1

Semi-empirical model for a hydraulic servo-solenoid valve

J. A. Ferreira*, F. Gomes de Almeida** and M. R. Quintas**

*Department of Mechanical Engineering, University of Aveiro, Portugal

**IDMEC – Pólo FEUP, University of Porto, Portugal

Abstract: High performance proportional valves, also called servo-solenoid valves, can be used

today in closed-loop applications that previously were only possible with servovalves. The valve

spool motion is controlled in close loop with a dedicated hardware controller that enhances the valve

frequency response and minimizes some nonlinear effects. Due to their lower cost and maintenance

requirements as well as increasing performance they can compete with servovalves in a large number

of applications. This paper describes a new semi-empirical modeling approach for hydraulic

proportional spool valves to be used in hardware-in-the-loop simulation experiments. The developed

models use either data sheet or experimental values to fit the model parameters in order to reproduce

both static (pressure gain, leakage flow rate and flow gain) and dynamic (frequency response) valve

characteristics. Valve behavior is divided into two parts: the static behavior and the dynamical

behavior. A parameter decoupled model, with a variable equation structure, and a flexible model,

with fixed equation structure, are proposed for the static part. Spool dynamics are modeled by a non-

linear second order system, with limited velocity and acceleration, the parameters being adjusted

using optimization techniques.

Keywords: fluid power, modeling, simulation, proportional valves

NOTATION

Aij, An, Ap pseudo sections

fn frequency of sine wave

Lv, La spool velocity and acceleration limits

0pK relative pressure gain at 0sx =

0qK flow gain 0sx =

k, γ, α, β, xt pseudo section parameters

k1, k2, k3, k4, k5 pseudo section parameters

Pi relative pressure at valve port I

∆Pij pressure drop between port i and port j

PL load pressure drop

PL relative load pressure

Pn nominal pressure drop

qij volumetric flow rate from port i to port j

QL load volumetric flow rate

qlk leakage volumetric flow rate

qlk0 leakage flow at 0sx =

Qn nominal volumetric flow rate

Q1, Q2 outlet ports volumetric flow rate

Qs, Qt source and tank volumetric flow rate

u normalized valve input, [ ]1,1u ∈ −

sx normalized valve spool position, [ ]1,1sx ∈ −

ωn natural angular frequency

ξ damping ratio

2

1 INTRODUCTION

Hydraulic hardware has experienced a large evolution during the last few years. The use of electronics and

microprocessors contributes to improving the dynamic performance of hydraulic components, namely of

proportional valves. In fact, advances in electrohydraulic proportional valves have increased their performance

very close to that of servovalves. Today, high performance proportional valves, in conjunction with electronic

control cards, can handle some closed-loop applications with lower cost and maintenance requirements.

The typical input to a servo-hydraulic control system is a force applied to a spool valve. This force will

cause the spool to change position and thereby change the orifice areas, which control system flow and pressure.

When the valve is of the proportional type this input force is usually produced by a solenoid. Disturbances to the

spool motion are highly undesirable and can be due to friction forces between valve spool and sleeve, static

pressure forces due to the spool geometry, or flow induced inertial forces. Advanced control strategies can be

used to obtain good spool dynamic characteristics over the whole valve operating conditions in order to

overcome its highly non-linear operation. This is achieved with the use of closed loop spool position control

performed by an electronic card. The controller minimizes the non-linear perturbation effects in the spool

movement and enhances the frequency response up to, typically, 200 Hz for a ±5% input signal variation on a

size 6 valve.

These types of proportional valves only work with their matched controller. This leads to the possibility

of using simpler models of spool motion without the non-linear hysteresis, friction or flow forces effects. The

structure of a general model of the valve plus controller card system is discussed in the following section.

Section 3 presents the model equations for pressure gain, flow gain and leakage flow. Section 4 describes two

static valve models with different pseudo section functions, the simulation results being presented in Section 5.

Section 6 presents the model for the spool dynamics and explains the optimization techniques used in the model

parameters estimation.

