Aalborg Universitet

Challenges with Respect to Control of Digital Displacement Hydraulic Units

Pedersen, Niels Henrik; Johansen, Per; Andersen, Torben O.

Published in:Modeling, Identification and Control (Online)

DOI (link to publication from Publisher):10.4173/mic.2018.2.4

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Publication date:2018

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Citation for published version (APA):Pedersen, N. H., Johansen, P., & Andersen, T. O. (2018). Challenges with Respect to Control of DigitalDisplacement Hydraulic Units. Modeling, Identification and Control (Online), 39(2), 91-105.https://doi.org/10.4173/mic.2018.2.4

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Modeling, Identification and Control, Vol. 39, No. 2, 2018, pp. 91–105, ISSN 1890–1328

Challenges with Respect to Control of DigitalDisplacement Hydraulic Units

N. H. Pedersen, P. Johansen, T. O. Andersen

Fluid Power and Mechatronic Systems, Department of Energy Technology, Aalborg University, Pontoppidanstraede111, 9220 Aalborg, Denmark. E-mail: nhp@aau.dk, pjo@aau.dk, toa@aau.dk

Abstract

This paper investigates the many complications arising when controlling a digital displacement hydraulicmachine with non-smooth dynamical behavior. The digital hydraulic machine has a modular constructionwith numerous independently controlled pressure chambers. For proper control of dynamical systems, amodel representation of the systems fundamental dynamics is required for transient analysis and controllerdesign. Since the input is binary (active or inactive) and it may only be updated discretely, the machinecomprises both continuous and discrete dynamics and therefore belongs to the class of hybrid dynamicalsystems. The study shows that the dynamical system behavior and control complexity are greatly de-pendent on the configuration of the machine, the operation strategy, and in which application it is used.Although the system has non-smooth dynamics, the findings show that simple continuous and discreteapproximations may be applicable for control development in certain situations, whereas more advancedhybrid control theory is necessary to cover a broader range of situations.

Keywords: Digital Displacement Units, Fluid Power, Control, Non-smooth System, Hybrid Systems

1 Introduction

Throughout the past decade, there has been an increas-ing interest in research and development of energy effi-cient fluid power systems for both conventional cylin-der drives Schmidt et al. (2015); Jarf et al. (2016) andpower take-off systems Hansen and Pedersen (2016);Hansen et al. (2014); Payne et al. (2007, 2005); Peder-sen et al. (2016a); Rampen (2006); Salter et al. (2002).A promising technology is the so called digital displace-ment machine (DDM), which besides providing greatenergy efficiency also yields superb scalability. For suc-cessful deployment of the technology, proper control ofthe machine is considered a key challenge to be solved.However, this is a complicated task due to the non-smooth dynamical behavior.

For most computer controlled systems, either contin-uous or discrete approximation models are fairly accu-rate in describing the system macro dynamics. Either

linear or non-linear control theory is then applied onthese models to ensure stability and obtain the desiredperformance. However, control of DDM’s is differentin various ways, since the triggering of a control up-date is angle dependent and not time dependent. Ad-ditionally, the dynamics are highly influenced by theconfiguration of the machine and how it is operated,as well as in which application it is used. Since controlof the digital displacement technology is a relativelyunexplored field, an important first step is to iden-tify the control challenges related to the non-smoothmachine behavior. This paper provides an overviewwith respect to development of model based dynami-cal analysis and feedback control design methods forthese machines. The investigation is based on a sin-gle module of a DDM with 5 cylinders as illustrated inFig. 1. It is seen that the machine comprises numerouscylinder pressure chambers being radially distributedaround an eccentric shaft. The flow to and from the

doi:10.4173/mic.2018.2.4 c© 2018 Norwegian Society of Automatic Control

Modeling, Identification and Control

3

Low pressure mainfold1

2 High pressure manifold

3 Low pressure valve

4 High pressure valve

5 Cylinder

6 Piston

7 Cylinder chamber

8 Eccentric

(Pressure chamber)

3

4

2

1

56

7

8

Figure 1: Illustration of the radial piston type digitaldisplacement machine with 5 cylinders.

pressure chambers is controlled by electro-magneticallyactuated on/off valves connecting each pressure cham-ber to the high and low pressure manifold. A moredetailed description of the digital displacement technol-ogy may be found in several publications Ehsan et al.(1997); Payne et al. (2005); Rampen (2010); Roemer(2014); Noergaard (2017); Johansen (2014); Johansenet al. (2015b).

Published state of the art control strategies for dig-ital displacement units is mostly limited to neglect-ing the transient behavior of the machine and utiliz-ing pulse density modulation or pulse width modula-tion techniques to determine the actuation sequencefor the pressure chambers Johansen et al. (2015a);Sniegucki et al. (2013). Another method where thedynamics has been neglected is through determinationof the actuation sequence of the individual chambersas function of the displacement reference and shaft an-gle through offline optimization Armstrong and Yuan(2006); Sniegucki et al. (2013); Song (2008); Heikkilaand Linjama (2013). However, none of these meth-ods includes the machine dynamics which may intro-duce transient problems with respect to stability andperformance. Dynamical models applicable for systemanalysis and controller design has been developed andgoverns both continuous, discrete and hybrid represen-tations Johansen et al. (2017); Pedersen et al. (2016b,2017a,b,c, 2018c); Sniegucki et al. (2013). However, thediscrete and continuous models are often only valid inlimited situations, while the hybrid models are oftenquite complex with respect to development of stabiliz-ing feedback control laws. This paper provides a clar-ification of the related control challenges in a broadrange of machine operations, machine designs and pos-sible target applications. A discussion of the applicable

control models and methods, as well as their limitationsis made. The findings of the paper should hence serveas a tool for determining suitable control strategies anddynamical control oriented models when utilizing dig-ital displacement machines.

