Motion Control of Hydraulic Actuators In the Presence ofDiscrete Pressure Rail Switching

Arpan Chatterjee and Perry Y. Li

Abstract— Conventional off-highway mobile machines arehydraulically actuated and have poor energy efficiencies. Thegoal to increase system efficiency and to reap the benefits of elec-trification have led to the creation of a novel Hybrid Hydraulic-Electric Architecture (HHEA) which could significantly increaseefficiency, decrease the sizes and cost of electrical components,and maintain control performance. The key concept is to utilizea set of common pressure rails to provide the majority of powervia the power dense hydraulics and to modulate that powerby small electric components, which are less power dense, forprecise control.

The paper presents the motion control strategy for thisarchitecture. The architecture presents a distinct challenge inthat as the energy minimization algorithm selects differentpressure rails, the system could experience large discrete jumpsin pressures that impact control performance, especially whenthe electric components are torque limited. To meet thischallenge, a passivity based back-stepping controller is used asthe nominal controller between pressure rail switching eventsand a separate transition controller is used to deal with thepressure rail switching events. Two transition controllers areproposed, one uses the electric motor torque as the input, andthe other also uses valve timing as a second input. For bothtransition controllers, a least norm control approach is used tosteer the system states to the exact desired states at the end ofthe short transition period (∼ 40 ms). The transition controllersare able to reduce the tracking error and the required electricmotor torque, and hence achieve better control performanceeven with smaller electric components.

I. INTRODUCTIONConventional heavy-duty off-highway vehicles used in the

construction, agricultural and turf industries are hydraulicallyactuated to take advantage of hydraulics’ unsurpassed powerdensity. However, the efficiencies of these machines are low(∼21% from engine to output power [1]) due to the useof throttling for control, components operating at partialdisplacements where they are less efficient, and the inabilityto recuperate energy from overrunning loads.

A novel system architecture that combines the power den-sity advantage of hydraulics with the efficiency and control-lability advantages of electric actuation has been proposedto increase efficiency [2]. This Hybrid Hydraulic-ElectricArchitecture (HHEA) (Fig. 1) uses a set of common pressurerails (CPRs) and an electric motor driven pump/motor tocontrol each degree-of-freedom. Here, the flow and hencethe actuator speed can be directly controlled via the electricmotor drive. At each moment, a pair of pressure rails forthe inlet to the pump/motor (PB) and for the return port

Authors are with the Department of Mechanical Engineering,University of Minnesota, Minneapolis, Minnesota 55455. Emails:{chatt116,lixxx099}@umn.edu. Please send all correspondence toP. Y. Li.

Fig. 1. Top: The hybrid hydraulic-electric architecture (HHEA) with 3services and 3 pressure rails at 0 MPa, 17.5 MPa, and 35 MPa. Theelectric generator/motor at the engine is optional. Bottom: Hydraulic-Electric Control Module (HECM) for a linear actuator.

of the hydraulic actuator (PA) is selected to provide ahydraulic force that is close to the desired actuator force.This way, the electric motor is responsible only to make upfor the difference. Consequently, the majority of the poweris transmitted hydraulically and the electric motors serveonly to modulate that power. Since electric machines tendto have much lower power density and higher cost thanhydraulics, the HHEA enables high power machines to reapthe benefits of electrification at lower cost and with smallercomponents. The HHEA is inherently throttle-less and iscapable of capturing regenerative energy from the load.Previous studies have shown that the system architecturecan potentially reduce energy consumption by over 55-65%compared to the baseline load sensing system for a varietyof off-road mobile machines [2].

This paper addresses the motion control aspect of theHHEA. Precise motion control is important for off-road

2020 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM), Boston, USA(Virtual Conference), July 6-9, 2020

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vehicles whose large part of utility depends on the machinesbeing dexterous and performing tasks exactly as commanded.The HHEA poses a unique challenge to motion control dueto the discrete pressure changes when the selected pressurerails for the pump/motor inlet and the actuator return switchfrom one selection to another in order to minimize systemefficiency or to keep the system within the torque capabilityof electric motor.

