Prediction-Based Control for Mitigation of Axial-Torsional

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Prediction-Based Control for Mitigation of Axial–Torsional Vibrations in a Distributed Drill-String System

Shabnam Tashakori, Gholamreza Vossoughi , Hassan Zohoor, and Nathan van de Wouw , Fellow, IEEE

Abstract— This article proposes a control strategy to stabilize the main sources of these unwanted vibrations, which leads the axial–torsional dynamics of a distributed drill-string system. to the reduction of drilling efficiency, system failure, and An infinite-dimensional model for the vibrational dynamics of rig downtime. The mitigation of these vibrations is hence of the drill string is used as a basis for controller design. In this article, both the cutting process and frictional contact effects utmost importance. are considered in the bit–rock interaction model. Moreover, Different approaches have been employed to model these models for the top-side boundary conditions regarding axial vibrational phenomena: lumped-parameter models [1]–[3], dis- and torsional actuation are considered. The resulting model is tributed parameter models [4]–[6], and neutral-type time-delay formulated in terms of neutral-type delay differential equations (NTD) models [7]–[9]. NTD models strike a favorable balance that involve constant state delays, state-dependent state delays, and constant input delays arising from the distributed nature between modeling accuracy and complexity of simulation, of the drill-string dynamics and the cutting process at the analysis, and control. Namely, these models are more accurate bit. Using a spectral approach, the stability and stabilizability than lumped-parameter models since they do not neglect the of the associated linearized dynamics are analyzed to sup- distributed nature of the drill-string dynamics. Moreover, for port controller design. An optimization-based continuous pole- such NTD models, a wide range of methods for stability analy- placement technique has been employed to design a stabilizing controller. Since the designed state-feedback control law needs sis and control are available, compared to models in terms of state prediction, a predictor with observer structure is proposed. partial differential equations (PDEs), while they involve less Both the controller and the predictor only employ top-side complexity in control design than distributed parameter models measurements. The effectiveness of the control strategy, in the [10]. Such an internal NTD model is obtained directly from presence of measurement noise, is shown in a representative case the distributed parameter model by neglecting the damping study. It is also shown that the controller is robust to parametric uncertainty in the bit–rock interaction. along the drill string. The NTD model was established in [7], [8], where the bit–rock interaction is modeled as a mere Index Terms— Continuous pole-placement method, distributed frictional contact, and modified in [9] with consideration of the dynamics, drill-string dynamics, neutral-type time-delay (NTD) model, prediction-based control, spectral approach, stability cutting process as well as the frictional contact in the bit–rock analysis. interaction. A review of mathematical modeling of axial and torsional self-excited drilling vibrations can be found in [11]. To study the bit–rock interaction in drilling systems, several I. INTRODUCTION friction models have been employed [12], [13]. However, it has RILLING systems are used for exploration and harvest- been shown in [14] that the bit–rock interaction consists of Ding of oil, gas, and geothermal energy and suffer from a cutting process and a frictional contact process, where the complex coupled dynamics that involve axial, torsional, and interaction between the drill-string dynamics and the delay lateral vibrations. Point contact between the drill string and nature of the cutting model leads to instability and self- the well-bore, mass imbalance, and downhole interactions are excited oscillations. This time-delay effect is known as the root cause of such self-excited axial–torsional vibrations. The Manuscript received September 18, 2020; revised February 27, 2021; aforementioned bit–rock interaction model has been employed accepted March 2, 2021. Manuscript received in final form March 10, 2021. Recommended by Associate Editor S. Pirozzi. (Corresponding author: to study the drill-string vibrations in [5] and [15]–[19]. In Gholamreza Vossoughi.) this article, we will combine the infinite-dimensional NTD Shabnam Tashakori, Gholamreza Vossoughi, and Hassan Zohoor are with modeling approach with such bit–rock interaction model to the Department of Mechanical Engineering, Sharif University of Technol- ogy, Tehran 11365-11155, Iran (e-mail: [email protected]; arrive at an NTD model capturing both the distributed nature [email protected]; [email protected]). of the drill string and the cutting process at the bit. Nathan van de Wouw is with the Department of Mechanical Engineering, The stability of vibrational models for drilling systems, Eindhoven University of Technology, 5600 MB Eindhoven, The Nether- lands, and also with the Department of Civil, Environmental and Geo- including the state-dependent delay (induced by the cutting Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: process in the bit–rock interaction), has been studied in [20]– [email protected]). [25] for finite-dimensional models and in [5] and [26] for Color versions of one or more figures in this article are available at https://doi.org/10.1109/TCST.2021.3065669. distributed models. For NTD-type models, the exponential Digital Object Identifier 10.1109/TCST.2021.3065669 stability is investigated in [27] for a torsional model with 1063-6536 © 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.

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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY considering only the frictional contact in the bit–rock interface. equations of motion are neutral-type delay differential equa- Here, we will pursue stability (and stabilizability) analysis for tions (NDDEs) with constant input delays and constant state the novel NTD model to be proposed. delays (both related to the wave propagation speeds along Vibrations of the drilling system are usually suppressed by the drill string) and a state-dependent state delay (induced adjusting surface-controlled variables (e.g., the hook force, by the bit–rock interaction). Then, a spectral approach is the top-drive torque, and the characteristics of the drilling employed to analyze the stability of the associated linearized fluid). The controller design for mitigation of drilling tor- dynamics to study the root causes of the steady-state vibrations sional vibrations has been studied in [28]–[31] for finite- and to serve as a basis for controller design. Moreover, with dimensional models and in [32]–[36] for infinite-dimensional an input transformation (i.e., precompensators), the system models. The suppression of drilling axial–torsional vibrations is rendered to be “formally stable” and “spectrally stabiliz- has been studied in [17], [24], and [37] for lumped-parameter able” [47] such that the stabilizability (by state feedback) models, in [38] for distributed parameter models, in [39] for of the system dynamics is guaranteed. Next, to design the a coupled PDE-ordinary differential equation (ODE) model, controller, the optimization-based continuous pole-placement and in [40]–[42] for NTD models. The Z-torque system of method [48] has been employed, aiming at eliminating both Shell [43] considers an infinite-dimensional model as a starting axial and torsional vibrations. Then, to render the control point and implements impedance matching in a practical way, strategy causal, a predictor with observer structure is designed but it still only considers the torsional dynamics. Besselink to complement the state-feedback controller. Note that the et al. [17] and Liu et al. [24] considered both the axial and generic controller and predictor design methodologies have torsional dynamics as a basis for controller design but employ been previously presented in the delay systems literature, but a simple lumped-parameter model and hence also ignore the in this article, they are employed for the first time to solve a rich dynamics of the drill string. Vromen et al. [28], [29] took stabilization control problem for drilling systems. multiple modes into account for the controller design but still The main contribution of this article is the design of only consider the torsional dynamics and a finite-dimensional a prediction-based state-feedback controller, mitigating the (discretized) model. Regarding the control of NTD models, coupled axial–torsional drill-string vibrations while dealing a controller is designed in [40] based on the Lyapunov theory, with the infinite-dimensional dynamics of the drill string aiming at suppressing stick-slip vibrations. Then, based on and the associated constant state delays and input delays simulation results, it is concluded that it also mitigates the and the state-dependent delay effect induced by the bit–rock axial vibrations. The design of a PID controller to avoid axial– interaction. Additional contributions of this article are that the torsional vibrations is presented in [41]. Since the downhole designed controller, first, guarantees asymptotic stability of the data are received with some time delay, a delayed feedback desired solution, second, is compatible with the top-drive actu- controller is also designed. In [42], the flatness property of ator limitations, third, only employs top-side measurements, the drilling system is used to design a feedback controller, fourth, is robust to measurement noise, and, fifth, is robust to which eliminates both axial and torsional vibrations. Note parametric uncertainty in the bit–rock interaction. Moreover, that the NTD models in [40]–[42] do not consider the bit– an illustrative case study is presented to show the effectiveness rock interaction model mentioned in [14], which is considered of both the controller and the predictor. essential in the root cause for drill-string vibrations. It is worth Existing control strategies come at many levels of com- mentioning that employing NTD models for controller design plexity (simple PI control, impedance matching, H∞-based serves as a worst case scenario since in-domain damping, dynamic controllers, and other wave-based control approaches which is neglected in these models, actually helps stabilization. [43], [49], [50]). Note that although the strategy introduced in Summarizing, many existing control strategies consider finite- this article is more involved than some previously presented dimensional models and hence ignore the inherent infinite- controllers, it is still feasible for practical implementation. The dimensional nature of the dynamics (see [17], [24]) and additional complexity is motivated by the fact that it provides many control strategies consider only the stabilization of the the possibility of dealing with the complex, coupled axial– torsional dynamics. Some examples are [4], [28], [29], and torsional and infinite-dimensional dynamics of the drill-string [36] and the industrial soft torque system [44]. system. This is important, first, as the interaction between the In this article, an infinite-dimensional NTD model is axial and torsional dynamics is key in the stability of the employed and extended to study coupled axial–torsional vibra- drilling system (see [14]) and, second, as higher order modes tions. In comparison with previous studies, first, both the can induce instability and a (simpler) control approach may cutting process and frictional contact effects are taken into not be able to mitigate drill-string vibrations (see [29], [51]). consideration in the bit–rock interaction model, and second, This article is organized as follows. Section II intro- realistic top-side boundary conditions are included. More duces the distributed model describing the axial and torsional specifically, the high inertia of the torsional top drive is drill-string dynamics. Subsequently, the associated linearized considered to regard the reflection of torsional waves at the dimensionless perturbation dynamics is presented. The sta- top drive. Furthermore, instead of the hook load, the axial bility properties of the open-loop system are analyzed in actuation is considered in terms of the velocity of the traveling Section IV. Moreover, the stabilizability of the system is block to which the drilling system is attached [45], [46]. studied. Section V presents the controller design. To make Hence, this article presents a novel, extended NTD model the resulting controller causal, a predictor is designed in for coupled axial–torsional drill-string dynamics. The resulting Section VI. Illustrative simulation results are presented in

TASHAKORI et al.: PREDICTION-BASED CONTROL FOR MITIGATION OF AXIAL–TORSIONAL VIBRATIONS 3

Section VII, where the robustness of the designed controller against parametric uncertainties is also analyzed. Finally, con- clusions are presented in Section VIII.

