} \leq (- λ_{min} (Q) - \frac{1}{2}) {‖ \tilde{X} ‖}^{2} + \frac{1}{2} {‖ P δ ‖}^{2} + e_{1} {\overset{\cdot}{e}}_{1} \\ + z_{1} (z_{2} + α_{1} + e_{1} + {\tilde{x}}_{2} - {\overset{\cdot}{y}}_{d}) \end{matrix} \end{matrix}

(36)

In order to ensure the derivative is negative, the virtual control law is set as

α_{1} = {\overset{\cdot}{y}}_{d} - c_{1} z_{1}

(37)

where $c_{1}$ is a positive constant to be designed.

Using $2 ab \leq a^{2} + b^{2}$ and Young’s inequality, we have

z_{1} {\tilde{x}}_{2} \leq \frac{1}{2} ({z_{1}}^{2} + {| {\tilde{x}}_{2} |}^{2})

(38)

z_{1} e_{1} \leq {z_{1}}^{2} + \frac{1}{4} {e_{1}}^{2}

(39)

So substituting equations (37)–(39) into equation (36), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = - {\tilde{X}}^{T} Q \tilde{X} + {\tilde{X}}^{T} P δ + e_{1} {\overset{\cdot}{e}}_{1} + z_{1} z_{2} + z_{1} e_{1} + z_{1} {\tilde{x}}_{2} - c_{1} {z_{1}}^{2} \\ \leq - [λ_{min} (Q) - \frac{1}{2}] ‖ \tilde{X} ‖^{2} + \frac{1}{2} {\tilde{x}}_{2}^{2} + \frac{1}{2} ‖ P δ ‖^{2} + e_{1} {\overset{\cdot}{e}}_{1} \\ + z_{1} z_{2} - (c_{1} - \frac{3}{2}) {z_{1}}^{2} + \frac{1}{4} {e_{1}}^{2} \\ \leq - [λ_{min} (Q) - 1] ‖ \tilde{X} ‖^{2} + \frac{1}{2} ‖ P δ ‖^{2} + e_{1} {\overset{\cdot}{e}}_{1} \\ + z_{1} z_{2} - (c_{1} - \frac{3}{2}) {z_{1}}^{2} + \frac{1}{4} {e_{1}}^{2} \end{matrix}

(40)

Step 2. The velocity error is $z_{2} = {\hat{x}}_{2} - β_{1}$ , derived from equations (12), (16), (21), and (22), its derivative can be expressed as

\begin{matrix} {\overset{\cdot}{z}}_{2} = {\overset{\cdot}{\hat{x}}}_{2} - {\overset{\cdot}{β}}_{1} \\ = D (v) + \hat{f} (\hat{x}) + K {\tilde{x}}_{1} - {\overset{\cdot}{β}}_{1} \\ = (u_{d} + ({\tilde{b}}_{mr} - \frac{u_{d} + {\hat{b}}_{mr}}{{\hat{m}}_{r}} {\tilde{m}}_{r}) δ + ({\tilde{b}}_{ml} - \frac{u_{d} - {\hat{b}}_{ml}}{{\hat{m}}_{l}} {\tilde{m}}_{l}) \\ (1 - δ) + ε_{d}) + {W^{*}}^{T} σ ({\hat{x}}_{2}) + ε' + K {\tilde{x}}_{1} - {\overset{\cdot}{β}}_{1} \\ = u_{d} + ({\tilde{b}}_{mr} - \frac{u_{d} + {\hat{b}}_{mr}}{{\hat{m}}_{r}} {\tilde{m}}_{r}) δ + ({\tilde{b}}_{ml} - \frac{u_{d} - {\hat{b}}_{ml}}{{\hat{m}}_{l}} {\tilde{m}}_{l}) \\ (1 - δ) + ε_{d} + W^{T} σ ({\hat{x}}_{2}) + {\tilde{W}}^{T} σ ({\hat{x}}_{2}) + ε' + K {\tilde{x}}_{1} - {\overset{\cdot}{β}}_{1} \end{matrix}

(41)

where $W^{*} = W + \tilde{W}$ , $ε' = ε^{*} - ε$ , and for any given small positive constants $\bar{ε}$ , it satisfies $| ε' | < \bar{ε}$ .