2 Simplified model analysis

Instead of deriving a complex model for investigationof the control challenges, a relatively simple model withidealized behavior is used. The model maintains all thesystem wide control challenges and includes the funda-mental dynamics of the machine. Although only sys-tem wide control is considered in this paper, valve con-trol related problems are important for proper machineoperation. The non-linear features during the switch-ing event has been studied in Bender et al. (2018b);Roemer et al. (2015a, 2014, 2013, 2015b) and the repet-itive nature of the machine has lead to several valvetiming control strategies Bender et al. (2018a); Brendiet al. (2017); Breidi (2016); Merrill (2012). The deriva-tion of the mathematical model of the macro dynamicsis based on the illustration of a single pressure chambershown in Fig. 2.

HPV

LPV

Pressure

Chamber

x

pH

p

pLPiston

Figure 2: Illustration of the individual pressure cham-ber and the configuration of the correspond-ing valves.

It is seen that a normally closed high pressure valve(HPV) and a normally open low pressure valve (LVP) isused, such that each pressure chamber is normally in anidling mode at low pressure. The piston displacement,x, is kinematically described as function of the shaftangle given by

xi (θi) = re (1− cos(θi)) (1)

where re is the eccentric radius, such that the pistonstroke length is x = 2 re. The local cylinder angle is

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Pedersen et.al., “Challenges with respect to Control of Digital Displacement Hydraulic Units”

given by

θi = θ +2 π

Nci i ∈ {0, . . . , Nc − 1} (2)

where Nc is the number of cylinders. The cylinderchamber volume, V , is then given as

Vi (θi) =Vd

2(1− cos(θi)) + V0

Vi (θ) =Vd

2sin(θi) θ

(3)

where V0 is the minimum chamber volume and Vd =2 re Ap is the displacement volume, with Ap being thepiston area. The pressure dynamics in one displace-ment chamber is governed by the continuity equationand is given to be

pi =βe

Vi

(QH,i −QL,i − Vi

)(4)

Where QH and QL are the flows through the high andlow pressure valve respectively and βe is the effectivebulk modulus. The oil-flows through the valves maybe described by the orifice equation given by

QL,i =xL

kf

√|pi − pL| sign (pi − pL)

QH,i =xH

kf

√|pH − pi| sign (pH − pi)

(5)

where kf is the valve flow coefficient and xL and xH arenormalized valve plunger positions. The torque fromeach pressure chamber is described by

τi =d Vi (θi)

dθpi =

Vd

2sin(θi) pi (6)

The dynamical behavior of a single pressure cham-ber is investigated by considering the pressure trajec-tories as function of the shaft angle in a polar phaseplot shown in Fig. 3. In the simulation, the pres-sure is initially equal to the high pressure, p = pH,during the down-stroke, θ = [0;π] and the initiallyopen HPV is closed xH : 1 → 0. This results in a de-pressurization and when p < pL the LPV is passivelyopened xL : 0 → 1. During the upstroke, θ = [π; 2π],the pressure is initially equal to the low pressure p = pL

and the initially open LPV is closed xL : 1 → 0. Thisresults in a pressurization and when p > pH the HPVis passively opened xH : 0 → 1. φmL is the latest an-gle that the LPV should be closed in order to obtainpassive opening of the HPV at top dead center (TDC).φmH is the latest angle that the HPV should be closedduring motoring for the LPV to be passively opened atbottom dead center (BDC). Similarly, φpL is the ear-liest angle that the pumping stroke may be initiated

60°120°

180°

240° 300°

0°

p

pL

pH

BDC TDC

mH

pL pH

mL

Figure 3: Phase portrait of the pressure dynamics asfunction of the angle. Red trajectory illus-trates a full motoring stroke and blue trajec-tory illustrates a full pumping stroke.

and φpH is the angle where the HPV is closed to endthe pumping stroke.

It is seen that the pressurization and de-pressurization are very fast as function of the angleexcept at BDC and TDC. Since the pressure is ap-proximately constant when the HPV is open, the flowthroughput may be simplified to be QH ≈ Vc, seenfrom (4), when considering the macro dynamics of thesystem. From the above analysis, the flow and torqueas functions of the displacement volume are given by

D(θ) =Vd

2sin (θ) xH QH = D θ τ = D p (7)

where xH = {0, 1} depending on whether the chamberis active or inactive. The displacement throughput isused in the following analysis to examine the operationprinciple and control challenges related to the digitaldisplacement machine. The control challenges for full-and partial-stroke operation are vastly different, sincethe system behavior is changed based on the chosenoperation strategy. Therefore, the control challengesfor the two operation strategies are investigated sepa-rately.

