# Using simulation software to simplify DSP-based Electro-Hydraulic Servo Actuator Designs: Part 2

With a clear idea of the basic electro-mechanical building blocks and their relationship to one another (Part 1), the use of simulation modelling tools makes it possible to derive the basic mathematical models of the hydraulic components, and more, importantly simulate and test them in a typical implementation and extract meaningful results that will be useful in coming up with a final design.

Tools such as the Matlab Simulink allow the developer to first derive a top level system design as shown in Figure A-1 below and then proceed to lower levels and derive models for each of the important elements in the design.

Figure
A-1: Top level system diagram of a typical electro hydraulic system. |

Modelling the Flow Control Servo
Valve

The two stage nozzle-flapper
servo-valve consists of three main parts: an electrical torque motor,
hydraulic amplifier, and valve spool assembly and is shown in detail in
Figure 8, below.

Figure
8. Valve Torque Motor Assembly (illustration courtesy of Moog) |

The torque motor consists of an armature mounted on a thin-walled sleeve pivot and suspended in the air gap of a magnetic field produced by a pair of permanent magnets.

When current is made to flow in the two armature coils, the armature ends become polarized and are attracted to one magnet pole piece and repelled by the other. This sets up a torque on the flapper assembly, which rotates about the fixture sleeve and changes the flow balance through a pair of opposing nozzles, shown in Figure 9, below. The resulting change in throttle flow alters the differential pressure between the two ends of the spool, which begins to move inside the valve sleeve.

Figure
9. Valve Responding to Change in Electric Input (illustration courtesy
of Moog) |

Lateral movement of the spool forces the ball end of a feedback spring to one side and sets up a restoring torque on the armature/flapper assembly.

When the feedback torque on the flapper spring becomes equal to the
magnetic forces on the armature the system reaches an equilibrium
state, with the armature and flapper centred and the spool stationary
but deflected to one side. The offset position of the spool opens flow
paths between the pressure and tank ports (P_{s}
and T), and
the two control ports (A and B), allowing oil to flow to and
from the
actuator.

Modelling the Torque Motor

For simplicity, the electrical characteristics of the servo-valve
torque motor may be modelled as a series L-R circuit, neglecting for
the time being any back-EMF effects generated by the load. The transfer
function of a series L-R
circuit is:

where L_{c} is
the inductance of the motor coil, and R_{c}
the combined resistance of the motor coil and the current sense
resistor of the servo amplifier. Values of inductance and resistance
for series and parallel winding configurations of the motor are usually
published in the manufacturer's data sheet.

The lateral force on the valve spool is proportional to torque motor current, but oil flow rate at the control ports also depends upon the pressure drop across the load.

Modelling the Valve Spool Dynamics

A servo-valve is a complex device which exhibits a high-order
non-linear response, and knowledge of a large number of internal valve
parameters is required to formulate an accurate mathematical model.
Indeed, many parameters such as nozzle and orifice sizes, spring rates,
spool geometry and so on, are adjusted by the manufacturer to tune the
valve response and are not normally available to the user.

Practically all physical systems exhibit some non-linearity: in the simplest case this may be a physical limit of movement, or it may arise from the effects of friction, hysteresis, mechanical wear or backlash. When modelling complex servo-valves, it is sometimes possible to ignore any inherent non-linearities and employ a small perturbation analysis to derive a linear model which approximates the physical system.

Such models are often based on classical first or second order differential equations, the coefficients of which are chosen to match the response of the valve based on frequency plots taken from the data sheet.

A simple first or second order model yields only an approximation to
actual behavior, however the servo-valve is not the primary dynamic
element in a typical hydraulic servo system and is generally selected
such that the frequency of the 90 degree phase point is a factor of at
least three higher than that
of the actuator.

For this reason it is usually only necessary to accurately model valve response through a relatively low range of frequencies, and the servo-valve dynamics may be approximated by a second order transfer function without serious loss of accuracy.

Figure
10. Typical Servo-valve Frequency Response Curve (illustration courtesy
of Moog) |

A typical performance graph for a high-responsive servo-valve is shown in Figure 10, above. Assuming a second order approximation is to be used, suitable values for natural frequency and damping ratio will need to be determined from the graph.

Natural frequency (_{v}) can be read fairly accurately from
the -3dB or 90 degree phase point of the 40% curve. Damping can be
determined from an estimate of the magnitude of the peaking present.
For an under-damped second order system, the damping factor (S_{v})
can be shown to be related to peak amplitude ratio (M_{v})
by
the formula

In this example, a reasonable estimate of peaking based on the 40% response curve would be about 1.5 dB, which corresponds to an amplitude ratio of about 1.189. A suitable value of damping determined iteratively from Equation 2 is about 0.48. Armed with these values, a simplified model of the servo-valve spool dynamics may be constructed.

The input to the model will be the torque motor current derived from Equation 1 in Part 1 normalized to the saturation current obtained from the datasheet, and the output will be the normalized spool position. Shown in Figure A-2 below is the Simulink model of the servo valve.

Figure
A-2. A Simulink model of the servo-valve |

Modelling Valve Flow-Pressure

The servo-valve delivers a control flow proportional to the spool
displacement for a constant load. For varying loads, fluid flow is also
proportional to the square root of the pressure drop across the valve.
Control flow, input current, and valve pressure drop are related by the
following simplified equation:

In the above equation, Q_{L}, is the hydraulic flow
delivered through the load actuator, Q
_{R} the rated valve
flow at a specified pressure drop P_{R}, and i*_{v}
is normalized input current. P_{R} is the pressure
drop across the valve given by P_{V} = P_{S}+
P_{T} + P_{L},
where P_{S}, P_{T}
and P_{L}
are system pressure, return line (tank)
pressure, and load pressure
respectively.

