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  • Modeling, Identification and Control, Vol. 39, No. 1, 2018, pp. 114, ISSN 18901328

    Iterative Learning Applied to HydraulicPressure Control

    P. H. Gytil 1 M.R. Hansen 1 G. Hovland 1

    1Mechatronics Group, University of Agder, N-4898 Grimstad, Norway.

    Abstract

    This paper addresses a performance limiting phenomenon that may occur in the pressure control ofhydraulic actuators subjected to external velocity disturbances. It is demonstrated that under certainconditions a severe peaking of the control error may be observed that significantly degrades the performanceof the system due to the presence of nonlinearities. The phenomenon is investigated numerically andexperimentally using a system that requires pressure control of two hydraulic cylinders. It is demonstratedthat the common solution of feed forwarding the velocity disturbance is not effective in reducing the peakingthat occurs as a result of this phenomenon. To improve the system performance, a combination of feedbackand iterative learning control (ILC) is proposed and evaluated. The operating conditions require that ILCbe applied in combination with a feedback controller, however the experimental system inherently suffersfrom limit cycle oscillations under feedback due to the presence of valve hysteresis. For this reason the ILCis applied in combination with a feedback controller designed to eliminate limit cycle oscillations basedon describing function analysis. Experimental results demonstrate the efficacy of the solution where thefeedback controller successfully eliminates limit cycle oscillations and the ILC greatly reduces the peakingof the control error with reductions in the RMS and peak-to-peak amplitude of the error by factors ofmore than 30 and 19, respectively. Stability of the proposed solution is demonstrated analytically in thefrequency domain and verified on the experimental system for long periods of continuous operation.

    Keywords: Hydraulic pressure control, peaking phenomenon, iterative learning control, limit cycles.

    1 Introduction

    Hydraulic actuators are used in a wide range of applica-tions, for example mobile machinery, offshore drillingand material handling. In general, hydraulic actua-tors are used whenever high power density is of majorimportance. Both open-loop and closed-loop controlsystems are common, depending on the system require-ments. Closed-loop control may be further divided intotwo categories: motion control and pressure control.Motion control refers to position and velocity control,whereas pressure control, in the context of this paper,refers to controlling the pressure difference across theactuator, which includes force control of linear actua-tors and torque control of rotary actuators.

    Hydraulic pressure control is a challenging and oneof the most studied problems within the field of hy-draulic control (Ledezma et al., 2015), often requiringsystem modifications or sophisticated control methodseven for simple control specifications (Baghestan et al.,2014). Several challenges unique to hydraulic pressurecontrol arise from the effects of an inherent feedbackpath in the systems dynamics coupling the pressuresin the chambers of the actuator to its velocity, com-monly known as the natural velocity feedback (Zhaoet al., 2004). When the controlled actuator force ortorque is applied to an environment whose natural fre-quency is not significantly above that of the valve ac-tuator combination, the systems frequency responsesuffers from an antiresonance at the natural frequency

    doi:10.4173/mic.2018.1.1 c 2018 Norwegian Society of Automatic Control

    http://dx.doi.org/10.4173/mic.2018.1.1

  • Modeling, Identification and Control

    of the environment followed by a resonant mode. Thisphenomenon makes high bandwidth tracking of pres-sure controlled actuators a difficult task that has beenstudied extensively, see for example (Zhao et al., 2004)and (Lamming et al., 2010).

    Another consequence of the natural velocity feed-back is the effect of external velocity disturbances inpressure controlled hydraulic actuators. Due to thenear incompressibility of hydraulic oil and the couplingbetween the pressures of the actuator and its velocity,even minor velocity disturbances may correspond tosignificant disturbances in the pressures of the actua-tor and thus in the controlled output (Esfandiari andSepehri, 2014). A common method of improving theperformance in situations where velocity disturbancesare measurable involves feed forwarding from the dis-turbance, see for example (Conrad and Jensen, 1987)and (Jiao et al., 2004). As will be seen in Sections 4 and5 however, a large abrupt peaking of the control errormay occur under certain operation conditions that isnot reduced by feed forwarding from the velocity dis-turbance and appears to be a previously unaddressedphenomenon.

    This paper examines the conditions under which thisphenomenon occurs and several factors that may re-duce or amplify the resulting peaking of the controlerror are identified. The relevance of the phenomenonfrom a system design perspective is discussed and theuse of iterative learning control (ILC) is proposed andevaluated as a potential solution for achieving satisfac-tory system performance. An additional contributionof the paper is the application of ILC in a closed-loopfashion to a system that inherently suffers from limitcycle oscillations under feedback.

    For systems performing repetitive tasks, ILC hasbeen shown to be capable of significantly improv-ing the systems performance (Blanken et al., 2017),with several industrial applications having been re-ported (Boeren et al., 2016). Originating from thefield of industrial robotics where industrial manipula-tors are often used to perform the same tasks repeti-tively (Wallen, 2011), ILC takes advantage of the repet-itive nature of the system in order to improve the per-formance iteratively. By recording the system perfor-mance each iteration, a learning algorithm updates afeed forward signal that is either sent to the plant in-put or modifies the set point or reference trajectory ofa conventional feedback controller (Longman, 2000).Figure 1 illustrates a simple example of an ILC algo-rithm where the output of the ILC is denoted u. Theblock G represents the dynamics of a plant and its feed-back controller. The reference input is r(t) and the out-put of the plant is denoted y(t), where the subscript kdenotes iteration. By recording the control error e(t)

    in each iteration and feed forwarding it to the next it-eration, the control error is reduced provided that thebehaviour of the system is deterministic. The ILC algo-rithm in this example may be thought of as a feed for-ward in the time domain, and a feedback P-controllerwith unity gain in the iteration domain. This outlinesthe basic mechanism of an ILC controller, where moresophisticated algorithms are found in the literature, seefor example (Wallen, 2011).