3 Control Challenges - Full stroke

Considering a full stroke operated DDM, the inputsmay be considered to be u = {1, 0,−1} correspondingto motoring, idling and pumping respectively. For ev-ery chamber the decision to either motor or idle is done

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Modeling, Identification and Control

once every revolution at φmL, while the decision to ei-ther pump or idle is done once every revolution at φpL.This angle dependent sampling yields an asynchronoussampling scheme for varying shaft speeds due to therelation

dθ = ω(t)dt↔ dt =1

ω(θ)dθ (8)

An illustration of the sampling scheme in the posi-tion domain and time domain for an arbitrary variablespeed DDM is illustrated in Fig. 4.

02 π5

4 π5

6 π5

8 π5

2 π

C1 C2 C3 C4 C5 C1D C5

θ

θ

θk

θk+1

θk+2

θk+3

θk+4

θk+5

θk

θk+1

θk+2

θk+3

θk+4

θk+5

ttk

tk+1

tk+2

tk+3

tk+4

tk+5

tk

tk+1

tk+2

tk+3

tk+4

tk+5

Figure 4: Displacement output response and controldecision scheme of a full stroke operatedDDM using both pumping and motoringstrokes. Black samples are for motoring andblue samples are for pumping.

It is seen how a varying speed machine results inan asynchronous sampling scheme in the time domain,while the sampling scheme is synchronous in the angledomain if the machine is to only operate as a pumpor motor. However, if the machine is to both motorand pump the sampling scheme is only periodic syn-chronous in the angle domain. Additionally, the inputscan only be chosen from um = {1, 0} for motoring sam-ples and up = {0,−1} for pumping samples, similar toa pulse density modulation technique used in electricalcircuit control. Furthermore, the impulse response isseen to be different, since the last part of the motoringstroke is not used, while the first part of a pumpingstroke is not used. Since a full stroke lasts for approxi-mately half of a revolution, it takes half of a revolutionto activate all the pressure chambers. Hence, the re-sponse angle from zero to maximum displacement isalso half of a revolution. As a result, both the controlinput frequency and response time is proportional tothe speed of the machine, while the displacement dis-cretization resolution is proportional to the number ofcylinders.

3.1 Four quadrant operation

An additional complication is introduced if the ma-chine is to operate in both directions of motion, which

may be illustrated by a steep change in the rotationspeed between positive and negative rotation as shownin Fig. 5.

ω

0θ

QH

0θ2 π 5π

22 π 3π

2π π

2π 3π

22π

θφ+mL φ−

mL φ−pL φ+

pL φ+mL

Figure 5: Illustration of response and sampling schemewith motion in both directions.

It is seen that a motoring decision is taken at φ+mL

ahead of TDC, but as the direction of motion ischanged, the stroke becomes a pumping stroke. As aresult, the subsequent decision is also for motoring, butsince the direction of motion is reverse, the samplingangle ahead of TDC is now at φ−mL = 2 π − φ+

mL. Thesame is illustrated for a change of direction during apumping stroke, which similarly results in two pump-ing samples subsequently. It is seen that the samplingscheme for a DDM operating in both directions isasynchronous in both the time and position domainand the position domain is further not monotonicallyincreasing.

4 Control Challenges - Partialstroke

The challenges with respect to control of a partial-stroke operated machine is very different than thoseidentified for full-stroke operation. A partial motoringstroke is simply made by closing the HPV earlier anda partial pumping stroke by closing the LPV later asillustrated in Fig. 6.

It is seen that the impulse response of a partial strokeoperated DD unit behaves different than that of a con-ventional dynamical system, since the input scales thewidth of the response rather than the amplitude. Theinput is the angle φmH for motoring and φpL for pump-ing, which controls the ratio where the given pressurechamber is active. The state changing input angle maybe linearly normalized by an input variable change,such that the input is a displacement fraction η = [0; 1].The input transformation is done by considering thecommitted displacement fraction for a motoring stroke

94

Pedersen et.al., “Challenges with respect to Control of Digital Displacement Hydraulic Units”

60°120°

180°

240° 300°

0°

p

pL

pH

BDC TDC

mH

pL

pH

mL

Figure 6: Phase portrait of the pressure dynamics asfunction of the angle. Red trajectory illus-trates a 70% partial motoring stroke and bluetrajectory illustrates a 70 % partial pumpingstroke.

by use of (3) and yields

ηm =V (φmH)

V (π)=

Vd

2 (1− cos(φmH))Vd

2 (1− cos(π))=

(1− cos(φmH))

2

(9)

The resulting input transformation for motoring andpumping is then given by

φmH(ηm) = acos(1− 2 ηm) η ∈ [0, ηm] (10)

φpL(ηp) = 2 π − acos(1− 2 ηp) η ∈ [0, ηp] (11)

where ηm and ηp are the maximum possible displace-ment fractions (corresponding to a full stroke) for mo-toring and pumping respectively. How the impulse re-sponse of the DD unit depends on the input is illus-trated in Fig. 7 for a motoring stroke, where ηm = 0.7corresponding to a displacement of 70 % of the maxi-mum displacement. It is seen that the state change ofa chamber from active to inactive may only be givenonce for each revolution (considering motoring modeonly) with conventional partial stroke. Therefore, thedecision frequency is still proportional to the speedof the machine, but the response may be smoothenedcompared to full stroke operation. The state chang-ing angle may however be updated continuously un-til the state change is carried out at the given angle.Considering, the input to be limited to only updateat φmL, when the stroke is initiated, the system be-comes a sample-and-hold system similar to that for