Consider the following Lyapunov function

\begin{matrix} V_{2} = V_{1} + \frac{1}{2} {z_{2}}^{2} + \frac{1}{2 γ} {\tilde{W}}^{T} \tilde{W} + \frac{1}{2 ρ_{1}} {\tilde{b}}_{mr}^{2} + \frac{1}{2 ρ_{2}} {\tilde{b}}_{ml}^{2} \\ + \frac{1}{2 ρ_{3}} {\tilde{m}}_{r}^{2} + \frac{1}{2 ρ_{4}} {\tilde{m}}_{l}^{2} \end{matrix}

(42)

where $γ, ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}$ are the positive parameters to be chosen, then we can obtain the differential equation of $V_{2}$

\begin{array}{l} {\dot{V}}_{2} = {\dot{V}}_{1} + z_{2} {\dot{z}}_{2} - \frac{1}{γ} {\tilde{W}}^{T} \dot{W} + \frac{1}{ρ_{1}} {\tilde{b}}_{m r} {\dot{\tilde{b}}}_{m r} + \frac{1}{ρ_{2}} {\tilde{b}}_{m l} {\dot{\tilde{b}}}_{m l} + \frac{1}{ρ_{3}} {\tilde{m}}_{r} {\dot{\tilde{m}}}_{r} + \frac{1}{ρ_{4}} {\tilde{m}}_{l} {\dot{\tilde{m}}}_{l} \\ = {\dot{V}}_{1} + z_{2} [(u_{d} + ({\tilde{b}}_{m r} - \frac{u_{d} + {\hat{b}}_{m r}}{{\hat{m}}_{r}} {\tilde{m}}_{r}) δ + ({\tilde{b}}_{m l} - \frac{u_{d} - {\hat{b}}_{m l}}{{\hat{m}}_{l}} {\tilde{m}}_{l}) (1 - δ) + ε_{d}) + W^{T} σ ({\hat{x}}_{2}) + {\tilde{W}}^{T} σ ({\hat{x}}_{2}) \\ + ε^{'} + K {\tilde{x}}_{1} - {\dot{β}}_{1}] - \frac{1}{γ} {\tilde{W}}^{T} \dot{W} + \frac{1}{ρ_{1}} {\tilde{b}}_{m r} {\dot{\tilde{b}}}_{m r} + \frac{1}{ρ_{2}} {\tilde{b}}_{m l} {\dot{\tilde{b}}}_{m l} + \frac{1}{ρ_{3}} {\tilde{m}}_{r} {\dot{\tilde{m}}}_{r} + \frac{1}{ρ_{4}} {\tilde{m}}_{l} {\dot{\tilde{m}}}_{l} \\ = {\dot{V}}_{1} + z_{2} [u_{d} + ε_{d} + W^{T} σ ({\hat{x}}_{2}) + ε^{'} + K {\tilde{x}}_{1} - {\dot{β}}_{1}] + \frac{1}{γ} {\tilde{W}}^{T} [γ z_{2} σ ({\hat{x}}_{2}) - \dot{W}] + \frac{1}{ρ_{1}} {\tilde{b}}_{m r} ({\dot{\tilde{b}}}_{m r} + ρ_{1} z_{2} δ) \\ + \frac{1}{ρ_{2}} {\tilde{b}}_{m l} [{\dot{\tilde{b}}}_{m l} + ρ_{2} z_{2} (1 - δ)] + \frac{1}{ρ_{3}} {\tilde{m}}_{r} ({\dot{\tilde{m}}}_{r} - \frac{u_{d} + {\hat{b}}_{m r}}{{\hat{m}}_{r}} ρ_{3} z_{2} δ) + \frac{1}{ρ_{4}} {\tilde{m}}_{l} [{\dot{\tilde{m}}}_{l} - \frac{u_{d} - {\hat{b}}_{m l}}{{\hat{m}}_{l}} ρ_{4} z_{2} (1 - δ)] \end{array}

(43)

So far, the control input signal and adaptive laws can be determined as

{\begin{matrix} u_{d} = - c_{2} z_{2} - z_{1} - K {\tilde{x}}_{1} - W^{T} σ ({\hat{x}}_{2}) + \frac{α_{1} - β_{1}}{τ} \\ \overset{\cdot}{W} = γ z_{2} σ ({\hat{x}}_{2}) - ϕ W \end{matrix}

