Robust tube MPC for suppressing stick-slip vibrations in drill string systems

Abstract

Addressing stick-slip vibrations in drill string systems is critical in drilling operations, as these vibrations can cause costly downtime and equipment damage. A four-degrees-of-freedom (4-DOF) model captures the dynamics of long vertical drilling systems, followed by a robust tube model predictive control (MPC) strategy to account for system uncertainties. The controller is designed to handle variations in key parameters, such as drill pipe inertia—affected by pipe length—and weight on bit—affected by axial vibrations—without relying on direct bit velocity measurements. Comparative analysis with sliding mode control (SMC) demonstrates that the proposed MPC achieves substantially faster settling times and lower mean squared error (MSE). Specifically, for the top driver, the settling time is reduced by 41% and MSE by 74% compared to SMC. Relative to proportional–integral–derivative (PID), the tube MPC improves MSE by 69% and reduces settling time by 40%. In addition, the linear quadratic tracking (LQT) and linear MPC controllers were compared with the proposed method. A sensitivity analysis further evaluates the influence of parameter uncertainties, with simulation results confirming the controller’s effectiveness in suppressing stick-slip oscillations and maintaining robust performance under uncertain conditions.

Keywords

drilling drill-well torsional vibration stick-slip tube MPC sliding mode controller

1. Introduction

Vertical drilling systems are essential components in the exploration and extraction of subsurface resources, ranging from oil and gas to geothermal energy. Unlike conventional drilling methods, vertical drilling allows for the creation of deep, precise wells, directly penetrating target formations. This precision not only maximizes resource recovery but also minimizes environmental impact by reducing the surface footprint. Drilling is a critical process in accessing natural resources that power industries and fuel economies worldwide. Without efficient drilling techniques, resource extraction would be far more costly, time-consuming, and environmentally detrimental. A key factor in the efficiency of vertical drilling is the control system, which ensures that the drill bit remains on the intended path, even when encountering unpredictable subsurface conditions. Advanced control systems employ real-time data analysis and feedback mechanisms to adjust drilling parameters, such as rotational speed, weight on the drill bit, and mud flow rate. By maintaining the vertical trajectory, these systems reduce deviation, enhance well integrity, and optimize the overall drilling process.

In this article, we will explore the technical aspects of vertical drilling systems and delve into the role of modern control technologies in improving drilling precision and efficiency. By understanding these elements, we can better appreciate the engineering challenges and innovations that continue to push the boundaries of subsurface exploration.

Figure 1 presents a drill bit that consists of multiple parts, mainly a drill collar, drill pipes, a rotary table, and a drill bit.

Figure 1.

Whole drill string dynamic.

In modeling this phenomenon, several approaches are considered. Saldivar et al. (2016) introduce four model types: Lumped Parameter, Partial Differential Equation (PDE), Neutral Type model, and Coupled PDE-ODE (Ordinary Differential Equation). Each of these models has its advantages. The lumped parameter model offers simplicity in analysis and is commonly used. For instance, in Yigit and Christoforou (2006), robust pole placement is employed to stabilize the system. In Mihajlovic et al. (2004), an experimental setup is used with a discontinuous static friction model to predict vibrations, which is validated through numerical and experimental analyses. Tang et al. (2015) investigate the occurrence of stick-slip at various rotary speeds, concluding that this phenomenon occurs when the speed is below 127 RPM.

The PDE model provides high accuracy but requires complex analysis, and its precision depends heavily on boundary conditions. In Sagert et al. (2013), a backstepping control approach is proposed, while Wang et al. (2020) employ adaptive output feedback control, utilizing adaptive methods and infinite-dimensional backstepping to handle unknown disturbances and anti-damping terms. Simulations validate the controller’s effectiveness, significantly reducing torsional vibrations. Similarly, in Tashakori et al. (2021), a prediction-based control strategy is introduced to stabilize the axial-torsional dynamics of a distributed drill string system using an infinite-dimensional model and continuous pole-placement techniques based on top-side measurements.

The Neutral Type model simplifies system analysis, but its accuracy also depends on boundary conditions. Article Wu et al. (2020) were the first to use this model, applying a simple robust controller with equivalent input disturbances. The Coupled PDE-ODE model facilitates the design of control strategies but is suitable only for specific strategies.

Two additional models exist. The first is the simplified wave model, which uses two first-degree PDEs instead of one second-degree wave equation. In Aarsnes et al. (2019), this model is used with an optimal controller via extremum seeking, and in Aarsnes and Shor (2018), presented a distributed model of a drill string to investigate torsional vibrations when the bit is off bottom. It characterizes the stick-slip limit cycle caused by distributed Coulomb friction and validates the model with field data from deviated wells. The study finds a close match between recorded and modeled data, highlighting the model’s effectiveness. In Aarsnes and Van de Wouw (2018) analyzed the stability of distributed drill string systems by deriving a minimal set of parameters for their axial-torsional dynamics. It uses these parameters to create stability maps, revealing complex dynamics not evident in lumped models. The findings aid in designing control strategies to mitigate vibrations 12 The second is the finite element method using spring-damper elements, where the number of spring-dampers represents the system’s degrees of freedom. The simplest version, a 2-DOF model, is explored in Yan et al. (2023), where a PID controller is presented, and in Monteiro and Trindade (2017), where a PI controller is implemented. Various bit-rock interaction models for this method will be discussed in the modeling section. In Chen and Allgöwer (1998), an LQG controller is implemented, showing that LQG offers a smoother response compared to H_∞ control for structured disturbances. In Doghmane et al. (2022), an H_∞ controller is introduced, demonstrating better performance in rejecting unstructured disturbances. Mayne et al. (2000) propose a dual-loop controller with an outer loop for fast response and an inner loop for damping vibration. Other controllers, such as the modified integral resonant controller (MIRC), are compared with sliding mode controllers (SMC) in MacLean et al. (2021), while Aarsnes et al. (2021) compare nonlinear model predictive controller (NMPC) with ZTorque, claiming that ZTorque has better performance.