II. DISTRIBUTED DRILL-STRING MODEL To the best of authors’ knowledge, expressing the oscillatory behavior of the drilling system by using the following wave equations was ﬁrst presented in [52], [53]: ∂2U ∂2U (s, t) = c2 (s, t) (1a) ∂s2 a ∂t2 ∂2θ ∂2θ (s, t) = c2 (s, t) (1b) ∂s2 t ∂t2 where U(s, t) and θ(s, t) are the axial and angular positions along the drill string with the spatial variable s ∈[0, L] (with L the drill-string length) and time t (see Fig. 1), respectively. Fig. 1. Schematic of the drill string. The wave velocity constants in (1) are given by ρ ρ TABLE I c = , c = (2) MODEL PARAMETERS [14], [40], [42] a E t G where ρ, E,andG are, respectively, the density, Young modulus, and shear modulus of drilling pipes (see Table I). Based on the above wave equation model, the following relations between the downhole and top-side velocities hold (see [54]): ∂U 1 ∂U 1 ∂U ∂U b (t − τ ) = top (t) + top (t) + top (t − τ ) ∂ a ∂ ∂ ∂ 2 a t 2 t ca s t 1 ∂U − top (t − τ ) ∂ 2 a (3a) ca s ∂θ 1 ∂θ 1 ∂θ ∂θ b (t − τ ) = top (t) + top (t) + top (t − τ ) ∂ t ∂ ∂ ∂ 2 t t 2 t ct s t 1 ∂θtop − (t − 2τt ) (3b) ct ∂s where

Utop(t) := U(0, t), Ub(t) := U(L, t) (4a)

θtop(t) := θ(0, t),θb(t) := θ(L, t). (4b)

The time delays in (3) (τa and τt ) are, respectively, the time required for the axial and torsional waves to travel from one extremity of the drill string to the other, which are deﬁned as

τa = ca L,τt = ct L. (5) These time delays can be assumed constant since the drill- string length is quasi-constant on the relevant vibrational time respectively, where and J are, respectively, the drill pipe scale. cross-sectional area and polar moment of area, MB and IB The boundary conditions for the axial and torsional dynam- specify the bottom hole assembly (BHA) inertial character- ics are given by istics, IT is the top-drive moment of inertia, and α and β characterize the viscous friction at the top drive (see Table I). ∂Utop (t) = VTB(t) (6a) The traveling block is an arrangement of pulleys or sheaves, ∂t whereby the drilling line is reeved. This block can move ∂U ∂2U E b (t) =−M b (t) − W(t) (6b) (axially) up and down freely and, together with the crown ∂ B ∂ 2 s t block and the drill line, facilitates lifting the drill string. The and axial velocity of the traveling block VTB (i.e., the feed rate) ∂θ ∂2θ ∂θ top top top [46] in (6a) and the top-drive torque u in (7a) are the control GJ (t) = I (t) + β (t) − u (t) (7a) T ∂s T ∂t2 ∂t T inputs, both exerted at the rig (see Fig. 1). ∂θ ∂2θ Let us introduce the following input transformation in GJ b (t) =−I b (t) − T (t) (7b) ∂s B ∂t2 support of the stabilizing controller design, pursued in

4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

˙ ˙ Section V: 1) Normal Cutting (d > 0, Ub > 0, and θb > 0): All ∂U the cutting and friction components of the bit–rock u (t) = αV (t) − E top (t) (8a) 1 TB ∂s interaction are nonzero, which leads to the following ∂2θ bit–rock interaction law: u (t) = u (t) − I top (t) (8b) 2 T T ∂ 2 t W(t) = εaζ d(t) + σal (14a) ( ) ( ) 1 1 where u1 t and u2 t are the new control inputs and = ε 2 + μγ 2σ . 2 2 T (t) a d(t) a l (14b) E(∂Utop)/(∂s)(t) and IT (∂ θtop)/(∂t )(t) are precompen- 2 2 sators. The axial precompensator E(∂U )/(∂s)(t) is avail- top > , ˙ < , θ˙ > able as a measured output by the saver sub force measure- 2) Bit Bouncing (d 0 Ub 0 and b 0): The friction 2 2 components of both WOB and TOB are zero since there ment. The torsional precompensator IT (∂ θtop)/(∂t )(t) is also available as the top-side acceleration can be obtained by is no contact between the bit and well-bottom. > , ˙ > , θ˙ < differentiating the top-side velocity, which is itself available 3) Reverse Rotation (d 0 Ub 0 and b 0): There by measurement. Given (8), the top-side boundary conditions is no cutting although the bit is in contact with the (6a) and (7a) can be rewritten in terms of new control inputs formation. Therefore, the cutting components are zero. < as follows: 4) Bit Off-Bottom (d 0): There is no contact between the bit and the formation, which leads to W = T = 0. ∂Utop ∂Utop E (t) = α (t) − u (t) (9a) The NTD model is obtained directly from the wave equation ∂s ∂t 1 model, given in (1), the top-side boundary conditions, given ∂θtop ∂θtop GJ (t) = β (t) − u (t). (9b) in (9), and bottom-side boundary conditions, given in (6b) and ∂s ∂t 2 (7b), by employing Riemann variables, defined by On the other extremity, i.e., at the bit, W and T are the resistive force and torque applied from the formation to the ϒa = t + cas,a = t − cas (15) bit, called weight-on-bit (WOB) and torque-on-bit (TOB), respectively (see Fig. 1). These are modeled by the following for the axial dynamics and by bit–rock interaction law [9], [24]: ϒ = + ,= − t t ct s t t ct s (16) ˙ Wc = εaζ R(d(t))H θb(t) (10a) for the torsional dynamics. Consequently, the solutions of W = σalH(d(t))H U˙ (t) (10b) f b undamped axial and torsional wave equations are decomposed 1 T = εa2R(d(t))H θ˙ (t) (10c) as follows in these Riemann variables: c 2 b ( , ) = (ϒ ) + ( ) 1 2 ˙ ˙ U s t fa a ga a (17a) T f = μγ a σlSign θb(t) H(d(t))H Ub(t) (10d) 2 θ( , ) = (ϒ ) + ( ) s t ft t gt t (17b) where the subscripts c and f denote the cutting compo- ∈{ , } nents and friction components of the bit–rock interaction, where fi and gi , i a t , are arbitrary functions correspond respectively. The parameters ε, a,ζ,σ,l,μ,andγ in (10) to uptraveling and downtraveling waves, respectively, with characterize bit and rock properties, defined in Table I. The subscripts a and t the axial and torsional dynamics. With the depth of cut d(t), used in (10), is given by use of (4), (5), and [15]–[17], the downhole velocities can be expressed as d(t) = n(Ub(t) − Ub(t − τn(t))) (11) ∂f ∂g ˙ ( ) = a ( + τ ) + a ( − τ ) τ ( ) Ub t t a t a (18a) for a n-blade bit, where the state-dependent delay n t is the ∂ϒa ∂a time that takes for the bit to rotate by the angle of 2π/n.The ∂f ∂g θ˙ ( ) = t ( + τ ) + t ( − τ ). delay τ (t) is state-dependent and governed by b t t t t t (18b) n ∂ϒt ∂t 2π θb(t) − θb(t − τn(t)) = . (12) Along the line of [9], the procedure of obtaining the axial and n torsional NTD models is explained in the following. Nonlinear functions R(.), H(.), and Sign(.) in (10) are defined as follows: A. Axial NTD Model y, y ≥ 0 R(y) = (13a) The axial boundary conditions, (9a) and (6b), can be, 0, y < 0 respectively, reformulated in terms of Riemann variables by 1, y ≥ 0 using (17a) as follows: H(y) = (13b) 0, y < 0 ∂f ∂g (Ec −α) a (t)−(Ec +α) a (t) =−u (t) a ∂ϒ a ∂ 1 (19a) sign(y), y = 0 a a Sign(y) = (13c) ∂ ∂ − , , = fa ga [ 1 1] y 0 Eca (t + τa) − (t − τa) =−MB U¨b(t)−W(t). ∂ϒa ∂a which describes the following cases of drilling [15]. (19b)

TASHAKORI et al.: PREDICTION-BASED CONTROL FOR MITIGATION OF AXIAL–TORSIONAL VIBRATIONS 5

Equations (19b) and (18a) can be solved for (∂fa)/(∂ϒa)(t + Substituting (25) into (24a) gives the NTD model for the τ ) (∂ )/(∂ )( −τ ) a and ga a t a . After time shifting, the following torsional dynamics as follows: equations hold: ˙ ˙ θ¨b(t) − Aθ¨b(t − 2τt ) =−Bθb(t) − ABθb(t − 2τt ) ∂fa 1 1 (t) − ( ) + ( − τ ) ∂ϒ T t AT t 2 t a IB IB 1 M 1 + Cu (t − τ ) (26) = − B U¨ (t − τ ) + U˙ (t − τ ) − W(t − τ ) 2 t b a b a a 2 E ca E ca where (20a) β − c GJ c GJ 2B ∂ = t , = t , = . ga A B C (27) (t) β + ct GJ IB β + ct GJ ∂ a Substituting the (linearized) bit–rock interaction law (14b), 1 M 1 = B U¨ (t + τ ) + U˙ (t + τ ) + W(t + τ ) . which is valid as we aim to (locally) stabilize a nominal b a b a a 2 E ca E ca drilling solution with the “normal cutting” regime, the depth (20b) of cut (11) in (26) gives

Substituting (20) into (19a) ultimately leads to θ¨b(t) − Aθ¨b(t − 2τt ) ¨ ( ) − ¯ ¨ ( − τ ) =−¯ ˙ ( ) − ¯ ¯ ˙ ( − τ ) ˙ ˙ 1 2 Ub t AUb t 2 a BUb t ABUb t 2 a =−Bθb(t) − ABθb(t − 2τt ) − μγ a σl(1 − A) 1 1 2IB − W(t) + AW¯ (t − 2τ ) a 1 2 MB MB − εa n(Ub(t) − Ub(t − τn(t))) 2IB + Cu¯ 1(t − τa) (21) 1 2 + Aεa n(Ub(t − 2τt ) − Ub(t − 2τt − τn(t − 2τt ))) where 2IB α − c E c E 2B¯ + Cu2(t − τt ). (28) A¯ = a , B¯ = a , C¯ = . (22) α + ca E MB α + ca E The NTD model, given in (23) and (28), is in the form of Substituting the (linearized) bit–rock interaction law (14a), an NDDE with several state delays and input delays. which is valid as we aim to (locally) stabilize a nominal Reformulating the PDE model into a DDE model beneﬁts drilling solution with the “normal cutting” regime, the depth the control design by opening up the opportunities for using of cut (11) in (21) gives controller synthesis tools for delay systems [10]. ¨ ( ) − ¯ ¨ ( − τ ) Ub t AUb t 2 a C. Dimensionless Perturbation Dynamics ˙ ˙ 1 =−B¯ Ub(t) − A¯ B¯ Ub(t − 2τa) − σal 1 − A¯ The steady-state response of the model in (23) and (28) for MB nominal constant inputs, u1s and u2s,isdeﬁnedas − 1 ε ζ ( ( ) − ( − τ ( ))) a n Ub t Ub t n t U (t) = U (t) := V t + U , U˙ = V (29a) MB b s 0 0 s 0 ˙ 1 θb(t) = θs(t) := 0t + θ0, θs = 0 (29b) + A¯εaζ n(Ub(t − 2τa) − Ub(t − 2τa − τn(t − 2τa))) MB where V and are the nominal penetration rate and nominal + ¯ ( − τ ). 0 0 Cu1 t a (23) rotational velocity of the bit, which physically represent the desired drilling response without vibrations. Substituting (29) B. Torsional NTD Model into (23) and (28) gives the following relations between the Reformulating the torsional boundary conditions, (9b) and nominal speeds and nominal constant control inputs, associ- (7b), in terms of Riemann variables (17b), respectively, gives ated with the steady-state response in (29): 1 ∂ft ∂g V B¯ 1 + A¯ =− εaζ n 1 − A¯ V τ (GJc − β) (t) − (GJc + β) t (t) =−u (t) 0 0 n0 t ∂ϒ t ∂ 2 (24a) MB t t ∂ ∂ − 1 σ − ¯ + ¯ ft gt al 1 A Cu1s (30a) GJct (t + τt ) − (t − τt ) =−IB θ¨b(t)−T (t). MB ∂ϒt ∂t 1 2 (24b) 0 B(1 + A) =− εa n(1 − A)V0τn0 2IB (∂ )/(∂ϒ )( + τ ) 1 Solving (24b) and (18b) for ft t t t and − μγ 2σ ( − ) + . (∂ )/(∂ )( − τ ) a l 1 A Cu2s (30b) gt t t t and shifting time leads to the following 2IB relations: This condition also leads to a constant depth of cut d0 and ∂ τ τ ft 1 IB ˙ 1 a constant delay n0 [nominal value for n in (23) and (28)] (t) = − θ¨b(t − τt )+θb(t − τt )− T (t −τt ) ∂ϒt 2 GJct GJct satisfying (25a) 2πV d = 0 (31) ∂g 1 I 1 0 t ( ) = B θ¨ ( + τ )+θ˙ ( + τ )+ ( + τ ) . 0 t b t t b t t T t t π ∂t 2 GJct GJct 2 τn0 = (32) (25b) n0

