Cortes, D.

Navarro Lopez, Eva

Navarro-Lopez, Eva M.

English

smooth (PWS) systems can be ineffective if an additional bifurcationanalysis is not made. Due to the presence of discontinuities,PWS systems present a wide variety of standard and non-standard bifurcations.Safe ranges of system and controller parameters can be establishedlest these bifurcations appear. This is studied by means of asimplified torsional model of an oilwell drillstring of three degrees offreedom (DOF). A PID-type controller is applied. The bifurcation analysisof the open-loop and the closed-loop systems is used to choose thecontroller parameters for which non-desired bit sticking situations areavoided. The control goal of driving the rotary velocities to a constantpositive value is achieved despite the existence of a sliding motion

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The University of Manchester ResearchParameters Controller Selection Through BifurcationAnalysis in a Piecewise-smooth SystemLink to publication record in Manchester Research ExplorerCitation for published version (APA):Navarro Lopez, E., Navarro-Lopez, E. M., Cortes, D., Bemporad, A. (Ed.), Bicchi, A. (Ed.), & Buttazzo, G. (Ed.)(2007). Parameters Controller Selection Through Bifurcation Analysis in a Piecewise-smooth System. In LectureNotes on Computer Science, LNCS 4416: Hybrid Systems Computation and Control, HSCC (1st ed., pp. 736-740).Springer Nature.Published in:Lecture Notes on Computer Science, LNCS 4416Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:09. Jun. 2022Parameters Controller Selection ThroughBifurcation Analysis in a Piecewise-smoothSystemEva M. Navarro-López1, and Domingo Cortés21 ETS de Ing. Industriales, Univ. de Castilla-La ManchaAv. Camilo José Cela s/n, 13071 Ciudad Real, Spainnavarro.lopez@gmail.com2 Centro de Inv. y Estudios Avanzados del IPN, Dpto. de Ing. EléctricaAv. IPN, 2508, Col. San Pedro Zacatenco, 07360 Mexico, D.F., Mexicodomingo.cortes@gmail.comAbstract. Classical linear and nonlinear control techniques applied topiecewise-smooth (PWS) systems can be ineffective if an additional bi-furcation analysis is not made. Due to the presence of discontinuities,PWS systems present a wide variety of standard and non-standard bi-furcations. Safe ranges of system and controller parameters can be es-tablished lest these bifurcations appear. This is studied by means of asimplified torsional model of an oilwell drillstring of three degrees offreedom (DOF). A PID-type controller is applied. The bifurcation anal-ysis of the open-loop and the closed-loop systems is used to choose thecontroller parameters for which non-desired bit sticking situations areavoided. The control goal of driving the rotary velocities to a constantpositive value is achieved despite the existence of a sliding motion.Keywords: Discontinuous systems, sliding motions, bifurcation analy-sis, oilwell drillstrings, dry friction, stick-slip oscillations.1 IntroductionHybrid systems are dynamical systems consisting of continuous-time and discr-ete-event dynamics. They are characterized by discontinuous changes in systemproperties. PWS systems are an interesting subclass of hybrid systems. An ex-ample of PWS system is the model of a conventional vertical oilwell drillstring.Oilwell drillstrings are systems exhibiting a wide variety of complex phenom-ena and non-desired oscillations. Stick-slip friction-induced oscillations and thepermanent stuck-bit situation are particulary harmful [1]. Several approacheshave appeared in order to model and control drillstring stick-slip oscillations.Most of them use lumped-parameter models of one DOF and two DOF (see[2] and references therein). In addition, in most of these works, no bifurcation This work has been partially supported by the Ministerio de Educación y Ciencia(MEC) of Spain under Ramón y Cajal research contract.A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, LNCS 4416, pp. 736–740, 2007.c© Springer-Verlag Berlin Heidelberg 2007Parameters Controller Selection Through Bifurcation Analysis 737analysis of system and controller parameters is made, and the influence of theweight on the bit (WOB) (a key drilling parameter) is not analysed.Recently in [2], an analysis of bifurcations and transitions between several bitdynamics is reported for an n-DOF drillstring. Changes in drillstring