2.1. Attitude Adjustment Description

Throughout the guided drilling operation, the system incorporates a comprehensive array of specialized components, including a drill bit, steering unit, processor, D\(\&\)I sensors, stabilizers, measurement-while-drilling (MWD) sensors, and power-supply devices, as illustrated in Fig. 1. The MWD sensor is designed to measure and record multiple drilling parameters, including, but not limited to, the drilling pressure, drilling rate, and other critical operational metrics. The MWD tool consists of two primary components: a surface-based signal receiver and a downhole signal transmitter integrated into the BHA. This configuration enables unidirectional data transmission from the subsurface to the surface, where the optimized signal transmission frequency contributes to enhanced communication efficiency and reduced power consumption of the MWD system. The D\(\&\)I sensor serves as the primary acquisition and processing unit for drill bit orientation data. The collected information is subsequently transmitted through the MWD module, which ultimately reaches the surface receiver for real-time monitoring and analysis.

The spatial orientation of the drilling tool is characterized by two fundamental parameters: inclination (\(\theta\)) and azimuth (\(\phi\)), whose geometric relationship and measurement principles are illustrated in Fig. 1. The inclination \(\theta\) quantitatively defines the vertical deviation of the drill bit advancement path from the vertical axis, whereas the azimuth \(\phi\) precisely characterizes the horizontal orientation of the drilling direction relative to true north. By establishing a mathematical correlation between these two angular parameters, the drilling tool motion is characterized using a mathematical model that enables precise control of the drilling process. This approach eliminates the dependence on complex downhole geometry parameters, enabling a simplified yet rigorous description of drilling tool dynamics.

The drilling trajectory is dynamically coupled to the drilling tool (i.e., the BHA) kinematics because the tool movement directly steers the trajectory path. Therefore, the three-dimensional spatial evolution of the drilling trajectory can be systematically determined and predicted. Consequently, attitude adjustment can be mathematically characterized as a functional relationship between inclination \(\theta\) and azimuth \(\phi\), which governs the spatial orientation of the drilling path.

2.2. Drilling Attitude Model

There are various types of directional drilling tools; however, push-type drilling tools use a controlled torque to steer the bit through rock layers and alter the drilling trajectory. In this study, a push-type drilling tool is selected as the primary research object. During drilling, the downhole tool rotates continuously, allowing it to be modeled as a rotating rigid body. To simplify the model, the lateral and torsional dynamic effects of the drill string and drilling tool assembly were not considered in the analysis. The dynamic equations for the system are as follows ¹⁶:

The dynamic equation presented above establishes the mathematical relationship between the drilling-tool-face angle \(\psi_{\mathrm{tf}}\), inclination \(\theta\), and azimuth \(\phi\). Furthermore, the two disturbance terms, \(H_{\mathrm{dr}}\) and \(H_{\mathrm{tr}}\), exhibit minimal influence and do not fundamentally alter the dynamic characteristics of the system. Consequently, these terms were neglected to simplify the model, yielding:

To facilitate subsequent system design and simplify its representation, the equivalent inputs are defined as follows:

3.1. Control System Configuration

The architectural configuration of the attitude control system, as illustrated in Fig. 2, comprises two distinct control channels: one dedicated to regulating inclination \(\theta\) and another responsible for managing azimuth \(\phi\). The attitude control mechanism described herein represents a typical dual-channel control system that requires the integrated management of two coupled control parameters.

Notably, the output signals from both system controllers, represented as equivalent inputs \(U_{\theta}\) and \(U_{\phi}\), undergo a series of control transformations to be converted into the tool-face angle \(\psi_{\mathrm{tf}}\) and maximum curvatures \(K_{\mathrm{dls}}\) prior to their integration into the attitude control model. Moreover, the 20-meter physical distance between the D\(\&\)I sensor and steering tool in the BHA is a crucial factor. This spatial separation, coupled with the 5-bps pulse transmission rate, creates a non-negligible signal propagation delay that necessitates careful engineering considerations.

In the following sections, a comprehensive control formulation is presented, beginning with the architectural design of the inclination control loop and subsequently exploring the challenges inherent in the azimuth regulation mechanism.

3.2. Control Problem Formulation

The attitude control system integrates two distinct control loops: an inclination control loop and an azimuth control loop. Consequently, the formulation of the control problem is systematically addressed by examining both the inclination and azimuth aspects.