6 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY which are obtained by substituting (29) into (11) and (12), follows: respectively. u tˆ − Au¯ tˆ − 2ˆτ In line with the model formulations in [14], [17], and [24], a =−¯ ˆ − ¯ ¯ ˆ − τ − ψ¯ ˆ − ˆ − τ ˆ the following scaled perturbed quantities are introduced by Nu t ANu t 2â u t u t ˆn t = + ψ¯ ¯ ˆ − τ − ˆ − τ − τ ˆ − τ using the characteristic time t∗ ct L and characteristic length A u t 2ˆ a u t 2â ˆn t 2â L∗ = L: − ¯ ϕ ˆ − ϕ ˆ − τ ˆ Q t t ˆn t + ¯ ¯ ϕ − τ − ϕ − τ − τ − τ U − U Q A tˆ 2â tˆ 2â ˆn tˆ 2â = b s u (33a) + ˆ − τ L∗ r1 t â (40a) ϕ = θb − θs (33b) ϕ ˆ − ϕ ˆ − τ t A t 2ˆt = ¯ 2 ( − ) r1 Cct L u1 u1s (33c) =−Nϕ tˆ − ANϕ tˆ − 2ˆτt − ψ u tˆ − u tˆ − τˆn tˆ 2 2 r2 = Cc L (u2 − u2s). (33d) + ψ ˆ − τ − ˆ − τ − τ ˆ − τ t A u t 2ˆ t u t 2ˆt ˆn t 2ˆt − Q ϕ tˆ − ϕ tˆ − τˆ tˆ Subsequently, the dimensionless perturbation dynamics is n + ϕ ˆ − τ − ϕ ˆ − τ − τ ˆ − τ given by QA t 2ˆt t 2ˆt ˆn t 2ˆt + r2 tˆ − τˆt (40b) u tˆ − Au¯ tˆ − 2ˆτ a where =−¯ ˆ − ¯ ¯ ˆ − τ − ψ¯ ˆ − ˆ − τ ˆ Nu t ANu t 2â u t u t ˆn t Q¯ =−ψδ¯ 0/2π, Q =−ψδ0/2π (41) + ψ¯ A¯ u tˆ − 2ˆτa − u tˆ − 2ˆτa − τˆn tˆ − 2ˆτa δ − ψν¯ τˆ tˆ − τˆ + ψ¯ A¯ν τˆ tˆ − 2ˆτ − τˆ with 0 the dimensionless nominal depth of cut defined as 0 n n0 0 n a n0 πν + − τ d0 2 0 r1 tˆ â (34a) δ0 = = . (42) L∗ ω ϕ tˆ − Aϕ tˆ − 2ˆτ 0 t Note that (40) is preferred to (34) since the former equations =−Nϕ tˆ − ANϕ tˆ − 2ˆτ − ψ u tˆ − u tˆ − τˆ tˆ t n do not depend explicitly on the state-dependent delayτ ˆn. + ψ ˆ − τ − ˆ − τ − τ ˆ − τ A u t 2ˆt u t 2ˆt ˆn t 2ˆt − ψν τ ˆ − τ + ψ ν τ ˆ − τ − τ D. Associated Linearized System Dynamics 0 ˆn t ˆn0 A 0 ˆn t 2ˆt ˆn0 State-dependent delay differential equations (SD-DDEs), + r2 tˆ − τˆt (34b) such as those in (40) with (36), are nonlinear as the state where the scaled time and scaled time delays are defined as argument is a function of the state itself [55]. Therefore, follows: by linearization, we mean obtaining an associated linear DDE with the same local stability features as the original nonlinear τ τ τ t a t n0 system. tˆ = , τâ = , τˆt = , τˆn0 = (35) t∗ t∗ t∗ t∗ The linearization of neutral-type SD-DDEs has been studied in [56], where it is shown that the exponential stability of the and the dimensionless state-dependent delayτ ˆ (t) in terms of n trivial solution of the associated linearized equation implies the perturbation coordinates can be obtained from the exponential stability of a constant steady-state solution of 2π the original NDDE. ϕ tˆ − ϕ tˆ − τˆ tˆ + ω τˆ tˆ = . (36) T =[ ]:= n 0 n n Introducing the state vector x x1 x2 x3 x4 [uu ϕϕ], (40) is rewritten as follows: Moreover, in (34) and (36), ω and ν are the scaled nominal 0 0 x tˆ − E x tˆ − τˆ − E x tˆ − τˆ velocities given by 1 1 2 2 = f x tˆ , x tˆ−τˆ , x tˆ − τˆ , x tˆ−τˆ , x tˆ−τˆ , x tˆ−τˆ 1 2 3 4 5 ω = ,ν= . 0 0t∗ 0 ct V0 (37) + B1r1 tˆ − hˆ1 + B2r2 tˆ − hˆ2 (43) where f =[f f f f ]T with Note that the prime () in (34) indicates differentiation with 1 2 3 4 respect to the scaled time tˆ and = ˆ f1 x2 t f =−Nx¯ tˆ − A¯ Nx¯ tˆ − τˆ − ψ¯ x tˆ − x tˆ − τˆ N¯ = Bc¯ L, N = Bc L 2 2 2 1 1 1 3 t t + ψ¯ ¯ − τ − − τ 1 1 A x1 tˆ ˆ1 x1 tˆ ˆ4 ψ¯ = ε ζ 2 2,ψ= ε 2 2 3. a nct L a nct L (38) − Q¯ x tˆ − x tˆ − τˆ MB 2IB 3 3 3 + Q¯ A¯ x tˆ − τˆ − x tˆ − τˆ Motivated by [24], (36) can be rewritten as 3 1 3 4 = ˆ f3 x4 t ϕ − ϕ − τ tˆ tˆ ˆn tˆ f4 =−Nx4 tˆ − ANx4 tˆ − τˆ2 − ψ x1 tˆ − x1 tˆ − τˆ3 τˆn tˆ − τˆn0 =− (39) ω + ψ ˆ − τ − ˆ − τ 0 A x1 t ˆ2 x1 t ˆ5 − Q x tˆ − x tˆ − τˆ sinceτ ˆn0 = 2π/nω0. Substituting (39) into (34), 3 3 3 the dimensionless perturbation dynamics can be rewritten as + QA x3 tˆ − τˆ2 − x3 tˆ − τˆ5 . (44)

TASHAKORI et al.: PREDICTION-BASED CONTROL FOR MITIGATION OF AXIAL–TORSIONAL VIBRATIONS 7

The matrices in (43) are defined as follows: ⎡ ⎤ 0100 ⎡ ⎤ ⎡ ⎤ ⎢−ψ¯ − N¯ − Q¯ 0 ⎥ 0000 0000 A = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ 0001⎦ ⎢ 0 A¯ 00⎥ ⎢ 0000⎥ E1 = , E2 = −ψ − Q − N ⎣ 0000⎦ ⎣ 0000⎦ ⎡ 0 ⎤ 0000 000A 0000 ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ψ¯ A¯ − A¯ N¯ Q¯ A¯ 0⎥ 0 0 A1 = ⎣ ⎦ ⎢ 1⎥ ⎢ 0⎥ 0000 B = ⎢ ⎥, B = ⎢ ⎥ (45) 1 ⎣ 0⎦ 2 ⎣ 0⎦ ⎡ 0000⎤ 0 1 000 0 ⎢ 000 0⎥ A = ⎢ ⎥ 2 ⎣ 000 0⎦ and the time delays are given by ψ − ⎡ A 0 QA⎤ AN 0000 ⎢ ⎥ τ = τ , τ = τ , τ = τ ( ) ψ¯ 0 Q¯ 0 ˆ1 2â ˆ2 2ˆt ˆ3 ˆn xt A = ⎢ ⎥ 3 ⎣ 0000⎦ τˆ = 2ˆτ + τˆ (x ), τˆ = 2ˆτ + τˆ (x ) 4 a n t 5 t n t ψ 0 Q 0 hˆ = τˆ , hˆ = τˆ . (46) ⎡ ⎤ 1 a 2 t 0000 ⎢ ⎥ ⎢ −ψ¯ A¯ 0 −Q¯ A¯ 0⎥ A4 = ⎣ ⎦ The notationτ ˆn(xt ) is simply indicating thatτ ˆn is a state- 0000 0000 dependent delay, where xt denotes the past function segment of ⎡ ⎤ the state variables, defined as xt (s) := x(t +s), s ∈[−τˆn, 0]. 0000 ⎢ ⎥ Based on [56], the linearized system associated with the = ⎢ 0000⎥. T A5 ⎣ ⎦ (50) constant solution x(tˆ) = x0 =[0000] is given by 0000 −ψ A 0 −QA 0 Summarizing, in this section, the associated linearized per- x tˆ − E1x tˆ − τ¯1 − E2x tˆ − τ¯2 turbation dynamics has been obtained. These dynamics [see = D f(x , x , x , x , x , x )x tˆ 1 0 0 0 0 0 0 (49)] are formulated in terms of NDDEs with constant state 5 delays and constant input delays. The latter is important + ( , , , , , ) ˆ − τ Di+1f x0 x0 x0 x0 x0 x0 x t ¯i in the context of control since it supports the application i=1 of the continuous pole-placement method [48] for controller + − ¯ + − ¯ B1r1 tˆ h1 B2r2 tˆ h2 (47) synthesis, which is applicable on linear time-invariant time- delay systems. where III. CONTROL PROBLEM FORMULATION In the drilling industry, it is desired to realize constant axial τ¯1 = τˆ1, τ¯2 = τˆ2, τ¯3 = τˆn(x0) and angular velocities during operation. Therefore, we design a τ = τ + τ ( ), τ = τ + τ ( ) state-feedback control law that drives the downhole velocities, ¯4 2â ˆn x0 ¯5 2ˆt ˆn x0 ˙ ˙ Ub(t) and θb(t), to constant (and positive) set-point values h¯1 = hˆ1, h¯2 = hˆ2. (48) V0 and 0, respectively. Note that this is achieved if the equilibrium x = 0 of (49) is stabilized. Toward this goal, we employ two control inputs, the dimensionless force r (tˆ) The constant delayτ ˆn(x0) is obtained by substitution of 1 and the dimensionless torque r (tˆ), which enter the dynamics x(tˆ) ≡ x0 in (36), which results inτ ˆn(x0) = 2π/nω0.Note 2 in a delayed fashion [see (49)]. that Di f in (47) indicates the derivatives of f with respect to the ith argument of f. Finally, the linearized nondimensional For drilling systems, there are different methods of data perturbation dynamics can be written in the following form: transmission from well-bottom to the top (mud-pulse teleme- try, acoustic waves, and so on). These downhole sensing methods introduce significant time delays and are subjected 2 5 to significant noise levels. Besides, the downhole sensors are − − τ = + − τ x tˆ Ei x tˆ ¯i A0x tˆ Ai x tˆ ¯i prone to failure due to harsh downhole conditions. Therefore, i=1 i=1 in this article, we only employ the top-side measurements, 2 i.e., the rate of penetration and angular velocity measured + Biri tˆ − h¯i (49) at the top-drive [(∂Utop)/(∂t) and (∂θtop)/(∂t), respectively] = i 1 and the data concerning the saver sub force and saver sub torque [(∂Utop)/(∂s) and (∂θtop)/(∂s), respectively]. Conse- where Ai := Di+1f(x0, x0, x0, x0, x0, x0) gives quently, (∂Ub)/(∂t)(t −τa) and (∂θb)/(∂t)(t −τt ) are available

