ADVANTAGES OF AN LQR CONTROLLER FOR STICK-SLIP AND BIT-BOUNCE MITIGATION IN AN OILWELL DRILLSTRING

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1 roceedings of the ASME 22 nternational Mechanical Engineering Congress & Exposition MECE22 November 9-5, 22, Houston, Texas, USA MECE ADVANTAGES OF AN LQ CONTOLLE FO STCK-SL AND BT-BOUNCE MTGATON N AN OLWELL DLLSTNG Md. Mejbahul Sarker Graduate Student Faculty of Engineering and Applied Science Memorial University St. John s, NL, Canada mms426@mun.ca D. Geoff ideout Associate rofessor Faculty of Engineering and Applied Science Memorial University St. John s, NL, Canada g.rideout@mun.ca Stephen D. Butt rofessor Faculty of Engineering and Applied Science Memorial University St. John s, NL, Canada sdbutt@mun.ca KEYWODS Drilling, vibration, lumped-segment model, stick-slip, bitbounce, bit-rock interaction, linear quadratic regulator, torsion spring-damper. ABSTACT Failure of oilwell drillstrings is very costly in terms of money and time. There are many reasons for drillstring failure, such as vibration, fatigue, and buckling. Stick-slip vibration has received considerable attention in recent years with increasing use of polycrystalline diamond compact (DC) bits in harder formations, and has motivated extensive research on this type of drillstring vibration. This paper addresses the advantages of a linear quadratic regulator (LQ) controller, compared to a spring-damper isolator, for stick-slip and bit-bounce mitigation in an oilwell drillstring. A bond graph model of a drillstring has been used for simulation that predicts axial vibration, torsional vibration, and coupling between axial and torsional vibration due to bit-rock interaction. NTODUCTON The boreholes of oil exploration and exploitation wells are typically drilled by means of a rock cutting tool (drill bit), which is attached at the end of a rather long drill string consisting of many smaller interconnected drillpipe sections, and driven by a speed controlled electrical drive. Due to large lengths and small cross sections of the drilling pipes, low tool inertia, and emphasized tool vs. rock bed friction, the overall drillstring electrical drive is prone to poorly damped torsional vibrations including stick-slip behavior. Stick-slip predominates when drilling with drag bits (especially with DC bits) and stick-slip oscillations induce large cyclic stresses, which may also excite severe axial and lateral vibrations in the bottom hole assembly (BHA). This can lead to fatigue problems, reduction of bit life, unexpected changes in drilling direction, and even failure of the drillstring. Since drilling is one of the most expensive operations in oil exploration and development, vibration control in the oil drilling process is required from an economic point of view. n practice, the drilling operator typically controls the surface-controlled drilling parameters, such as the weight on bit (WOB), drilling fluid flow through the drill pipe, the drillstring rotational speed and the density and viscosity of the drilling fluid to optimize the drilling operations. n particular, the only means of controlling vibration with current monitoring technology is to change either the rotary speed or the weight on bit. Historically, the experience of drillers has revealed that increasing the rotary speed, decreasing the weight on bit, modifying the drilling mud characteristics and introducing an additional friction at the bit etc. are effective strategies to suppress stick-slip motion [-6]. Trade-offs must be managed, however. For example, mitigating stick-slip by reducing weight on bit might result in a lower rate of penetration. ncreasing rotary speed to reduce stick-slip might bring other vibration modes into resonance. The effectiveness of ad hoc changing of drilling parameters is also dependent on the skill of the operator. ecently linear quadratic regulator controllers have been discussed [, 3] as a means of improving multiple types of vibration such as stick-slip and bit-bounce motion simultaneously. This paper presents the improvements of using an LQ controller instead of a torsional spring-damper isolator near the top drive system for stick-slip and bit-bounce mitigation in an oilwell drillstring. assive spring-damper isolators must be tunable given that drillstring properties are a function of its Copyright 22 by ASME