D 10 0.05 0.2 0.4 0.6 0.8 0.95ηm

φmH(ηm)

Vd

2

0 π θ

Figure 7: Impulse response as function of the input,ηm = 0.7 in conventional partial strokeoperation.

full-stroke. However, the impulse response is now time-varying, where the number of subsequent samples be-ing made during the length of the impulse responsevaries as function of the input. This is illustrated inFig. 8, where it is shown that 50% displacement duringone revolution may be obtained with vastly differentactivation patterns.

0 2 π5

4 π5

6 π5

8 π5

2 π

C1 C2 C3 C4 C5 C1

θ

D

θ

0 2 π5

4 π5

6 π5

6 π5

2 π

C1 C2C3 C4 C5 C1

θ

D

θ

Figure 8: Flow/torque output response and control de-cision scheme of a partial stroke operatedDDM, using only motoring strokes.

The top shown activation pattern corresponds to theoutput of a pulse width modulation strategy, while thebottom shown pattern is one arbitrary pattern of in-finitely many, yielding an average of 50% displacementduring a full revolution. Since the flow is highest atthe mid-point between TDC and BDC, the most se-vere flow spikes are also expected to occur if the valvesare switched here. The flow level also has an impacton the energy cost when switching, which however isminor since the pressure across the valve is low whenswitching.

95

Modeling, Identification and Control

4.1 Sequential partial stroke

If the valves are designed to be able to open againsthigh pressure difference, the sequential partial strokemethod may be used to allow for multiple state changesduring a revolution as illustrated in Fig. 9.

DVd

2

0 π θ

Figure 9: Impulse response as function of the input,ηm = 0.7, with sequential partial stroke.

It is seen that a 70 % stroke has been split into twosections, but in theory the displacement may be splitinto many more parts. With sequential partial strokeit is also possible to utilize the last part of the motor-ing stroke and first part of a pumping stroke, sincethe valves may be actively opened and closed inde-pendent of the angle. With this strategy, the statechanging input may be updated continuously and isonly constrained by the time it takes to open and closethe valves. Therefore, the response may become evenmore smooth with this strategy. However, opening theHPV against high pressure difference (against the so-lution trajectories shown in Fig. 6) requires a substan-tially amount of energy and larger valves with higheractuation power is necessary Noergaard (2017). Eventhough the input may be updated continuously it isstill binary, such that the output is in discrete levelsbased on the shaft angle as shown in Fig. 10. Only

0 2 π5

4 π5

6 π5

8 π5

2 π

C1 C2 C3 C4 C5 C1C5

θ

D

D

θ

Figure 10: Illustration of the displacement combina-tions as function of the shaft angle with 5cylinders.

motoring combinations are illustrated, but the num-ber of combinations may be increased by including thepossibility to pump. This does however mean that sev-eral cylinders will work against each other to lower thecombined throughput. Contrary to conventional dis-crete input systems, it is seen that the discrete levelsare changing as function of the angle.

4.2 Discussion of operation methods

The investigation clearly identifies that the dynamicalbehavior and control complications of the digital dis-placement machine is greatly influenced by the chosenoperation strategy. A summarization of the dynami-cal behavior and control challenges is provided in Tab.1. To give an overview of when to use which opera-tion strategy, based on the mentioned advantages anddisadvantages the following may be summarized:

• Full stroke: Is considered the favorable choicefor high-speed machine operation, where energyefficiency is important. Also it is considered themost simple strategy, since state changes are doneat fixed angles.

• Partial stroke: Is considered the favorable choicefor low-speed machine operation, where bothtracking performance and energy efficiency is im-portant.

• Sequential partial stroke: Is considered the fa-vorable choice for very low-speed machine opera-tion, where control tracking performance is impor-tant and energy efficiency is of less importance.

Even though the advantages and disadvantages of eachmethod are speed dependent, it may also be beneficialto use a full stroke strategy at low speed if trackingperformance is not very important, but simplicity andenergy efficiency is. Similar, it may be beneficial to usea partial stroke strategy at high speed, if tracking per-formance is very important, but simplicity and energyefficiency is not.

Based on the above analysis it may be favorable toutilize a combination of full-stroke and partial-strokeoperation if the machine is to operate in a wide rangeof speeds. However, a combination of full and partialstroke operation is simply a partial stroke operation,since a full stroke is also a possible choice when utiliz-ing a partial stroke strategy. Using the sequential par-tial stroke strategy, where several state changes may bemade during each stroke also covers the conventionalpartial stroke strategy, since a possible choice is to onlydo one state change or none at all (full stroke). Thisdoes however not mean that it is advantageous to usethe sequential partial stroke independent of the opera-tion speed and application, since the strategy requires

96

Pedersen et.al., “Challenges with respect to Control of Digital Displacement Hydraulic Units”

Machine featuresPressure andmechanicaldynamics

Non linear continuous differential equationsx = f(x

(

Control inputtype

Binary (active or inactive)