    +1()

    -

    +

    -

    +

    ()+1()

    +

    +1 = ()

    Figure 1: Basic mechanism of an ILC configuration.

    The rest of the paper is organized as follows. Sec-tion 2 describes the experimental system and its con-trol requirements. Modelling of the system is presentedin Section 3 and an investigation into the aforemen-tioned peaking phenomenon is conducted in Section4. Sections 5 and 6 concern the control designs to beevaluated with experimental results given in Section7. Lastly, a discussion and conclusions are found inSection 8.

    2 System Description

    The experimental system is a test rig designed forstudying friction phenomena and friction compensa-tion methods in hydraulic cylinders, see Figure 2. Theprimary component of the test rig is the main cylinderwith a piston diameter of 125 mm, which is controlledby means of a servo valve and may translate freely inthe vertical direction. In addition, two hydraulic cylin-ders connected in parallel, referred to as the rotationcylinders, may be used to rotate the piston rod and pis-ton of the main cylinder without affecting the extensionor retraction of the main cylinder. As the main cylin-der is actuated in the vertical direction, the rotationcylinders simply follow the main cylinder as they glidefreely on low-friction sliding tracks. This arrangementconstitutes a non-model based friction compensationmethod that was studied in (Ottestad et al., 2012).

    Further, two additional cylinders, referred to as theload cylinders, connected in parallel hydraulically areoriented opposite and attached to the main cylindermechanically. A second servo valve, connected to the

    2

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    load cylinder circuit may be used to control the pres-sure differential across the load cylinders and thus theforce applied to the main cylinder. The purpose of theload cylinders is to simulate the application of externalforces applied to the main cylinder. Due to their me-chanical connection, any actuation of the main cylinderforces the load cylinders to move as well and thereforerepresents an external velocity disturbance to the loadcylinder circuit.

    Main Cylinder

    Load CylinderSliding Track

    Rotation Cylinder

    SOLIDWORKS Student Edition. For Academic Use Only.

    Figure 2: Overview of the experimental system.

    The work presented in this paper concerns the de-sign of a force controller for the load cylinder circuit,where for the application under consideration the loadcylinders are required to maintain a constant force of10 kN in the presence of external velocity disturbancesgenerated by the main cylinder. In particular, the loadcylinders should be capable of maintaining their forceset point with a reasonable accuracy as the main cylin-der is actuated in a sinusoidal velocity profile with afrequency of 0.1 Hz and amplitudes of up to 40 mm,where the worst case situation, i.e. an amplitude of 40mm is considered here. The remaining control hard-ware consists of a CompactRIO from National Instru-ments that is connected to the servo valves and theinstrumentation of the system, used for data acquisi-tion and the implementation of control algorithms inthe LabVIEW programming environment.

    The rest of the paper concerns the control design ofthe load cylinder circuit for which the use of the rota-

    tion cylinders is not relevant and therefore not consid-ered here.

    3 Modelling

    In this section a linear transfer function is derived forthe load cylinder circuit for use in the control design.First, the governing equations of the system are estab-lished. Next, the nonlinear equations are linearized andcombined to describe the dynamics from servo valve in-put to the output force FLC .

    Due to the large size of the fluid volumes in the maincylinder chambers, the flexibility of the main cylinderhas to be accounted for. The load cylinders are there-fore modelled as applying their output force to a mass-spring-damper system as shown in Figure 3. The massm refers to the lumped mass of all parts that may beaccelerated by the load cylinder circuit, the spring stiff-ness k describes the stiffness of the main cylinder andthe damping coefficient d is due to the presence of vis-cous damping.

    + +

    Figure 3: Modelling of the load cylinder circuit.

    The load cylinder circuit is equipped with hydraulicpressure sensors in both the piston-side and rod-sidechambers, however no force transducer is available.For this reason the friction of the load cylinder cir-cuit, which is negligible in comparison to the requiredoutput force, is not included in the estimation of theforce applied by the load cylinder circuit.

    The fluid volumes whose pressures are denoted p1and p2 are referred to as the first and second controlvolumes, respectively. The volume flows of the servovalve are described by the orifice equation (Merritt,

    3

  • Modeling, Identification and Control

    1967). For extension (u 0):

    Qv1 = Cd Ad0 u

    2

    (ps p1) (1)

    Qv2 = Cd Ad0 u

    2

    (p2 pT ) (2)

    For retraction (u < 0):

    Qv1 = Cd Ad0 u

    2

    (p1 pT ) (3)

    Qv2 = Cd Ad0 u

    2

    (ps p2) (4)

    The volume in each control volume is described by:

    V1 = VL1 +A1 y (5)V2 = VL2 +A2 (h y) (6)

    Applying the continuity equation to each control vol-ume:

    Qv1 = A1 y +V1 p1 (7)

    Qv2 = A2 y V2 p2 (8)

    From Pascals law, neglecting friction:

    FLC = p1 A1 p2 A2 (9)

    Applying Newtons second law to the mass m:

    m y = p1 A1 p2 A2 d y k y (10)

    Linearizing the orifice equation using a Taylor seriesexpansion and ignoring higher order terms:

    Qv1 = Kqu1 uKqp1 p1 (11)Qv2 = Kqu2 u+Kqp2 p2 (12)

    The linearization coefficients for extension are givenby (u 0):

    Kqu1,ext = Cd Ad0

    2

    (ps p1ss) (13)

    Kqu2,ext = Cd Ad0

    2

    (p2ss pT ) (14)