The proposed motion control strategy is to utilize twodifferent controllers: 1) for the duration between consecutivepressure rail transitions, and 2) for the short time whenthe pressure rails undergo discrete switching. A passivitybased back-stepping controller is used for a former, whilean open-loop least-norm controller is used for the latter. Thispartition is necessary because the electric motor has limitedtorque capability so that it will not be able to completelycompensate for the discrete pressure changes due to railswitches. The passivity based back-stepping control is similarto that in [3] where a hydraulic-transformer is used forcontrol. Control in the presence of discrete pressure switchesis also encountered in the control of the switched modehydraulic transformer [4]. There, a bumpless transfer witha delayed action was adopted. The solution proposed in thepresent paper is more comprehensive.

The rest of the paper is organized as follows. Section IIdescribes the HHEA and our control objectives. The nominalcontroller is briefly presented in Section III. Two versionsof the transition controllers are presented in section IV.Simulation results are presented in section V. Concludingremarks are given in section VI.

II. SYSTEM DESCRIPTION AND CONTROL OBJECTIVES

In the HHEA, there is set of N common pressure railseach at a different nominal pressure (PR1, . . . , PRN ). Therails were fed by a common pump/motor and the pressureat each rail is regulated by a hydraulic accumulator. Foreach degree-of-freedom, there is a hydraulic-electric controlmodule (HECM) that combines hydraulic power from thepressure rails with electric power to actuate the linear degree-of-freedom (See Fig. 1-bottom). Ignoring pressure and inertiadynamics for the moment, if the pressure rails with pressuresPRi and PRj are selected for the inlet of the pump/motor PBin the HECM and the return of the hydraulic actuator PA,and the electric motor torque is Tm, the resultant actuatorforce and speed are

Fact = PRiAcap − (PRj +2π

DTm)Arod

= (PRiAcap − PRjArod)︸ ︷︷ ︸Frail

−Arod2π

DT︸ ︷︷ ︸

Felect

(1)

x = − D

2πArodω (2)

where D is the pump/motor displacement, Acap and Arodare the cap and rod side piston areas, Frail and Felect arethe portions of the actuator force provided by the selectedpressure rails and by the electric motor. There are N2

options of Frail based on different selections of pressure

rails. Thus, from (1), by choosing Frail close to the desiredFact, the electric motor torque can be minimized (and theelectric motor can be downsized). In general, the choice of(PRi, PRj) is to minimize the energy loss globally.

In the presence of pressure and inertia dynamics, thesystem dynamics are:

Mx = PcapAcap − ProdArod − FL (3)

Prod =β

Vrod(x)(Q+Arodx) (4)

Q =D

2πω (5)

Jω =(PB − Prod)D

2π+ Tm (6)

where FL(t) is the load on the actuator, Pcap = PA and Prodare the actuator chamber pressures, β is the bulk modulus ofthe fluid, Vrod(x) := Vo−Arodx is the rod side fluid volume,J is the inertia of the electric motor and the pump/motor,and PB is the inlet pressure of the pump/motor. PA (i.e.Pcap) and PB are connected to the selected pressure rails sothat PA, PB ∈ {PR1, . . . , PRN}. Because these selectionscan change instantaneously, PB(t) and Pcap(t) can undergodiscrete jumps. In between rail transitions, PB and Pcap areconstant.

The control objective is to utilize the motor torque TM totrack the desired position xd(t) trajectory specified by theoperator. It is assumed that xd(t), xd(t), xd(t) and as wellas the load on the actuator FL(t) are known. In addition, thedesired rail selections at any time are assumed to have beendetermined by another controller concerned with maximizingenergy saving. However, the controller is expected to havesome flexibility in terms of the exact time when the switchingoccurs.

III. NOMINAL CONTROL: PASSIVITY BASEDBACK-STEPPING

In between rail transitions, a non-linear feedback controlis used for trajectory tracking. Since a hydraulic actuatorhas been shown to be a passive two port system [5], apassivity based back-stepping control design that makesuse of the natural energetic interaction is used. An energybased Lyapunov function is used to establish stability andconvergence. By relying on the inherent physical structure,stability does not require the control to cancel specific terms,and there are fewer controller parameters to be tuned. Similarto the situation in [6], the controller proposed in [5] whereflow Q in (5) is the input has to be extended to use electricmotor torque TM in (6) as the input.