2 length. Non-linear energy sinks (NES) [9, 2] are a potential means of attenuating vibration over a wide frequency range, but would likely require physical realization of a large rotary inertia. n this paper, a bond graph model of an oilwell drillstring has been used to compare LQ control and a tunable linear isolator. The model includes a bit-rock interaction submodel that predicts axial vibration, torsional vibration, and coupling between axial and torsional vibration. OLWELL DLLNG SYSTEM MODELNG The rotary drilling system being modeled consists of drillpipes, the drillcollar assembly (made up of heavier collar pipes) and the drill bit at the end of the collar assembly and the rock (formation). Drilling fluid is circulated in the drillpipe and the annular space between the drillpipe and the wellbore. The drilling fluid is characterized by the flow rate developed by the mud pumps. The top of the drillstring is subject to a tension force, applied through the surface cables. otary motion is applied by an armature-controlled motor, through a gear box, to the rotary table via the kelly (a square, hexagonal or octagonal shaped tubing that is inserted through and is an integral part of the rotary table that moves freely vertically while the rotary table turns it). The essential components of the oilwell drilling system and the necessary geometry used for the model are shown in Fig.. A lumped-segment approach is used in the axial and torsional dynamic model. n the lumped segment approach, the system is divided into number of inertias, interconnected with springs [7-9]. The accuracy of the model depends on the number of elements considered; however, in contrast to a modal expansion approach [7], the analytic model shapes and natural frequencies need not be determined. Both axial and torsional submodels have a total of 2 segments to capture the first eight axial and torsional natural frequencies of the whole drillstring. A physical schematic of the lumpedsegment models is shown in Fig. 2. Figure : Oilwell drilling system (adapted from [2]) n the axial submodel, hydraulic forces are included at the top of the drill collar and bottom of the drillstring to capture the effect of drilling mud density and buoyancy. Hydrodynamic damping, due to drilling fluid circulation in the drill pipe and the annular space, is considered in the drill pipe and collar model []. Figure 2: hysical schematic of (a) axial segments and (b) torsional segments n the torsional model, the drill pipe and drill collar dynamic models consider viscous damping which results from the contact between drillstring surfaces and drilling fluid []. A quasi-static rock-bit model, which provides coupling between axial and torsional drillstring dynamics, is used instead of a computationally intensive and difficult-to-parameterize complete dynamic representation. The model equations are based on the bit-rock model in [3-4], and are discussed in []. The bond graph model of the rotary drillstring is shown in Appendix A. The reader is referred to [, 7] for more details on the bond graph modeling method. LNEA QUADATC EGULATO A linear quadratic regulator (LQ) has been designed to control the torsional dynamics of the system. LQ is a wellknown design technique that provides optimal feedback gains. n order to determine LQ gains, a performance index is required. A performance index is the integral over time of several factors which are to intended to be minimized. The iccati equation is solved to calculate optimal linear gains. n order to reduce the dimension of the state vector and to minimize the number of states that must be physically measured or estimated, a simplified lumped parameter torsional model (Fig. 3) is used instead of taking 2 segments. The overall method of designing a controller using LQ technique is discussed in [, 3]. The necessary equations for the controller design are shown in Appendix B. n order to design the controller, a five-state (motor current, rotary table speed, rotary table displacement, drill bit speed, and bit displacement) simplified model is used. The resulting controller gains are then applied to the high order model for simulation. Fig. 4 in Appendix B shows the controller block diagram connection to the bond graph plant 2 Copyright 22 by ASME