    Kqp1,ext =Cd Ad0 uss2 (ps p1ss)

    (15)

    Kqp2,ext =Cd Ad0 uss

    2 (p2ss pT )(16)

    The linearization coefficients for retraction are givenby (u < 0):

    Kqu1,ret = Cd Ad0

    2

    (p1ss pT ) (17)

    Kqu2,ret = Cd Ad0

    2

    (ps p2ss) (18)

    Kqp1,ret = Cd Ad0 uss

    2 (p1ss pT )(19)

    Kqp2,ret = Cd Ad0 uss2 (ps p2ss)

    (20)

    Taking Laplace transforms and combining Eqs. (7)-(12), the following transfer function is derived describ-ing the dynamics from servo valve input to the force ap-plied by the load cylinders using block diagram meth-ods:

    GLC(s) =FLC(s)

    u(s)= Gv(s) Ghyd(s) (21)

    where:

    Ghyd(s) =n3 s3 + n2 s2 + n1 s+ n0

    d4 s4 + d3 s3 + d2 s2 + d1 s+ d0(22)

    and:

    n3 = A1Kqu1V2m+A2Kqu2V1m (23)

    n2 = A1Kqu1V2d+A2Kqu2V1d

    +A1Kqp2Kqu12m+A2Kqp1Kqu2

    2m (24)

    n1 = A1Kqu1V2k +A2Kqu2V1k

    +A1Kqp2Kqu12d+A2Kqp1Kqu2

    2d (25)

    n0 = A1Kqp2Kqu12k +A2Kqp1Kqu2

    2k (26)

    and:

    d4 = V1V2m (27)

    d3 = V1V2d+Kqp1V2m+Kqp2V1m (28)

    d2 = V1V2k +A21V2 +A

    22V1 +Kqp1Kqp2

    2m

    +Kqp1V2d+Kqp2V1d (29)

    d1 = A21Kqp2

    2 +A22Kqp12 +Kqp1Kqp2

    2d

    +Kqp1V2k +Kqp2V1k (30)

    d0 = Kqp1Kqp22k (31)

    The servo valve dynamics are modelled as a secondorder transfer function from valve input command tooutput spool position u:

    Gv(s) =u(s)

    uref (s)=

    112vs2 + 2vv s+ 1

    (32)

    The parameters of the operating point considered asthe worst case in terms of stability have been identified

    4

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    Table 1: Overview of system parameters.

    Parameter Value UnitKqu1 1.0467 103 m3/sKqu2 1.1217 103 m3/sKqp1 2.05 1012 m3/(s Pa)Kqp2 1.53 1012 m3/(s Pa) 109 Pak 2.4 107 N/mm 98.4 kgd 500 N s/mA1 0.025 m

    2

    A2 0.013 m2

    V1 2.359 104 m3V2 2.361 104 m3v 0.85 -v 91.4 Hz

    in (Gytil, 2017) as given by Table 1 and are used inthe control design together with the linearized model.

    Additionally, for the numerical investigations pre-sented in the following section, a simulation model ofthe experimental system is constructed using libraryelements found in the commercially available softwarepackage SimulationX as shown in Figure 4. In thismodel the velocity control of the main cylinder is con-sidered ideal and for this reason the motion of the maincylinder is implemented as a preset that simply forcesthe load cylinders to move with the sinusoidal velocityprofile of the main cylinder. As in the mathematicalmodel, the elasticity of the main cylinder fluid columnis represented by a mass-spring-damper system. In ad-dition, five orifices, not shown in Figure 4 are used tomodel the leakage flows of the servo valve which areassumed to be laminar and modelled based on the nullleakage of the servo valve that was measured in (Gytil,2017).

    4 Nonlinear Peaking Phenomenon

    This section investigates a performance limiting phe-nomenon that may be observed in pressure controlledhydraulic actuators subjected to external velocity dis-turbances. As will be seen in Section 5, the load cylin-der circuit suffers from a severe peaking of the con-trol error at the instances where the velocity distur-bance from the main cylinder forces the load cylindersto change direction. In this paper this is referred to asthe nonlinear peaking phenomenon and is reported andinvestigated here using an asymmetrical actuator. Thisappears to be a previously unaddressed phenomenon,

    Figure 4: Numerical simulation model constructed us-ing SimulationX.

    although it may be observed in the experimental workof (Klausen and Trdal, 2015) for a symmetrical ac-tuator. In (Jiao et al., 2004) on the other hand, theabsence of this phenomenon may be observed both nu-merically and experimentally also using a symmetri-cal actuator, despite the presence of external veloc-ity disturbances that periodically force the actuatorto change direction, indicating that the appearance ofthe nonlinear peaking phenomenon is dependent uponthe system configuration. This section presents nu-merical results demonstrating some of the factors thatdetermine whether or not this phenomenon occurs fora given system configuration as well as several factorsthat may amplify the magnitude of this peaking.

    Using the numerical simulation model developed inthe previous section, Figure 5 shows the control errorof the load cylinder circuit using the PI controller ofSection 5 in the presence of a sinusoidal velocity dis-turbance with an amplitude of 10 mm and a frequencyof 0.1 Hz for a force set point of 0 kN. The solid line inFigure 5 indicates the control error for feedback controlalone and the dashed line indicates the control errorfor feedback combined with a fixed-gain feed forward(ff) from the velocity disturbance. Observe in Figure 5that for this sinusoidal velocity disturbance, the con-trol error appears sinusoidal, as may also seen in thenumerical and experimental work of (Jiao et al., 2004).Furthermore, a significant reduction in the control er-ror is achieved by feed forwarding the velocity distur-bance, where the peak-to-peak amplitude of the controlerror is reduced from 521 N to 25 N after adding thefeed forward term to the controller.