0

1

Operation strategy features

Full stroke Partial StrokeSequential Partial

stroke

Valve typeand

requirements

Only capable ofopening against lowpressure difference

Only capable ofopening against lowpressure difference

Capable of openingagainst high pressure

difference

u

Controlupdate rate

Proportional to theshaft speed and onceevery revolution at afixed angle for both

pumping andmotoring

Proportional to theshaft speed and onceevery revolution at a

variable angle forboth pumping and

motoring

Continuous, butconstrained by the

actuation time

θ

Pump Motor

t

Displacementresolution

Proportional to thenumber of cylindersover one revolutionand non-existentinstantaneously

Continuous over onerevolution andnon-existent

instantaneously

Continuous over onerevolution, but

proportional to thenumber of cylindersinstantaneously withdiscrete levels beingdependent on the

angle

2

D

N

Revolution

θ

θ1

Instantaneously

Chamberimpulse

response time

Proportional to theshaft speed (half of a

revolution)

Scaled by the input,up to half of a

revolution. Theimpulse response

time is hence scaledby the input and

shaft speed.

The time betweeneach actuation

decision

D

0π

θResponse angle

Machineresponse time

(0 to maxdisplacement)

Proportional to theshaft speed (half of a

revolution)

Proportional to theshaft speed (up to

half of a revolution)

Instantaneously(limited by theactuation time)

D

0 2

Pumping vsmotoring

Impulse response isdifferent (first part ofpumping is not used

and last part ofmotoring is not used)

Impulse response isdifferent (first part ofpumping is not used

and last part ofmotoring is not used)

Similarly 0

D

θMotor

Pump

π

Change ofrotationdirection

Pumping becomesmotoring and vice

versa. Angle ofcontrol decision

changes.

Pumping becomesmotoring and vice

versa. Angle ofcontrol decision

changes.

Pumping becomesmotoring and vice

versa

t

t

Dφ+mL

φ−mL

θ

Energy costand pressure

spikes

Very low due to nopressure difference

across the valve andno flow at switching

Dependent on theangle where the

chamber isdeactivated. Largestat mid-point between

BDC and TDC

Dependent on theangle where the

chamber isdeactivated. Largestat mid-point between

BDC and TDC.

θ0 π

D

CheapVery cheap

Very cheap

CheapMedium Medium

Expensive

Table 1: Summarization of the dynamical behavior and control challenges of digital displacement units.97

Modeling, Identification and Control

large active valves capable of opening against high pres-sure and the control complexity may be severely in-creased if the optimal control choice is e.g. only fullstrokes.

5 Control model development

For successful deployment of the digital displacementmachine in a wide range of application it is beneficialto be able to analyze and design feedback controllersindependently of operation. This is considered a com-plex task, especially since the dynamics is changingsignificantly depending on operation. Instead of devel-oping a very complex control oriented model capable ofhandling every situation, it is considered beneficial todevelop control oriented models for specific situationsindividually. In certain situations, simple dynamicalapproximation models may be sufficient for describingthe systems macro dynamics.

5.1 Continuous approximation

Even though the non-smooth dynamical system is dis-cretely controlled, a continuous approximation may beapplicable in certain situations and results in a signif-icantly simpler model for analysis and control designPedersen et al. (2018a). The continuous approximationis illustrated for a full stroke operated pulse-densitymodulated (PDM) machine and a partial stroke pulse-width modulated (PWM) machine, where the inputcorresponds to a duty cycle. The step response forvarious duty cycles (normalized displacement fractionreferences), D∗ = {0.1 0.2 0.4 0.6 0.8 1}, is shown inFig. 11. Both responses for a machine with 5 cylin-ders and one with 20 cylinders is shown to illustratethe applicability of the method. It is seen that for ahigh number of pressure chambers, the response resem-bles a continuous response with ripples in the positiondomain. However, for low displacement fractions, thecontinuous approximation is seen to be quite poor for afull stroke PDM strategy, while it remains reasonablyaccurate with a partial stroke PWM strategy. The re-sponse may be fitted to a continuous transfer functionby use of a spatial domain Laplace transformation de-fined as

F (s) = L{f(θ)} =

∞∫0

e−s θ f(θ)dθ (12)

Fitting the response dynamics to a second order systemresults in the transfer function given by

G(s) =DD∗ = K

ω2n

s2 + 2 ζ ωn s+ ω2n

e−s θd (13)

Figure 11: Step response for various displacementsusing delta-sigma modulation and pulse-density modulation.

where θd is the delay angle due to the decision beingmade ahead at φmL ahead of TDC, where the activemotoring stroke starts. Converting the transfer func-tion to the time domain by (8) yields a time-invariantsystem dependent on the machine speed. Therefore,if linear control is to be applied, a linearization pointfor the shaft speed must be chosen. In the time do-main the bandwidth of the system is proportional tothe shaft speed and to ensure stability in the case oflinear control, the speed should be greater than the lin-earization speed. For high speed operation the dynam-ics of the machine is possibly significantly faster thanthe dynamics of the high pressure line and mechanicalsystem it is connected to and may therefore possiblybe neglected during control design. This is coherentwith the non-smooth ripples in the response not beingseen in the mechanical response for high mass inertiasystems. It should however be clear that the method isvery inaccurate for a low number of cylinders and forfull stroke operation also at low displacements, wherethis method should not be applied. It is also impor-tant to consider that it is a discrete sampled system,so a continuous approximation is only valid up to acertain frequency where the sampling effect becomessignificant.