2 VALVE MODELS

The main use of hydraulic valve models is to preview the behavior of hydraulic systems in order to improve

controller and overall system performance. According to (1), it is important that for a given application the

relative merits of different control schemes can be evaluated, being the computer simulation one of the best

evaluation tools. Supporting this idea is (2), who states that a simulated environment is the cheapest and fastest

way to test control algorithms. Hardware-in-the-loop techniques (3) can then be used to test and adjust

parameters of real controllers by controlling the real time simulation of a hydraulic system. This technology

provides a way for testing control systems over the full range of operating conditions, including failure modes.

The feasibility of a real time simulation of a hydraulic system depends on its complexity and the performance of

the computing hardware used.

Several proportional valve models can be found in literature (4-16). Some of these describe complex

theoretical models of spool valves, some use computational fluid dynamics to model flow forces, others model

3

the effects of the spring characteristics, spool friction and valve geometry in the steady and dynamic valve state.

However the real time simulation of these complex theoretical models is, at present time, unfeasible. Another

difficulty is the identification of the valve geometric structure and physical parameters needed to parameterise

the models, information rarely provided by the manufacturers.

Other types of spool valve models are the semi-empirical models. These require less computing power,

and their parameters are more easily adjusted by simple numerical calculations using measured characteristic

curves or manufacturer data. These models capture the most influential static and dynamic behaviours, using

well-known mathematical functions, and are independent of the manufacturers design differences between the

modeled valves.

Valve chamber volumes are very small and thus the compressible flows are usually lumped to the line and

chamber actuator models (4). The valve volumetric flow rate can then be modelled by algebraic equations. Also,

if the transitional flows are negligible at each instant during the spool position displacement, the output flow is

the same as that obtained in steady state for the same spool position and pressure conditions (17). Therefore,

valve models may be split into two main blocks with a serial connection. The first block, where the electrical

reference spool position signal to the controller card is the input and the actual spool position is the output,

models the non-linear spool position dynamics. The second block describes the static behaviour and relates the

volumetric flow rate through the valve with the spool position and port pressures.

3 FLOW RATE MODELS

The proposed static models intend to reproduce the valve static characteristics (data sheet or experimental) as

the relative pressure gain, the leakage flow rate and flow gain, predominantly near the central spool position.

The valve under consideration has four control sections modulated by spool position as shown in Fig. 1. The

valve orifice areas versus relative spool position ( [ ]1,1sx ∈ − ) are modeled with pseudo section functions,

Aij( sx ), as in (14). Typical plots of the pseudo section functions of a matched and symmetrical valve are

presented in Fig. 2.

Q1 Q

2

P2P1

Qt

PtPsPt

q1t

qs1 qs2q2t

∆P1t

xs

Spool

Qs

∆Ps1 ∆Ps2 ∆P2t

Port1 Port2

Ports Portt Fig. 1 Five-port, four way valve diagram

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

xs [-1,1]

Sec

tion

( l/m

in⋅ b

ar-1

/2)

As2(xs) = A1t(xs) As1(xs) = A2t(xs)

Fig. 2 Pseudo Sections as function of the spool position

4

It must be emphasised that, in the proposed models, the volumetric flow rate is always assumed turbulent,

being the laminar flow regions implicitly modeled by the pseudo section functions Aij( sx ). Thus, the volumetric

flow rate, qij, crossing a control section from port i to port j is written as follows (4, 5, 7):

( ) ( )sij ij ij ijq sign P A x P= ∆ ⋅ ⋅ ∆ (1)

where ij i jP P P∆ = − is the pressure drop between the two ports and ( )ijsign P∆ is the sign of ( )i jP P− .

High performance proportional valves, as servovalves, are usually designed with matched and

symmetrical control orifices. This option leads to maximum loop gain and load stiffness when driving a

symmetrical actuator (4). The valve has matched orifices if As1( sx )=A2t( sx )=Ap( sx ) and

As2( sx )=A1t( sx )=An( sx ); symmetry means Ap( sx )=An(- sx ). These relations imply that a single pseudo section

function is needed to characterize the valve static behavior and, in this case, manufacturer data is usually

sufficient to define this function. Nevertheless the proposed model structure and pseudo section functions may

be used in the general unmatched and asymmetrical case. Then, four pseudo section functions Aij( sx ) must be

found from a set of experiments needed to individually characterize each of the four control orifices (14).