8 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

(indirectly) due to the use of relations (3a) and (3b), respec- always transform an NDDE to DDAE, without introducing tively. Subsequently, the dimension-less measured output vec- any additional characteristic root. Hence, the analysis can be T tor y(tˆ) =[y1 y2]:=[u (tˆ − τâ)ϕ(tˆ − τˆt )] is expressed done based on the NDDEs and the transformation to DDAEs as follows: is performed whenever it is needed (e.g., when using TDS- STABIL [58] MATLAB package to find the stability-relevant y tˆ = C x tˆ − τˆ + C x tˆ − τˆ (51) 1 a 2 t characteristic roots). where 0100 0000 A. Stability Analysis C = , C = . (52) 1 0000 2 0001 The equilibrium x = 0 of (49) [or X = 0 for the equivalent formulation (54)] is exponentially stable if and only if all the Note that u (tˆ − τˆD) and ϕ (tˆ − τˆD) are also available for any characteristic roots are located in the left-half complex plane givenτ ˆD ≥ max(τâ, τˆt ). The main controller goal is to stabilize the linearized dimen- (LHP) and away from the imaginary axis [59]. sionless perturbation dynamics (49) by only using the available A retarded-type delay differential equation (RDDE) always measurements (the output y(tˆ) (51) and its delayed versions). has a finite number of characteristic roots in the right-half To facilitate controller synthesis, the spectrum of the open-loop complex plane (RHP), whereas an NDDE may have an infinite system is analyzed in Section IV, which supports the stability number of characteristic roots in the RHP or on the imaginary analysis and the assessment of stabilizability properties. The axis [59]. Such properties can effectively be analyzed using controller design is pursued in Section V, under the premise Proposition 1 below. In support of Proposition 1, the spectral of stabilizability. abscissa of the delay difference equation associated with (49) and defined as follows: IV. STABILITY ANALYSIS AND STABILIZABILITY 2 In this section, a spectral approach [57] is employed to x tˆ − Ei x tˆ − τ¯i = 0 (56) analyze the stability of the presented linearized drilling dynam- i=1 ics to study the root causes for drill-string vibrations and to is expressed as serve as a basis for controller design. Although the eigenvalue- based framework presented in [57] is developed for a class of cD := sup{Re(λ) : det D(λ) = 0} (57) DDEs called delay differential-algebraic equations (DDAEs), where D(λ) is the characteristic matrix of the delay differ- NDDEs, such as those in (49), can also be analyzed in this ence equation, which is given by framework. ( ) =[( ) ( )]T 2 By introducing a new state vector X tˆ z tˆ x tˆ , −λτ (λ) := − ¯i where the variable z(tˆ) is expressed as D I Ei e i=1 2 ⎡ ⎤ 1000 z tˆ = x tˆ − Ei x tˆ − τ¯i . (53) ⎢ −λτ¯ ⎥ ⎢ 01− Ae¯ 1 00⎥ i=1 = . (58) ⎣ 0010⎦ −λτ Equation (49) can be reformulated as a DDAE as follows: 0001− Ae ¯2 5 2 Proposition 1: [57] There exists a sequence {λ } ≥ of E X tˆ = A X tˆ + A X tˆ − τ + B r tˆ − h¯ k k 1 0 0 i ¯i i i i characteristic roots satisfying i=1 i=1 (54) lim Re(λk ) = cD, lim Im(λk ) =∞. (59) k→∞ k→∞ where the matrices are defined as follows: Remark 1: We observe that not only there exists a sequence E = I 0 , A = 0 A0 of characteristic roots with real parts converging to cD and 0 00 0 −II imaginary parts tending to infinity (see Proposition 1), but ⎧ ⎪ also there exists such sequences for each member of the set ⎪ 0 Ai ={ (λ) : (λ) = } ⎨⎪ , for i = 1, 2 CD Re det D 0 . Clearly, given (57), the one 0 −Ei B A = B = i , i = 1, 2. mentioned in Proposition 1 is the right-most one. i ⎪ i 0 < ⎪ 0 Ai Therefore, if cD 0, all such sequences of characteristic ⎩⎪ , otherwise 00 roots are located in the LHP, which ensures that only a finite number of characteristic roots can potentially lie in the (55) RHP. This feature plays an important role in the analysis of The inverse transformation (from DDAE to NDDE) can take stabilizability of the system as pursued in Section IV-B. place by differentiating the variable in (53). The equivalence of the spectrum of an NDDE and its corresponding DDAE B. Stabilizability depends on how one transforms from one description to An NTD system is exponentially stabilizable by state feed- the other. When this operation involves differentiation of an back if and only if it is “formally stable” and “spectrally equation, additional dynamics is introduced, which is reflected stabilizable” [47]. A system is called “formally stable” if it by additional characteristic roots at zero. However, one can has at most a finite number of characteristic roots in the RHP,

TASHAKORI et al.: PREDICTION-BASED CONTROL FOR MITIGATION OF AXIAL–TORSIONAL VIBRATIONS 9

i.e., if cD < 0. The importance of the formal stability property The state-feedback control law is designed as follows to in the scope of the existence of a stabilizing state-feedback compensate for the input delays in (49): controller can be understood as follows. Since a state-feedback ˆ = ˆ + ¯ + ˆ − τ + ¯ controller does not affect the delay difference equation in r1 t K11x t h1 K12x t ¯1 h1 + ˆ − τ + ¯ (56), an open-loop system that is not formally stable leads K13x t ¯3 h1 (66a) to an infinite number of unstable characteristic roots in the ˆ = ˆ + ¯ + ˆ − τ + ¯ r2 t K21x t h2 K22x t ¯2 h2 closed-loop system. Note that the input transformation (8) was + K x tˆ − τ¯ + h¯ (66b) introduced to render the system formally stable. 23 3 2 The property of “spectral stabilizability” of the system can with Kij, i ∈{1, 2} and j ∈{1, 2, 3}, the gain matrices. be investigated using the following proposition. Note that in the proposed control law (66), both the axial Proposition 2: [47] An NTD system of the form and torsional inputs, r1 and r2, are influenced by the related F F F axial and torsional time delays (or axial and torsional wave q˙(t) − Ei q˙(t − τi ) = Ai q(t − τi ) + Bi u(t − τi ) speeds), respectively. Subsequently, the closed-loop tracking i=1 i=1 i=1 error dynamics is given by (60) 2 5 − − τ = A + A − τ with the state vector q(t) ∈ Rn, input vector u(t) ∈ Rm,and x tˆ Ei x tˆ ¯i 0 x tˆ i x tˆ ¯i (67) time delays i=1 i=1 where 0 <τ1 <τ2 < ···<τF A = + + , A = + is spectrally stabilizable if 0 A0 B1 K11 B2 K21 1 A1 B1 K12 A = A + B K , A = A + B K + B K rank λ I − Eˆ (λ) − Aˆ (λ), Bˆ (λ) = n ∀λ|Re(λ) ≥ 0 (61) 2 2 2 22 3 3 1 13 2 23 A4 = A4, A5 = A5. (68) where F F F The following structure is imposed on the gain matrices Kij −λτ −λτ −λτ i i i to, first, alleviate the computational burden of the optimization- Eˆ = Ei e , Aˆ = Ai e , Bˆ = Bi e . (62) i=1 i=1 i=1 based tuning of these gains and, second, avoid using delayed downhole angular and axial positions that are not available For a system of the form (60) that is “formally stable” and by measurement (note that there is no sensor at the bit that “spectrally stabilizable,” the stabilizability is assured by a measures the bit position): state-feedback control law of the following form [60]: H = k k k , = k k k K11 1 2 3 0 K21 7 0 8 9 u(t) =− Ki q(t − αi ) (63) = , = K12 0 k4 00 K22 000k10 i=1 K13 = k5 0 k6 0 , K23 = k11 0 k12 0 (69) where Ki are the gain matrices and , = ,..., F where ki i 1 12, represent the nonzero gain elements. With these gain matrices, the state-feedback control law (66) αi = mijτ j , i = 1,...,H, mij ∈ Z, H ∈ N. (64) j=1 can be written explicitly as follows: For the system under study, the spectral stabilizabilty can ˆ = ˆ + ¯ + ˆ + ¯ + ϕ ˆ + ¯ r1 t k1u t h1 k2u t h1 k3 t h1 be analyzed, using the rank condition given in (61) with the + k4u tˆ − τ¯1 + h¯1 following matrices: + ˆ − τ + ¯ + ϕ ˆ − τ + ¯ k5u t ¯3 h1 k6 t ¯3 h1 (70a) 2 5 2 = + ¯ + ϕ + ¯ + ϕ + ¯ −λτ¯i −λτ¯i −λh¯i r2 tˆ k7u tˆ h2 k8 tˆ h2 k9 tˆ h2 Eˆ = Ei e , Aˆ = A0 + Ai e , Bˆ = Bi e . + ϕ ˆ − τ + ¯ i=1 i=1 i=1 k10 t ¯2 h2 (65) + k11u tˆ − τ¯3 + h¯2 + k12ϕ tˆ − τ¯3 + h¯2 . (70b) A quantitative stability analysis and analysis of stabiliz- In (70), the terms with the argument tˆ+h¯ 1 and tˆ+h¯ 2 are future ability are pursued for an exemplary drill-string system in states since h¯1, h¯2 > 0. The terms with the argument tˆ−τ¯3 +h¯ 1 Section VII. Under the premise of stabilizability, we pursue the and tˆ − τ¯3 + h¯2 are also future states becauseτ ¯3, i.e., the design of a stabilizing state-feedback controller in Section V. delay induced by the cutting process at the bit, is smaller than h¯1 and h¯2, i.e., the delays induced by the axial and torsional V. C ONTROLLER DESIGN METHODOLOGY wave propagation in the drill string. These terms render the In this section, the continuous pole-placement approach [48] control law noncausal. To arrive at a causal implementation, is employed to design a state-feedback controller, aiming at this control law will be combined with a predictor, which stabilizing (49) and (50). In particular, the objective of this will be proposed in Section VI. Moreover, there are two other state-feedback controller is to place all roots in the LHP, delayed terms in (70), u (tˆ− τ¯1 + h¯ 1) and ϕ (tˆ− τ¯2 + h¯2) (i.e., which, in turn, results in an asymptotically stable closed-loop u (tˆ− τâ) and ϕ (tˆ− τˆt ), respectively), which are available by system. top-side measurements, as discussed in Section III.