5.2 Discrete approximation

To capture the discrete control effect of a full strokeoperated digital displacement machine, a discrete ap-proximation of the machine dynamics may be appliedin several situations Johansen et al. (2017); Pedersen

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Pedersen et.al., “Challenges with respect to Control of Digital Displacement Hydraulic Units”

et al. (2017a,b, 2018d). The displacement throughputbetween samples is given by the change in chambervolume divided by the constant sampling angle givenas

∆D[k] =∆V [k]

θs=

(V [k + 1]− V [k]) N

2 π(14)

where θs is the sampling angle and N is the number ofcylinders. The discrete approximation of the displace-ment response is shown in Fig. 12 for a DDM with 15cylinders.

k k+1

k+2

k+3

k+4

k+5

k+6

k+7

k+8

k+9

D

Sample

d0

d1

d2d3

d4

d5

d6

d7

Chamber

responseDiscrete

approximation

Figure 12: Discretization of displacement response fora motoring stroke.

The discrete response is seen to be fairly accurate fora high number of cylinders. The initial delay is againdue to the decision being taking at φmL ahead of TDC.The discrete transfer function becomes that given by

G(z) =Du

=d7 z

7 + d6 z6 + · · ·+ d1 z + d0

z7(15)

where u = {0, 1} depending on whether an active orinactive stroke is taken. Utilizing a pulse density mod-ulator introduces a high non-linearity to the controlsystem due to the quantizer given by

u[k] =

{1 if D∗[k] ≥ 0.5

0 if D∗[k] < 0.5(16)

where D∗[k] is the displacement fraction reference.Since the flow and torque are functions of the speed andpressure respectively given by (7), the output trans-fer function is non-linear. Additionally, for a variablespeed machine the sampling scheme is asynchronous,why a transformation to the position domain by (17)is necessary to obtain a synchronous sampled system.

dx

dθ=

1

ω(θ)

dx

dt(17)

However, this transformation is seen to yield anothernon-linearity due to the speed being varying. Sincenon-linear discrete control is often quite complex, sim-pler linear approximations are often applied, whichhowever limit the range of applicability. By lineariz-ing around a steady-state speed, the operation speedshould be greater than the linearization speed to ensurestability. The common method of describing a quan-tizer linearly is to simply consider the quantizer as anadditive noise input, u[k] = D∗[k] + n[k]. This corre-sponds to a duty-cycle ratio approximation, where n[k]is the noise input with intensity I ∈ [0; 0.5]. By omit-ting the noise term, the resulting response is shown inFig. 13 for an input of D∗[k] = 1/3. The discretizationis not shown for simplicity of illustration.

k k+1

k+2

k+3

k+4

k+5

k+6

k+7

k+8

k+9

k+10

D

Sample

Non-linear

responseLinear

approximation

Figure 13: Discretization of displacement response.

It is seen that the linear approximation results inan angle average approximation, which is significantlymore smooth than the non-linear physical response.Since the linear discrete approximation is poor at lowdisplacement, it is not very accurate if the machineshould both pump and motor. Furthermore, the pump-ing and motoring impulse responses are different, andif the method should both allow for pumping and mo-toring, synchronous sampling is required, which in-troduces additional delay to the system. The linearmethod hence includes the sampling effect and to someextent the ripples in the response, but the methodrange of validity is limited to machines with a highnumber of cylinders and high displacements, similar tothe continuous approximation. Additionally, the direc-tion of motion must not change since a monotonicallyincrease in the sampling domain (angle or time) is re-quired when applying discrete control methods.

5.2.1 Partial stroke strategy

It has been identified that partial stroke and sequentialpartial stroke methods are great for low speed opera-tion, where tracking performance is quite poor for afull stroke strategy. However, for a partial stroke op-

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Modeling, Identification and Control

eration, the energy cost of switching and the result-ing pressure spikes are greatly influenced by the an-gle where the active stroke is stopped. With respectto control, the optimal solution may include trajec-tory tracking, energy efficiency, pressure spikes andfrequency content, which is hardly achieved with astatic control law. Therefore, model predictive con-trol (MPC) is considered to have a high potential forcontrol of a partial stroke operated DDM. MPC doeshowever require a discrete model representation of thesystem dynamics Sniegucki et al. (2013); Pedersen et al.(2018c). For partial stroke operation, only a part of thefull stroke is used, which introduces additional compli-cations seen from a partial stroke discrete approxima-tion shown in Fig. 14.

k k+1

k+2

k+3

k+4

k+5

k+6

k+7

k+8

k+9

D

Sample

Chamber

responseDiscrete

approx.Full-stroke

response

Full-stroke

approx.

Figure 14: Discretization of displacement response,where η = 0.5 corresponding to half of astroke. A full stroke η = 1, with u =0.5 yields the angle average committed dis-placement as if η = 0.5 and u = 1.