(44)

in order to deal with the dead-zone, the adaptive laws are employed as

{\begin{matrix} {\overset{\cdot}{\tilde{b}}}_{mr} = a_{1} {\hat{b}}_{mr} - ρ_{1} z_{2} δ \\ {\overset{\cdot}{\tilde{b}}}_{ml} = a_{2} {\hat{b}}_{ml} - ρ_{2} z_{2} (1 - δ) \\ {\overset{\cdot}{\tilde{m}}}_{r} = a_{3} {\hat{m}}_{r} + \frac{u_{d} + {\hat{b}}_{mr}}{{\hat{m}}_{r}} ρ_{3} z_{2} δ \\ {\overset{\cdot}{\tilde{m}}}_{l} = a_{4} {\hat{m}}_{l} + \frac{u_{d} - {\hat{b}}_{ml}}{{\hat{m}}_{l}} ρ_{4} z_{2} (1 - δ) \end{matrix}

(45)

where the positive constants $c_{2}, a_{1}, a_{2}, a_{3}, a_{4}$ will be designed later.

From equations (40), (44), and (45), equation (43) can be rearranged as

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - [λ min (Q) - 1] ‖ \tilde{X} ‖^{2} + \frac{1}{2} ‖ P δ ‖^{2} + e_{1} {\overset{\cdot}{e}}_{1} \\ - (c_{1} - \frac{3}{2}) {z_{1}}^{2} - c_{2} {z_{2}}^{2} + z_{2} ε_{d} + z_{2} ε' + \frac{1}{4} e^{2} \\ + \frac{ϕ}{γ} {\tilde{W}}^{T} W + \frac{a_{1}}{ρ_{1}} {\tilde{b}}_{mr} {\hat{b}}_{mr} + \frac{a_{2}}{ρ_{2}} {\tilde{b}}_{ml} {\hat{b}}_{ml} + \frac{a_{3}}{ρ_{3}} {\tilde{m}}_{r} {\hat{m}}_{r} \\ + \frac{a_{4}}{ρ_{4}} {\tilde{m}}_{l} {\hat{m}}_{l} \end{matrix}

(46)

Using inequality $2 ab \leq a^{2} + b^{2}$ , we have

z_{2} ε' \leq \frac{1}{2} ({z_{2}}^{2} + {ε'}^{2})

(47)

z_{2} ε_{d} \leq \frac{1}{2} ({z_{2}}^{2} + {ε_{d}}^{2})

(48)

According to Young’s inequality, we have

{\tilde{b}}_{mr} {\hat{b}}_{mr} = {\tilde{b}}_{mr} (b_{mr} - {\tilde{b}}_{mr}) \leq \frac{1}{2} ({b_{mr}}^{2} - {\tilde{b}}_{mr}^{2})

(49)

{\tilde{b}}_{ml} {\hat{b}}_{ml} = {\tilde{b}}_{ml} (b_{ml} - {\tilde{b}}_{ml}) \leq \frac{1}{2} ({b_{ml}}^{2} - {\tilde{b}}_{ml}^{2})

(50)

{\tilde{m}}_{r} {\hat{m}}_{r} = {\tilde{m}}_{r} (m_{r} - {\tilde{m}}_{r}) \leq \frac{1}{2} ({m_{r}}^{2} - {\tilde{m}}_{r}^{2})

(51)

{\tilde{m}}_{l} {\hat{m}}_{l} = {\tilde{m}}_{l} (m_{l} - {\tilde{m}}_{l}) \leq \frac{1}{2} ({m_{l}}^{2} - {\tilde{m}}_{l}^{2})

(52)

Substituting equations (47)–(52) into equation (46) results in

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - [λ_{min} (Q) - 1] ‖ \tilde{X} ‖^{2} + \frac{1}{2} ‖ P δ ‖^{2} + e_{1} {\overset{\cdot}{e}}_{1} + \frac{1}{4} {e_{1}}^{2} \\ + \frac{1}{2} {ε'}^{2} + \frac{1}{2} {ε_{d}}^{2} + \frac{ϕ}{γ} {\tilde{W}}^{T} W - (c_{1} - \frac{3}{2}) {z_{1}}^{2} \\ - (c_{2} - 1) {z_{2}}^{2} - \frac{a_{1}}{2 ρ_{1}} {\tilde{b}}_{mr}^{2} - \frac{a_{2}}{2 ρ_{2}} {\tilde{b}}_{ml}^{2} - \frac{a_{3}}{2 ρ_{3}} {\tilde{m}}_{r}^{2} \\ - \frac{a_{4}}{2 ρ_{4}} {\tilde{m}}_{l}^{2} + \frac{a_{1}}{2 ρ_{1}} {b_{mr}}^{2} + \frac{a_{2}}{2 ρ_{2}} {b_{ml}}^{2} + \frac{a_{3}}{2 ρ_{3}} {m_{r}}^{2} + \frac{a_{4}}{2 ρ_{4}} {\tilde{m}}_{l}^{2} \end{matrix}

(53)

In order to show the overall controller structure, a diagram of the configuration of the stabilized platform control scheme proposed above is shown in Figure 4.