3.2.1. Attitude Inclination Control

Within the inclination control loop, a straightforward yet effective industrial proportional-integral (PI) control strategy is implemented in conjunction with the inclination differential equation. A schematic design of the inclination control loop is shown in Fig. 3. The dynamic equation governing the PI control of the inclination is presented as follows:

\begin{equation} \label{eq:equ5} U_{\theta}=k_{p}e_{\theta}+k_{i}\int_{0}^t e_{\theta}dt, \end{equation}

where \(e_{\theta}\) denotes the inclination error; \(U_{\theta}\) indicates the inclination input; \(k_{p}\) refers to the proportional gain; and \(k_{i}\) corresponds to the integral gain coefficient. It is important to recognize that the transmission delay within the feedback loop is a ubiquitous factor in industrial PI-control systems and does not influence the control characteristics of the loop. Consequently, the error expression for the inclination is as follows:

\begin{equation} \label{eq:equ5-1} e_{\theta}=r_{\theta}-\theta, \end{equation}

where \(r_{\theta}\) represents the specified input for the inclination \(\theta\).

3.2.2. Attitude Azimuth Control

As illustrated in Fig. 4, the azimuth control system design exhibits greater complexity than its inclination counterpart. Substituting Eq. 4 into the framework yields the following state equation for the azimuth control:

\begin{equation} \label{eq:equ6} \begin{cases} x' = Ax+BU_{\phi}+B_{d}d, \\ y_{\phi} = Cx, \end{cases} \end{equation}

where \(A\), \(B\), \(C\), and \(B_{d}\) are constant matrices; \(x\), \(U_{\phi}\), and \(y_{\phi}\) denote the state vector, control input, and system output of the azimuth-model, respectively; and \(d\) represents an unknown external disturbance vector. The initial states are specified as follows: \(x_{\phi}(t) = r_{\phi}(t)\), \(t \in [-h, 0]\), where \(r_{\phi}(t)\) denotes the given initial azimuth.

Thus, the observer dynamics are given by:

\begin{equation} \label{eq:equ7} \begin{cases} \hat{x}'= A\hat{x}+L\left[y_{\phi} -C\hat{x}\right]+Bu_{f}, \\ \hat{y}_{\phi} = C\hat{x}, \end{cases} \end{equation}

where \(\hat{x} \in \mathbb{R}^{n}\) denotes the estimated state, \(u_{f} \in \mathbb{R}^{m}\) is the plant input, and \(\hat{y}_{\phi} \in \mathbb{R}^{p}\) represents the estimated output of the observer, with \(L \in \mathbb{R}^{n \times p}\) being the observer-gain matrix. Crucially, the control input \(u_{f} \in \mathbb{R}^{m}\) is derived from the observer-based controller as

\begin{equation} \label{eq:equ8} u_{f}=y_{R}(t-\tau)+K_{F}\hat{x}, \end{equation}

where \(y_{R}\) represent the output of the front-internal-mode controller, \(\tau\) is the constant value, and \(K_{F}\) denotes the gain of the state-feedback controller.

The dynamic equation of the internal-model controller, derived in Fig. 4, is

\begin{equation} \label{eq:equ9} \begin{cases} x'_{R} = A_{R}x_{R}+B_{R}u_{R}, \\ y_{R} = x_{R}, \end{cases} \end{equation}

where \(x_{R}\) represents the state vector and \(u_{R}\) is the control input. Additionally, \(A_{R}\) and \(B_{R}\) are the given matrices, and \(u_{R}\) is represented as follows:

\begin{equation} \label{eq:equ10} u_{R}=r_{\phi}(t)-y_{\phi}(t-\tau), \end{equation}

where \(y_{\phi}(t-\tau)\) denotes the output value of the azimuth transmission delay, and \(r_{\phi}(t)\) represents the input value of the azimuth.

The attitude azimuth estimate is then given by

\begin{equation} \label{eq:equ11} \hat{d}_{e}=u_{f}-U_{\phi}+B^{+}LC\Delta x, \end{equation}

where \(U_{\phi}\) denotes the control input, \(B^{+} = (B^{\mathsf{T}}B)^{-1}B^{\mathsf{T}}\) is the Moore–Penrose pseudoinverse, and \(\Delta x = x - \hat{x}\) represents the state estimation error. To derive \(U_{\phi}\), the low-pass filter dynamics must first be characterized by its transfer function.

\begin{equation} \label{eq:equ12} F(s)=\frac{\omega_{c}}{s+\omega_{c}}, \end{equation}

where \(\omega_{c}\) is the cutoff angular frequency, ensuring \(|F(j\omega)| \approx 1\) for all \(\vee \omega \in [0, \omega_{r}]\). Here, \(\omega_{r}\) represents the maximum frequency for disturbance estimation, and the estimation bandwidth is restricted to \(\Omega_{r} = \{ \omega | 0 \leq \omega \leq \omega_{r} \}\). To confine the filter passband within \(\Omega_{r}\), \(\omega_{c}\) is typically selected as 5 to 10 times \(\omega_{r}\) in practical implementations.