In this partial stroke example, the displacement isseen to be committed over less samples, such that thetransfer function order is reduced to

G(z) =Du

=d4 z

4 + d3 z3 + · · ·+ d1 z + d0

z4(18)

Partial stroke operation hence results in an inputdependent time-variant system, where the input tothe transfer function is unity. This model is onlyobtainable if the stroke ratio is determined at samplek, where the stroke is initiated. If the stroke ratio maybe updated continuously until the pressure chamberis deactivated, the impulse response and hence thediscrete model is unknown when the stroke is initiatedby closing the LPV at φmH. It is possible to applymodel predictive control to such system, but it doesnot fulfill the requirements of classical MPC, where theinput scales the magnitude of the response and not themodel it self. To enable classical linear MPC control,an angle average approximation may me made, since

a 50 % stroke, η = 0.5 with unity input, u = 1, has thesame integrated displacement as a full stroke, η = 1,with input u = 0.5. This angle average approximationis also shown in Fig. 14 and is seen to smoothenthe response and be decreasingly accurate for lowerdisplacement fractions. However, the angle averageapproximation is quite conservative with respect tostability, since the slowest response with the largestphase angle is a full stroke (highest order model),but the gain is seen to be lowered simultaneously. Itshould be evident, that for a fixed sampling angle, thetime between samples is inverse proportional to themachine speed and to solve the optimization problemin real time constrains the maximum speed.

Similarly to partial stroke operation, MPC has ahigh potential for sequential partial stroke operation,since energy efficiency and pressure ripples becomeincreasingly important with an increased number ofswitchings. MPC for a sequential partial stroke oper-ated DDM is very similar to conventional MPC controlof continuous dynamical systems, since the input maybe updated at every sample Pedersen et al. (2018e).However, the input is limited to be a discrete set ofdisplacements, which requires optimization algorithmscapable of handling such problem. The methods en-able both motoring and pumping strokes, since the op-timal combination of all combinations of active andinactive pressure chambers is considered. The methoddoes however not allow for operation in both direc-tions, since the sampling domain must increase mono-tonically.

5.3 Hybrid dynamical system

Since the non-smooth dynamics of the digital displace-ment machine is of hybrid nature, hybrid system the-ory is able to describe both the continuous and dis-crete dynamics of the system Pedersen et al. (2017c,2018b); Sniegucki et al. (2013). Neither continuousnor discrete approximations are able to describe themachine dynamics if the machine is to operate in bothdirections. Therefore, hybrid dynamical system theoryis necessary to capture the dynamics, especially sincestability is of high concern at low speeds where the con-trol update rate is low in full and partial stroke oper-ation. Furthermore, the approximation methods haveshown poor accuracy for a low number of cylinders,low displacements and low speeds, where hybrid the-ory might be necessary to ensure stability and obtainthe desired transient behavior. Several hybrid formu-lations exist, which are able to describe the dynamicalbehavior (e.g. hybrid automaton, impulsive system,switched system), where each has their own advantagesand disadvantages. Although, hybrid dynamical sys-

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tem models are very intuitive, hybrid control theory isquite complex even for simple systems. Since a discreteevent occurs every time a valve is opened or closed (orevery time the pressure chamber is activated or deacti-vated) for every cylinder, the hybrid dynamical modelof the digital displacement machine is not considereda simple system. Control theory for hybrid systemsis an undergoing research topic, but stabilizing feed-back control theory does exist, e.g. control Lyapunovfunction theory Teel et al. (2015); Goebel et al. (2009);Brogliato (2016).

6 Control challenges related toapplication and design

The control challenges when operating a system com-prising digital displacement machines is highly depen-dent on the application where it is used. Furthermore,due to the modular construction of the machine, thecontrol challenges are also greatly dependent on howthe machine is designed and how it is connected to theindividual target application.

6.1 Fluid Power Transmission

One application with very high potential for use ofDDM’s is a transmission system for converting lowspeed high torque power into high speed low torquepower or vice versa. An illustration of a digital fluidpower transmission is shown in Fig. 15. A ring camdesign is used to increase the number of strokes perrevolution of the slow rotating machine.

D1

D2

Figure 15: Illustration of a fluid power transmissionsystem comprising two digital displacementmachines.

Having multiple digital displacement machineswhich dynamics are directly influencing each other,the control challenges are further complicated. Thehigh speed machine is seen to have 5 cylinders andthe low speed machine to have 10 cylinders and 8cam ring lobes for a total of 80 strokes per revolution.This means that for two digital displacement machines,

where one is solely pumping and the other is solely mo-toring in a full stroke mode, the combined control inputscheme is only periodic synchronous in the position do-main even for fixed speed as illustrated in Fig. 16.

θmθk+4

θk+3

θk+2

θk+1

θk

θk+5

θpθk

θk+10

θk+20

θk+30

θk+40

θk+50

θk+60

θk+70

θk+80

Figure 16: Input decision scheme for a digital hydraulictransmission with a high and a low speeddigital displacement machine.

It is seen that if the number of strokes per revolu-tion of the two machines are vastly different, a largeamount of information is required to describe the com-bined sampling scheme in one revolution. Several drivetrains need to operate in four quadrants, where eachmachine can both pump and motor and rotate in bothdirections (vehicle transmission etc.). Based on theidentified control complications, it is clear that thecontrol challenges of such system become significantlymore complex and increasingly complicated if it is toutilize partial stroke operation also.