3 Spring stiffness, k (Nm/rad) model. The control voltage necessary to keep the torsional vibrations zero while maintaining a desired bit and rotary table speed is given by V C = V ref K K 2 rt ω d t K 3 rt ω d K 4 rt K 5 ( ω d ) () compliance and inertia) increase. For a particular C s (7 Nms/rad) value the range of possible values of K s for which stick slip does not occur has been determined, using the bond graph system model, for different drilling depths as shown in Fig. 4. Setting the drive stiffness outside this range will not mitigate stick-slip. For different drilling depths the gain matrix K is calculated by using MATLAB and the gains vs. depth curves (Fig. 5) are shown in Appendix B Kmax Drilling depth vs Spring stiffness Curve ALTENATVE CONTOL SCHEMES n the literature, numerous solutions have been presented to control stick-slip oscillations, such as robust µ-synthesis controller [], H controller [], genetic algorithm optimized controller [2], D-OSKL controller [3], torque estimatorbased controller [4], and modeling error compensation based controller [5]. Many such controllers have practical limitations. However, one system that has achieved real-world acceptance is the soft torque rotary system (STS) [6-8]. STS is a torque feedback at the top of the drillstring which makes the system behave in a softer way rather than as a fixed heavy flywheel, so that the torsional waves arriving at the surface are absorbed, breaking the harmful cycling motion. The STS increases the system damping to the extent that rotational speeds will not drop to levels where there is a risk of the bottom hole assembly (BHA) sticking. Therefore, the feedback system, which acts on the rotary drive s speed input, modifies the speed of the motor such that the vibrational energy is optimally extracted from the drillstring. The effect of this feedback circuit, in practice fully implemented by electronics, is to emulate a parallel combination of a torsional spring and damper in series with an ideal motor as shown in Fig. 3. Figure 3: (a) Conventional or Normal (no STS) drilling [8], (b) STS schematic [8], (c) STS virtual mechanical elements [8], (d) Bond graph model of the STS virtual elements. The STS must be tuned by giving values of K s (drive stiffness in Nm/rad) and C s (drive damping in Nms/rad) [7]. The parameters must change as the drillstring length (and thus Kmin Drilling depth Figure 4: Drive spring stiffness (K s ) vs. drilling depth curve for a particular drive damping (C s = 7 Nms/rad) SMULATON ESULTS Table in Appendix A summarizes all relevant parameters that are used in the simulation. The main objective of the current simulations is to study the theoretical performance of an LQ controller compared to a torsional spring-damper (or virtual spring-damper as in the STS system) on the mitigation of stick-slip and bit-bounce vibrations in an oilwell drillstring. The simulation results for a drilling depth 42 m, where drill pipe and collar lengths are 4 m and 2 m, are shown below in Figs Fig. 5 shows the full model simulation results in the case of conventional drilling when the desired rotary table speed is 5 rad/sec (42 rpm) with 75 kn applied WOB. Though the motor appears to maintain the rotary table speed as desired, the bit experiences large speed fluctuations indicative of stick-slip. Also at the same time the torque at surface experiences large fluctuations consistent with stick-slip [7-8]. When the input torque grows sufficiently to overcome static friction and the bit releases, bit speed approaches the axial vibration critical speed range that is discussed in []. Bit-bounce then occurs as demonstrated in Fig. 5 where dynamic WOB periodically becomes zero. Fig. 6 shows the response of the model when LQ control is activated at the simulation time of 4 seconds, for the case of 75 kn applied WOB and a desired speed of 5 rad/sec (42 rpm). As can be seen, when LQ controller is active the stickslip vibration is controlled and a smooth drilling condition is achieved. That means the drill bit is rotating with constant desired speed and the torque at the surface becomes constant. At the same time the controller eliminates high dynamic force and bit-bounce, as a result of the axial-torsional coupling at the bit-rock interface. 3 Copyright 22 by ASME