dynamicsare studied through variations in three parameters: 1) the WOB, 2) the rotaryspeed at the top-rotary system, 3) the torque given by the surface motor.This paper follows [2] and sums up new results. On the one hand, the bitsticking problems are related to the existence of a sliding motion when the bitvelocity is zero. On the other hand, the presence of multiple Hopf bifurcations inthe vicinity of the standard equilibrium point when velocities are greater thanzero is studied. A PID-type control which overcomes the existing sliding motionand drives the rotary velocities to a desired value is proposed. Controller param-eters are chosen lest non-desired system transitions appear. These non-desiredsituations are permanent stuck bit, bit stick-slip oscillations and other periodicmotions. A lumped-parameter torsional PWS model with three DOF includ-ing the bit-rock interaction is considered. This model is more general than thetorsional lumped-parameter models of one and two DOF previously proposed.2 A Torsional Discontinuous Model of the DrillstringThree main parts of a conventional vertical oilwell drillstring are considered: 1)the rotating mechanism in the surface (inertia Jr), 2) the set of drill pipes whichform a long pipeline (inertia Jp), 3) the bottom-hole assembly (BHA), whichconsists of the drill collars, the stabilizers, a heavy-weight drill pipe and the bit(inertia Jb). Hereinafter, the BHA will be also referred to as bit. A simplifiedtorsional model is proposed in which the inertias are connected to each other bylinear springs with torsional stiffness (kt, ktb) and torsional damping (ct, ctb).The following system state vector is defined:x = (ϕ̇r, ϕr − ϕp, ϕ̇p, ϕp − ϕb, ϕ̇b)T = (x1, x2, x3, x4, x5)T, (1)with ϕi, ϕ̇i (i ∈ {r, p, b}) the angular displacements and angular velocities ofdrillstring elements, respectively. At the top-drive system, a viscous dampingtorque is considered (Tar = crx1). Tm = u is the torque coming from the electricalmotor at the surface, with u the control input. Tb(x5) = Tab(x5)+Tfb(x5) is thetorque on the bit with Tab = cbx5 approximating the influence of the mud drillingon the bit behaviour. Tfb(x5) is the friction modelling the bit-rock contact, andTfb(x5) = WobRb[μcb + (μsb − μcb) exp− γbvf |x5|]sgn(x5), (2)with Wob > 0 the WOB, Rb > 0 the bit radius; μsb , μcb ∈ (0, 1) the static andCoulomb friction coefficients associated with Jb, 0 < γb < 1 and vf > 0. Finally,the drillstring behaviour is described by the following equations:ẋ1 =1Jr[−(ct + cr)x1 − ktx2 + ctx3 + u] , ẋ2 = x1 − x3,738 E.M. Navarro-López and D. Cortésẋ3 =1Jp[ctx1 + ktx2 − (ct + ctb)x3 − ktb x4 + ctb x5] , ẋ4 = x3 − x5, (3)ẋ5 =1Jb[ctb x3 + ktb x4 − (ctb + cb)x5 − Tfb(x5)] ,or in a compact form: ẋ(t) = Ax(t) + Bu + Tf(x(t)), where A, B are constantmatrices depending on system parameters and Tf is the torque on the bit.3 Transitions and Bifurcations in the Open-Loop SystemTwo key dynamical characteristics of system (3) determine its behaviour: 1)the existence of a sliding regime which implies the existence of a sliding surfacelocally attractive; 2) the existence of a unique standard equilibrium point whenvelocities are greater than zero, which is locally asymptotically stable dependingon the values of Wob, the torque u and the rotary velocity at the equilibrium.The existence of a sliding motion (see [3]) in system (3) is directly related todifferent bit sticking phenomena, mainly, stick-slip motion and the permanentstuck-bit situation. System (3) is a PWS system of the form,ẋ ={f+(x, Wob, u) = Ax + Bu + Tf(x)|Tfb=T+fbif x5 > 0,f−(x, Wob, u) = Ax + Bu + Tf(x)|Tfb=T −fbif x5 < 0,(4)where T +fb and T−fb are Tfb(x5) for x5 > 0 and x5 < 0, respectively. Let Σ :={x ∈ IR5 : x5 = 0} be the switching manifold, Σ̃ ⊂ Σ the sliding region withΣ̃ = {x ∈ Σ : |ktb x4 + ctb x3| < WobRbμsb} and x̃ the quasiequilibrium pointexisting on Σ, which can be shown to be asymptotically stable. If x5 > 0 then thesystem is described by ẋ = f+(x, Wob, u) and has a unique standard equilibriumpoint x such that f+(x, Wob, u) = 0, which is the solution of the set of equations:x1 = x3 = x5 > 0, u − (cr + cb)x5 − T +fb (x5) = 0, x2 =h(x5, u)kt, x4 =h(x5, u)ktb,with h(x5, u) =[crT+fb (x5) + cbu]/(cr + cb) and u > WobRbμsb > 0.Three main kinds of bit dynamical behaviours can be identified: 1) stick-slipat x5, that