The most usual case of a matched and symmetrical valve will be considered throughout this paper. Under

this assumption and considering the nomenclature presented in Fig. 1, valve flow rate equations may be written

as:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 1 1

2 2 2

1 1 1

2 2 2

sps s s

sns s s

snt t t

spt t t

q sign P A x P

q sign P A x P

q sign P A x P

q sign P A x P

= ∆ ⋅ ⋅ ∆ = ∆ ⋅ ⋅ ∆ = ∆ ⋅ ⋅ ∆ = ∆ ⋅ ⋅ ∆

(2)

Continuing the reference to the valve diagram presented in Fig. 1, the tank flow, Qt , the source flow, Qs,

and the outlet flows Q1 and Q2 may be formulated as follows:

1 2

1 2

1 1 1

2 2 2

s s s

t t t

s t

t s

Q q q

Q q q

Q q q

Q q q

= + = + = − = −

(3)

When the valve is connected to a symmetrical actuator the outlet ports flow rates are equal, i.e. Q1 = Q2.

If the tank pressure is, without loss of generality, assumed equal to zero (or Pt is considered as a datum pressure

for all other pressures) the port pressures are related by (18):

1 2sP P P= + (4)

The most important operating point is the central spool position because servo-systems usually operate

near this region. The flow gain at this position affects the open loop gain constant while the characteristics of the

pressure gain and the leakage flow are closely linked to servo-system stiffness. These static characteristics of the

valve are the base for parameter calculation of the static model. In the following sections the equations for the

flow gain, the pressure gain and leakage flow of a centered spool valve are presented.

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3.1 Flow gain of a centered spool valve

The flow gain is defined as the derivative of the load flow, under null load pressure, at the middle position

( 0sx = ):

00s

Lq

x

QK

xs =

∂=

∂ (5)

In the valve diagram of Fig. 1 if ports 1 and 2 are connected with a null resistance and, to simplify the

notation, assuming ∆Pij always positive, i.e. ( ) 1ijsign P∆ = , the volumetric load flow rate, LQ , results into:

( ) ( )1 1 1 1s sp ns t s tLQ q q A x P A x P= − = ⋅ ∆ − ⋅ ∆ (6)

Assuming Pt=0, or considering Pt a datum pressure, the load flow is:

( ) ( )1 1s sp s nLQ A x P P A x P= ⋅ − − ⋅ (7)

where Pi is the relative pressure at port i referred to the Pt datum pressure.

As ports 1 and 2 are connected with a null resistance,

( )1 2 , 2s

sLPP P Q x= = ∀ (8)

so,

( ) ( )2 2s s

s sp nLP PQ A x A x= ⋅ − ⋅ (9)

3.2 Pressure gain

The load pressure is defined as the pressure difference between ports 1 and 2 ( 1 2LP P P= − ). The relative load

pressure is then defined as sL LP P P= . The relative pressure gain is defined as the derivative of the relative

load pressure at the middle spool position under no flow condition ( )0LQ = :

00s

Lp

x

PK

xs =

∂=

∂ (10)

Using (4) the port pressures can be expressed as functions of LP :

( )

( )

1 2

2 1

12

12

ss L

ss L

PP P P P

PP P P P

= − = + = − = −

(11)

In the valve diagram of Fig. 1, if ports 1 and 2 are externally closed in such a way that the load flow is

zero, the internal flow rates will be equal two by two ( 1 1t sq q= and 2 2t sq q= ). Then, using (4) and (8) leads to

the following relationship:

( ) ( ) ( ) ( )1 12 2s s

s sp nL LP PA x P A x P⋅ − = ⋅ + (12)

Subsequently, solving (12) for LP , the relative load pressure is given by:

6

( ) ( )

( ) ( )