A 3-DOF model is used in Mendil et al. (2021), which compares four controllers (PI, PID, Hybrid PI, and Hybrid PID), showing stabilization within 10 seconds. A new cascade sliding mode control architecture is proposed in Ghasemi and Song (2017). A 4-DOF model is explored in Talbi et al. (2023), where an optimal hybrid fractional-order fuzzy logic controller is compared with PID, fractional-order PID, and SMC. In Huang et al. (2024), a fuzzy comprehensive evaluation method is proposed to identify bit-rock interaction, while Fouad et al. (2011) propose an SMC to reduce top-driver motor effort. Sadeghimehr et al. (2021) introduce an SMC with a Smith predictor, and Huang et al. (2018) compare the fourth-fifth order Runge-Kutta method with field data. Article Van Veggel et al. (2015) propose robust output feedback, and in Laib and Gharib (2024) proposed cascade feedforward neuro-fuzzy PID control for this system.

Some articles, such as Xie et al. (2022), do not specify the model explicitly but propose three architectures for active stick-slip vibration control. In others, like Cheng et al. (2020), more than 4-DOF models are used, with Cheng et al. (2020) employing a 20-DOF model and a gain-scheduled controller for control. The conventional controllers face several limitations:

• Extended Settling Time: These controllers generally require a longer period to stabilize the system after a disturbance or change in conditions, impacting the efficiency of the drilling process.

• Limited Constraint Handling: They do not adequately address input constraints, such as limiting the system from achieving negative velocity, which is crucial in maintaining operational safety and stability within desired bounds.

• Sensitivity to Disturbances: These controllers lack robustness in the presence of varying weight-on-bit (WOB) and fluctuations in drill pipe inertia, making the system more susceptible to performance degradation under real-world, dynamic drilling conditions.

To cope with these limitations Tube MPC controller designed and, the main contributions of this article are as follows:

• Considering the 4-DOF Drill String Modeling: Smaller versions of this model were unable to accurately represent the system, while larger models posed computational challenges and did not significantly improve precision.

• Designing a Robust Tube MPC Strategy: To address the limitations and sensitivity to disturbances observed in previous work, a Tube MPC strategy was developed.

• Performance Comparison and Improvement over SMC with and without Uncertainty: As demonstrated, the proposed approach outperformed the SMC, achieving significant improvements in performance.

• Performance Comparison with Traditional Industrial Controllers: Manual control methods may induce stick-slip phenomena due to the inherent structural characteristics of the system.

In this article, the spring-damper model is employed and uncertainties in inertia are considered to address unstructured disturbances. This is achieved by implementing a robust model predictive controller (MPC) that accounts for uncertainties. In the following sections, we present our model, explore the interaction between the rock and bit, and finally, propose the controller, demonstrating its performance through simulations.

2. The mathematical model

2.1. Drill string model

Figure 2 at the top is the top drive inertia, J_r, followed by the equivalent elements for drill pipes, J_p, drill collars, J_l, and the bit inertia, J_b. In this model, the torsional stiffness of the springs is represented by k_rp, k_ph, and k_hb, and torsional damping by b_rp, b_ph, and b_hb. The angular displacement and angular velocity of each element are denoted as θ_i and $\dot{θ_{i}}$ , with i = r, p, l, b. The control input is the applied torque from the top drive, T_m. The viscous damping torque, $T_{a r} = c_{r} \dot{θ_{r}}$ , relates to the lubrication of the mechanical elements of the top-rotary system, where c_r is the viscous damping coefficient. Additionally, the torque at the bit is a nonlinear function of the bit velocity. The equations of the drill string system are expressed in these terms. Figure 2 shows the model dynamic of drill string.

\begin{matrix} {\ddot{θ}}_{r} = & \frac{- b_{r p}}{J_{r}} ({\dot{θ}}_{r} - {\dot{θ}}_{p}) - \frac{k_{r p}}{J_{r}} (θ_{r} - θ_{p}) + \frac{T_{m} - T_{a_{r}} ({\dot{θ}}_{r})}{J_{r}}, \\ {\ddot{θ}}_{p} = & \frac{b_{r p}}{J_{p}} ({\dot{θ}}_{r} - {\dot{θ}}_{p}) + \frac{k_{r p}}{J_{p}} (θ_{r} - θ_{p}) \\ - \frac{b_{p h}}{J_{p}} ({\dot{θ}}_{p} - {\dot{θ}}_{l}) - \frac{k_{p h}}{J_{p}} (θ_{p} - θ_{l}), \\ {\ddot{θ}}_{l} = & \frac{b_{p h}}{J_{l}} ({\dot{θ}}_{p} - {\dot{θ}}_{l}) + \frac{k_{p h}}{J_{l}} (θ_{p} - θ_{l}) \\ - \frac{b_{h b}}{J_{l}} ({\dot{θ}}_{l} - {\dot{θ}}_{b}) - \frac{k_{h b}}{J_{l}} (θ_{l} - θ_{b}), \\ {\ddot{θ}}_{b} = & \frac{b_{h b}}{J_{b}} ({\dot{θ}}_{l} - {\dot{θ}}_{b}) + \frac{k_{h b}}{J_{b}} (θ_{l} - θ_{b}) - \frac{T_{b} ({\dot{θ}}_{b})}{J_{b}} . \end{matrix}

(1)

Figure 2.

4-DOF drill string model.

2.2. Bit-rock interaction

Models like Simscape, Karnopp’s model, and the Coulomb friction model enable realistic simulations of bit-rock interactions in drilling. Simscape modeling in MATLAB aids in optimizing bit design, predicting drilling performance, and refining control strategies, supporting risk assessment and operator training through virtual testing. Karnopp’s model introduces exponentially decaying friction, while the Coulomb model captures torque dynamics by simulating transitions between static and dynamic friction—both essential for understanding drill-bit behavior in complex drilling environments.