4 Figure 5: High stick-slip vibrations with bit-bounce at 5 rad/sec (42 rpm) rotary table speed and 75 kn applied WOB Figure 7: al spring-damper system unable to eliminate stick-slip and bit-bounce vibrations at 5 rad/sec (42 rpm) table speed and 75 kn WOB Figure 6: Stick-slip and bit-bounce eliminated by LQ control at 5 rad/sec (42 rpm) table speed and 75 kn applied WOB Fig. 7 shows the response of the model when a torsional springdamper system is used, for the case of 75 kn applied WOB and a desired speed of 5 rad/sec (42 rpm). The torsional spring-damper system with the assigned parameters should be unable to eliminate stick-slip vibration at the desired speed. By increasing the desired speed to 24 rad/sec (23 rpm) in Fig. 8, the torsional spring-damper system becomes able to eliminate stick-slip vibration. From the simulation results in Fig. 5-8, an LQ controller can suppress stick-slip vibrations at lower desired speeds than can a torsional spring-damper system. This indicates a theoretical advantage of an LQ controller over torsional spring-damper isolators. Figure 8: Stick-slip and bit-bounce vibrations eliminated by torsional spring-damper system at 24 rad/sec (23 rpm) table speed and 75 kn applied WOB ADVATAGES OF LQ CONTOLLE Stick-slip occurs at a rotary speed below a certain threshold value. Fig. 9 shows the threshold phenomena of stickslip vibrations. The threshold value depends on system parameters such as design of the drillstring, mud, bit, BHA and weight on bit (WOB). Fig. and show the threshold rotary speed for different applied WOB for conventional drilling, drilling with torsional spring-damper system near the rotary table, and drilling with the LQ controller. Simulation results show that for a particular applied WOB the LQ controller gives the lowest magnitude of the threshold rotary speed. At higher WOB the difference in the threshold rotary speed between the LQ controller and torsional spring-damper 4 Copyright 22 by ASME

5 Applied WOB,(kN) Applied WOB,(kN) system increases, and it indicates that at higher WOB, and notwithstanding certain practical implementation issues to be discussed later, an LQ controller can increase the no stick-slip zone significantly compared to a torsional spring-damper system Figure 9: Threshold speed for stick-slip vibration Threshold otary Speed vs. Applied WOB Curve Stick-slip resent Zone Drilling with LQ Controller Drilling with Spring-Damper ncreasing Safe Zone Conventional Drilling Stick-slip Not resent Zone Threshold otary speed (rad/sec) Figure : Threshold rotary speed vs. applied WOB curve for different operating conditions at 22 m depth. During drilling, the LQ controller requires: (i) motor current, (ii, iii) rotary table rotary speed and displacement, (iv-v) bit rotary speed and displacement. Except for the bit speed and displacement, all other quantities in the controller can be measured. The bit speed measurement (and calculation of bit rotary displacement through integration) requires downhole equipment that is expensive and at this point not typically used in well drilling because the information is not needed if a controller is not used. Bit speed measurement is the biggest challenge preventing LQ and other sophisticated controller implementations, as discussed also in [2-3, 5]. The virtual spring-damper of the STS system requires only measurement of motor current, giving it an economic and implementation advantage at present. The additional potential benefits of LQ are expected to motivate drillers to eventually use advanced downhole measurement tools, to enable such control. The additional cost of instrumentation would be justified by even smoother drilling and fewer tool failures. CONCLUSONS Self-excited stick-slip oscillations in oilwell drillstrings are largely suppressed by the application of LQ control. Therefore, it is possible to drill smoothly at very low speeds which are otherwise not possible without LQ control. t has been shown that the advantages of using LQ control increase with higher applied WOB. The performance of LQ control for mitigation of stick-slip decreases with increasing depth. t nonetheless retains an advantage compared to a system with a spring-damper isolator. This should motivate the use of LQ controllers in future when practical challenges in measuring required state variables for LQ control are addressed by advances in downhole measurement