is, the trajectory enters and leaves repeatedly the sliding mode;2) permanent stuck bit, i.e., x(t) ∈ Σ̃, ∀t; 3) the trajectory converges to thestandard equilibrium x. The bit moves with a constant positive velocity (thethird situation) when x̃ is far away enough from the boundaries of Σ̃, which isaccomplished when u is greater enough than WobRbμsb .The local stability of x must be also assured. Periodic orbits around it due tothe presence of Hopf bifurcations (HB) can be avoided. Ranges of the WOB andvelocities at equilibrium for which a HB may appear can be identified, they inter-sect the values for which stick-slip oscillations are present. The stability regionof x corresponds to low Wob and high enough values of the rotary velocities.Parameters Controller Selection Through Bifurcation Analysis 7394 Linear Control to Eliminate Non-desired TransitionsThe control goal is to eliminate bit sticking problems and establish a stablestandard equilibrium point in which the angular velocities are all equal to adesired positive velocity. This is accomplished by means of the following control:u = K1 x6 + K2(Ω − x1) + K3(x1 − x5) + u∗, (5)with x6 =∫ t0 [Ω − x1(τ)] dτ , ẋ6 = Ω − x1, Ω > 0 the desired rotary velocity,Ki positive constants and u∗ = WobRbμsb > 0 the minimum value of u for thesystem trajectory to cross the boundary of Σ̃. This value of u∗ prevents the bitfrom sticking when control (5) is initially switched on. The closed-loop systemhas a unique standard equilibrium point xc with the form,xc,1 = xc,3 =xc,5 = Ω, xc,2 =h(Ω)kt, xc,4 =h(Ω)ktb, xc,6 =1K1[crΩ + h(Ω) − u∗] ,where h(Ω) =[cbΩ + T +fb (Ω)]. The dynamical changes introduced by control (5)in the open-loop system (3) are mainly: 1) the standard equilibrium point has theangular velocities equal to Ω; 2) the sliding motion is maintained, however, thereis no quasiequilibrium point on the switching manifold, and the permanent stuck-bit situtation is eliminated; 3) periodic orbits (including stick-slip oscillations)may still arise in the system.Stick-slip oscillations appear in the closed-loop system due to the existence of asliding region Σ̃ locally attractive and a standard equilibrium which can becomeunstable or whose domain of attraction can be reduced due to the variation of thecontroller parameters, Ω and Wob. The analysis of the zero crossings of the realparts of the pairs of the complex eigenvalues of the Jacobian at xc of the closed-loop system (referred to as ZC) is a good approach for studying the changingstability properties of xc. The ZC points might imply a Hopf bifurcation and bethe origin of periodic orbits. The extension of the parameters region where noZC point is present, is indicative of the domain of attraction of xc. When thedomain of attraction of xc increases, stick-slip motion may disappear.Extensive simulations of the closed-loop system lead to establish safe rangesof controller parameters Ki in order to avoid bit sticking problems and the un-stability of the equilibrium xc. The main conclusion is that for typical operationvalues of Ω (between 10 and 14 rad/s), stick-slip oscillations disappear in theclosed-loop system and trajectories converge to xc with low enough values ofK3 (K3 <= Ka3 ) despite variations in Wob. For K3 > Ka3 , stick-slip oscillationswill disappear if K2 > K3 and K2 − K3 is high enough. In this case, the higherK2 is, the bigger the (Ω, Wob)-region without ZC points is. K1 does not implysignificant changes in the curves (Ω, Wob) corresponding to ZC points whichdelimit safe parameters regions. The value of K1 influences the transient systemresponse: the higher K1 is, the higher the overshooting in the velocities is.Other non-standard bifurcations typical in PWS systems with sliding motionshould be studied. The analysis shown could be successfully applied to other dis-continuous mechanical systems exhibiting stick-slip oscillations and dry friction.740 E.M. Navarro-López and D. CortésReferences1. Brett, J.F.: The genesis of torsional drillstring vibrations. SPE Drilling EngineeringSeptember (1992) 168–1742. Navarro-López, E.M., Cortés, D.: Avoiding harmful oscillations in a drillstringthrough dynamical analysis. Journal of Sound and Vibration (to appear)3. Utkin, V.I. Sliding Modes in Control Optimization. Springer-Verlag, Berlin (1992)

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