Referring to (3)-(6), control design proceeds from assum-ing successively Prod, Q, and finally TM as the inputs. Ateach step, the Lyapunov function for proving stability issuccessively extended to include additional states by addingto it the energy associated with that state. The procedure issummarized below.

Let e = x − xd be the tracking error and assuming x isthe control input, the control law

r := xd − λpe; λp > 0 (7)

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leads to the stable dynamics e = −λpe. Next assume thatProd is the control input. The extra state to be included is:ev := x− r. The control law and Lyapunov function and itsderivative are:

Prod,d =1

Arod[PcapAcap − FL −Mr +Kvev +Kpe]

(8)

V2 :=1

2Me2v +

1

2Kpe

2 (9)

V2 := −Kve2v − λpKpe

2 + PArod (10)

where P := Prod − Prod,d is the pressure error. Thus, ifP → 0, then e→ 0, and ev → 0.

Next we assume that the output flow of the pump/motorQ is the control input. The control law and the Lyapunovfunction are:

Qd = Arodr +Vrod(x)Pd

β− λ3P (11)

V3 := V2 + Vrod(x)WV (P , Prod,d) (12)

where WV (P , Prod,d) is the energy density associated withcompressing the fluid from Prod,d to Prod [7]. It can beshown [7] that when λ3 is sufficiently large, V3 satisfies:

V3 :≤ −[e ev P

]M

eevP

+ Φv(P )Q (13)

where M is positive definite, Q = Q − Qd and Φv(P ) :=P + Wv(P , Prod,d) is the hydraulic effort (conjugate tohydraulic flow Q). Note that the desired Qd can be referredto the desired pump/motor speed ωd via

Qd =D

2πωd

Finally, with the actual control input Tm as the control input,the control law and Lyapunov function are:

Tm =D

2π(Prod − PB − P )− λ4ω (14)

V4 := V3 +1

2Jω2 (15)

V4 ≤ −[e ev P

]M

eevP

− λ4 ω2 (16)

where ω = ω−ωd. This shows that e, ev , P , ω all convergeto 0 exponentially.

IV. TRANSITION CONTROL: LEAST NORM CONTROL

The nominal controller guarantees good tracking perfor-mance under nominal conditions. However, when the pres-sure rails switch, pressure error can instantaneously becomevery large. As seen from (14), it demands a significanttorque input that far exceeds the capability of the electricmotor/drive. This is especially significant when both pressure

rails are switched at the same time 1. Recall that thepremise of the system architecture is that it requires onlysmall electric machines. One can saturate the torque at itsmaximum level and hope for the best. This unfortunatelyleads to quite significant increase in tracking error and ittakes a long time for the controller to recover (see Fig. 2 foran example).

In this section, we design a controller during a shorttransition period (about 40ms) during which the effect of thepressure rail changes on the motion tracking is minimizedwhile keeping the control input to a realizable value. Afterthe transition period, control is handed back to the nominalcontroller. In this particular approach, the transition controlaims to reach a state with zero error at the end of thetransition.

A. Least Norm Control

The rod side chamber volume Vrod(x) is assumed toremain constant during the very short transition period. Thestate dynamics in (3)-(6) thus become a linear time-invariantsystem:

X = AX +BP

FL(t)Pcap(t)PB(t)

+BUTM (t) (17)

where

X =

xx

Prodω

; A =

0 1 0 0

0 0 −Arod

M 0

0 βArod

Vrod0 βD

2πVrod

0 0 − D2πJ 0

BP =

0 0 0

− 1M

Acap

M 00 0 00 0 D

2πJ

, BU =

0001J

Here, we assume that the time courses of Pcap(t) = PA(t)and PB(t) are known. In particular PB(t) switches instan-taneous to the new selected rail pressure at the transitiontime, whereas Pcap(t) will switch after a slight delay (to bedetermined).

Given the initial state X(t0) at the beginning of thetransition period, and the known time courses of PB(t) andPcap(t), our goal is to use motor torque TM (t) to controlthe state to the desired values Xf at the end of the transitionperiod tf . A computationally efficient way for computing thecontrol is via least norm control - i.e. the control that satisfiesthe requirement and minimizes the L2 norm of TM (·).

1For continuous actuator load and if both the prior and next rail selectionsare within the electric motor torque capability, the electric motor should havesufficient capability to cope with the rare instances when only one pressurerail switches so that a separate transition control may not be necessary.