technology. The implementation of the high-order model in commercial software that allows block diagrams to be superimposed on bond graphs greatly facilitated inclusion of the coupled axial and torsional degrees of freedom due to bit-rock interaction, along with the controller Threshold otary Speed vs. Applied WOB Curve Stick-slip resent Zone Drilling with LQ Controller Drilling with Spring-Damper Conventional Drilling ACKNOWLEDGMENTS This work was done at the Advanced Exploration Drilling Technology Laboratory at Memorial University in St. john s, Canada. Financial support was provided by Husky Energy, Suncor Energy, Newfoundland and Labrador esearch and Development Corporation, and Atlantic Canada Opportunities Agency under AF contract number ncreasing Safe Zone Stick-slip Not resent Zone Threshold otary speed (rad/sec) Figure : Threshold rotary speed vs. applied WOB curve for different operating conditions at 42 m depth. NOMENCLATUE C rt = Equivalent viscous damping coefficient, N-m-s/rad = Current, A J = Drillstring mass moment of inertia, kg-m 2 J k = nertia of kelly, kg-m 2 J m = nertia of motor. kg-m 2 J rt = nertia of rotary table, kg-m 2 K m = Motor constant, V-s K t = al stiffness, N-m/rad L = Motor inductance, H 5 Copyright 22 by ASME

6 n = Gear ratio m = Armature resistance, Ω C v = Viscous damping coefficient, N-m-s/rad rt = otary table speed, rad/sec rt = otary table angular displacement, rad EFEENCES [] Sarker, M.M., ideout, D.G., & Butt, S.D. (22) Dynamic Model of an Oilwell Drillstring with Stick-Slip and Bit-Bounce nteraction Submitted to th nternational Conference on Bond Graph Modeling and Simulation. Genoa, taly, July 8-, 22. The Society for Modeling and Simulation nternational, San diego, CA, USA. [2] Leine,.., Van Campen, D. H., & Keultjes, W. J. G. (22). Stick-slip Whirl nteraction in Drillstring Dynamics. Journal of Vibration and Acoustics, 24(2), [3] Yigit, A. S., & Christoforou, A.. (2). Coupled al and Bending Vibrations of Actively Controlled Drillstrings. Journal of Sound Vibration, 234(), [4] Yigit, A. S., & Christoforou, A.. (26). Stickslip and Bit-bounce nteraction in Oil-well Drillstrings. Journal of Energy esources Technology, 28(4), [5] Cobern, M. E., & Wassell, M. E. (24). Drilling Vibration Monitoring and Control System. National Gas Technologies Conference hoenix, AZ, 8- February. [6] res/mwd/drilling_dynamics_sensors_opt_br.ashx [7] Karnoop, D. C., Margolis, D. L., & osenberg,. C., (999). System Dynamics; Modeling and Simulation of Mechatronics Systems, 3 rd ed., John Wiley & Sons, nc., New York. [8] Eronini,. E., Somerton, W. H., & Auslander, D. M. (982). A Dynamic Model for otary ock Drilling. Journal of Energy esources Technology, 4(2), 8 2. [9] ao, S.S. (995). Mechanical Vibrations, 3 rd ed., Addison-Wesley ublishing Company, New York. [] Karkoub, M., Zribi, M., Elchaar, L., & Lamont, L. (2). obust μ-synthesis Controllers for Suppressing Stick-slip nduced Vibrations in Oil Well Drill Strings. Multibody System Dynamics, 23(2), [] Serrarens, A. F. A., van de Molengraft, M. J. G., Kok, J. J., & van den Steen, L. (998). H Control for Suppressing Stick-slip in Oil Well Drillstrings. Control Systems, EEE, 8(2), 9 3. [2] Karkoub, M., Abdel-Magid, Y. L., & Balachandran, B. (29). Drill-string al Vibration Suppression Using GA Optimized Controllers. Journal of Canadian etroleum Technology, 48(2), [3] Canudas-de-Wit, C., ubio, F.., & Corchero, M. A. (28). D-OSKL: A New Mechanism for Controlling Stick-slip Oscillations in Oil Well Drillstrings. EEE Transactions on Control Systems Technology, 6(6), [4] avkovic, D., Deur, J., & Lisac, A. (2). A Torque Estimator-based Control Strategy for Oilwell Drill-string al Vibrations Active Damping ncluding an Auto-tuning Algorithm. Control Engineering ractice, 9(8), [5] uebla, H., & Alvarez-amirez, J. (28). Suppression of Stick-slip in Drillstrings: A Control Approach Based on Modeling Error Compensation. Journal of Sound and Vibration, 3(4-5), [6] Womer, K. A., Torkay, D.., Villanueva, G.., Geehan, T., Brakel, J., irovolou, D., et al. (2). esults of July 5, 2 ADC Stick-Slip Mitigation Workshop. SE/ADC Drilling Conference and Exhibition. Amsterdam, The Netherlands: Society of etroleum Engineers. [7] [8] Kriesels,. C., Keultjes, W. J. G., Dumont,., Huneidi,., Owoeye, O. O., & Hartmann,. A. (999). Cost Savings through an ntegrated Approach to Drillstring Vibration Control. SE/ADC Middle East Drilling Technology Conference. Abu Dhabi, United Arab Emirates: Society of etroleum Engineers. [9] Younesian, D. et al. (2). Application of the Nonlinear Energy Sink Systems in Vibration Suppression of ailway Bridges. roc. ASME th Biennial Conf. on Eng. Sys. Design and Analysis, July 2-4, stanbul, Turkey. [2] Ahmadabadi, Z. N., and Khadem, S.E. (22). Nonlinear vibration control of a cantilever beam by a nonlinear energy sink. Mech. and Machine Theory 5, pp Copyright 22 by ASME