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Next, we design the control gains ki in (69) by an optimiza- Based on the closed-loop system dynamics in (67), the pre- tion method for pole placement. The underlying optimization dictor is designed as follows: problem is considered as a procedure of finding controller 2 5 gains for which the spectral abscissa (i.e., the real part of ω tˆ − Ei ω tˆ − τ¯i = A0ω tˆ + Ai ω tˆ − τ¯i the right-most characteristic root) of the closed-loop system is i=1 i=1 strictly negative. Hence, the objective function is formulated − P ω tˆ − τˆp − τˆm −x tˆ − τˆm as follows: (73) F(ki ) = sup{Re(λ) : det N (λ) = 0} (71) where ω(tˆ) is the estimate of the future states x(tˆ + τˆp) for where (λ) is the characteristic matrix of the closed-loop N any given dimensionless timeτ ˆp withτ ˆm = max(τâ, τˆt ).As system (67), which is which is founded defined as follows: discussed in Section III, only delayed downhole velocities are 2 5 available (by using top-side measurements). Thus, the follow- −λτ¯i −λτ¯i N (λ) := λ I − Ei e − A0 − Ai e . (72) ing structure is considered for the predictor gain matrix P (in i=1 i=1 order to avoid using delayed downhole position measurements Asymptotic stability is guaranteed when the objective function that are not available in practice): ⎡ ⎤ (71) is negative. In particular, the continuous pole-placement 0 p1 00 method aims at designing ki such that the optimization prob- ⎢ ⎥ = ⎢ 0 p2 00⎥ lem min F(ki ) is solved with the stopping criterion F(ki ) ≤ δ P ⎣ ⎦ (74) k 000p3 i δ< for a given 0. 000p4 Since the objective function in (71) is a nonsmooth function where p , i = 1,...,4, are the nonzero elements. Intuitively of the controller parameters, common optimization algorithms i speaking, in (73) with the gain matrix (74), x(tˆ + τˆ ) is are not applicable [59]. Instead, the HANSO1 method [61] has p estimated by ω(tˆ) by using x (tˆ − τˆ ) and x (tˆ − τˆ ) (i.e., been employed, which is founded on the BFGS2 and gradient 2 m 4 m u(tˆ−τˆ ) and ϕ(tˆ−τˆ ), respectively), which are both available sampling methods [62]. It is noteworthy to mention that in m m at time tˆ (see Section III). this article, the PSO3 [63] method has been used in order to Introducing the prediction error search globally for a proper initial guess of the controller gains in order to avoid convergence to local minima. e tˆ = ω tˆ − x tˆ + τˆp (75)

VI. PREDICTOR DESIGN METHODOLOGY the prediction error dynamics is given by The designed feedback controller in (70), presented in 2 5 Section V, is noncausal as it requires the knowledge of future e tˆ − Ei e tˆ − τ¯i = A0 e tˆ + Ai e tˆ − τ¯i states. To make such controller causal, the use of a proper i=1 i=1 predictor is essential. The predictor design for linear time- − Pe tˆ − τˆp − τˆm . (76) delay systems has been taken into consideration in the series of By design of the predictor gain matrix P, the equilibrium studies [64]–[68] for systems, including different combinations = of state delays, (one or several) input delays, and neutral solution e 0 of (76) can be rendered an asymptotically stable terms. However, the implementation of such ideas includes solution of (76), which implies the asymptotic convergence of ω(tˆ) x(tˆ + τ ) approximating some integral terms, which makes the closed- to ˆp . loop system unstable, since the approximation introduces The predictor gains pi in (74) can be designed by an new unstable eigenvalues. The implementation complexity is optimization-based method for pole placement. The underlying explained thoroughly in [69] and [70], where the design of optimization problem is considered as a procedure of finding a low-pass filter is suggested to overcome such a problem. predictor gains for which the spectral abscissa of the predic- Another approach proposes to estimate the future states in tion error dynamics is strictly negative. Hence, the objective an asymptotic way, instead of approximation, by using an function is formulated as follows: observer-like structure (see [71]–[73]). FP (pi ) = sup{Re(λ) : det P (λ) = 0} (77) Here, based on the work in [71]–[73], a predictor with an (λ) observer-like structure is presented to estimate the future states where P is the characteristic matrix of the prediction error asymptotically. The formulation presented in [71]–[73] can dynamics (76), which is defined as follows: be applied directly to systems that, first, have a single input 2 −λτ¯i and, second, the prediction time is equal to the input delay. P (λ) := λ I − Ei e − A0 However, the main underlying idea adapted to be applicable to i=1 systems with several inputs and including state delays (which 5 −λτ¯i −λ(τˆp+τˆm ) may make the prediction time a combination of state delays − Ai e + Pe . (78) and input delays, like those in this article). i=1

1Hybrid algorithm for nonsmooth optimization. Asymptotic stability is guaranteed when the objective function 2Broyden–Fletcher–Goldfarb–Shanno. (77) is negative. In particular, the continuous pole-placement 3 Particle swarm optimization. method aims at designing pi such that the optimization

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Fig. 2. Series connection of the prediction error dynamics and the closed- loop dynamics.

problem min FP (pi ) is solved with the stopping criterion pi FP (pi ) ≤ δ for a given δ<0. Remark 2: Note that even for P = 0, (76) is asymptotically stable. Hence, the design of P is geared toward rendering the Fig. 3. Open-loop spectrum. error dynamics in (76) faster than those in (67). For nonzero prediction error, the state-feedback control law (66) is given by and the fact that (82) is asymptotically stable for e = 0, it holds that (82) is input-to-state stable with the prediction ˆ = ω ˆ + ˆ − τ + ¯ + ω ˆ r1t K01 1t K11xt ¯1 h1 K31 2t (79a) error e as an input. Therefore, the total closed-loop dynamics r2 tˆ = K02ω3 tˆ + K22x tˆ − τ¯2 + h¯2 + K32ω4 tˆ (79b) is asymptotically stable based on the fact that it consists of a series connection of the asymptotically stable dynamics in where ω (tˆ), i = 1,...,4, are the estimates of the states x(tˆ+ i (76) and the input-to-state stable dynamics in (82) (see Fig. 2). τˆpi) with prediction times

τˆp1 = h¯1, τˆp2 = h¯1 − τ¯3 LLUSTRATIVE IMULATION ESULTS τˆp3 = h¯2, τˆp4 = h¯2 − τ¯3. (80) VII. I S R In this section, we present a representative case study to Given (75), (79) can be written as follows: illustratively show the effectiveness of the proposed con- ˆ = ˆ + ¯ + ˆ − τ + ¯ trol approach. In Section VII-A, the characteristics of the r1 t K01x t h1 K11x t ¯1 h 1 open-loop dynamics are investigated (stability and stabi- + K31x tˆ − τ¯3 + h¯1 + K01e1 tˆ + K31e2 tˆ (81a) lizability properties), and in Section VII-B, the controller ˆ = ˆ + ¯ + ˆ − τ + ¯ r2 t K02x t h2 K22x t ¯2 h 2 design is performed and the resulting closed-loop performance + K32x tˆ − τ¯3 + h¯2 + K02e3 tˆ + K32e4 tˆ (81b) is analyzed. where ei , i = 1,...,4, are the prediction errors at times tˆ+τˆpi. Subsequently, the closed-loop dynamics is given by A. Analysis of the Open-Loop Dynamics 2 For a 117-m long-drill-string with the parameter values x tˆ − Ei x tˆ − τ¯i given in Table I, the TDS-STABIL MATLAB package [58] is i=1 employed to find the stability-relevant characteristic roots of = = 5 the infinite-dimensional model in (49) for r1 r2 0, using = A0 x tˆ + Ai x tˆ − τ¯i (54) and (55). The open-loop spectrum is shown in Fig. 3, i=1 which illustrates that the system is intrinsically unstable since + ˆ − ¯ + ˆ − ¯ there are some characteristic roots in the RHP, depicted in red. B1 K01e1t h1 B1 K31e2t h1 Similar results have been obtained in the literature (see [14], + B2 K02e3 tˆ − h¯2 + B2 K32e4 tˆ − h¯2 (82) [17], [20], [25] for lumped-parameter models and [5], [26] which is the dynamics in (67) with additional input terms for infinite-dimensional models), which show that for realistic related to the prediction errors. parametric settings, the nominal solution is generally unstable. The following proposition states the conditions under which The behavior of the system (21) and (26) with the nonlinear the proposed predictor-based control strategy indeed stabilizes bit–rock interaction law (10) is shown in Fig. 4 when constant 3 3 the closed-loop system dynamics. control inputs, u1 = 16 × 10 Nandu2 = 18 × 10 N·m, are Proposition 3: Consider the drill-string dynamics in (49), employed. As shown, in the absence of any controller, severe (50). Let the controller in (79) be designed such that F(ki )< stick-slip and bit-bouncing occur, which can lead to system 0, with F(ki ) given in (71). Moreover, let the predictor in failure. (73) be designed such that FP (pi )<0, with FP (pi ) defined As explained in Section IV, it is essential to investigate in (77). Then, the closed-loop dynamics (76) and (82) is the exact place (i.e., real value) of the right-most vertical asymptotically stable. asymptote of the spectrum, as shown in Fig. 3. The spectral Proof: First, FP (pi )<0 implies that the prediction error abscissa of the delay difference equation, given in (56), dynamics (76) is asymptotically stable. Second, F(ki )<0 is cD −0.0022, which shows that the asymptote is located implies that error dynamics in (67) (for zero prediction error) in the LHP (see Proposition 1). The other asymptote, visible is asymptotically stable. By the grace of linearity of (82) in Fig. 3, is located on −0.1510, which is the other root

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Fig. 4. Open-loop dynamics. Fig. 5. Velocity references. of det D(λ) = 0 (see Remark 1). Now, we can conclude that, ﬁrst, there is only a ﬁnite number of characteristic roots in the RHP and, second, the roots are not accumulated on the imaginary axis. Therefore, the system is formally stable [47]. It is worth restating that the formal stability property is achieved as a result of the input transformation (8). Moreover, the spectral stabilizability of the system can be investigated based on Proposition 2. Since for all six characteristic roots with positive real parts (see red dots in Fig. 3), the rank condition in (61) holds, and it can be concluded that the system is indeed spectrally stabilizable. Therefore, the stabilizability of the presented drilling system (49) is assured by delayed state feedbacks of the form Fig. 6. Variation of spectral abscissa for different time delaysτ ˆn. Note that τ ∈[ . , . ] (θ˙ ) ∈[ . , ] given in (63). ˆn 0 4220 1 7956 for b ref 2 35 10 .