2.3. Simscape modeling

In some articles, such as Chen and Allgöwer (1998) and Doghmane et al. (2022), the model is implemented using Simscape, as shown in Figure 3. Simscape modeling of bit-rock interaction in MATLAB enables realistic simulations that optimize bit design, predict performance, and enhance control, improving safety, efficiency, and cost-effectiveness in drilling.

Figure 3.

Bit-rock interaction in Simscape Chen and Allgöwer (1998).

2.4. Karnopp’s model

This method includes an exponentially decaying friction term, as shown below:

T_{b} = d_{b} \dot{θ_{b}} + T_{f b} (X)

(2)

T_{f b} (x) =

\{\begin{cases} T_{e b} (x) i f | \dot{θ_{b}} | < D_{v}, | T_{e b} | \leq T_{s b} \\ T_{s b} = s i g n (T_{e b} (x)) i f | \dot{θ_{b}} | < D_{v}, | T_{e b} | > T_{s b} \\ T_{c b} s i g n (\dot{θ_{b}}) i f | \dot{θ_{b}} | < D_{v} \end{cases}

(3)

The parameters in this equation describe the forces and conditions influencing torque during bit-rock interaction in drilling. T_fb(x) represents the frictional torque at the bit, and T_eb(x) is the elastic torque caused by the deformation of the drill string. The rotational velocity of the bit, $\dot{θ_{b}}$ , determines the interaction dynamics, with D_v acting as the velocity threshold distinguishing between stick-slip and dynamic motion. T_sb is the static friction torque, which defines the point at which the bit breaks free from sticking, while T_cb is the dynamic friction torque that controls the bit’s behavior during rotation in the slip regime. The sign function ensures the torque direction corresponds to the bit’s rotational movement. Together, these parameters model the complex transitions between static and dynamic friction, crucial for understanding bit-rock interactions in drilling systems. For complete equations refer to Talbi et al. (2023); Fouad et al. (2011); Yan et al. (2023); MacLean et al., 2021; Mayne et al., 2000; Monteiro and Trindade, 2017.

Coulomb friction model: This equation characterizes the torque generated by the interaction between the drill bit and the rock using an adapted Coulomb friction model as follows:

μ_{b} (\dot{θ_{b}}) = μ_{c b} + (μ_{s b} - μ_{c b}) e^{- (\frac{γ_{b}}{v_{f}}) | \dot{θ_{b}} |}

(4)

T_{f b} (\dot{θ_{b}}) = W_{o b} R_{b} μ_{b} (\dot{θ_{b}}) sgn (\dot{θ_{b}})

(5)

The key variables include the rotational velocity of the bit, ${\dot{θ}}_{b}$ , which directly influences the frictional torque. The weight on the bit, W_ob, defines the axial force exerted on the drill bit, while R_b specifies the bit’s radius. The dynamic and static friction coefficients between the bit and rock are denoted as μ_cb and μ_sb, respectively. Additionally, γ_b governs the rate at which friction decays as velocity increases, and v_f is the characteristic velocity that determines the speed of transition between static and dynamic friction.

2.5. Tube MPC

Tube Model Predictive Control (MPC) has emerged as a groundbreaking approach within control theory, addressing the challenges of uncertainties in dynamic systems with novel strategies. Its core principle involves wrapping the predicted system trajectories within a “tube,” offering a robust shield against disturbances and model imperfections. This tube-based strategy not only enhances resilience against external factors but also ensures a level of guaranteed performance within a restricted zone around the nominal path. More than just stabilizing the system, Tube MPC supports optimal performance in the presence of uncertainty, redefining the boundaries of modern control strategies.

As optimization methods advance, Tube MPC continues to play a leading role in precision-driven and adaptive control solutions, finding applications in industries ranging from aerospace to automotive. Its innovative framework paves the way for the next generation of control systems, merging cutting-edge theory with real-world application. The associated cost function can be expressed as:

J_{0} (X (0), U_{0}) = p (X) + \sum_{k = 0}^{N - 1} q (X (k), U (k))

(6)

In this formulation, p (x_N) represents a Lyapunov function, often written as p (x_N) = X^TPX, while q (X(k), U(k)) defines a quadratic cost given by q (X(k), U(k)) = X^TQX + U^TRU. The set U₀ refers to the sequence of inputs over the control horizon from k = 0 to N − 1. Generally, the optimal control solution for MPC problems can be formulated as:

\begin{matrix} J_{0}^{*} (X (0)) = \min J_{0} (X (0), U_{0}) \\ s.t. X_{k + 1} = A X_{k} + B U_{k} \\ X_{k} \in X, U_{k} \in U \\ |\frac{d U}{d t}| \leq constant \\ X_{N} \in X_{f} \\ X_{0} = X (0) \end{matrix}

(7)

Here, N represents the control horizon, and the sets $X, U a n d X_{f}$ are polyhedral regions that constrain the system’s state and control variables.

Tube MPC offers significant advantages over LQG and SMC for control of complex and constraint-limited systems such as drilling rigs. As a robust control strategy, Tube MPC explicitly accounts for bounded disturbances and incorporates both state and input constraints directly into its optimization framework. In contrast, neither LQG nor SMC inherently supports constraint handling, which limits their applicability in safety-critical systems where operational bounds must be strictly respected. SMC, while robust to matched uncertainties, suffers from the well-known chattering phenomenon, which can induce mechanical wear and excite unmodeled dynamics in drilling equipment. Moreover, SMC focuses primarily on driving the system to its sliding surface without predictive capabilities or multi-objective optimization, resulting in suboptimal performance in applications requiring both robustness and efficiency. LQG, although optimal in the linear quadratic sense under Gaussian noise assumptions, is sensitive to modeling inaccuracies and lacks explicit mechanisms to handle physical constraints. In contrast, Tube MPC not only maintains robustness but also enables simultaneous optimization of performance objectives—such as minimizing torsional vibration, improving rate of penetration, and reducing energy consumption—while guaranteeing constraint satisfaction. Its predictive nature allows it to anticipate future disturbances and adjust control inputs proactively, resulting in smoother actuator commands and reduced mechanical stress compared to the discontinuous control actions typical of SMC. These combined features make Tube MPC a superior choice for high-value, safety-critical systems where both robustness and performance optimization are paramount.