2 2

2 2s sp n

Ls sp n

A x A xP

A x A x−

=+

(13)

3.3 Leakage flow

The valve connections used for leakage flow (qlk) measurements are the same as above, for pressure gain

measurements. Analyzing qlk for 0xs ≥ :

1 2 1 2t s s tq q q q qlk = + = + (14)

( ) ( ) ( ) ( )1 2 1 2s s s sn n s p s plkq A x P A x P P A x P P A x P= ⋅ + ⋅ − = ⋅ − + ⋅ (15)

Using (11) and (15) the leakage flow can be expressed as a function of the relative load pressure, LP , as:

( ) ( ) ( ), 2 12s

s snlk lk L LPq q x P A x P= = ⋅ + (16)

In a similar way the leakage flow rate can be expressed as function of the pseudo section ( )pA xs :

( ) ( ) ( ), 2 12s

s splk lk L LPq q x P A x P= = ⋅ − (17)

At the central position, i.e. 0, 0s Lx P= = and the leakage flow is:

( ) ( )00, 2 0 2s

s nlk L lks

Pq x P q Ax = = = ⋅ (18)

The leakage flow at a positive spool position, *sx , far from the central position may be found assuming

that 1LP = at *sx . Using this assumption in (16) leads to the following equation for qlk:

( )*2 sn slkq A x P= ⋅ ⋅ (19)

4 PSEUDO SECTION FUNCTIONS

Two different approaches are proposed for the construction of pseudo sections functions. The first approach

uses a variable equation structure based on the exponential function. The main characteristic of this approach is

the algebraic relation that results between the model parameters and the static valve characteristics. The other

approach uses a fixed equation structure based on the hyperbolic function. The parameters are found solving a

non-linear system of equations.

It is interesting to look at the possibility of adjusting the three static characteristics (pressure gain, flow

gain and leakage flow rate), simultaneously, at the central position.

At the central spool position, the equation ( )( ), 0s s L L Lx x Q P Q= = defines an implicit function between

LP and LQ and equation (20) can be stated for the four-way valve in consideration.

L L Ls sL

Q Q Px xP

∂ ∂ ∂= −

∂ ∂∂ (20)

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The flow-relative pressure coefficient, L LQ P−∂ ∂ , can be found by differentiation of the QL equation.

From (7) and (11) an expression for QL may be written as:

( ) ( ) ( ) ( )1 12 2s s

s sp nL L LP PQ A x P A x P= − − + (21)

Deriving QL in relation to LP and using (16) and (17) to substitute the pseudo sections, ( )nA xs and

( )pA xs , leads to:

1 14 1 1

L lk

L L L

Q qP P P

∂ = − + ∂ − + (22)

When 0sx = , under either flow gain or pressure gain measurement conditions, 0LP = . So, using (20)

and (22) the relation between flow gain, relative pressure gain and leakage flow rate at the middle spool position

results into:

00 0

2s s

L lk L

s sx x

Q q Px x= =

∂ ∂=

∂ ∂ (23)

This means that the model can simultaneously adjust only two of the three static characteristics. The user

must chose, depending on the application, which characteristics should be adjusted. This result is independent of

the structure of the pseudo sections functions used.

4.1 Pseudo sections with variable equation structure

In this case the pseudo section functions have two different regions that change at a transition point, tx :

( )2 ; 1

; 1

sk xst

sn

s s t

e x xA x

x x x

γ

α β

− ⋅ ⋅ − ≤ ≤= − ⋅ + − ≤ < −

; (24a)

( )2

; 1

; 1s

s stsp k x

s t

x x xA x

e x x

α β

γ⋅

⋅ + < ≤= ⋅ − ≤ ≤

(24b)

with , , tk , , xγ α β +∈ and tx ∈ ]0, 1[.

The static characteristics used for parameter calculation are: nominal flow (Qn) at a valve pressure drop

equal to the nominal pressure Pn; relative pressure gain; leakage flow or flow gain at middle spool position.