    Next, the force set point is changed from 0 kN to 10kN, while all other parameters of the simulation model

    5

  • Modeling, Identification and Control

    and the velocity disturbance are kept the same. Fig-ure 6 shows the control error for this situation. Observethe change in the overall shape of the control error bothwith and without the use of feed forward. In Figure 6an abrupt peaking of low magnitude is observed whenthe external velocity disturbance forces the load cylin-ders to change direction. Under these conditions, thecontrol error reaches a peak-to-peak amplitude of 797N using feedback and 226 N using feedback combinedwith feed forward, where the feed forward gain was ad-justed slightly for optimal performance after changingthe force set point.

    0 10 20 30 40

    Time [s]

    -300

    -200

    -100

    0

    100

    200

    300

    Con

    trol

    Err

    or [N

    ]

    FeedbackFeedback+ff

    Figure 5: Force set point of 0 kN.

    0 10 20 30 40

    Time [s]

    -500

    -400

    -300

    -200

    -100

    0

    100

    200

    300

    400

    500

    Con

    trol

    Err

    or [N

    ]

    FeedbackFeedback+ff

    Figure 6: Force set point of 10 kN.

    0 10 20 30 40

    Time [s]

    -1500

    -1000

    -500

    0

    500

    1000

    1500

    Con

    trol

    Err

    or [N

    ]

    FeedbackFeedback+ff

    Figure 7: Force set point of 10 kN and a valve dead-band of 0.5%.

    Observe that although the feed forward is able toimprove the overall accuracy, the peaking that occursas the load cylinders change directions appear to be ofthe same magnitude. This is the phenomenon referredto in this paper as the nonlinear peaking phenomenon.The reason for the appearance of the nonlinear peakingphenomenon in Figure 6 after increasing the force setpoint becomes clear upon inspecting how the change inthe force set point affects the states of the system: inFigure 5, the velocity disturbance and force set pointfor the current system configuration results in the servovalve operating only about one side of valve null, in thiscase u < 0. This is determined by a number of factorssuch as the magnitude of the velocity disturbance, therequired pressure differential resulting from the forceset point, the type of actuator (symmetrical or asym-metrical), leakages, the size of the servo valve and thesupply pressure. In Figure 6 on the other hand, theoperating conditions force the servo valve to operateabout valve null when the load cylinders are chang-ing directions. As is clear from Eqs. (13)-(20), thisresults in abrupt changes in the parameters of the sys-tem as the valve goes from positive to negative spoolstroke and vice versa. When the controlled output ispressure, the effect of any abrupt changes in the flowgain of the system is intuitively expected to be muchmore severe than in position or velocity servos due tothe continuity equation and the near incompressibilityof hydraulic oil, in other words only a minute increasein the flow through the valve may lead to a drasticchange in the pressure differential across the actuatorin a small amount of time. The effect of abrupt changesin the parameters of a system under linear feedback isequivalent to that of an additional external disturbancethat must be attenuated by the feedback (Horowitz,1963), which makes it clear why a feed forward fromthe velocity disturbance is not capable of reducing themagnitude of this peaking of the control error.

    Using the numerical simulation model it is foundthat the magnitude of the peaking may be amplified orreduced by adjusting the parameters of Eqs. (13)-(20),as well as the parameters of Eqs. (7)-(8). Further-more, the magnitude and shape of the peaking may beaffected in varying degrees by nonlinearities commonlyfound in hydraulic servo systems such as hysteresis,backlash, variable gain in the flow characteristics ofthe servo valve and valve deadband. Figure 7 showsthe control error under conditions identical to that ofFigure 6 with the exception of the introduction of aminute valve deadband of 0.5%, which may occur inpractice even for critically lapped servo valves (Mer-ritt, 1967). In Figure 7 the control error has increasedto 2816 N and 2185 N without and with the use offeed forward, respectively. Again, as in Figure 6, the

    6

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    use of feed forward from the velocity disturbance ap-pears effective in reducing the control error, except atthe instances where the actuator is forced to changedirections. Although valve deadband may be easily becompensated for, compensation of the other aforemen-tioned nonlinearities is more difficult to achieve andtypically constitute uncertain parameters that may beproblematic to identify or measure in practice.

    0 5 10 15 20 25 30 35 40

    Time [s]

    -5000

    0

    5000

    Con

    trol

    Err

    or [N

    ]

    ExperimentSimulation

    Figure 8: Initial comparison.

    0 5 10 15 20 25 30 35 40

    Time [s]

    -6000

    0

    5000

    Con

    trol

    Err

    or [N

    ]

    ExperimentSimulation

    Figure 9: Variable gain characteristics introduced.

    0 5 10 15 20 25 30 35 40

    Time [s]

    -5000

    0

    5000

    Con

    trol

    Err

    or [N

    ]

    ExperimentSimulation

    Figure 10: Variable gain characteristics adjustedfurther.

    At this point it is clear that depending upon a num-ber of parameters which may be difficult to predictbeforehand, the nonlinear peaking phenomenon may

    or may not occur in a hydraulic pressure servo sub-jected to external velocity disturbances and could po-tentially be amplified by the presence of nonlinearitiescommonly found in hydraulic systems. Care shouldbe taken to investigate the effects of such nonlineari-ties when attempting to predict the performance of asystem to be designed based on numerical simulations.Figure 8 shows a comparison between the initially ex-pected performance of the experimental system basedon numerical simulations and the actual performancein the presence of a sinusoidal velocity disturbance withan amplitude of 40 mm and a frequency of 0.1 Hz with-out the use of velocity feed forward for a force set pointof 10 kN using the PI controller of Section 5. Based onthe simulations in Figure 8, a minor peak is expected asthe load cylinders are forced to change directions. Onthe experimental system on the other hand, a muchlarger peak along with two minor peaks are observedthat significantly degrade the performance of the sys-tem.