2.6. Stability and feasibility analysis

In this section, the control policy v, using feedback gain K, is designed to keep the actual system trajectory close to the nominal trajectory, aiming to meet system constraints while tracking a reference. Key assumptions include bounded disturbances within a convex set, controllability of the system, and accessible state measurements. The system’s error dynamics—defined as the deviation between actual and nominal states—are contained within a robust positively invariant (RPI) set, which forms a “tube” around the nominal path. Stability and recursive feasibility are ensured through conditions that maintain bounded state trajectories over time. Supporting lemmas confirm recursive feasibility when the cost function is initially feasible, and stability is achieved by a local stabilizing controller within a defined terminal region. Stability of the closed-loop system is further supported by a Lyapunov inequality involving the weight matrices Q, R, and an augmented cost matrix Q*, which collectively guarantee system performance and stability.

2.6.1. Nominal plant

The nominal disturbance-free system is described:

x_{k + 1}^{*} = A x_{k}^{*} + B u_{k}

(8)

2.6.2. Assumption 1

The dynamics of the system can be described as follows:

x_{k} = e_{k} + x_{k}^{*}

(9)

Where

x_{k}^{*}

is the nominal dynamics of the system and e is the bounded error of the model, which is as follows:

‖ e ‖ < Δ (x)

(10)

2.6.3. Assumption 2

Disturbance ω belongs to a closed, bounded, and connected set (convex polyhedron):

Ω = {d \in R^{n} : ‖ ω ‖ \leq η}

(11)

2.6.4. Assumption 3

The control policy v can be defined as follows:

v_{k} = u_{k}^{*} + K (x, x^{*})

(12)

In the above equation,

K (x, x^{*})

is an ancillary feedback law which stabilizes the nominal system and is constructed as follows:

K (x, x^{*}) = K (x_{k} - x_{k}^{*})

(13)

Where K is an offline linear feedback controller. Based on the control input v, the real trajectory is kept close to the nominal trajectory.

2.6.5. Assumption 4

The pair (A, B) is controllable,and the state of the system is known at each sample time.

Control Objective: The main objective of this paper is to propose a controller such that the controlled system satisfies the plant constraints despite the disturbance and tracks the reference trajectory.

Control Input: According to assumption 3, the control input can be designed as follows:

u_{k} = K (x_{k} - x_{k}^{*}) + v_{k}

(14)

The error of the system can be defined as:

e_{k + 1} = x_{k + 1} - x_{k + 1}^{*}

(15)

By substituting 1 and 8 in 15, the error between the nominal and disturbed plant is given by:

e_{k + 1} = (A + B K) e_{k} + ω_{k}

(16)

If the feedback control gain K is chosen in a way that matrix (A + BK) is Hurwitz, then the evolution of the error term e_k remains bounded. Consequently, the actual trajectory x_k will reside in a neighborhood around the predicted trajectory $x_{k}^{*}$ . This neighborhood can be visualized as a tube of trajectories Zanelli et al. (2018). To constrain the error term e_k, the concept of a robust positively invariant (RPI) set is employed Gonzalez Villarreal et al. (2018).

2.7. Bound maximum error

As per control theory, it becomes evident that there exists a set ɛ such that the error term e_k remains within it indefinitely. If the set ɛ is robustly positively invariant (RPI) for the system equation (16), and the initial conditions x₀ and $x_{0}^{*}$ satisfy $e_{0} = x_{0} - x_{0}^{*}$ within ɛ, then the trajectory of the uncertain system will be constrained.

Controlled by equation (14) ensures that $x_{k} \in x_{k}^{*} Φ E$ for any possible realization of disturbances. This concept is often referred to as the tube of trajectories.

Definition 1

A set ɛ is considered a robust positively invariant (RPI) set for the uncertain system equation (11) if the matrix product (A + BK) remains bounded by the matrix (A + BK)ɛ ⊕Ω ⊆ɛ.

2.8. Recursive feasibility and closed-loop stability

Definition 2

Ultimate Boundedness refers to a property of a closed-loop system where the trajectories of the state variables remain bounded over time, regardless of the initial conditions. This property is essential for ensuring the stability of feedback control systems. When a system is ultimately bounded, it guarantees that the state variables do not escape to unbounded regions, even if disturbances or uncertainties are present.

For conventional Tube MPC, the following lemma establishes a foundational theoretical framework to guarantee both recursive feasibility and closed-loop stability. This lemma serves as a cornerstone for understanding the robustness properties of MPC, ensuring that feasible control inputs are always available while maintaining long-term stability of the closed-loop system.

\dot{V} (x_{k}) \leq 0

(17)

Lemma 1

Recursive feasibility of MPC for the linear model is guaranteed if there exists a feasible solution for the cost function at k = 0.

Lemma 2

Consider a local stabilizing controller $K (x, x^{*})$ operating within a robust terminal region. This controller ensures that for all $x \in X_{f}$ , $K (x, x^{*}) \in U$ , and the following inequality holds:

Furthermore, if the controller $K (x, x^{*})$ is chosen as $K (x, x^{*}) = K z$ , we arrive at the Lyapunov equation involving the weight matrices Q, R, and P:

{(A + B K)}^{T} P + P (A + B K) \leq - Q^{*}

(18)

where Q* = Q + K^TRK represents an augmented cost matrix. This formulation ensures both stability and performance in the closed-loop system.