The equations that can be used for parameter calculation of the above pseudo section functions will now

be presented. From (13) and (24) the relative load pressure near the middle spool position is given by:

( )tanh ; -s s

s s

k x k xs st tL k x k x

e eP k x x x xe e

⋅ − ⋅

⋅ − ⋅−= = ⋅ ≤ ≤+

(25)

So, the relative pressure gain (10) at the origin is:

0pK k= (26)

From (13) and (17) the leakage flow can be expressed as:

8

( ) ( )( ) ( )( )2 2

2 ss sn plks sp n

Pq A x A xA x A x

= ⋅+

(27)

For - st tx x x≤ ≤ the pseudo sections are ( ) 2sk x

spA x eγ⋅

= ⋅ and ( ) 2sk x

snA x eγ− ⋅

= ⋅ . When these are

substituted into (27) the leakage flow becomes

( ) ( )

22 2

22

coshs ss s

lk k x k x s

P Pq

k xe eγ γ

γ γ⋅ − ⋅= =

⋅+ (28)

It should be noted that the function 1/cosh(.) has the typical pattern of a real leakage flow curve. The

maximum leakage flow occurs at the central spool position. The leakage flow at the origin is directly related

with the parameter γ:

02 slkq Pγ= (29)

From the load flow equation (9), and for - st tx x x≤ ≤ , the load flow is given by:

2 22 2

s sk x k xs s

LP PQ e eγ γ

⋅ − ⋅= ⋅ ⋅ − ⋅ ⋅ (30)

So, the flow gain is:

0 2s

qPK k γ= ⋅ ⋅ (31)

The other equations used for parameter calculation are based on the continuity of the pseudo-section

( )spA x and its derivative, at a certain transition point (xt), and on the nominal flow definition. The system of

equations (32) should be solved for the α, β and xt parameters. The nominal flow (Qn) at nominal pressure (Pn),

that is available as manufacturers catalog information or can be measured for a specific valve, defines the flow

far from the origin as ( ) 0snA x ≈ for 1sx = .

( )

2

2

1 1

continuity at

continuity of the derivate at 2

nominal flow definition

t

t

s s

k x

t t

k x

t

sn nx x

x e x

k e x

Q x P

α β γ

α γ

α β

⋅

⋅

= =

⋅ + = ⋅ = ⋅ = ⋅ +

(32)

An interesting property of the pseudo sections given by (24) is the direct relation between their

parameters and the static valve characteristics, namely the pressure gain (k) and the maximum leakage flow (γ).

Assuming that relative pressure gain should always be reproduced, model parameters may be computed from

(26), (32) and (29) or (31) for the replication at null of, respectively, leakage flow or flow gain.

4.2 Pseudo sections with fixed equation structure

Hyperbolic functions were used for this type of pseudo sections. The main reason to use hyperbolic functions is

the possibility of characterizing two well-defined asymptotes. For a symmetrical valve these functions are:

( ) [ ]2 25 51 2 3 4 3 4; 0, 1,1s s s s s s snA x k x k k x k x k k x k x k x= ⋅ + + ⋅ + ⋅ + ⋅ + ⋅ + ≥ ∀ ∈ − (33a)

( ) [ ]2 25 51 2 3 4 3 4; 0, 1,1s s s s s s spA x k x k k x k x k k x k x k x= − ⋅ + + ⋅ − ⋅ + ⋅ − ⋅ + ≥ ∀ ∈ − (33b)

9

with ik ∈ ℜ .

The following equations, relating the valve static characteristics and the ki parameters of ( )snA x and

( )spA x , can be used for parameter calculation. The nominal flow (Qn) and nominal pressure (Pn), can be used

to adjust the flow rate at 1sx = :

( ) 51 2 3 41sn

sp xn

Q A x k k k k kP == = − + + − + (34)

Assuming PL = 0, an expression for QL , valid for any [ ]1,1sx ∈ − , may be obtained by the simultaneous

use of (9) and (33):

( ) ( )2 25 51 3 4 3 422s

s s s s s sLPQ x k x k x k x k k x k x k= − ⋅ + ⋅ − ⋅ + − ⋅ + ⋅ + (35)

So, the flow gain at the central position is:

51 4 00 5

22

s

sLq

s x

Q k k k P Kx k=

∂ + = − = ∂ (36)

Using (13) and (33) the relative pressure gain near the middle spool position is given by:

51 4 05 520

2

s

Lp

s x

P k k kK

x k k k=

∂ ⋅ += − =

∂ + (37)

The simultaneous use of (18) and (33a) leads to:

( )520 2 slkq P k k= ⋅ ⋅ + (38)

that may be used to adjust the leakage flow at the middle spool position.