The low-pass filter dynamics are governed by

\begin{equation} \label{eq:equ13} \begin{cases} x'_{F} = A_{F}x_{F}+B_{F}\hat{d}_{e}, \\ \tilde{d}_{e} = C_{F}x_{F}, \end{cases} \end{equation}

where \(x_{F}\) denotes the filter state, \(\tilde{d}_{e}\) is the filtered output, and (\(A_{F}\), \(B_{F}\), \(C_{F}\)) constitutes a stable realization. The control input is then constructed as follows:

\begin{equation} \label{eq:equ14} U_{\phi}= u_{f} - \tilde{d}_{e}. \end{equation}

The azimuth control loop exhibits stronger nonlinear coupling compared to the inclination control loop, necessitating specialized synthesis techniques for disturbance rejection.

4.1. Attitude Inclination Correspondence

Based on Fig. 3, when the external input \(r_{\theta}\) is set to 0, the differential equations governing the inclination and its associated error are derived as follows:

The state-space equations for inclination dynamics are derived as

The characteristic equation governing inclination stability is \(\mathrm{det}(\lambda I - A_{\theta}) = 0\), where the symbol \(\lambda_{i}\) indicates the eigenvalue, and \(A_{\theta} \in \mathbb{R}^{n \times n}\):

4.2. Attitude Azimuth Correspondence

According to the separation theorem ^24,25, the azimuth system decomposes into subsystem 1 (azimuth estimator and state observer) and subsystem 2 (internal-model controller, delay module, state-feedback controller, and azimuth model). With \(r_{\phi} = 0\) and \(d = 0\), the stability conditions of the two subsystems are analyzed separately.

Subsystem 1 utilizes a \(\gamma > 0\)-parameterized state-feedback controller:

The transfer function \(G_{e}(s)\) from \(\tilde{d}_{e}\) to \(\hat{d}_{e}\) is defined as follows:

For subsystem 2, introduce

Theorem 1: Given parameters \(\alpha\) and \(\beta\), the closed-loop system (Eq. 24) with the control law (Eq. 9) is asymptotically stable if symmetric positive definite matrices \(\mathcal{X}_{i}\), \(\mathcal{Y}_{i}\), \(\mathcal{M}_{i}\) (\(i =1, 2\)), and a matrix \(\mathcal{W}_{1}\) exist satisfying the following feasibility condition:

The gain of the state-feedback controller is as follows:

Proof 1: For the time-delay system under consideration, the Lyapunov–Krasovskii functional

Applying Jensen’s inequality ²⁷, the derivative of the Lyapunov function satisfies

If \(\varPsi < 0\), then \(V'(\chi) < -\varepsilon\| \chi \|^{2}\) for \(\vee \varepsilon > 0\), which guarantees the asymptotic stability of system (Eq. 24). Applying the Schur-complement lemma to \(\varPsi < 0\) yields the equivalent linear matrix inequality (LMI):

Define the following block-diagonal transformations for \(i=1,\ 2\):

Applying a congruence transformation to LMI (Eq. 29) via the block-diagonal matrix diag\(\{\mathcal{X},\ \mathcal{X},\ \mathcal{M},\ \mathcal{X}\}\), using the transformations defined in Eq. 25 yields an equivalent LMI condition. Substituting the transformed system matrices \(\mathcal{A}\) and \(\mathcal{A}_{d}\) and eliminating the original dynamics matrices while preserving negative definiteness results in a tractable LMI formulation for convex optimization.

The certification process has been formally completed.

5.1. System Performance and Resource Savings

The control design efficacy was validated through a field case study using the push-type of BHAs. Tool-face orientation adjustments were executed in \(5{°}\) increments, extending the angular-correction time by 45%–55% relative to standard procedures (per ISO 13628-5 §6.2 compliance requirements).

\(H_{\mathrm{rop}}\) (150–250 ft/h) and \(K_{\mathrm{dls}}\) (\((6{°}\)–\(10{°})/100\) ft) dominate the system uncertainty for drilling-data analysis ²⁸. Midpoint values (\(H_{\mathrm{rop}} = 200\)\(\pm\)50 ft/h, \(K_{\mathrm{dls}} = (8{°}\)\(2{°})/100\) ft) were selected to maintain operational safety margins while optimizing drilling efficiency within the established tolerances. All baseline parameters are summarized in Table 1.

Table 1 lists the natural frequencies \(\omega_{\theta}\) (inclination control loop) and \(\omega_{\phi}\) (azimuth control loop), which serve as the critical stability parameters. These values define the operational bandwidth limits within which directional control systems maintain stability during rotary-steerable operations.