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The solution to the linear system (17) is:

X(tf ) = Φ(tf , t0)X(t0)+

∫ tf

t0

Φ(tf , τ)BP

FL(τ)Pcap(τ)PB(τ)

dτ+

∫ tf

t0

Φ(tf , τ)BUTM (τ)dτ︸ ︷︷ ︸Lr[TM (·)]

(18)

where Φ(t2, t1) is the transition matrix of A, Lr : <[t0,tf ] →<4 denotes the reachability map. Defining

∆x := Xf−Φ(tf , t0)X(t0)−∫ tf

t0

Φ(tf , τ)BP

FL(τ)Pcap(τ)PB(τ)

dτ(19)

the least norm control problem is to find TM :

minTM (·)

∫ tf

0

T 2M (τ)dτ s.t. ∆x = Lr[TM (·)]

Since the system (17) can be shown to be completelycontrollable from TM , the reachability map Lr is surjectiveand the least norm control is given by [8]:

TM (·) = L∗r(LrL

∗r)

−1∆x (20)

where L∗r is the adjoint of Lr. Explicitly, it is written as:

TM (τ) = BTUΦT (tf , τ)G−1∆x (21)

G =

∫ tf

t0

Φ(tf , τ)BUBTUΦT (tf , τ)dτ (22)

where G is the reachability Grammian.

B. Scaling for different transitions

The least norm control solution in (19) and (21) can becomputed efficiently for different rail switch conditions. In(21), the transition matrix Φ(tf , τ) and the Grammian Gcan be precomputed. Furthermore, if PB(t) is a step changeand Pcap(t) is a delayed step change with a known delay,∆x in (19) can be expressed as a linear function of X(t0),Xf , the new value of PB , and the old and new values ofPcap. Therefore, the least-norm control at any time TM (τ)can be computed by multiplying a precomputed kernel witha scaling matrix.

Although the rod side chamber volume Vrod(x) can be as-sumed to remain constant during the transition time, it differsfor different transitions. Hence the precomputed kernel needsto be parameterized by the fluid volume.

C. Transition control with motor torque and valve-timing

The transition control strategy in section IV-A assumesinstantaneous changes from the pressure of one pressure railto another, and that the electric motor torque TM is the onlycontrol input. In reality, during rail switching, the switchingvalve connected to the rail prior to the switch has to be closedand the valve connected to the rail to be connected has tobe opened. These valve transitions take time during whichthe relevant chamber pressure transitions from one pressureto another. This reality suggests that valve timing, especially

on the capside of the actuator, can be a second control inputduring transition.

To utilize this secondary control input, we partition thetransition control into two sub-problems: 1) the electricmotor torque TM is used to control the pump/motor speed ωand the rod side chamber pressure Prod; and 2) the capsidevalve timing is used to more precisely control the actuatorposition and velocity x and x.

1) Rod-side pressure control: Assuming the piston posi-tion and velocity are indeed precisely controlled, the rod-sidedynamics are:[Pω

]=

[0 βD

2πVrod−D2πJ 0

] [Pω

]+

[βArod

Vrodxd(t)

D2πJPB(t)

]+

[01J

]TM

(23)

Similar to the least norm control for the full system in sectionIV-A, TM (·) is computed as the least norm control that steersthe initial states from (P (t0), ω(t0)) to the desired values att = tf with the least L2 effort.

2) Cap-side control: With consideration of the valve dy-namics during switching, the actuator and cap side chamberpressure dynamics are given by:

Mx = Pcap(t)Acap − Prod(t)Arod − FL(t) (24)

Pcap =β

Vcap(x)(Qvalve −Acapx) (25)

Qvalve = kvxv1(t)sign(P−A − Pcap)

√|P−A − Pcap|

+ kvxv2(t)sign(P+A − Pcap)

√|P+A − Pcap| (26)

Here Vcap(x) := Vco−Acapx is the capside fluid volume, P−A

and P−A are the pressures of the previous and next selected

rail pressures, xv1 and xv2 are the valve openings for theconnections to these rails, and kv is the valve constant forboth valves.