    -50 -40 -30 -20 -10 0 10 20 30 40 50

    Spool Stroke [%]

    -20

    -15

    -10

    -5

    0

    5

    10

    15

    20

    Val

    ve F

    low

    [l/m

    in]

    Ideal CurveModified Curve

    Figure 11: Variable gain characteristics utilized in Fig-ure 9.

    Numerical investigations indicated that out of theaforementioned factors and nonlinearities, the in-creased magnitude of the peak observed experimentallycombined with the overall shape of the control errorcould only be explained for the system under consid-eration by imperfections in the curve describing therelation between the spool stroke of the valve and itsoutput flow for a fixed pressure differential across thevalve. Modifying this curve in the numerical simulationmodel as shown in Figure 11, where minor fluctuationsin the linearity of the curve have been introduced, acloser match between numerical and experimental datais achieved as seen in Figure 9, where the qualitativeshape is matched quite well with a slight mismatch inthe amplitudes. Adjusting the curve slightly and mod-ifying each metering edge of the valve individually, theresults shown in Figure 10 were obtained, where the

    7

  • Modeling, Identification and Control

    amplitudes are now closer matched, however with thelargest peaks slightly out of phase. Possibly a bet-ter match could be achieved by further adjustments,however even minor modifications of the curve in Fig-ure 11 can result in large changes in the control error,which makes obtaining a perfect match rather difficult.The results of Figures 9 and 10 are however sufficientto demonstrate the likelihood that the amplification ofthe control error on the experimental system is largelydue to imperfections in the flow-spool stroke curve ofthe valve, particularly near valve null. Although thereis a slight null shift present in the curve of Figure 11,this is not the cause of the amplification of the con-trol error and similar results may be obtained with acurve that does not have such a null shift. Attempt-ing to compensate for the null shift does not affect themagnitude nor the shape of the control error.

    In addition to ensuring valve operation only on oneside about valve null, i.e. u > 0 or u < 0, whichdepends upon a number of factors that typically donot constitute a part of the design freedom, numericalinvestigations based on the model used here revealedthat for certain system configurations a sinusoidal con-trol error with the absence of the nonlinear peakingphenomenon may be achieved by sufficient crossportleakage, i.e. from p1 to p2, even for operating condi-tions where the valve is operating about null. Froma design perspective, including an adjustable crossportorifice between the lines of the actuator could thereforebe advantageous in the design of pressure controlledhydraulic actuators subject to external velocity distur-bances. Such an orifice could possibly also be used forsome system configurations to shift the operating pointof the valve away from null.

    In the following sections, the capability of ILC toreduce or eliminate the peaking of the control error thatoccurs as a result of the nonlinear peaking phenomenonis evaluated.

    5 Control Design I: InitialEvaluation

    As mentioned previously, the control specifications arefor the load cylinders to apply a constant force in thepresence of velocity disturbances caused by the maincylinder. In particular, the load cylinders should becapable of maintaining a force of 10 kN when the maincylinder is actuated with a sinusoidal velocity profileof frequency of 0.1 Hz with an amplitude of 40 mm.The minimum acceptable accuracy for this applicationis an accuracy of 500 N, where greater accuracies arepreferred. Initially a PI controller optimized for dis-turbance rejection using MATLABs pidtune was eval-

    uated on the experimental system. Due to the resonantmode resulting from the presence of a flexible environ-ment it was found favourable to include a first orderfilter in the design, which increased the achievable gainand bandwidth of the controller.

    Time [s]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    Con

    trol

    Err

    or [N

    ]

    -300

    -200

    -100

    0

    100

    200

    300

    Figure 12: Limit cycle oscillations in the absence of ex-ternal velocity disturbances.

    0 10 20 30 40 50

    Time [s]

    -4500

    0

    4500

    Con

    trol

    Err

    or [N

    ]

    Figure 13: Performance in the presence of velocity dis-turbance using feedback.

    0 10 20 30 40 50

    Time [s]

    -3000

    0

    3000

    Con

    trol

    Err

    or [N

    ]

    Figure 14: Performance in the presence of velocity dis-turbance using feedback and feed forward.

    Figure 13 shows the performance of the PI controllerin the presence of the velocity disturbances generatedby the sinusoidal motion of the main cylinder. Aspointed out previously, the effects of even minor ve-locity disturbances can be quite severe due to the nearincompressibility of hydraulic oil and the natural ve-locity feedback as is observed in Figure 13. The pres-ence of the nonlinear peaking phenomenon discussed

    8

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    in Section 4 is also evident, where a severe peaking ofthe control error is observed as the load cylinders areforced to change directions, resulting in a peak-to-peakamplitude of the control error of 7238 N.

    The velocity disturbance caused by the main cylinderis measurable, for this reason a fixed gain feed forwardfrom the velocity disturbance was evaluated in com-bination with the PI controller. Figure 14 shows theresulting control error where it is seen that althoughthe feed forward significantly improves the overall ac-curacy, the peaking that occurs as a result of the non-linear peaking phenomenon appears unaffected, result-ing in a peak-to-peak amplitude of the error of 4501N.An explanation for the large magnitude of this peakingwas presented in Section 4.