B. Controller Design and Closed-Loop Performance Analysis 1) The system matrices of the error dynamics for time- varying reference remain time-invariant as these are not The design procedure of the state-feedback controller, aim- affected by the nominal solution. ing at stabilizing (49), has been presented in Section V. 2) Although the state-dependent delayτ ˆ is time-varying In the following simulation study, we introduce the fol- n along the time-varying angular reference trajectory in lowing smoothened references to deal with the actuator (83b), it can be shown that for any (constant) delay limitations (in particular, high-frequency content and large τˆ induced by the angular velocities in the reference, overshoots in the control inputs should be avoided in n the spectral abscissas of both the closed-loop dynamics practice): and prediction error dynamics are always negative (see exp(0.5373t − 8) Fig. 6). U˙ = V (83a) b ref 0 1 + exp(0.5373t − 8) 3) The reference (and hence also the state-dependent delay along the nominal solution) evolves on a much slower exp(0.5373t − 8) time scale than the dominant dynamics of the drill-string θ˙ = ( − 2.35) + 2.35. (83b) b ref 0 1 + exp(0.5373t − 8) system. Introducing such reference design has also been suggested Therefore, the control design methodology presented in V is extended as follows. With xref(tˆ) := earlier in [28] and [33]. As shown in Fig. 5, the drill string [ ( ) ( )ϕ ( )ϕ ( )] uref tˆ uref tˆ ref tˆ ref tˆ the desired reference trajectory, has been ﬁrst accelerated to a constant angular velocity, ˙ the tracking errorx ˜ := [u˜ u˜ ϕ˜ ϕ˜ ] is introduced as i.e., (θb)ref(t = 0) = 2.35 rad/s, with the bit off bottom, ˙ follows: i.e., (Ub)ref(t = 0) = 0 m/s. Then, both the axial and angular velocities are increased simultaneously to the desired x˜ tˆ = x tˆ − xref tˆ . (84) operating conditions (V0 = 0.002 m/s and 0 = 10 rad/s, respectively). Remark 3: Although the references are time-varying, this The total control scheme is shown in Fig. 7. The control law does not disqualify the control design methodology pre- is composed of two terms as follows: sented in Section V, which was designed for linear ˆ = ˆ + ˆ time-invariant error dynamics, based on the following r1t r1ct r1 f t (85a) facts. r2 tˆ = r2c tˆ + r2 f tˆ (85b)

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Fig. 7. Total closed-loop block diagram.

where rif, i = 1, 2, are the feedforward terms to ensure that the desired trajectory is a solution of the closed-loop system Fig. 8. Spectrum of the prediction error dynamics. r tˆ = u tˆ + h¯ − Au¯ tˆ − τ + h¯ 1 f ref 1 ref ¯1 1 + ψ¯ u tˆ + h¯ + Nu¯ tˆ + h¯ + Q¯ ϕ tˆ + h¯ ref 1 ref 1 ref 1 − ψ¯ Au¯ tˆ − τ + h¯ + A¯ Nu¯ tˆ − τ + h¯ ref ¯1 1 ref ¯1 1 − ψ¯ ˆ − τ + ¯ − ¯ ϕ ˆ − τ + ¯ uref t ¯3 h1 Q ref t ¯3 h1 + ψ¯ ¯ ˆ − τ + ¯ Auref t ¯4 h1 (86a) r tˆ = ϕ tˆ + h¯ − Aϕ tˆ − τ + h¯ 2 f ref 2 ref ¯2 2 + ψu tˆ + h¯ + Qϕ tˆ + h¯ + Nϕ tˆ + h¯ ref 2 ref 2 ref 2 − ψ Au tˆ − τ + h¯ + ANϕ tˆ − τ + h¯ ref ¯2 2 ref ¯2 2 − ψ ˆ − τ + ¯ − ϕ ˆ − τ + ¯ uref t ¯3 h2 Q ref t ¯3 h2 + ψ Auref tˆ − τ¯5 + h¯2 (86b) which are designed such that Fig. 9. Axial prediction errors for different prediction times. ⎧ ⎫ ⎪ 0 ⎪ ⎨ ⎬ 2 r1 f tˆ − h¯1 = x tˆ − Ei x tˆ − τ¯i ⎪ 0 ⎪ ref ref ⎩ ⎭ i=1 r2 f tˆ − h¯2 5 − A0xref tˆ − Ai xref tˆ − τ¯i . (87) i=1

Note that we use the fact that xref(tˆ) is known apriori,and hence, the delays in the control inputs as in (49) can be compensated for as far as the feedforward is concerned. The terms ric, i = 1, 2, in (85) are given in (79), while here,x ˜ [deﬁned in (84)] is used in the feedback actions, and Fig. 10. Torsional prediction errors for different prediction times. ω(tˆ) is the estimate of the tracking errorx ˜(tˆ+ τˆp) [instead of x(tˆ + τˆp)]. Fig. 8 shows the spectrum of the prediction error dynamics are obtained: (76) with p = p = p = p = 0.01. As depicted, all the 1 2 3 4 =− . , =− . , =− . × −6 characteristic roots lie in the LHP, which ensures the stability k1 994 4 k2 85 5 k3 6 4 10 −4 of the prediction error dynamics (76). k4 = 9, k5 = 342.5, k6 = 8.3 × 10 7 The prediction errors are shown in Figs. 9 and 10 for k7 =−9.1 × 10 , k8 =−3.5, k9 =−78.7 the axial and torsional dynamics, where eu(tˆ), eu (tˆ), eϕ (tˆ), 7 k10 =−4.3, k11 = 4.2 × 10 , k12 = 2.8 (88) and eϕ (tˆ) are the prediction errors of tracking errorsu ˜(tˆ + τˆp), u˜ (tˆ + τˆp), ϕ(˜ tˆ + τˆp),and˜ϕ (tˆ + τˆp), respectively. These which leads to the closed-loop spectrum shown in Fig. 11. It ﬁgures depict an asymptotic convergence to zero prediction illustrates that the closed-loop system is exponentially stable error for all prediction times for a given initial condition of since all closed-loop characteristic roots lie in the LHP. Note the predictor [5% difference compared tox ˜(0)]. Note that Figs. that the vertical asymptotes of the spectrum remain unchanged 9 and 10 are dimension-less. since the controller does not affect the neutral terms. The latter By employing HANSO and PSO algorithms to design the fact emphasizes the importance of the formal stability analysis controller gains, given in (69), the following controller gains presented in Section IV.

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Fig. 14. Physical actuation variables.

Fig. 11. Spectrum of the tracking error dynamics.

Fig. 15. Closed-loop axial behavior of the nonlinear system.

Fig. 12. Closed-loop axial behavior. from the precompensators, which can be computed using the following relations: ∂ ∂ ∂ Utop fa ga (t) = ca (t) − (t) (89) ∂s ∂ϒa ∂a (∂ )/(∂ϒ )( ) (∂ )/(∂ )( ) where fa a t and ga a t are given in (20), and ∂2θ ∂2f ∂2g top (t) = t (t) + t (t) (90) ∂ 2 2 2 t ∂ϒt ∂t (∂2 )/(∂ϒ2)( ) (∂2 )/(∂2)( ) where ft t t and ga a t can be obtained by differentiating (25). As shown in Fig. 14, in the steady-state motion, the feed rate VTB is equal to the rate of penetration, Fig. 13. Closed-loop angular behavior. i.e., 0.002 m/s. Moreover, the maximum value of the top-drive 4 torque uT (t) is around 2 × 10 N·m, which is an acceptable value regarding the torque limitation of the top drive. Note The closed-loop system behavior is shown in Figs. 12 and that with signiﬁcantly heavier top-drives, either the maximum 13 for the axial and torsional dynamics, which shows that value of the uT (t) may cross the acceptable limit or increased indeed the bit velocities track the desired references. Here, high-frequency contents in uT (t) might be observed. the bit is considered to have a zero axial velocity and 1.4rad/s Next, the designed controller is applied to the nonlinear angular velocity for −(2τt + τn)

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Fig. 18. Variation of spectral abscissa with (a) varying ζ and ﬁxed ε and Fig. 16. Closed-loop angular behavior of the nonlinear system. (b) varying ε and ﬁxed ζ .