Using (19) new relations can be stated for the leakage flow and leakage flow derivative at a spool position

far from the origin, where the condition 1LP = may be assumed:

( ) ( )2 51 2 3 41 2Ls s s sslk Pq x P k x k k x x k k= = ⋅ ⋅ + + ⋅ + ⋅ + (39)

3 41 21 53 4

22

2L

slks

s s sP

q k x kP k

x k x x k k=

∂ ⋅ + = ⋅ + ∂ ⋅ + ⋅ + (40)

If the leakage flow characteristic is available, a measure at a position * 0sx > may be used; otherwise the

leakage and its derivative at 1sx = can be set very small or zero.

Assuming that relative pressure gain should always be reproduced, model parameters may be computed

from (34), (37), (39), (40) and (36) or (38) for the replication at null of, respectively, flow gain or leakage flow.

5 SIMULATION RESULTS

This section presents some simulation results for both variable and fixed structure pseudo sections. The static

valve characteristics that can be adjusted are the nominal flow rate, the pressure gain, and the leakage flow rate

or the flow gain at the origin.

Valve manufacturers usually provide the following set of information: load pressure characteristic (for

null load flow); an estimation of the maximum leakage flow rate; the nominal flow rate at the nominal pressure

10

difference; an estimation of the flow gain near the origin (typically between 50% and 200% of the nominal flow

gain); Bode diagrams for different amplitudes of spool displacement. The parameters for the valve models may

then be adjusted through data sheet information or experimental data. A proportional valve from Vickers@,

model KBSDG4V-3 (19), is used as an example of parameter calculation. The relative load pressure, volumetric

flow rate and leakage flow rate plotted functions can be used to extract the model’s parameters (nominal flow

rate, flow gain, pressure gain and leakage flow rate at central spool position). All the simulations presented

bellow use the following measured characteristic values:

Qn = 25.5 l/min and 0qK = 28 l/min at Pn = 35 bar; 0pK = 36.5 and qlk0 =1.36 l/min at Ps = 70 bar.

Using the variable structure approach for the pseudo section functions and choosing to adjust the pressure

gain and the maximum leakage flow rate, that is, using (26), (29) and (32), the parameter values are:

2 236.5; 0.1149; 4.241; 6.877 10 ; 3.858 10tk xγ α β− −= = = = ⋅ = ⋅ .

Figure 3 shows values of ( )snA x and ( )spA x obtained using (16) and (17) with experimental data of

relative load pressure and leakage flow, for 0.6 0.6LP− < < (this interval of confidence results from the

uncertainties on pressure and flow measurements). The figure also shows pseudo sections computed by the fixed

structure model, adjusting the leakage flow using (39) and a measured leakage flow rate at 0.04sx = . A very

good agreement between the measured and modeled pseudo sections may be observed.

-0.06 -0.04 -0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

xs [-1,1]

Pse

udo

sect

ion

( l/m

in⋅b

ar-1

/2)

An(xs) Ap(xs) Real An(xs) Real Ap(xs)

Fig. 3 Pseudo sections area versus spool position

Figures 4, 5 and 6 present the simulation results. The leakage flow rate and the pressure gain were

measured and simulated for 70 sP bar= . In load flow rate experiments the pressure drop in each orifice was

constrained to 35 P bar∆ = . The use of the fixed structure functions to adjust the same parameters, that is,

using (34), (37), (38), (39) and (40), leads to identical results.