    Additionally it was found that the load cylinder cir-cuit inherently suffers from limit cycle oscillations un-der feedback control, as seen in Figure 12 where a limitcycle with a peak-to-peak amplitude of approximately400 N is observed for a constant set point in the ab-sence of external velocity disturbances. The systemwas also found to limit cycle with a similar amplitudeusing PID controllers, lag compensators, lead compen-sators and lag-lead networks. A simple P controller didnot result in limit cycle oscillations, however failed toprovide any control over the system with the controlledoutput simply drifting to an arbitrary value in the ab-sence of velocity disturbances. In the presence of veloc-ity disturbances, the error under P control reaches itsmaximum attainable value dictated by pressure reliefvalves placed in the load cylinder circuit that for safetyreasons limit the maximum line pressures. These valvesare to be closed under normal operating conditions andwere therefore not included in the mathematical modelof the system.

    6 Control Design II: IterativeLearning Control

    In general, ILC may be applied alone in an open-loopfashion, or in a closed-loop fashion where the ILC al-gorithm adjusts the set point or reference trajectory ofa feedback controller (Longman, 2000). For the sys-tem under consideration, in the absence of a feedbackcontroller, the velocity disturbance generated by themain cylinder is sufficient for the control error to growuntil the pressures of the load cylinder circuit reachlevels that cause the safety relief valves to activate.The control error then reaches values above 50 kN andthe relief valves of the load cylinder circuit are contin-uously activated. Based on this it was concluded thatthe application of ILC to the system in an open-loopfashion is not feasible and that ILC must be applied

    in combination with a feedback controller capable ofmaintaining a reasonable initial accuracy and prevent-ing the activation of the safety relief valves of the loadcylinder circuit. As demonstrated in Section 5 however,the experimental system inherently suffers from limitcycle oscillations under feedback control. This couldpotentially lead to instability issues due to interactionsbetween the ILC learning algorithm and the limit cy-cles. Furthermore, even if a stable solution could beachieved in the presence of limit cycles, with the pres-ence of limit cycle oscillations having a peak-to-peakamplitude of 400 N, achievement of a satisfactory ac-curacy is not a reasonable expectation. For these rea-sons, a feedback controller is first designed specificallyto eliminate limit cycle oscillations, and then appliedin combination with ILC.

    6.1 Describing Function Based FeedbackController

    Numerical investigations traced the cause of the limitcycle oscillations back to the electromagnetic hysteresisof the servo valve. In this section a feedback controlleris designed to eliminate limit cycle oscillations due tovalve hysteresis based on the describing function of hys-teresis. In (Mougenet and Hayward, 1995), the authorsproposed based on the general shape of the describingfunction of hysteresis that its effect may be viewed asa pure phase delay at the crossover frequency and usedthis to design a feedback controller that successfullyeliminated limit cycle oscillations due to valve hystere-sis. A similar approach is taken here, where empha-sis is also placed on the magnitude increase resultingfrom the hysteresis nonlinearity. The design approachis therefore to maximize the distance to the criticalpoint for all frequencies near and after crossover in or-der to prevent intersections with the critical point dueto the hysteresis nonlinearity. For a review on limitcycle prediction using describing functions and the de-scribing function of hysteresis, see (Merritt, 1967) and(Franklin et al., 2015).

    The controller is designed using the Nichols plot,where the distance to the critical point at any givenfrequency is easily observed. Figure 15 shows the re-sulting feedback controller design, where a higher orderintegrating controller has been designed using a PI con-troller as a starting point. By cascading multiple leadterms a large radial distance from the critical point isachieved at all frequencies after crossover. The intro-duction of lead terms brings the resonant mode closerto the critical point, to counteract this effect a com-plex pole and a complex zero were also placed in thecontroller to keep the resonant mode at an appropriatedistance from the critical point. In this case, the design

    9

  • Modeling, Identification and Control

    process led to a ninth order controller. Possibly thecontroller order could be reduced using reduction algo-rithms, however this was not found to be necessary forimplementation on the CompactRIO. This controlleris referred to as the describing function based (DFB)feedback controller and its capability to eliminate limitcycle oscillations is evaluated experimentally in Section7.

    -360 -315 -270 -225 -180 -135 -90 -45-100

    -80

    -60

    -40

    -20

    0

    20

    40

    60

    Open-Loop Phase (deg)

    Ope

    n-Lo

    op G

    ain

    (dB

    )

    Figure 15: Nichols plot of the DFB feedback controller.

    6.2 Iterative Learning Feed Forward

    The ILC update law considered here is given by(Norrlof, 2000):

    uk+1(t) = uk(t) + L(q) ek(t) (33)

    where q is the discrete time-shift operator and L(q) isa learning filter. The learning filter used in this paperis given by (Wallen, 2011):

    L(q) = k q (34)

    where k and are tunable parameters. The parame-ter k is referred to as the learning gain and is typicallychosen to be less than unity (Longman, 2000). For = 0 the physical interpretation of equation (34) is asfollows: at any given discrete point in time, the ILCfeed forward is updated by a term that is proportionalto the control error observed at the corresponding timestep in the previous iteration. For > 0 the interpre-tation is similar, except that the ILC feed forward isupdated by a term proportional to the control errorat time steps ahead of the corresponding time stepin the previous iteration. In this manner, the controlalgorithm may be interpreted as a PD-controller withrespect to iteration.

    A sufficient condition for monotonic convergence ofthe ILC is given by (Wallen, 2011):

    |1G(ei )L(ei )| < 1 (35)

    for all [, ], where refers to the sample rateand G describes the dynamics of the plant. Equation(35) is referred to here as the stability criterion andmay be evaluated graphically on the Nyquist plot byimposing that the Nyquist contour of G(ei )L(ei )must be contained inside a unit circle centered at unityin the complex plane (Longman, 2000).