6.2 Direct cylinder drive

Another application where digital displacement ma-chines has a large potential is in direct cylinder drivesfor large inertia systems, since they may potentiallyincrease the energy efficiency significantly. A concep-tional illustration of a multiple direct hydraulic cylin-der drive is shown in Fig. 17.

3m 3m 3m

Digital Displacement Machine

M

Figure 17: Conceptional drawing of a multiple cylinderdigital displacement direct cylinder drive.

It is seen that one digital displacement machine isbeing powered by a motor and is used to control sev-eral hydraulic cylinders independently. In the illus-trated concept, each module of the DDM is connected

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Modeling, Identification and Control

to either the rod or piston side of a cylinder for sep-arate metering control. Development of a dynamicmodel allowing for analysis and control of such sys-tem is complicated, since the dynamics of the wholesystem is coupled through the mechanical shaft. Sincethe input for each main cylinder chamber may onlybe updated every fourth sample, the sampling schemeis again asynchronous. In theory, the connections be-tween the chambers of the digital hydraulic machineand the main cylinder chamber could be reconfiguredin many different combinations (e.g. connecting addi-tional chambers to a main cylinder with a high load),which would change the dynamical behavior.

6.3 Alternative machine design concepts

An alternative concept is to utilize multiple indepen-dent inlets/outlets for each pressure chamber, requiringadditional high pressure valves. The concept is sim-ilar to the digital hydraulic power management sys-tem (DHPMS) developed by Heikkila and Linjama(2013). This concept directly increases the control up-date rate for each input to the main cylinder cham-bers and thereby possibly improves the control per-formance. The multiple inlet/outlet digital hydrauliccylinder drive concept is illustrated in Fig. 18.

M3m

Figure 18: Conceptional drawing of a digital hy-draulic direct cylinder drive with multipleinlets/outlets.

Even though the control update rate is doubled,the effective update rate is not, since multiple highpressure valves should not be opened simultaneously.This would result in a short circuit where the pressureis not controllable due to the differential cylinder andwill thus lead to unwanted dynamical problems. Withthis concept the control input consists of multipledecisions, being pump to rod or piston or idle duringup-stroke and motor to either piston or rod side oridle during down-stroke. Taking multiple decision isconsidered to further increase the control complexity,where classical discrete and continuous methods arenot applicable.

With respect to control tracking performance, a highnumber of small cylinder chambers is beneficial, butit is much more costly than using a few larger cylin-ders. An alternative design that may be beneficial isto utilize multiple sized cylinders. Using such designwith a full stroke operation strategy, the displacementdiscretization resolution is significantly improved with-out a huge increase in production cost. An illustrationof a DDM with variable sized displacement chambersand its corresponding sampling scheme and impulse re-sponses for motoring strokes is shown in Fig. 19.

3 large

sized

chambers

5 medium

sized

chambers7 small

sized

chambers

C1

C1

C5

C1

C2 C3

C4

C5

C6C7

D

0 π2

π 3 π2

2 π

C1 C2 C3 C1

C5C4C3C2C1C5

C7C6C5C4C3C2C1C7

θ

θ0 π2

π 3 π2

2 π

Figure 19: Illustration of a DDM with variable sizeddisplacement chambers.

It is seen that the control update scheme is only pe-riodic synchronous in the angle domain, while the im-pulse response is time varying dependent on the sizeof the chamber. As a result, this alternative conceptintroduces further complications with respect to modelbased dynamical analysis and control. Combining thedifferent concepts: e.g. a multiple digital hydraulicdirect cylinder drive with multiple sized displacementchambers and multiple inlets/outlets makes the dy-namical control challenges tremendous, especially if itis to operate in all four quadrants and using both par-tial and full stroke methods.

To summarize the outcome of the investigation, it isidentified that the control challenges and complexity ishighly dependent on the machine configuration, oper-ation strategy and the application it is used in. Forthe more complex systems, hybrid dynamical theoryis considered necessary, but the development of sta-bilizing feedback control laws is very challenging and

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remains an undergoing research topic.

7 Conclusion

This paper identifies many of the challenges related tomodel based feedback control of digital displacementmachines. It is found that the challenges are highlydependent on the chosen operation strategy of the ma-chine, as well as how the machine is configured andin which application it is used. Since the continuousdynamics of each pressure chamber is activated or de-activated discretely, the system dynamics belongs tothe class of hybrid systems. Since control of hybriddynamical systems is quite complex and remains an ac-tive research field, simpler continuous and discrete ap-proximations have been investigated and deemed validin certain situations. However, to be able to performproper feedback stabilizing control and obtain the de-sired transient performance in a broad range of situ-ations, more advanced control theory is required, e.g.hybrid control. The paper hence provides an overviewof the many unsolved problems related to control ofdigital displacement machines, which is considered akey elements for successful deployment of the technol-ogy in a broad range of hydraulic applications.

Acknowledgments

This research was funded by the Danish Council forStrategic Research through the HyDrive project at Aal-borg University, at the Department of Energy Tech-nology (case no. 1305-00038B). The work is made incooperation with the research council of Norway, SFIoffshore mechatronics, project number 237896/O30.

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