3. Simulations and results

In this section, the proposed modeling uses the variables in Table 1, as presented in Sadeghimehr et al. (2021). The uncertainties in the weight on bit and drill pipe inertia are taken into account. The length of the well can cause variations in the drill pipe, and the weight on bit may fluctuate due to axial vibrations. Therefore, in this simulation, a 20% uncertainty is considered for the drill pipe, and a 5% uncertainty is considered for the weight on bit. In the following results of proposed controller with SMC and PID compared. Finally, to demonstrate the Tube MPC controller’s robustness to time delays, we evaluate its performance under two distinct delay scenarios. It should be noted that, as mentioned earlier, stick-slip does not occur when the velocity reference is above 120 RPM. Therefore, a velocity reference of 105 RPM is used in this analysis.

Table 1.

Model design parameters Sadeghimehr et al. (2021).

Symbol	Parameter	Value	Unit
k_rp	Spring stiffness	698.063	Nm/rad
k_ph	Spring stiffness	1080	Nm/rad
k_hb	Spring stiffness	907.48	Nm/rad
J_r	Rotary table inertia	930	kg.m²
J_p	Drill pipe inertia	2782.25	kg.m²
J_l	Drill collars inertia	750	kg.m²
J_b	Drill bit inertia	471.9698	kg.m²
b_rp	Damping factor	139.6126	Nms/rad
b_ph	Damping factor	190	Nms/rad
b_hb	Damping factor	181.49	Nms/rad
v_f	Characteristic velocity	1	m/s
R_b	Radius of the bit	0.1555	m
W_ob	Weight on bit	97347	N
μ_sb	Coefficient of static friction	0.8	No Unit
μ_cb	Coefficient of dynamic friction	0.5	No Unit
γ_b	Friction decay rate	0.9	1/s

3.1. Assumption

• In these simulations, all states are assumed to be measurable. For a noisy system, however, a Luenberger observer or a Kalman filter can be designed using measurements of the rotary table position and velocity.

3.2. Manual control

In this control method, a constant input of 30,000 is applied to the drill-bit system. As shown in Figure 4, the system becomes stuck due to torsional vibrations, causing all velocities to converge to zero.

Figure 4.

Velocity of the top driver and drill bit under manual control with an input torque of 30,000 N⋅ m.

3.3. Tube MPC, SMC, and PID controllers

For designing these controllers, the following assumptions are considered:

• All states are assumed to be measurable.

• No measurement delay is considered.

• The top driver velocity is constrained to remain non-negative during control for Tube MPC.

In this section a normal Tube MPC with the following constrains considered

\begin{matrix} | x_{1} |, | x_{3} |, | x_{5} |, | x_{7} | \leq 20 \\ \frac{d (T_{m})}{d t} \leq 1 0^{4} \\ 0 \leq T_{m} \leq 4 * 1 0^{4} \end{matrix}

For the SMC controller, the input proposed in Sadeghimehr et al. (2021) has been used.

\begin{align} u = (c_{r} + c_{t}) x_{1} + k_{t} x_{2} - c_{t} x_{3} + J_{r} {\dot{x}}_{1}^{d} \\ - J_{r} \frac{k_{1}}{k_{2}} \frac{c_{t}}{J_{p}} e_{2} - k_{2} (η + k_{2} γ | s_{1} |) sign (s_{1}) \end{align}

(19)

Since the controller was originally tuned for an input delay system, it did not provide a satisfactory response with the parameters used in the referenced article. Therefore, the parameters were refined to achieve a better response. The updated parameters are as follows:

k_{1} = 5, k_{2} = 3, η = 1000, γ = 0.205

and the PID controller parameters are given as follows:

P = 911, I = 198, D = - 5.54 * 1 0^{3}, T_{f} = 6.27

(20)

Based on the results presented in Figure 5 and summarized in Tables 2 and 3, the proposed controller demonstrates superior performance by achieving reduced MSE, shorter settling time, and faster rise time compared to alternative control strategies. Moreover, as illustrated in Figure 6, the PID and SMC controllers generate higher magnitude control inputs, indicating their inability to adhere to actuator constraints. Notably, the PID controller also exhibits pronounced overshoot in the drill bit velocity response, which is undesirable from a performance standpoint.

Figure 5.

Comparison of Tube MPC, SMC, PID, and LQT controllers without uncertainties. (a) Comparison of top driver velocity performance of controllers in the absence of system uncertainty; (b) Comparison of the Bit angular velocity performance of controllers in the absence of system uncertainty.

Table 2.

Comparison of top driver criteria without uncertainty.

	MSE	Settling time	Rising time
PID	2.99	44.73	12.48
Tube MPC	0.91	27.11	0.55
SMC	3.69	45.91	1.1711
Linear MPC	0.96	28.66	0.57
LQT	11.87	63.04	21.17

Table 3.

Drill bit driver criteria comparison without uncertainty.

	MSE	Settling time	Rising time
PID	10.63	55.16	9.36
Tube MPC	5.34	32.03	5.11
SMC	7.57	48.84	5.75
Linear MPC	5.45	38.58	5.19
LQT	18.77	63.07	21.27

Figure 6.

Comparison of control inputs for tube MPC, SMC, PID, and LQT controllers without uncertainties.

3.4. J_p uncertainty

Based on Figure 7, the control performance criteria presented in Tables 4 and 5 are very close to those obtained in the nominal (uncertainty-free) case. Moreover, the proposed controller achieves a lower MSE, as well as shorter rise and settling times, compared to the other controllers. Additionally, as shown in Figure 8, similar to the previous section, the PID controller exhibits a significantly larger overshoot than the other controllers.

Figure 7.

Comparison of Tube MPC, SMC, PID, and LQT controllers with 20% uncertainty on J_p. (a) Comparison of top driver velocity performance of controllers with 20% uncertainty on J_p; (b) Comparison of the Bit angular velocity performance of controllers with 20% uncertainty on J_p.

Table 4.

Top driver criteria comparison with 20% J_p uncertainty.

	MSE	Settling time	Rising time
PID	3.02	55.52	12.30
Tube MPC	0.98	27.11	0.55
SMC	3.53	45.91	1.1711
Linear MPC	0.99	28.48	0.57
LQT	11.84	62.37	20.30

Table 5.