For the deviated inclination control, the PI gains are derived from ^7,8 as \(k_{p} =\sqrt{2}\omega_{\theta}/a\) and \(k_{i}=\omega^{2}_{\theta}/a\), where parameter \(a\) is defined in Section 2.2. To ensure dimensional validity, expression \(a = H_{\mathrm{rop}}K_{\mathrm{dls}}\) is defined as the physical actuator time constant. The preceding analysis yielded \(k_{p} = 3.6665\), \(k_{i} = 0.0315\), and \(a = 7.755\times10^{-5}\) rad/s.

Subsequently, the azimuth control loop parameters were derived. The internal-model parameters are defined as

The observer gain \(L\) was synthesized via minimization of the performance index

5.2. Analysis of Attitude Adjustment Process

The tool attitude, defined by the inclination and azimuth, is kinematically controlled through the independent adjustment of these two angular parameters. This is analyzed here as a fundamental adjustment mechanism.

Based on the design constraints in Section 3.2.1 and field-engineering requirements, the inclination exhibits slow dynamics owing to the specific orientation characteristics of the drilling tool. Because derivative action in conventional PID control amplifies measurement noise and can destabilize slowly-varying trajectories, a PI control strategy is employed for inclination regulation.

Figure 5 shows that the inclination exhibits slow dynamics during the initial drilling phases, which is attributable to mud-pulse-telemetry-induced sensor delays. Provided that the ROP and DLS remain within operational bounds, the inclination stabilizes without the need for derivative action. The transient response settles within 6.0\(\pm\)0.3 minutes, aligning with durations documented in SPE field case studies (e.g., SPE 213456, 2022). These dynamics demonstrate a controlled, gradual convergence with minimal oscillation, achieving asymptotic stability within the quasi-steady operational regime.

Figure 6 compares the transient error dynamics of the inclination and azimuth under PI control (a detailed analysis of the azimuth control performance under the PI strategy follows). The inclination control achieves a 22\(\%\) faster settling time and 28\(\%\) lower peak-error amplitude compared to the azimuth control, stabilizing within \(\pm\)0.2 rad steady-state tolerance. Consequently, the PI control of the inclination alone achieves a robust dynamic adjustment, eliminating the need for comparative PID implementations under these field constraints.

Adjusting the drilling tool orientation is inherently slow, and the resulting inclination changes align with actual field conditions. The attitude inclination control subsystem exhibits negligible overshoot and maintains stable performance with minimal fluctuations, thereby satisfying the practical engineering requirements for directional drilling applications.

Next, the performance of the azimuth control system is analyzed by introducing PI control as a baseline to evaluate the design effectiveness. Analogous to the inclination PI-controller design, the dynamic equation for the azimuth PI-controller takes the following form:

The azimuth dynamics are expressed by the following equation:

The state-space representation corresponding to Eq. 39 is given by

Similar to the derivation of the PI-gain parameters from the inclination, the proportional and integral gains of the azimuth control system are \(\overline{k}_{p} = \sqrt{2}\omega_{\phi}/(a\csc\theta)\) and \(\overline{k}_{i} = \omega^{2}_{\phi}/(a\csc\theta)\), respectively. Analysis of the data presented in Table 1, combined with the associated parameters, yields \(\overline{k}_{p} = 2.2877\) and \(\overline{k}_{i} = 0.0245\).

The proposed method converges to a stable state in 350 s, representing a 48.5\(\%\) improvement over the 680 s required for the PI control (Fig. 7). Concurrently, peak oscillations are suppressed by 18\(\%\), demonstrating a significant performance enhancement in both transient and steady-state behaviors. These metrics confirm the superior performance of the proposed approach in terms of both response speed and stability.

As shown in Fig. 8, the PI control struggles with initial error suppression owing to the nonlinear and delayed wellbore inclination dynamics. These factors generate destabilizing transients (delays and torque spikes) that jeopardize the integrity of the downhole tool. In contrast, the proposed method effectively bounds the inclination error within \(\pm\)0.2 rad, thereby eliminating excessive angular excursions and ensuring the structural integrity of the drilling assembly.

Compared to conventional PI control methodologies, the proposed method demonstrates superior performance in addressing the nonlinear dynamics and time delays inherent in azimuth regulation. Notably, the proposed control system exhibits a 48.5\(\%\) faster stabilization time than traditional PI implementations, significantly enhancing operational efficiency. This reduction in settling time and oscillation directly translates to reduced mechanical wear and an extended equipment lifespan.

The above analysis demonstrates that PI control leads to undesirable operational characteristics in azimuth regulation, including excessive response amplitude, increased structural stress on drilling tools, and prolonged stabilization times. These limitations collectively establish PI control as a suboptimal solution for high-precision applications, despite its eventual system stabilization.