Stability analysis

According to equation (25), we can get its time derivative

{\overset{\cdot}{e}}_{1} = {\overset{\cdot}{β}}_{1} - {\overset{\cdot}{α}}_{1} = - \frac{e_{1}}{τ} + B_{1} (\cdot)

(54)

According to Shojaei et al. (2019), $B_{1} (\cdot) = B_{1} (z_{1}, e_{1}, y_{d}, {\overset{\cdot}{y}}_{d}, {\overset{\cdot\cdot}{y}}_{d}) = - {\overset{\cdot}{α}}_{1} = c_{1} {\overset{\cdot}{z}}_{1} - {\overset{\cdot\cdot}{y}}_{d}$ , which is a continuous bounded function, and there is a maximum absolute value M.

Using Young’s inequality, we can get that

e_{1} {\overset{\cdot}{e}}_{1} \leq - \frac{{e_{1}}^{2}}{τ} + | e_{1} B_{1} (\cdot) | \leq - \frac{{e_{1}}^{2}}{τ} + \frac{1}{2} {e_{1}}^{2} + \frac{1}{2} M^{2}

(55)

\frac{ϕ}{γ} {\tilde{W}}^{T} W \leq - \frac{ϕ}{2 γ} {\tilde{W}}^{T} \tilde{W} + \frac{ϕ}{2 γ} {W^{*}}^{T} W^{*} \leq - \frac{ϕ}{2 γ} ‖ \tilde{W} ‖^{2} + \frac{ϕ}{2 γ} {W^{*}}^{T} W^{*}

(56)

Substituting equations (55) and (56) into equation (53) results in

\begin{matrix} \overset{\cdot}{V} \leq - [λ_{min} (Q) - 1] ‖ \tilde{X} ‖^{2} + \frac{1}{2} ‖ P δ ‖^{2} - (c_{1} - \frac{3}{2}) {z_{1}}^{2} \\ - (c_{2} - 1) {z_{2}}^{2} - (\frac{1}{τ} - \frac{1}{4}) {e_{1}}^{2} - \frac{a_{1}}{2 ρ_{1}} {\tilde{b}}_{mr}^{2} \\ - \frac{a_{2}}{2 ρ_{2}} {\tilde{b}}_{ml}^{2} - \frac{a_{3}}{2 ρ_{3}} {\tilde{m}}_{r}^{2} - \frac{a_{4}}{2 ρ_{4}} {\tilde{m}}_{l}^{2} + \frac{1}{2} M^{2} + \frac{1}{2} ε'^{2} \\ + \frac{1}{2} {ε_{d}}^{2} + \frac{ϕ}{2 γ} {W^{*}}^{T} W^{*} - \frac{ϕ}{2 γ} ‖ \tilde{W} ‖^{2} \\ \leq - [λ_{min} (Q) - 1] ‖ \tilde{X} ‖^{2} - \frac{ϕ}{2 γ} ‖ \tilde{W} ‖^{2} - (c_{1} - \frac{3}{2}) {z_{1}}^{2} \\ - (c_{2} - 1) {z_{2}}^{2} - (\frac{1}{τ} - \frac{1}{4}) {e_{1}}^{2} \\ - \frac{a_{1}}{2 ρ_{1}} {\tilde{b}}_{mr}^{2} - \frac{a_{2}}{2 ρ_{2}} {\tilde{b}}_{ml}^{2} - \frac{a_{3}}{2 ρ_{3}} {\tilde{m}}_{r}^{2} - \frac{a_{4}}{2 ρ_{4}} {\tilde{m}}_{l}^{2} + Π \end{matrix}

(57)

where

\begin{array}{l} Π = \frac{1}{2} {‖ P δ ‖}^{2} + \frac{1}{2} ε'^{2} + \frac{1}{2} {ε_{d}}^{2} + \frac{ϕ}{2 γ} {W^{*}}^{T} W^{*} \\ + \frac{1}{2} M^{2} + \frac{a_{1}}{2 ρ_{1}} {b_{m r}}^{2} + \frac{a_{2}}{2 ρ_{2}} {b_{m l}}^{2} + \frac{a_{3}}{2 ρ_{3}} {m_{r}}^{2} + \frac{a_{4}}{2 ρ_{4}} {m_{l}}^{2} \end{array}