Various open loop and closed loop methods can be appliedto control the actuator position x(t) using valve openingsxv1(t) and xv2(t) as control input. In general, we seek xv1(t)and xv2(t) to be monotonic. In the results below, an openloop approach is used in which the valve opening profiles arepredetermined. A closed loop approach, e.g. passivity basedback-stepping, can also be used.

V. RESULTS AND DISCUSSION

The control strategy is illustrated for a desired trajectoryand a rail transition in which PA(t) switches from 30MPato 40MPa, and PB(t) switches from 0MPa to 20MPa(see insets of Fig. 2).

Figure 2 shows the tracking result with the back-steppingcontrol with and without the electric motor torque being sat-urated at ±350 N-m. Note that the back-stepping controlleris able to track the desired trajectory except at the instancesof rail transition. Without saturation, the tracking error is ofthe order of 0.2mm but at the expense of very high torque( 4000 N-m). When the motor torque is limited, 7mm peakerror occurs and the tracking is not recovered for 100 ms.

To demonstrate the transition control, we focus on thesample transition in Fig. 2. Transition control using least

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Fig. 2. Trajectory tracking performance and torque input using the nominalback-stepping controller only using a desired trajectory and a samplepressure switch: with and without torque saturation at ±350Nm.

Fig. 3. Performance for different transition times - least norm control usingonly motor torque as input. Time delay of the PA switching is 8 ms.

norm method with only motor torque TM as control inputis considered first. Figures 3 and 4 show the performanceand torque requirements for different transition duration tf ’s.For shorter tf , the error decreases but torque requirementincreases.

As mentioned earlier, delaying the time when the cap-siderail switch takes place allows time for the rod-side pressureto build towards the desired pressure, thus reducing the errorduring the transition. Figure 5 investigates this influence. Fordelays between 7-9 ms, the required torque magnitude is theleast. Hence, 8 ms delay is used in the control in Figs. 3-4.

The transition control using both motor torque and cap-side valve timing are investigated next. Figures 6-7 show the

Fig. 4. Torque inputs for different transition times - least norm controlusing only motor torque as input. Time delay of the PA switching is 8 ms.

Fig. 5. Torque input for various cap-side delays - least norm control usingonly motor torque as input.

performance of this approach and the valve opening profilesand the resultant cap-side pressures. The transition time iskept at 40ms, which is a good trade-off between performanceand torque input. Notice that the tracking error has beensignificantly reduced to less than 0.3 mm compared to the 1-1.5mm in Fig. 3. Despite the improvement, Fig. 8 shows thatmotor torque requirement is slightly less than the transitioncontrol that uses motor torque as the only control input.

Figure 9 compares the tracking error using the back-stepping controller with motor torque saturated at ±350Nm,and the combination of back-stepping and the least normtransition controller using only motor torque as control input.The transition controller is able to reduce the peak errors andensure that error is zero at the end of the transition and thenit hands over the control to the back-stepping controller. Thetransition controller also makes sure that the transition timeis short and as it can be seen the transition time is 40 mswith the combination of the two controllers while it is 100mswith just the back-stepping controller.

VI. CONCLUSION

In the proposed hybrid hydraulic-electric architecture foroff-road mobile machines, switching of the common pressurerails are fundamental to realizing the premise of transmittingthe majority of the power via hydraulics while small electric

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Fig. 6. Transition control performance using both motor torque and valvetiming as control inputs.

Fig. 7. Cap side pressure change and valve profiles for the transition controlusing both motor torque and valve timing as control inputs.

machines are used to modulate that power. This paper haspresented an approach of dealing with the adverse effect oncontrol performance when pressure rails are switched. Whilea small amount of tracking error is tolerated during transition,a linear least norm transition controller ensures that thetracking error returns to zero at the end of the transitionwhen the nominal back-stepping controller takes over again.A hardware in the loop (HIL) setup is being constructed tovalidate the proposed approach experimentally.

ACKNOWLEDGEMENTSThis material is based upon work supported by the Depart-

ment of Energy, Office of Energy Efficiency and RenewableEnergy (EERE) under grant: DE-0008384.

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Fig. 8. Motor torque for the transition control using both motor torqueand valve timing as control inputs.

Fig. 9. Tracking error and control Input comparison of back-steppingcontroller and the combination of back-stepping and least norm transitioncontroller using electric motor torque as input.

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