Drill bit driver criteria comparison with 20% J_p uncertainty.

	MSE	Settling time	Rising time
PID	10.65	54.56	9.45
Tube MPC	5.62	33.60	5.18
SMC	7.43	48.57	5.77
Linear MPC	5.63	39.36	5.24
LQT	18.67	62.33	21.24

Figure 8.

Comparison of Control Inputs for Tube MPC, SMC, PID, and LQT Controllers with 20% uncertainty for J_p W_ob uncertainty.

3.5. MPC parameters tuning

Based on Table 6, any increase or decrease in the weights results in a higher MSE or leads to system instability; therefore, the MPC controller parameters are selected as follows:

\begin{matrix} W e i g h t s = [1 0^{3} 1 0^{3} 1 0^{3} 1 0^{3} 1 0^{3} 1 0^{9} 1 0^{9} 1 0^{9}], T_{s} = 0.1 \end{matrix}

Table 6.

MSE comparison for different MPC weight parameters.

Furthermore, according to the results in Table 7, the prediction horizon is selected as N_p = 100 and the control horizon as N_c = 50. Deviations from these values—either increasing or decreasing the horizons—lead to suboptimal performance: shorter horizons increase the mean squared error (MSE) at both the top driver and drill bit, while longer horizons only impose higher computational costs without delivering noticeable performance improvements.

Table 7.

Effect of prediction and control horizons on MSE in MPC.

Prediction horizon	Control horizon	Top-driver MSE	Drill-bit MSE
50	10	1.0598	5.6729
100	10	0.9206	5.3942
100	50	0.9150	5.3413
500	50	1.0318	5.3064
500	100	0.9294	5.3461
500	250	0.9790	5.4922
1000	500	0.9553	5.4102

3.6. W_ob uncertainty

Based on Figure 9, the control performance criteria in Tables 8 and 9 show that the proposed controller outperforms its counterpart in terms of MSE, rise time, and settling time. It is worth noting that, due to the presence of oscillations, the settling time cannot be precisely determined; therefore, the reported values correspond to the case without uncertainty. Regarding oscillations, although the Tube MPC produces larger oscillations at the top drive, it results in smaller oscillations at the drill bit. As shown in Figure 10, the SMC exhibits greater oscillations in the control inputs, whereas the PID controller produces a larger peak.

Figure 9.

Comparison of Tube MPC, SMC, PID, and LQT controllers with 5% uncertainty on W_ob. (a) Comparison of top driver velocity performance of controllers with 5% uncertainty on W_ob; (b) Comparison of the Bit angular velocity performance of controllers with 5% uncertainty on W_ob.

Table 8.

Comparison of top driver criteria with 5% W_ob uncertainty.

	MSE	Settling time	Rising time	Oscillation amplitude
PID	3.09	44.73	12.47	0.3
Tube MPC	1.11	27.11	0.49	0.7
SMC	3.89	45.91	0.89	1.4
Linear MPC	1.13	28.48	5.24	0.7
LQT	12.26	63.04	20.19	0.68

Table 9.

Comparison of drill bit criteria with 5% W_ob uncertainty.

	MSE	Settling time	Rising time	Oscillation amplitude
PID	11.59	55.16	9.56	1.23
Tube MPC	5.87	33.60	5.14	0.94
SMC	7.49	48.57	6.12	1.23
Linear MPC	5.90	39.36	1.01	1.01
LQT	19.90	63.07	21.90	1.14

Figure 10.

Comparison of control inputs for Tube MPC, SMC, PID, and LQT controllers with 5% uncertainty for W_ob.

3.7. LQT controller and tuning

For the LQT controller, the augmented system shown in equation (21) was employed. The weight of the integrator was set to 10⁹. Increasing this weight leads to unbounded controller gains, while decreasing it results in a significant increase in the MSE, reaching values of approximately 40.

\begin{aligned} u (t) & = [\begin{matrix} - k_{1} & k_{2} \end{matrix}] [\begin{matrix} x (t) \\ q (t) \end{matrix}] \\ \Rightarrow [\begin{matrix} \dot{x} (t) \\ \dot{q} (t) \end{matrix}] & = [\begin{matrix} A - B k_{1} & B k_{2} \\ - C & 0 \end{matrix}] [\begin{matrix} x (t) \\ q (t) \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] r \end{aligned}

(21)

k_{1} = 1 0^{4} * [0.1521 0.3559 3.1719 3.3756 2.151

(22)

- 0.2673 1.5582 - 3.4643]

k_{2} = - 3.1624 * 1 0^{4}

Based on Monteiro and Trindade (2017), the LQG controller was unable to achieve satisfactory performance in the presence of unstructured perturbations. Consequently, although the control criteria were evaluated, no comparison with other controllers was presented due to the limited performance of this method. The system parameters, summarized in Table 10, were obtained through a trial-and-error procedure, and a sensitivity analysis was performed. It should also be noted that the weighting matrices differ from those used in the MPC controller, since applying identical weights in the LQG design leads to unbounded controller gains.

Table 10.

MSE comparison for different LQT weight parameters.

Weights	Top-driver MSE	Drill-bit MSE
[10 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 1]	11.87	18.77
[10² 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[1 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10³ 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10¹ 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁶ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁴ 10⁴ 10⁷ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁵ 10⁷ 10⁴ 10⁷ 1]	11.88	18.79
[10 10² 10⁵ 10³ 10⁷ 10⁴ 10⁷ 1]	11.87	18.79
[10 10² 10⁵ 10⁴ 10⁸ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁴ 10⁶ 10⁴ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁴ 10⁷ 10⁵ 10⁷ 1]	11.88	18.80
[10 10² 10⁵ 10⁴ 10⁷ 10³ 10⁷ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁸ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁶ 1]	Infinite gain	Infinite gain
[10 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 10]	11.88	18.79
[10 10² 10⁵ 10⁴ 10⁷ 10⁴ 10⁷ 0.1]	Infinite gain	Infinite gain

3.8. PID controller and tuning

For the PID controller, the general form given in equation (23), which includes a high-pass filter on the derivative term as an implementation capability, has been considered. The corresponding coefficients are presented in equation (20). The Bode diagram of the closed-loop system is shown in Figure 11. Although the phase margin (PM) is relatively small, it’s because to get better rise time and faster performance, as illustrated in Figure 6. Further reduction of the PM could result in unstable system, which is undesirable for system performance.