7 AENDX A OTAY DLLNG SMULATON MODEL AND DATA Dynamic Model Cable_derrick_C Cable_Damping al Dynamic Model C Kelly_Swivel Kelly_Table_Motor_nertia otary_damping Motor_inductance esistance Sf Fixed_support Swiv el otary_table TF Gear_atio GY Motor_constant Motor Se nput_voltage ipe_axial4 ipe_axial3 ipe_axial2 ipe_axial ipe_axial Drillipe_Tor Drillipe_Tor Drillipe_Tor2 Drillipe_Tor3 Drillipe_Tor4 ipe_axial5 ipe_axial6 ipe_axial7 ipe_axial8 ipe_axial9 Drillipe_Tor9 Drillipe_Tor8 Drillipe_Tor7 Drillipe_Tor6 Drillipe_Tor5 Collar_axial4 Collar_axial3 Collar_axial2 Collar_axial F Collar_axial ntegrate DrillCollar_Tor DrillCollar_Tor DrillCollar_Tor2 DrillCollar_Tor3 DrillCollar_Tor4 Collar_axial5 Collar_axial6 Collar_axial7 Collar_axial8 Collar_axial9 Cosine DrillCollar_Tor9 DrillCollar_Tor8 DrillCollar_Tor7 DrillCollar_Tor6 DrillCollar_Tor5 Bit_Mass Se Bit_Weight Se F2_Hydraulic MSf Bit_otation Bit_nertia Flow_Excitation C MSe ock_c TOB ock-bit Model Figure 2: Bond graph model of rotary drilling system Table : Data used in rotary drilling simulation Drillstring data Cable and derrick spring constant 9.3e+6 N/m Swivel and derrick mass 73 kg Kelly length 5 m Kelly outer diameter.379 m Kelly inner diameter.825 m Drill pipe outer diameter. m (4 in) Drill pipe inner diameter.848 m (3.34 in) Drill collar length 2 m Drill collar outer diameter.7 m (6.75 in) Drill collar inner diameter.57 m (2.25 in) Drill string material Steel Wellbore diameter.2 m Drill bit-rock data Bit type DC (Single cutter) Drill bit diameter.2 m (7.875 in) Drill bit mass 65 kg ock stiffness.6e+9 N/m ock damping.5e+5 N.sec/m Surface elevation amplitude s. Bit factor, b Cutting coefficient ξ, C, C 2,.35e-8, -.9e-4 Frictional coefficient μ, α,β, γ & ν.6, 2,, &. Threshold force, W fs N Hydraulic data Mud fluid density, ρ m 98 kg/m 3 Mud flow rate, Q Q m + Q a sin(qt) Mean mud flow rate, Q m.22 m 3 /sec Mud flow pulsation amplitude, Q a.2 m 3 /sec Freq. of variation in mud flowrate, q 25.3 rad/sec Equivalent fluid viscosity for fluid 3e-3 a.sec resistance to rotation μ e Weisbach friction factor outside drill.45 pipe or collar, α a Weisbach friction factor inside drill.35 pipe or collar, α p Motor data L, K m, n and m,.5 H, 6 V/s, 7.2 and. Ω 7 Copyright 22 by ASME

8 AENDX B LQ CONTOL MODEL DESGN AND GANS CUVES CONTOLLE EQUATONS The state space equation of the simplified model in Fig. is X = AX + BV C (6) where X, A, and B are the state vector, coefficient, and input matrices, respectively: A = m L nk m L nk m J k +J rt +n 2 J m C rt J k +J rt +n 2 J m K t J k +J rt +n 2 J m K t J C v J (7) Damping J_torsion _comp C (a) Jrt + n^ Jm Motor_inductance X T = [ rt rt rt ] (8) B T = [ L The performance index: C = 2 ] (9) (x T Qx + rv 2 C ) dt (2) The resulting optimal control input (rotary table motor voltage) can be written as V C = V ref K(x x d ) (2) The gain matrix, K can be written as K = r B T (22) The algebraic iccati equation is given by The chosen r and Q are A T + A r BB T + Q = (23) MSe TOB TF /n Gear box (b) Motor Constant Figure 3: (a) hysical schematic of model used for control design. (b) Bond graph torsional model using simplified lumped parameter model Model DrillBit_otation ntegrate otary_table otary_damping Kelly_Table_Motor_nertia al Model otary_damping otary_table otary_table_dis TF TF Desired_table_speed GY Km Motor_inductance GY GY Desired_otar_Dis Motor esistance Motor K K2 Se Vc_nput_Voltage esistance nput_voltage MSe Q = r = 95 (25) (26) MSf C ock_c Cosine MSf Bit_nertia MSe TOB DrillCollar_Tor9 Bit_otation Bit_otar_Dis Figure 4: LQ controller block diagram connection to the bond graph plant model. K3 K4 K5 8 Copyright 22 by ASME

9 Gain, K5 Gain, K2 Gain, K4 Gain, K Gain, K Figure 5: Gains vs. depth curves for LQ controller 9 Copyright 22 by ASME