    In order to provide robustness towards unmodelledeffects such as sensor noise and model uncertainty inthe high frequency range, the control error is filtered bya second-order butterworth lowpass filter before sent tothe learning filter. This stops the learning process forfrequencies above the cutoff frequency of the filter andas a result the convergence criterion has to be fulfilledonly for frequencies up to the cutoff frequency (Long-man, 2000). For the system at hand it was found bytrial and error that a cutoff frequency of 10 Hz wassufficient to provide long term stability without reduc-ing the convergence rate of the ILC noticeably. Thelearning gain k was selected as k = 0.8, keeping inmind that k is typically selected as less than unity andthat larger values of k represent a more aggressive con-troller (Longman, 2000). The tunable parameter wasthen adjusted to fulfill the convergence criterion. It wasfound that = 5 is sufficient to satisfy the convergencecriterion up to and beyond the cutoff frequency andthus provide a stable ILC configuration. For detailson selecting the parameters of the ILC learning filtersee (Wallen, 2011) and (Norrlof, 2000). The feedbackcontroller and ILC update law are both implementedusing a sample rate of 10 ms.

    0 0.5 1 1.5 2

    -1

    -0.5

    0

    0.5

    1

    Real Axis

    Imag

    inar

    y A

    xis

    Figure 16: Graphical evaluation of the stability crite-rion for ILC applied to the DFB feedbackcontroller.

    Figure 16 shows the graphical evaluation of the con-vergence criterion using the Nyquist plot, where thegrey circle indicates a unit circle centered at unity. The

    10

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    Nyquist contour is contained within the unit circle forfrequencies up to about 14 Hz, meaning that the se-lected cutoff frequency of 10 Hz is sufficient to achievea stable ILC configuration. From equation (33) it isseen that the ILC requires storing of the arrays ek anduk in between sampling periods. For the selected sam-ple rate and the system specifications being periodicwith a period of 10 seconds this requires the storing oftwo arrays each containing 1000 elements in betweensampling, which is well within the memory capabilitiesof the CompactRIO. The ILC algorithm is then imple-mented on the CPU of the CompactRIO together withthe DFB feedback controller.

    7 Experimental Results

    Figure 17 shows the control error for a constant setpoint using the DFB feedback controller in the absenceof velocity disturbances, compare with Figure 12.

    It is seen that the DFB feedback controller effectivelyeliminates limit cycle oscillations in a steady-state situ-ation and the only fluctuations of the control error aredue to sensor noise. In Figure 18 the step response ofthe DFB feedback controller is shown where a referencestep change from 9 kN to 10 kN occurs at the 10 secondmark. With the exception of minor remnants that maybe observed in the transient response for the first twoseconds after the step change, limit cycle oscillationshave been eliminated in the transient response as well.These minor remnants indicate that the distance to thecritical point should not be decreased any further andthus prevents more aggressive tuning of the feedbackcontroller. In addition to the feedback controller, afixed-gain feed forward from the velocity disturbanceis also implemented. Figure 19 shows the control errorof this configuration in the presence of the velocity dis-turbance generated by the main cylinder without theapplication of ILC. The peak-to-peak amplitude andRMS of the control error are 9197 N and 1842 N, re-spectively. The PI controller discussed in Section 5achieved a peak-to-peak control error of 4501 N whencombined with fixed-gain feedforward, which indicatesthat the performance of the DFB feedback controller issomewhat conservative. This is however a natural re-sult of the design approach that was taken in order toeliminate limit cycle oscillations, as increasing the dis-tance to the critical point reduces the achievable per-formance as described by Bodes integral of feedback(Lurie and Enright, 2012).

    Next, the ILC algorithm is activated. The initialconvergence after activation of the ILC is shown inFigure 20 over two minutes, where the ILC is activatedshortly before the 20 second mark. In Figure 20 boththe RMS and the peak-to-peak amplitude of the control

    error are seen to decrease monotonically each iteration.After six minutes and forty seconds, the control errorappears to have converged to its final accuracy.

    Time [s]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    Con

    trol

    Err

    or [N

    ]

    -300

    -200

    -100

    0

    100

    200

    300

    Figure 17: Elimination of limit cycle oscillations,steady-state.

    0 5 10 15 20 25 30

    Time [s]

    8500

    9000

    9500

    10000

    10500F

    orce

    [N]

    Figure 18: Elimination of limit cycle oscillations, tran-sient response.

    0 10 20 30 40 50

    Time [s]

    -5500

    0

    5500

    Con

    trol

    Err

    or [N

    ]

    Figure 19: Performance in the presence of velocitydisturbance.

    Figure 21 shows the control error over eight itera-tions after convergence where k0 indicates the iterationwhere zero minutes have passed since convergence. Theaccuracy is maintained within a 250 N band for themajority of the time, with a reduction in the peak-to-peak amplitude of the control error by a factor of morethan 19 and a reduction in the RMS of the error by afactor of more than 30.

    The DFB-ILC combination achieves long term sta-bility and maintains the converged accuracy as time

    11

  • Modeling, Identification and Control

    grows large. Figure 22 shows the control error twohours after convergence of the ILC where the the con-verged accuracy is still maintained. This correspondsto the ILC having been active for more than 750 it-erations, demonstrating the long term stability of theDFB-ILC combination. Inspecting the control error atother points in time, errors similar to that of Figures21 and 22 are observed.

    Time [s]0 20 40 60 80 100 120 140

    Con

    trol

    Err

    or [N

    ]

    -5000

    0

    5000

    Figure 20: Initial convergence of the ILC.

    Iteration [-]

    k0 k0+1 k0+2 k0+3 k0+4 k0+5 k0+6 k0+7 k0+8

    Con

    trol

    Err

    or [N

    ]

    -400

    -300

    -200

    -100

    0

    100

    200

    300

    400

    Figure 21: Control error after convergence of the ILC.