VIII. CONCLUSION A distributed model has been employed to study the coupled axial–torsional vibrations in drilling systems. Here, first, both the cutting process and frictional contact have been taken into consideration in the bit–rock interaction, and second, realistic top-side boundary conditions are included. The resulting equations of motion are NDDEs with state-dependent state delays, constant state delays, and constant input delays. For the associated linear system, the stability has been analyzed Fig. 17. Robust behavior in the presence of parametric uncertainties. by using a spectral approach to study the root causes of the steady-state vibrations and to serve as a basis for controller design. It is shown that the drilling system is intrinsically approach is robust against the measurement noise. Another unstable but stabilizable by (delayed) state feedback. Under source of uncertainty, which plays a role in the scope of the the premise of stabilizability, the optimization-based contin- current work, is the parametric uncertainty, which is studied uous pole-placement method has been employed to stabilize in Section VII-C. both axial and torsional dynamics, using the velocity of the traveling block and the top-drive torque as control inputs. The designed state-feedback controller deals with several time C. Robustness Analysis Against Parametric Uncertainty delays corresponding to the oscillatory behavior of the dynam- Drilling systems are subjected to different types of uncer- ics, and the bit–rock interaction. Moreover, it only employs tainties. In the scope of the current work, the most important top-side measurements. To make the controller causal, a state source of uncertainty is given by the parameters in the bit– predictor with observer structure has also been designed. The rock interaction model (due to uncertainties in rock properties effectiveness and robustness of both the controller and the and bit wear) [74], [75]. Therefore, in this section, parametric predictor have been shown by illustrative simulation results. uncertainty in the bit–rock interaction is considered in order to have a more realistic description of the system and to study REFERENCES ζ the robustness of the designed controller. For this purpose, [1] E. M. Navarro-López, “An alternative characterization of bit-sticking (the cutter face inclination number) and ε (the rock intrinsic phenomena in a multi-degree-of-freedom controlled drillstring,” Nonlin- specific energy) in (14) are considered uncertain. ear Anal., Real World Appl., vol. 10, no. 5, pp. 3162–3174, Oct. 2009. [2] Y. Liu, J. P. Chávez, R. De Sa, and S. Walker, “Numerical and The variation of the closed-loop spectral abscissa for dif- experimental studies of stick–slip oscillations in drill-strings,” Nonlinear ferent values of the parameters ε and ζ isshowninFig.17 Dyn., vol. 90, no. 4, pp. 2959–2978, Dec. 2017. (0 <ζ <1andε>0). As illustrated, the spectral abscissa [3] Z. Huang, D. Xie, B. Xie, W. Zhang, F. Zhang, and L. He, “Investigation of PDC bit failure base on stick-slip vibration analysis of drilling string remains negative for a wide range of these parameters. This system plus drill bit,” J. Sound Vib., vol. 417, pp. 97–109, Mar. 2018. feature, along with the formal stability of system, assures the [4] D. Bresch-Pietri and M. Krstic, “Adaptive output feedback for oil drilling robustness of the designed controller in terms of stabilizing stick-slip instability modeled by wave PDE with anti-damped dynamic boundary,” in Proc. ACC, Jun. 2014, pp. 386–391. the system while these parameters are uncertain. As shown [5] U. J. F. Aarsnes and N. van de Wouw, “Dynamics of a distributed drill in Fig. 18, with fixed ε and ζ varying between 0 and 1, string system: Characteristic parameters and stability maps,” J. Sound the closed-loop spectral abscissa remains negative. However, Vib., vol. 417, pp. 376–412, Mar. 2018. ζ ε> [6] U. J. F. Aarsnes and N. van de Wouw, “Axial and torsional self- with fixed , the closed-loop will be unstable when excited vibrations of a distributed drill-string,” J. Sound Vib., vol. 444, 2.02 × 109 Pa which corresponds to an extremely hard rock pp. 127–151, Mar. 2019. formation. Note that we took the range on ε larger in Fig. 18 [7] B. Saldivar, S. Mondié, J.-J. Loiseau, and V. Rasvan, “Stick-slip oscil- ε lations in oillwell drilstrings: Distributed parameter and neutral type in comparison to Fig. 17 to find the critical value of that retarded model approaches,” IFAC Proc. Volumes, vol. 44, no. 1, makes the closed-loop system unstable. pp. 284–289, Jan. 2011.

16 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

[8] I. Boussaada, H. Mounier, S.-I. Niculescu, and A. Cela, “Analysis of [31] S. Tashakori and M. Fakhar, “Suppression of torsional vibrations in drilling vibrations: A time-delay system approach,” in Proc. 20th Medit. drilling systems by using the optimization-based adaptive backstepping Conf. Control Autom. (MED), Jul. 2012, pp. 610–614. controller,” Int. J. Mech. Control, vol. 20, no. 1, pp. 105–110, 2019. [9] S. Tashakori, G. Vossoughi, H. Zohoor, and E. A. Yazdi, “Modification [32] D. Bresch-Pietri and M. Krstic, “Output-feedback adaptive control of of the infinite-dimensional neutral-type time-delay dynamic model for a wave PDE with boundary anti-damping,” Automatica, vol. 50, no. 5, the coupled axial–torsional vibrations in drill strings with a drag bit,” pp. 1407–1415, May 2014. J. Comput. Nonlinear Dyn., vol. 15, no. 8, Aug. 2020. [33] U. J. F. Aarsnes, F. Di Meglio, and R. J. Shor, “Avoiding stick slip [10] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE vibrations in drilling through startup trajectory design,” J. Process Systems. Cambridge, MA, USA: Birkhäuser Boston, 2009. Control, vol. 70, pp. 24–35, Oct. 2018. [11] B. Saldivar, S. Mondié, S.-I. Niculescu, H. Mounier, and I. Boussaada, [34] M. B. Saldivar, S. Mondié, and J. J. Loiseau, “Reducing stick-slip “A control oriented guided tour in oilwell drilling vibration modeling,” oscillations in oilwell drillstrings,” in Proc. 6th Int. Conf. Electr. Eng., Annu. Rev. Control, vol. 42, pp. 100–113, 2016. Comput. Sci. Autom. Control (CCE), Nov. 2009, pp. 1–6. [12] N. Mihajlović, A. A. van Veggel, N. van de Wouw, and H. Nijmeijer, [35] C. Sagert, F. Di Meglio, M. Krstic, and P. Rouchon, “Backstepping and “Analysis of friction-induced limit cycling in an experimental drill-string flatness approaches for stabilization of the stick-slip phenomenon for system,” J. Dyn. Syst., Meas., Control, vol. 126, no. 4, pp. 709–720, drilling,” IFAC Proc. Volumes, vol. 46, no. 2, pp. 779–784, 2013. Dec. 2004. [36] M. Barreau, F. Gouaisbaut, and A. Seuret, “Practical stability analysis of [13] E. M. Navarro-López and E. Licéaga-Castro, “Non-desired transi- a drilling pipe under friction with a PI-controller,” IEEE Trans. Control tions and sliding-mode control of a multi-DOF mechanical system Syst. Technol., vol. 29, no. 2, pp. 620–634, Mar. 2021. with stick-slip oscillations,” Chaos, Solitons Fractals, vol. 41, no. 4, [37] A. S. Yigit and A. P. Christoforou, “Stick-slip and bit-bounce interaction pp. 2035–2044, Aug. 2009. in oil-well drillstrings,” J. Energy Resour. Technol., vol. 128, no. 4, [14] T. Richard, C. Germay, and E. Detournay, “A simplified model to explore pp. 268–274, Dec. 2006. the root cause of stick–slip vibrations in drilling systems with drag bits,” [38] D. Bresch-Pietri and F. Di Meglio, “Prediction-based control of linear J. Sound Vib., vol. 305, no. 3, pp. 432–456, Aug. 2007. input-delay system subject to state-dependent state delay–application to [15] X. Liu, N. Vlajic, X. Long, G. Meng, and B. Balachandran, “Nonlinear suppression of mechanical vibrations in drilling,” IFAC-PapersOnLine, motions of a flexible rotor with a drill bit: Stick-slip and delay effects,” vol. 49, no. 8, pp. 111–117, 2016. Nonlinear Dyn., vol. 72, nos. 1–2, pp. 61–77, Apr. 2013. [39] B. Saldivar, S. Mondié, and J. C. Á. Vilchis, “The control of drilling [16] X. Liu, N. Vlajic, X. Long, G. Meng, and B. Balachandran, “State- vibrations: A coupled PDE-ODE modeling approach,” Int. J. Appl. Math. dependent delay influenced drill-string oscillations and stability analy- Comput. Sci., vol. 26, no. 2, pp. 335–349, Jun. 2016. sis,” J. Vib. Acoust., vol. 136, no. 5, Oct. 2014, Art. no. 051008. [40] B. Saldivar, S. Mondié, J. J. Loiseau, and V. Rasvan, “Suppressing axial-torsional coupled vibrations in drillstrings,” J. Control Eng. Appl. [17] B. Besselink, T. Vromen, N. Kremers, and N. van de Wouw, “Analysis Inform., vol. 15, no. 1, pp. 3–10, 2013. and control of stick-slip oscillations in drilling systems,” IEEE Trans. Control Syst. Technol., vol. 24, no. 5, pp. 1582–1593, Sep. 2016. [41] I. Boussaada, A. Cela, H. Mounier, and S.-I. Niculescu, “Control of drilling vibrations: A time-delay system-based approach,” in Proc. 11th [18] S. K. Gupta and P. Wahi, “Global axial–torsional dynamics during rotary Workshop Time-Delay Syst., 2013, pp. 226–231. drilling,” J. Sound Vib., vol. 375, pp. 332–352, Aug. 2016. [42] B. Saldivar, T. Knüppel, F. Woittennek, I. Boussaada, H. Mounier, [19] S. K. Gupta and P. Wahi, “Criticality of bifurcation in the tuned and S. I. Niculescu, “Flatness-based control of torsional-axial coupled axial–torsional rotary drilling model,” Nonlinear Dyn., vol. 91, no. 1, drilling vibrations,” IFAC Proc. Volumes, vol. 47, no. 3, pp. 7324–7329, pp. 113–130, Jan. 2018. 2014. [20] C. Germay, V. Denoël, and E. Detournay, “Multiple mode analysis of [43] S. Dwars et al., “Recent advances in soft torque rotary systems,” the self-excited vibrations of rotary drilling systems,” J. Sound Vib., presented at the SPE/IADC Drilling Conf. Exhib., London, U.K., 2015. vol. 325, nos. 1–2, pp. 362–381, Aug. 2009. [44] J. D. Jansen and L. van den Steen, “Active damping of self-excited [21] C. Germay, N. Van de Wouw, H. Nijmeijer, and R. Sepulchre, “Nonlinear torsional vibrations in oil well drillstrings,” J. Sound Vib., vol. 179, drillstring dynamics analysis,” SIAM J. Appl. Dyn. Syst., vol. 8, no. 2, no. 4, pp. 647–668, Jan. 1995. pp. 527–553, Jan. 2009. [45] E. Cayeux, “On the importance of boundary conditions for real- [22] B. Besselink, N. van de Wouw, and H. Nijmeijer, “A semi-analytical time transient drill-string mechanical estimations,” presented at the study of stick-slip oscillations in drilling systems,” J. Comput. Nonlinear IADC/SPE Drilling Conf. Exhib., Fort Worth, TX, USA, 2018. Dyn., vol. 6, no. 2, Apr. 2011, Art. no. 021006. [46] U. J. F. Aarsnes, O. M. Aamo, and M. Krstic, “Extremum seeking for [23] K. Nandakumar and M. Wiercigroch, “Stability analysis of a state real-time optimal drilling control,” in Proc. Amer. Control Conf. (ACC), dependent delayed, coupled two DOF model of drill-stringvibration,” Jul. 2019, pp. 5222–5227. J. Sound Vib., vol. 332, no. 10, pp. 2575–2592, May 2013. [47] J. J. Loiseau, M. Cardelli, and X. Dusser, “Neutral-type time-delay [24] X. Liu, N. Vlajic, X. Long, G. Meng, and B. Balachandran, “Coupled systems that are not formally stable are not BIBO stabilizable,” IMA axial-torsional dynamics in rotary drilling with state-dependent delay: J. Math. Control Inf., vol. 19, nos. 1–2, pp. 217–227, Mar. 2002. Stability and control,” Nonlinear Dyn., vol. 78, no. 3, pp. 1891–1906, [48] W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time- Nov. 2014. Delay Systems: An Eigenvalue-Based Approach. Philadelphia, PA, USA: [25] A. Depouhon and E. Detournay, “Instability regimes and self-excited SIAM, 2007. vibrations in deep drilling systems,” J. Sound Vib., vol. 333, no. 7, [49] W. R. Tucker and C. Wang, “On the effective control of torsional vibra- pp. 2019–2039, Mar. 2014. tions in drilling systems,” J. Sound Vib., vol. 224, no. 1, pp. 101–122, [26] U. J. F. Aarsnes and O. M. Aamo, “Linear stability analysis of self- Jul. 1999. excited vibrations in drilling using an infinite dimensional model,” [50] E. Kreuzer and M. Steidl, “Controlling torsional vibrations of drill J. Sound Vib., vol. 360, pp. 239–259, Jan. 2016. strings via decomposition of traveling waves,” Arch. Appl. Mech., [27] B. Saldivar, S. Mondié, J.-J. Loiseau, and V. Rasvan, “Exponential vol. 82, no. 4, pp. 515–531, Apr. 2012. stability analysis of the drilling system described by a switched neu- [51] U. J. F. Aarsnes, J. Auriol, F. Di Meglio, and R. J. Shor, “Estimat- tral type delay equation with nonlinear perturbations,” in Proc. 50th ing friction factors while drilling,” J. Petroleum Sci. Eng., vol. 179, IEEE Conf. Decis. Control Eur. Control Conf. (CDC-ECC), Dec. 2011, pp. 80–91, Aug. 2019. pp. 4164–4169. [52] J. J. Bailey and I. Finnie, “An analytical study of drill-string vibration,” [28] T. Vromen et al., “Mitigation of torsional vibrations in drilling systems: J. Eng. Ind., vol. 82, no. 2, pp. 122–127, May 1960. A robust control approach,” IEEE Trans. Control Syst. Technol., vol. 27, [53] I. Finnie and J. J. Bailey, “An experimental study of drill-string vibra- no. 1, pp. 249–265, Jan. 2019. tion,” J. Eng. Ind., vol. 82, no. 2, pp. 129–135, May 1960. [29] T. Vromen, N. van de Wouw, A. Doris, P. Astrid, and H. Nijmeijer, [54] D. Bresch-Pietri and M. Krstic, “Adaptive output-feedback for wave “Nonlinear output-feedback control of torsional vibrations in drilling PDE with anti-damping–application to surface-based control of oil systems,” Int. J. Robust Nonlinear Control, vol. 27, no. 17, drilling stick-slip instability,” in Proc. 53rd IEEE Conf. Decis. Control, pp. 3659–3684, 2017. Dec. 2014, pp. 1295–1300. [30] C. Lu, M. Wu, X. Chen, W. Cao, C. Gan, and J. She, “Torsional vibration [55] T. Insperger, G. Stépán, and J. Turi, “State-dependent delay in regenera- control of drill-string systems with time-varying measurement delays,” tive turning processes,” Nonlinear Dyn., vol. 47, nos. 1–3, pp. 275–283, Inf. Sci., vol. 467, pp. 528–548, Oct. 2018. Dec. 2006.