11

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-1

-0.5

0

0.5

1

1.5

xs [-1,1]

qlk

(l/m

in),

PL

[-1,1

]

PL (sim) PL (real) qlk (sim) qlk (real)

Fig. 4 Leakage flow rate and pressure gain near middle position: [ ]6%, 6%sx −∈

-0.06 -0.04 -0.02 0 0.02 0.04 0.06-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xs [-1,1]

QL

(l/m

in)

QL (sim) QL (real)

Fig. 5 Load flow rate near middle position: [ ]6%, 6%sx −∈

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30

xs [-1,1]

QL

(l/m

in)

QL (sim) QL (real)

Fig. 6 Load flow rate for [ ]100%,100%sx −∈

Figures 7, 8 and 9 present the simulation results obtained when using the fixed structure model approach

for the pseudo sections and choosing to adjust the pressure gain and the flow gain near the origin. In this case

1 0slk xq = = and 1

0s

lk

s x

qx =

∂=

∂ was assumed. Using (34), (36), (37), (39) and (40) results into the following

model parameters:

3 2 251 2 3 42.142; 5.818 10 ; 4.604; 5.554 10 ; 1.534 10 ;k k k k k

− − −= − = ⋅ = = − ⋅ = ⋅

The use of the variable structure functions to adjust the same parameters, that is, using (26), (31) and (32),

leads to identical results.

12

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-1

-0.5

0

0.5

1

1.5

xs [-1,1]

qlk

(l/m

in),

PL

[-1,1

]

PL PL (real) qlk qlk (real)

Fig. 7 Leakage flow rate and pressure gain near middle position: [ ]6%, 6%sx −∈

-0.06 -0.04 -0.02 0 0.02 0.04 0.06-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xs [-1,1]

QL

(l/m

in)

QL (sim) QL (real)

Fig. 8 Load flow rate near middle position: [ ]6%, 6%sx −∈

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

20

30

xs [-1,1]

QL

(l/m

in)

QL (sim) QL (real)

Fig. 9 Load flow rate for [ ]100%,100%sx −∈

Both modeling approaches present similar behaviors. The main differences are the model structure

(variable or fixed) and the processing effort needed before and during simulation. The computation of the ki

parameters of the fixed structure model involves the solution of a system of five non-linear equations. Instead,

only one non-linear equation must be solved in the variable structure model case; the solution of the other

equations can be found algebraically. During simulation the fixed structure model requires a constant computing

power with a smaller supreme value than the variable structure one, as required by real time applications.

6 SPOOL MOTION MODEL

The presented model is oriented for control experiments. As hysterisis for this type of valve is usually less then

0.5% and other disturbances such as spool friction or flow forces are minimized by the spool position close loop

controller, the most important behavior to be modeled is the frequency response of the spool position in regard

13

to the input signal. A typical Bode diagram for the Vickers@ proportional valve is shown in Fig. 10, for different

motion amplitudes around the central spool position.

Fig. 10 Typical frequency response for ±5%, ±25% and ±50% of maximum spool stroke (with courtesy of Eaton Corporation)

The amplitude frequency response is modeled with a second order system with acceleration and velocity

saturation, as presented in Fig. 11. The phase lag is adjusted with a delay as in (14) and (18).

Lv s 1

s 1

La-Lv

Lv

Velocity limit

∆t -La

La

Acceleration limit

2ξ ωn

ωn 2ξ xs

u

Fig. 11 Dynamic model of spool position with velocity and acceleration limits

6.1 Amplitude gain adjust

The model parameters (natural frequency, ωn, damping ratio, ξ, velocity limit, Lv, and acceleration limit, La) are

estimated by an optimization algorithm based on a least squares cost function, F. In the present example, the

model was simulated over a frequency range from 10 to 300 Hz in steps of 10 Hz. The parameters were

optimized using the three Bode amplitude plots provided by the manufacturer (5%, 25% and 50% of maximum

amplitude).

A variable frequency (and amplitude) sine wave (u (t)) was applied to the input of the spool position

dynamic model:

( ) ( )sin 2 ninu t A f tπ= ⋅ ⋅ ⋅ (41)

where the integer n ∈ [1, 30], fn=10.n and t is the time.