G (s) = P + \frac{I}{s} + D * \frac{s}{T_{f} * s + 1}

(23)

Figure 11.

Bode diagram of the closed-loop system with gain and phase margins for stability analysis.

3.9. Unstructured time delays

The Tube MPC controller exhibits inherent robustness against unstructured time delays. According to Shen et al. (2013), the worst-case delay in transmitting downhole data to the surface is approximately 5 ms. To demonstrate this robustness, Figure 12 presents the drill-bit velocity response when a 1-s delay is imposed on both drill collar and drill-bit measurements. Furthermore, Figure 13 illustrates the controller’s performance under multiple heterogeneous delays, where we assume 0.3 s for drill pipe data, 0.6 s for drill collar data, and 0.9 s for drill-bit data. These scenarios highlight the controller’s capability to maintain acceptable performance despite significant operational delays.

Figure 12.

Drill-bit velocity response under a fixed 1-s measurement delay.

Figure 13.

Drill-bit velocity response under heterogeneous variable delays.

4. Conclusions

In this article, a four-degrees-of-freedom model is developed, and a robust tube MPC strategy is proposed to mitigate stick-slip oscillations in the drill string system. This approach effectively manages system uncertainties, such as variations in drill pipe inertia—impacted by pipe length—and changes in weight on the bit due to axial vibrations. Simulation results highlight the controller’s robustness, demonstrating its capacity to stabilize the system under these uncertain conditions. The proposed controller contributes to preventing stick-slip by actively limiting the rotary table torque, thereby avoiding the development of excessive differences in rotational speed or angular position along the drill string. By constraining these discrepancies, the controller effectively suppresses the onset of stick-slip dynamics. Compared to an SMC, the proposed controller achieves faster stabilization, reducing settling time by 41% at the top driver and 25% at the drill bit, with MSE reductions of 74% and 28%, respectively. The tube MPC also lowers control input peaks and reduces oscillations by 50% in the top driver and 35% in the drill bit under uncertainty. In comparison with a PID controller, the proposed controller achieves, in the top driver, a 69% improvement in MSE, a 40% reduction in settling time, and a 96% improvement in rise time; in the drill bit, it achieves a 50% lower MSE, a 42% faster settling time, and a 45% faster rise time. Furthermore, while the PID controller achieves 56% lower oscillations at the top driver, the tube MPC achieves 24% lower oscillations at the drill bit. Due to the very poor performance of the LQT controller, its results were not compared with those of the proposed method. Future research could focus on addressing periodic disturbances—commonly encountered in systems with rotating components such as motors—and on exploring explicit MPC approaches and validate the proposed control schemes through a co-simulation platform or laboratory implementation.

Footnotes

ORCID iDs

Mahdi Kahrizi

Amirhossein Nikoofard

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

References

Aarsnes

UJF

Shor

(2018) Torsional vibrations with bit off bottom: modeling, characterization and field data validation. Journal of Petroleum Science and Engineering 163: 712–721. https://doi.org/10.1016/j.petrol.2017.11.024

Aarsnes

UJF

van de Wouw

(2018) Dynamics of a distributed drill string system: characteristic parameters and stability maps. Journal of Sound and Vibration 417: 376–412. https://doi.org/10.1016/j.jsv.2017.12.002

Aarsnes

UJF

Aamo

Krstic

(2019) Extremum seeking for real-time optimal drilling control. In: 2019 American Control Conference (ACC). IEEE, pp. 5222–5227.

Aarsnes

UJF

Wang

(2021) Feasibility study of nonlinear model predictive control in mitigating stick slip in drilling. IOP Conference Series: Earth and Environmental Science 621(1): 012150. https://doi.org/10.1088/1755-1315/621/1/012150

Chen

Allgöwer

(1998) A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34(10): 1205–1217. https://doi.org/10.1016/s0005-1098(98)00073-9

Cheng

, et al. (2020) Modeling and control of drill-string system with stick-slip vibrations using LPV technique. IEEE Transactions on Control Systems Technology 29(2): 718–730. https://doi.org/10.1109/tcst.2020.2978892

Doghmane

Kidouche

Djezzar

, et al. (2022) Observer-Based H∞ controller design for high frequency stick-slip vibrations mitigation in drill-string of rotary drilling systems. Vibrations 5(2): 264–289.

Fouad

Abdel-Magid

Alhammadi

, et al. (2011) An optimized chattering free sliding mode controller to suppress torsional vibrations in drilling strings. In: Proceedings of the 2011 IEEE GCC Conference and Exhibition (GCC). IEEE, pp. 573–576.

Ghasemi

Song

(2017) Control of a vertical drilling system using a cascade sliding mode controller. In: 2017 American Control Conference (ACC). IEEE, pp. 5830–5835.

10.

Gonzalez Villarreal

Rossiter

Shin

, et al. (2018) Laguerre-based adaptive MPC for attitude stabilization of quad-rotor. In: Proceedings of the 2018 UKACC 12th International Conference on Control (CONTROL). IEEE, pp. 360–365.

11.

Huang

Xie

, et al. (2018) Investigation of PDC bit failure base on stick-slip vibration analysis of drilling string system plus drill bit. Journal of Sound and Vibration 417: 97–109. https://doi.org/10.1016/j.jsv.2017.11.053

12.

Huang

Zhou

, et al. (2024) Drill-String vibration analysis based on rock strength fuzzy comprehensive evaluation and bit–rock interaction parameter identification. IEEE Transactions on Industrial Electronics 72: 1–10.

13.

