Aerospace and Electronic Systems Magazine September 2017 - 23

Zhu et al.
The dynamic behavior of a servo valve can be simplified as a second-order system

VALVE CONTROL PBW SYSTEM MODELING
By Newton's second law, the motion equation of the hydraulic actuator is
d2y
dy
Ap ( pL − pR ) = mt 2 + Bp
+ f p + FL
dt
dt

(17)

where y is the position of actuator, defining the movement from
left to right is positive and the middle point is zero point, pL is the
pressure in the left chamber, pR is the pressure in the right chamber,
mt is load mass, Bp is viscous damping coefficient, and fp is friction.
Ignoring the pressure loss in pipes, and assuming that temperature and bulk modulus of oil are constant, by continuity equation
of fluid, the pressure in the left chamber is
VL dpL
dy
= qL − Ap
− Cip ( pL − pR ) − Cep pL
dt
β e dt

(18)

where VL is the pressure of the left chamber, βe is the equivalent
bulk modulus, qL is the flow rate that gets into the left chamber, Cip
is the internal leakage coefficient, and Cep is the external leakage
coefficient.
The pressure in the right chamber is
VR dpR
dy
= qR + Ap
− Cip ( pR − pL ) − Cep pR
dt
β e dt

(19)

where VR is pressure of the right chamber and qR is the flow rate
that gets into right chamber.
The volume of left and right chambers is given by
VL = V0 + Ap y

(20)

VR = V0 − Ap y

(21)

where V0 is initial volume of left and right chambers.
For valve control and the new PBW system, when normalized
displacement of servo valve spool xv ≥ 0, the flow rate of actuator
and system output can be expressed by
qs = qL = CdWxv sign( ps − pL )

qR = CdWxv sign( pb − pR )

2

ρ

2

ρ

ps − pL

pR − pb

2

ρ

qs = qR = CdWxv sign( ps − pR )

SEPTEMBER 2017

pL − p b

2

ρ

ps − pR

uc

s + 2ξsωns s + ωns 2
2

uc = sat ( K p (r − y ) / ymax )

1

sat ( x) =  x
−1


x ≥1
1 > x > −1
x ≤ −1

(28)

where x is the input variable of the saturation function.
For the constant pressure variable displacement pump, the output flow rate of the pump is
qp = Dp npη pvup

(29)

where qp is the output flow rate of pump, Dp is the rated displacement of the pump, np is rated speed, ηpv is volumetric efficiency
and, assuming that the efficiency is constant in all working conditions, up is the normalized positive control input of displacement.
In the pump, the system pressure is compared with the set value
pset, and the variable displacement actuator automatically regulates
the displacement of a pump under the action of pressure difference.
This process can be expressed by a simplified first-order system
up =

1
(1 − ps pset )
Ta s + 1

(23)

Tinput = Dpup ( ps − pb ) η pm + T0

(25)

(27)

where Kp is the proportional coefficient, r is the set point, ymax is
the maximum displacement of the actuator, and sat is the saturation
function expressed by

where Ta is time constant of variable displacement actuator.
The input torque of the pump is

(24)

(26)

where ξs is the damping coefficient and ωns is the natural angular
frequency.
A normalized proportional controller as follows is used to control the position of the actuator

(22)

where qs is the output flow rate of the system and it is also the input
flow rate of the servo valve, Cd is the flow rate coefficient of the
servo valve, W is the area gradient of the throttle area, pb is the back
pressure of the system, and ρ is the density of hydraulic oil.
And when xv < 0, the flow rate can be expressed by
qL = CdWxv sign( pL − pb )

xv =

(30)

(31)

where Tinput is the input torque of the pump, ηpm is the mechanic
efficiency of the pump, and T0 is the torque when flow rate is 0.
The input power of the pump is
Pinput = Tinput np (2π / 60)

(32)

where Pinput is the input power of the pump, np is the speed of the
pump, and ηpm is the mechanical efficiency of the pump.
In a value control system, a hydraulic accumulator is installed
in the outlet of a pump to provide instantaneous flow rate and reduce pressure oscillation. The output rate of the system is

IEEE A&E SYSTEMS MAGAZINE

23



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