The parameters $Q, c_{1}, c_{2}, τ$ should meet the following requirements to make the system converge

[λ_{min} (Q) - 1] > 0, (c_{1} - \frac{3}{2}) > 0, (c_{2} - 1) > 0, (\frac{1}{τ} - \frac{1}{4}) > 0

(58)

Let

c = min {2 \frac{[λ_{min} (Q) - 1]}{λ_{max} (P)}, 2 (c_{1} - \frac{3}{2}), 2 (c_{2} - 1), 2 (\frac{1}{τ} - \frac{1}{4}), ϕ}

(59)

Consequently, equation (57) can be expressed as

\overset{\cdot}{V} \leq - cV + Π

(60)

The differential solution of equation (55) can be obtained as follows

0 \leq V \leq \frac{Π}{c} + [V (0) - \frac{Π}{c}] e^{- ct}

(61)

Equation (61) shows all the signals in this stability system are semi-global uniformly ultimately bounded (SGUUB), and the tack error $| θ - θ_{d} |$ of toolface angle satisfies the inequality

| y - y_{d} | \leq \sqrt{2 V (0) e^{- ct} + \frac{Π}{c}}

which means the error is bounded, and we can select appropriate parameters to have corresponding control law, so that the tracking error converges to a very small neighborhood of zero. In addition, the value of $Π / c$ is related to the selection of parameters, if the parameters are properly, the error can be small enough, but it will result in a large input. Therefore, in the actual downhole design, we need to weigh the relationship between error and input signal to choose the appropriate control parameters.

Simulation results

Control effects

This section will research on real-time simulation experience of control system using Simulink toolkit in MATLAB 2019 to verify above designed controller in high accuracy tracking control ability of the stabilized platform with friction torque and input dead-zone and to provide a theoretical basis for practical engineering. We will conduct simulation experiments according to the parameters of the stabilized platform and the friction torque characteristics measured by Huo et al. (2009), which are listed in Table 1.

Table 1.

Model parameters of stabilized platform for rotary steerable system.

Parameter	Value	Parameter	Value
$J$	0.025 kg m²	$r_{0}$	0.0675 m
$K_{PWM}$	3.44	$r_{1}$	0.0255 m
$K_{W}$	5.74	$r_{2}$	0.015 m
$K_{E}$	0.22	$r_{3}$	0.015 m
$C_{m}$	3.82	$L$	1 m
$C_{e}$	0.4	$n_{0}$	20 rad/s
$R$	12.5Ω	$f_{μ 1}$	0.002
$G$	350 N	$f_{μ 2}$	0.005
$G_{2}$	20 N	$α$	0.25 rad
$G_{3}$	15 N	$μ$	0.5
$r$	0.055 m

Suppose that u is the output of a dead-zone described by

u = D (v) = {\begin{matrix} v - 1, v \geq 1 \\ 0, - 1 < v < 1 l \\ 0.9 (v - 1), v \leq - 1 \end{matrix}

In simulation verification, five nerve nodes were selected to RBFNN, the center and width are chosen as $c = [- 1, - 0.5, 0, 0.5, 1]$ , $b = 0.5$ . Adaptive law parameters of RBFNN are $γ = 0.2, ϕ = 0.5$ , the initial value of neural weight estimation is $W (0) = 0$ . Besides, to reduce the feedback errors to zero and ensure the system stability, the feedback gains are designed as $c_{1} = 70, c_{2} = 100$ . The observer gain is chosen as

K = [\begin{matrix} k_{1} \\ k_{2} \end{matrix}] = [\begin{matrix} 10 \\ 400 \end{matrix}]

to overcome model errors. The dead-zone adaptive law parameter is set to $a_{1} = 1, a_{2} = 2, a_{3} = 1, a_{4} = 1$ , $ρ_{1} = 2, ρ_{2} = 4, ρ_{3} = 6, ρ_{4} = 2$ , and the initial conditions are ${\hat{b}}_{mr} = 0.5, {\hat{b}}_{ml} = 0.5$ , ${\hat{m}}_{r} = 0.5, {\hat{m}}_{l} = 0.5$ . Set the initial state value as $[θ_{0} ω_{0}] = [0.2 0]$ and give the command toolface angle trajectory as $θ_{d} = \sin t$ , model error as $e = \cos (t)$ .