Laib

Gharib

(2024) Design of an intelligent Cascade control scheme using a hybrid adaptive Neuro-Fuzzy PID controller for the suppression of drill string torsional vibration. Applied Sciences 14(12): 5225. https://doi.org/10.3390/app14125225

14.

MacLean

JDJ

Vaziri

Aphale

, et al. (2021) Feedback control method to suppress stick-slip in drill-strings featuring delay and actuation constraints. The European Physical Journal Special Topics 230(18): 3627–3642. https://doi.org/10.1140/epjs/s11734-021-00228-4

15.

Mayne

Rawlings

Rao

, et al. (2000) Constrained model predictive control: stability and optimality. Automatica 36(6): 789–814. https://doi.org/10.1016/s0005-1098(99)00214-9

16.

Mendil

Kidouche

Doghmane

(2021) Hybrid sliding PID controller for torsional vibrations mitigation in rotary drilling systems. Indonesian Journal of Electrical Engineering and Computer Science 22(1): 146–158. https://doi.org/10.11591/ijeecs.v22.i1.pp146-158

17.

Mihajlovic

Van Veggel

van de Wouw

, et al. (2004) Analysis of friction-induced limit cycling in an experimental drill-string system. Journal of Dynamic Systems, Measurement, and Control 126(4): 709–720. https://doi.org/10.1115/1.1850535

18.

Monteiro

HLS

Trindade

(2017) Performance analysis of proportional-integral feedback control for the reduction of stick-slip-induced torsional vibrations in oil well drillstrings. Journal of Sound and Vibration 398: 28–38. https://doi.org/10.1016/j.jsv.2017.03.013

19.

Sadeghimehr

Nikoofard

Khaki Sedigh

(2021) Predictive-based sliding mode control for mitigating torsional vibration of drill string in the presence of input delay and external disturbance. Journal of Vibration and Control 27(21–22): 2432–2450. https://doi.org/10.1177/1077546320960995

20.

Sagert

Di Meglio

Krstic

, et al. (2013) Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling. IFAC Proceedings Volumes 46(2): 779–784. https://doi.org/10.3182/20130204-3-fr-2033.00126

21.

Saldivar

Mondié

Niculescu

, et al. (2016) A control oriented guided tour in oilwell drilling vibration modeling. Annual Reviews in Control 42: 100–113. https://doi.org/10.1016/j.arcontrol.2016.09.002

22.

Shen

Zhang

L-T

Cui

S-L

, et al. (2013) Delay pressure detection method to eliminate pump pressure interference on the downhole mud pressure signals. Mathematical Problems in Engineering 2013(1): 797549. https://doi.org/10.1155/2013/797549

23.

Talbi

Laib

Gharib

(2023) Stick-slip stabilization in oil well drill-strings via optimal hybrid fractional order fuzzy logic control scheme. Journal of Low Frequency Noise, Vibration and Active Control 42(2): 911–926. https://doi.org/10.1177/14613484221135863

24.

Tang

Zhu

Shi

, et al. (2015) Study of the influences of rotary table speed on stick-slip vibration of the drilling system. Petroleum 1(4): 382–387. https://doi.org/10.1016/j.petlm.2015.10.004

25.

Tashakori

Vossoughi

Zohoor

, et al. (2021) Prediction-based control for mitigation of axial–torsional vibrations in a distributed drill-string system. IEEE Transactions on Control Systems Technology 30(1): 277–293. https://doi.org/10.1109/tcst.2021.3065669

26.

Van Veggel

Dai

C-H

van de Wouw

, et al. (2015) Robust output-feedback control to eliminate stick-slip oscillations in drill-string systems. IFAC-PapersOnLine 48(6): 266–271.

27.

Wang

Tang

S-X

Krstic

(2020) Adaptive output-feedback control of torsional vibration in off-shore rotary oil drilling systems. Automatica 111: 108640. https://doi.org/10.1016/j.automatica.2019.108640

28.

Cheng

, et al. (2020) Disturbance estimator and smith predictor-based active rejection of stick–slip vibrations in drill-string systems. International Journal of Systems Science 51(5): 826–838. https://doi.org/10.1080/00207721.2020.1744046

29.

Xie

Zhang

(2022) Modelling and controlling of drill string stick slip vibrations in an oil well drilling rig. Journal of Petroleum Science and Engineering 216: 110759.

30.

Yan

Tian

Meng

, et al. (2023) Stick–Slip vibration characteristics study of the drill string based on PID controller. Energies 16(23): 7902. https://doi.org/10.3390/en16237902

31.

Yigit

Christoforou

(2006) Stick-slip and bit-bounce interaction in oil-well drillstrings. Journal of Energy Resources Technology 128(4): 268–274. https://doi.org/10.1115/1.2358141

32.

Zanelli

Horn

Frison

, et al. (2018) Nonlinear model predictive control of a human-sized quadrotor. In: Proceedings of the 2018 European Control Conference (ECC), Limassol, Cyprus, June 12-15 2018, 1542–1547.

Robust tube MPC for suppressing stick-slip vibrations in drill string systems

Abstract

Keywords

1. Introduction

2. The mathematical model

2.1. Drill string model

2.2. Bit-rock interaction

2.3. Simscape modeling

2.4. Karnopp’s model

2.5. Tube MPC

2.6. Stability and feasibility analysis

2.6.1. Nominal plant

2.6.2. Assumption 1

2.6.3. Assumption 2

2.6.4. Assumption 3

2.6.5. Assumption 4

2.7. Bound maximum error

2.8. Recursive feasibility and closed-loop stability

3. Simulations and results

3.1. Assumption

3.2. Manual control

3.3. Tube MPC, SMC, and PID controllers

3.4. J p uncertainty

3.5. MPC parameters tuning

3.6. Wob uncertainty

3.7. LQT controller and tuning

3.8. PID controller and tuning

3.9. Unstructured time delays

4. Conclusions

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

References

3.4. J_p uncertainty

3.6. W_ob uncertainty