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[56] F. Hartung, “Linearized stability for a class of neutral functional differ- Gholamreza Vossoughi received the Ph.D. degree ential equations with state-dependent delays,” Nonlinear Anal., Theory, from the Department of Mechanical Engineering, Methods Appl., vol. 69, nos. 5–6, pp. 1629–1643, Sep. 2008. University of Minnesota, Minneapolis, MN, USA, [57] W. Michiels, “Spectrum-based stability analysis and stabilisation of in 1992. systems described by delay differential algebraic equations,” IET Control Ever since, he joined the Faculty of Mechani- Theory Appl., vol. 5, no. 16, pp. 1829–1842, Nov. 2011. cal Engineering, Sharif University of Technology, [58] W. Michiels. TDS-Stabil: A MATLAB Tool for Fixed-Order Sta- Tehran, Iran. He was the Director of the Manufac- bilization Problems for Time-Delay Systems. [Online]. Available: turing Engineering and Applied Mechanics Division, http://twr.cs.kuleuven.be/research/software/delay control/stab the Graduate Dean and the Chairman of the Depart- [59] W. Michiels and S.-I. Niculescu, “Stability, control, and computation ment of Mechanical Engineering, and the Dean of for time-delay systems,” in An Eigenvalue-Based Approach. Advances Graduate Studies at the Sharif University of Technol- in Design and Control, vol. 27, 2nd ed. Philadelphia, PA, USA: SIAM, ogy. His research interests include bio-inspired robotics, microrobotic systems, 2014. man–machine interface, exoskeletal and haptic systems, mechatronics, and [60] C. I. Byrnes, M. W. Spong, and T.-J. Tarn, “A several complex variables control systems. He has published more than 100 peer-reviewed journal approach to feedback stabilization of linear neutral delay-differential articles in these fields. systems,” Math. Syst. Theory, vol. 17, no. 1, pp. 97–133, Dec. 1984. [61] HANSO: Hybrid Algorithm for Non-Smooth Optimization. [Online]. Available: https://cs.nyu.edu/overton/software/hanso/ [62] J. V. Burke, A. S. Lewis, and M. L. Overton, “A robust gradient sampling algorithm for nonsmooth, nonconvex optimization,” SIAM J. Optim., vol. 15, no. 3, pp. 751–779, Jan. 2005. Hassan Zohoor was born in 1945. He received [63] R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” the Ph.D. degree from Purdue University, West Swarm Intell., vol. 1, no. 1, pp. 33–57, Jun. 2007. Lafayette, IN, USA, in 1978. [64] V. L. Kharitonov, “An extension of the prediction scheme to the case He was the Founder, the President, and a Devel- of systems with both input and state delay,” Automatica, vol. 50, no. 1, oper of principal codes and regulations at Payame pp. 211–217, Jan. 2014. Noor University, Tehran, Iran; the Acting President [65] B. Zhou, “Input delay compensation of linear systems with both state of Al-Zahra University, Tehran; and the President of and input delays by nested prediction,” Automatica, vol. 50, no. 5, Shiraz University, Shiraz, Iran. He spent his post- pp. 1434–1443, May 2014. doctoral time at Purdue University. He is currently [66] V. L. Kharitonov, “Predictor based stabilization of neutral type systems an Academician of the IR Iran Academy of Sci- with input delay,” Automatica, vol. 52, pp. 125–134, Feb. 2015. ences (IAS) and a Professor Emeritus at the Sharif [67] V. L. Kharitonov, “Prediction-based control for systems with state and University of Technology, Tehran, Iran. He is also the Head of the Higher several input delays,” Automatica, vol. 79, pp. 11–16, May 2017. Education Committee, Iranian National Commission for UNESCO. He is the [68] B. Zhou and Q. Liu, “Input delay compensation for neutral type time- author or the coauthor of over 550 scientific articles and holds five patents. He delay systems,” Automatica, vol. 78, pp. 309–319, Apr. 2017. has supervised over 220 graduate theses. He was assigned as the Coordinator [69] S. Mondié and W. Michiels, “Finite spectrum assignment of unstable of the Preamble Section of the Mentioned Drafting Committee from 2016 to time-delay systems with a safe implementation,” IEEE Trans. Autom. 2017. Control, vol. 48, no. 12, pp. 2207–2212, Dec. 2003. Dr. Zohoor received the Mustafa Prize, the Award of Distinguished Professor [70] V. L. Kharitonov, “Predictor-based controls: The implementation prob- in the IR Iran in 1993, the Lasting Personalities Award in Iran in 2001, and lem,” Differ. Equ., vol. 51, no. 13, pp. 1675–1682, Dec. 2015. the Allameh Tabatabaei Award, special award of National Elite Foundation, [71] G. Besancon, D. Georges, and Z. Benayache, “Asymptotic state predic- in 2013. He has been elected as an expert by UNESCO, Asia Pacific Region, tion for continuous-time systems with delayed input and application to and appointed as a member of the Drafting Committee of a Global Convention control,” in Proc. Eur. Control Conf. (ECC), Jul. 2007, pp. 1786–1791. on the Recognition of Higher Education Qualifications at UNESCO since [72] V. Léchappé, E. Moulay, and F. Plestan, “Dynamic observation- 2016. He has been the Chair of the Scientific Committee since 2014, and a prediction for LTI systems with a time-varying delay in the input,” Section Editor of Scientia Iranica (Transaction B, Previous Elsevier Editorial in Proc. IEEE 55th Conf. Decis. Control (CDC), Dec. 2016, System). He was the Secretary of the IAS. pp. 2302–2307. [73] I. Estrada-Sánchez, M. Velasco-Villa, and H. Rodríguez-Cortés, “Prediction-based control for nonlinear systems with input delay,” Math. Problems Eng., vol. 2017, pp. 1–11, Oct. 2017. [74] T. G. Ritto, C. Soize, and R. Sampaio, “Non-linear dynamics of a drill- string with uncertain model of the bit–rock interaction,” Int. J. Non- Nathan van de Wouw (Fellow, IEEE) was born Linear Mech., vol. 44, no. 8, pp. 865–876, Oct. 2009. in 1970. He received the M.Sc. (Hons.) and Ph.D. [75] T. G. Ritto and R. Sampaio, “Stochastic drill-string dynamics with degrees in mechanical engineering from the Eind- uncertainty on the imposed speed and on the bit-rock parameters,” Int. hoven University of Technology, Eindhoven, The J. Uncertainty Quantification, vol. 2, no. 2, pp. 111–124, 2012. Netherlands, in 1994 and 1999, respectively. He was with Philips Applied Technologies, Eindhoven, and the Netherlands Organization for Applied Scientific Research, Delft, The Netherlands, in 2000 and 2001, respectively. He was a Visiting Professor at the University of California at Santa Barbara, Santa Barbara, CA, USA, from 2006 to 2007, The University of Melbourne, Melbourne, VIC, Australia, from 2009 to 2010, and the University of Minnesota, Minneapolis, MN, USA, from Shabnam Tashakori was born in 1990. She 2012 to 2013. He is currently a Full Professor with the Department of received the Ph.D. degree in mechanical engineering Mechanical Engineering, Eindhoven University of Technology, and an Adjunct from the Sharif University of Technology, Tehran, Full Professor with the University of Minnesota. He has authored a large Iran, in 2019. number of journal and conference papers and the books Uniform Output She was a Visiting Researcher with the Regulation of Nonlinear Systems: A Convergent Dynamics Approach with A. Eindhoven University of Technology, Eindhoven, V. Pavlov and H. Nijmeijer (Birkhauser, 2005) and Stability and Convergence The Netherlands, in 2018. She is currently the of Mechanical Systems with Unilateral Constraints with R. I. Leine (Springer- Chair of Management Board (COB) with Raha, Verlag, 2008). Tehran, startup company, and a Founding Member Dr. van de Wouw was a recipient of the IEEE Control Systems Technology of the Rahesh Innovation Center, Shiraz, Iran. Award for the development and application of variable-gain control techniques Her current research interests include (non)linear for high-performance motion systems in 2015. He is also an Associate dynamics and control with applications in the drilling industry and impact Editor of Automatica and the IEEE TRANSACTIONS ON CONTROL SYSTEMS force reconstruction. TECHNOLOGY.

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