The output spool position ( sx (t)) was then band pass filtered, in order to exclude harmonics, and used to

compute the output gain, Gsn, in dB:

1020 logout

in

AGs

A = ⋅

(42)

where Aout is the amplitude of the (fundamental) sinusoidal output spool position.

This gain was then compared with the manufacturer gain at the same frequency and amplitude (Grn). The

cost function F was then computed. As most of the time valve action takes place around the central position, a

weight of four was applied to the 5% amplitude quadratic error when computing the cost function value:

14

( ) ( ) ( ) ( )30 30 302 2 2

1 1 15% 25% 50%

, , , 42 2 2

in in in

n n n n n nn v a

n n nA A A

Gs Gr Gs Gr Gs GrF L Lω ξ= = == = =

− − −= ⋅ + +∑ ∑ ∑ (43)

Numerical methods were used to generate new model parameters leading to the minimization of F and a

new iteration, through (41) to (43), was taken. The iteration loop was paused when model parameters converged

to a minimization set.

Different model complexities were tested such as using velocity saturation only and both velocity and

acceleration saturation. Several non-linear limiter shapes, based on continuous and variable structure functions,

were also tried in the model presented in Fig. 11. The best results were obtained with a limiter based on the

arctangent function for the velocity and a pure saturation limiter for the acceleration. It is interesting to note that

the best velocity limiter is in action from small velocities but acceleration is better handled by an abrupt

saturation limit. All limiters have unitary gain at the origin.

The parameter set ( -1 -1 -21007 ; 0.48; 125 ; 81184 n v arad s L s L sω ξ= ⋅ = = = ) minimizes the cost

function for the selected model. The simulation results, presented as dotted lines in Fig. 12, show that amplitude

artifacts due to non-modeled dynamics are mostly present for frequencies higher then 200Hz.

Fig. 12 Data sheet and simulated (dotted lines) bode diagrams for amplitude and phase lag response

6.2 Phase adjust

The phase characteristics were adjusted using a pure time delay. The approach taken for delay estimation was

similar, that is, a set of simulations was used to compute a least squares cost function and then iterated until the

delay that minimizes the cost function was found. From the results ( 47.625 10t s−∆ = ⋅ ), presented in Fig. 12, it

may be concluded that a good matching is achieved for the 5% amplitude characteristic (inside the valve

frequency range). However, the other phase characteristics show that the model has place for improvement,

mainly in the high frequency range.

7 CONCLUSIONS

A semi-empirical model for today’s high performance proportional valves was proposed. The model describes

the behaviour of the whole hydraulic valve package (valve, spool position transducer and electronic controller

15

card). Two decoupled sub-models are used. The static sub-model reproduces the flow through the valve as a

function of the spool position and pressure drops. Valve flow was assumed to be always turbulent being the

laminar flow regions implicitly modelled by the pseudo sections functions. Two different approaches were

followed for the pseudo section functions: a variable equation structure, with decoupled parameters, based on

the exponential function and a fixed equation structure based on the hyperbolic function. The dynamic sub-

model describes the spool displacement when an input signal is applied.

The semi-empirical model presented in this paper has some advantages over the physical models that may

be found in the literature: it is easier to tune and has lower computational needs. Valve parameters of the static

sub-model may be computed directly from manufacturer data or experimental measurements. The parameters of

the dynamic sub-model are determined by well-known optimisation techniques using the Bode diagrams offered

by the manufacturer.

The parameterisation example shown in this paper is for a Vickers@ valve model. Nevertheless, other

manufacturer valves, namely a NG6 Servo solenoid valve from Bosch@, were tested with identical qualitative

results in both static and dynamic sub-models. The model accurately reproduces flow gain, pressure gain,

leakage flow and the amplitude Bode diagram up to 200Hz. The phase response still has room for improvement,

mainly at high frequencies.

ACKNOWLEDGMENTS

This work was funded by FCT under the programme POCTI.

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