Based on these parameters, the actual trajectory and track error of toolface angle are shown in Figures 5 and 6. Figure 7 shows the input control signal through the dead-zone $D (v)$ and its estimate $u_{d}$ . According to Figure 8, it can be seen the toolface observation, and the state observer realizes compensation effect well. The experimental results confirm that the proposed algorithm (29) achieve excellent tracking performance in practical stabilized platform motion control system. Figure 9 shows the approximation effect of RBFNN on the friction torque that it can track and compensate it well. Figure 10 shows the proposed dead-zone adaptive parameters are updated during the simulation to compensate the dead-zone in the system, and these parameters are bounded, making the control signal bounded, which is applicable for implementation.

Comparative simulation results

In order to better illustrate the effectiveness of the proposed control law and the reason of designing the controller for the actual stabilized platform, the traditional PID controller, DSC controller without NNs state observer, and the adaptive neural network dynamic surface control (ANNDSC) controller proposed in this paper are compared. Traditional PID controller is widely used in practical engineering because of its simple and reliable characteristics, we select the three gains as k_p = 800, k_i = 0.5, k_d = 500 by error-and-try method. As for DSC controller, we choose the same parameters as ANNDSC for the sake of fairness and the designed control law is

\begin{matrix} α_{1} = - c_{1} z_{1} + y_{d}, u = \frac{RJ}{C_{m}} \\ (\frac{(K_{PWM} K_{W} K_{E} C_{m} + C_{e} C_{m})}{RJ} x_{2} + \frac{C_{m}}{RJ} T_{f} - c_{2} z_{2} - z_{1} + \frac{α_{1} - β_{1}}{τ}) \end{matrix}

Figures 11 and 12 shows the tracking errors under the three controllers, it can be seen that ANNDSC controller has the best performance among the three in not only amplitude but also final tracking errors. Even in the case of high parameters, the error of PID controller only converges to 0.02, and although the error decreases quickly under the control of DSC controller, the final error is not as good as ANNDSC controller.

Control effects under parameter variations

Considering the actual downhole operation, due to temperature and drilling fluid, which will cause parameters such as moment of inertia, load resistance changes, and in fact, the friction term considered in this paper is not all the friction torque, there are other friction interferences not considered. The friction torque is double, the resistance and moment of inertia are reduced by 20% and the disturbance $e = 2 \cos (3 t)$ . To verify the robustness of the control scheme, simulation experiments are carried out under the same control parameters.

It can be seen from Figures 13 and 14 that the control effect of toolface angle is not greatly affected when the parameter perturbations and errors increase under ANNDSC controller, the friction torques can still be well-compensated and the control input signal does not increase significantly. However, the control errors of the PID controller and DSC controller are affected to varying degrees. Under the parameter perturbation, the amplitude and magnitude of the error increase, but the error under the controller ANNDSC has no obvious change. The traditional PID controller can only work under one set of parameters, and has no adaptability to the unknown environment and changing parameters in the well, so it cannot meet the control effect at all. The DSC method needs precise object, friction model, and so on, which is difficult to achieve in the well. And it is doubt that we have errors in the controlled object, and the friction items are not considered well. The effect of such a controller in the actual underground is not satisfactory. Since the ANNDSC controller employed both the RBFNN to approach unknown friction model and the state observer to reduce model error, then the adaptability and robustness are enhanced which is enough to cope with changing downhole conditions.

Conclusion

In order to achieve high-speed and high-precision tracking control of stabilized platform in rotary steerable system with input dead-zone, friction torque, and modeling error. The structure, friction torque, and dead-zone actuator of the stable platform are analyzed, the controlled object is established, and an NN DSC strategy based on state observer is proposed, in which NN adaptive law is used to handle unknown friction parametric uncertainty and the state observer to attenuate unmodeled disturbance, and it results in an excellent asymptotic tracking performance even under parametric perturbation. All signals in the system are proved to be SGUUB by the Lyapunov function, and the superior performance was verified by comparative simulations, which provides a certain reference for the engineering practice. In the future, we will focus on the research of real-time experiments in order to better reveal the actual performance of the controller.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key R&D Program of China (2019YFC0312303-05) and the National Natural Science Foundation of China (51875489).

ORCID iD

Shanshan Huang

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