    Iteration [-]

    k120 k120+1 k120+2 k120+3 k120+4 k120+5 k120+6 k120+7 k120+8

    Con

    trol

    Err

    or [N

    ]

    -400

    -300

    -200

    -100

    0

    100

    200

    300

    400

    Figure 22: Control error two hours after convergence ofthe ILC.

    It should be noted that the peaking of the control er-ror resulting from the nonlinear peaking phenomenonthat is observed in Figure 20 appears to completely dis-appear as the ILC converges. The maximum amplitudeof the converged control error seen in Figures 21 occurswhen the cylinders are moving with their maximum ve-locity, whereas the nonlinear peaking phenomenon oc-curs when the cylinders are close to zero velocity. In

    other words, the nonlinear peaking phenomenon is nolonger the limiting factor in terms of performance, andit may be concluded that its effects have been elimi-nated by the application of ILC.

    8 Discussion and Conclusions

    In this paper a performance limiting phenomenon thatmay occur in the pressure control of hydraulic actu-ators was presented and investigated. It was shownthat depending upon the system configuration and op-erating conditions, a severe peaking of the control er-ror may be observed in the presence of external veloc-ity disturbances. Numerical investigations showed thatthe occurrence of such a phenomenon depends upon anumber of factors that may be difficult to predict for agiven system without access to experimental data. Thesignificance of the phenomenon from a design perspec-tive and possible ways of avoiding it were discussed.For situations where the peaking of the control errorcannot be avoided, ILC was proposed and evaluated asa solution for improving the systems performance. Forthe system under consideration, application of ILC inan open-loop fashion was not feasible due to the pres-ence of external velocity disturbances, necessitating theapplication of ILC in combination with a feedback con-troller. The experimental system was found to inher-ently suffer from limit cycle oscillations under feedbackcontrol due to the presence of valve hysteresis. For thisreason a feedback controller was designed for the elimi-nation of limit cycle oscillations based on the describingfunction of hysteresis and applied in combination withthe ILC.

    Experimental results demonstrate the elimination oflimit cycle oscillations and long term stability of theproposed solution, evaluated here for more than 750 it-erations of continuous operation throughout which theILC learning algorithm remained active. Upon con-vergence of the ILC the maximum amplitude of thecontrol error remains less than 0.3 kN, with the RMSand peak-to-peak of the error having been reduced byfactors of 30 and 19, respectively. Complete conver-gence was achieved in 40 iterations, however an accu-racy greater than 0.5 kN was achieved already after 23iterations.

    Several accounts of ILC applied to hydraulic motioncontrol problems have appeared in the literature, seefor example (Lingjun et al., 2014), (Chen and Zeng,2003), (Daley et al., 2004) and (Zhao et al., 2005),where the achieved reduction in the control error varysignificantly from one application to another, even forsimilar or identical ILC algorithms. The reported con-vergence rates in these applications range from any-where between 15 and 100 iterations, also for similar

    12

  • Gytil et.al., Iterative Learning Applied to Hydraulic Pressure Control

    algorithms. Presumably the achievable improvementusing a given ILC algorithm depends upon a numberof factors such as the system configuration, tuning ofthe ILC algorithm, whether applied in an open-loopfashion or a closed-loop fashion as well as the feedbackcontroller when applied in a closed-loop fashion.

    Previous investigations of ILC applied to hydraulicpressure control problems on the other hand appear tobe limited, where a previous application suitable forcomparison has been located in (Wang et al., 2015),where ILC was applied to improve the tracking accu-racy of a pressure controlled hydraulic actuator in theabsence of external velocity disturbances. Limit cycleoscillations were not present and the absence of veloc-ity disturbances allowed the application of ILC bothin an open-loop and closed-loop fashion, the latter re-sulting in the better performance. Improved trackingaccuracy was achieved for both low and high frequencytracking, where the high frequency tracking achievedan accuracy of 0.5 kN, improving the accuracy by afactor of four over a conventional PID controller after25 iterations. This is comparable to the results pre-sented in this paper, with the exception of the factorby which ILC improved the accuracy. This differenceis likely due to a larger initial error resulting from thepresence of the nonlinear peaking phenomenon as wellas limit cycle oscillations which necessitated the use ofa rather conservative feedback controller.

    In summary the results presented here compare wellto that of previous applications both in terms of accu-racy and convergence rate, the novelty of the resultspresented here being the application of ILC to elimi-nate the effects of the nonlinear peaking phenomenonand the application of ILC in a closed-loop fashion to asystem that inherently suffers from limit cycle oscilla-tions under feedback. By applying ILC in combinationwith the DFB feedback controller both the effects ofthe nonlinear peaking phenomenon and limit cycle os-cillations were eliminated and stability was achievedwith a satisfactory accuracy for long periods of con-tinuous operation. For the system under considerationthe convergence rate is also satisfactory. Faster con-vergence may possibly be achieved using a strongerfeedback controller, however preliminary attempts toaccomplish this were not successful due to the appear-ance of limit cycles when increasing the gain of the con-troller or decreasing the distance to the critical point.Further investigations into feedback controller designfor the elimination of limit cycle oscillations in valvecontrolled hydraulic actuators will be the topic of fu-ture research.

    Acknowledgments

    The research presented in this paper has received fund-ing from the Norwegian Research Council, SFI OffshoreMechatronics, project number 237896.

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    IntroductionSystem DescriptionModellingNonlinear Peaking PhenomenonControl Design I: Initial EvaluationControl Design II: Iterative Learning ControlDescribing Function Based Feedback ControllerIterative Learning Feed Forward

    Experimental ResultsDiscussion and Conclusions

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