Maximum power is transferred to the load when P_{L}
= 2/3 P_{S},
and since the most widely used supply pressure is 3,000 psi, it is
common practice to specify rated valve flow at P = 1,000 psi
(approximately 70 bar). The static relationship between valve pressure
drop and load flow is often presented in manufacturer's datasheets as a
family of curves of normalized control flow against normalized load
pressure drop for different values of valve input current as shown in
Figure 11, below.

Figure
11. Servo-valve Flow-pressure curves (illustration courtesy of Moog) |

The horizontal axis is the load pressure drop across the valve,
normalized to 2/3 of the supply pressure. The vertical axis is output
flow expressed as a percentage of the rated flow, Q_{R}.
The
valve orifice equation
is applied separately for the two control ports to obtain expressions
for oil flow into each of the two actuator chambers. Since load flow is
defined as the flow through the load: Q_{L} = Q_{A} =
-Q_{B}

Figure
A-3: Simulink model of the Hydraulic Actuator |

As shown in the Simulink model in Figure
A-3, above, the inputs are
command voltage from the amplifier, supply and return oil pressures
from the hydraulic power supply (P_{S} and
P_{T}),
and
load pressures from the actuator chambers (P_{A}
and P_{B}).
Outputs are the flows to each side of the piston (Q_{A}
and Q_{B}),
and the load flow (Q_{L}).

Modelling the Linear Actuator

Cylinder Chamber
Pressure. The relationship between valve control flow and
actuator chamber pressure is important because the compressibility of
the oil creates a "spring" effect in the cylinder chambers which
interacts with the piston mass to give a low frequency resonance. This
is present in all hydraulic systems and in many cases this abruptly
limits the usable bandwidth. The effect can be modelled using the flow
continuity equation from fluid mechanics which relates the net flow
into a container to the internal fluid volume and pressure.

The left hand side of the equation is the net flow delivered to the chamber by the servo valve. The first term on the right hand side is the flow consumed by the changing volume caused by motion of the piston, and the second term accounts for any compliance present in the system. This is usually dominated by the compressibility of the hydraulic fluid and it is common to assume that the mechanical structure is perfectly rigid and use the bulk modulus of the oil as a value for .

Mineral oils used in hydraulic control systems have a bulk modulus in the region of 1.4 x 109 N/m. The above previous equation can be re-arranged to find the instantaneous pressure in chamber A as follows:

Piston
Dynamics. Once the two chamber pressures are known, the net
force acting on the piston (F_{P})
can be computed by multiplying by the area of the piston annulus (A_{P}) by the differential
pressure across it.

**F**_{P} = (**P**_{A} - **P**_{B})
**A**_{P}

An equation of forces for piston motion can now be established by
applying
Newton's second law. For the purposes of this analysis, it
will be assumed that the piston delivers a force to a linear spring
load with stiffness K_{L},
which will allow us to investigate the load capacity of the actuator
later. The effects of friction (F_{f})
between the piston and the oil seals at the annulus and end caps will
also be included. The resulting force equation for the piston is shown
below and may be modelled in Simulink using two integrator blocks.

The total frictional force depends on piston velocity, driving force
(F_{P}), oil temperature and possibly
piston position. One method of modelling friction is as a function of
velocity, in which the total frictional force is divided into static
friction (a transient term present as
the actuator begins to move),
Coulomb friction (a constant force dependent only on the direction of
movement), and viscous friction (a
term proportional to velocity).
Assuming that viscous and Coulomb friction components dominate,
frictional force (F_{f}) can be modelled as

where viscous and Coulomb friction coefficients are denoted by F_{v0} and F_{c0}
respectively. Frictional effects are notoriously difficult to measure
and accurate values of these coefficients are unlikely to be known, but
order of magnitude estimates can sometimes be made from relatively
simple empirical tests.

One test which can yield useful information is to subject the system to a low frequency, low amplitude sinusoidal input signal, and plot the output displacement over one or two cycles. A low friction system should reproduce the input signal, but the presence of friction will tend to flatten the tops of the sine wave as the velocity falls to a level blow that necessary to overcome any inherent Coulomb friction.

In actuators fitted with conventional PolyTetraFluoroEthylene(PTFE)-based bearings, friction is related fairly linearly to supply pressure and oil temperature and care should be taken to conduct testing under representative conditions.

In a first analysis, leakage effects in the actuator are sometimes neglected, however this is an important factor which can have a significant damping influence on actuator response. Leakage occurs at the oil seals across the annulus between the two chambers and at each end cap, and is roughly proportional to the pressure difference across of the seal. Including leakage effects, the flow continuity equation for chamber A

is similar with appropriate changes of sign. It is a relatively simple matter to modify the Simulink model to compute the instantaneous chamber leakages and subtract them from the total input flow.

Modelling the hydraulic power supply

The behavior of the hydraulic power supply described in Part 1 of this series may be
modelled in the same way as the chamber volumes: by applying the flow
continuity equation to the volume of trapped oil between the pump and
servo-valve. In this case, the input flow is held constant by the
steady speed of the pump motor, and the volume does not change. The
transformed equation is

This equation takes into account the load flow (Q_{L})
drawn from the supply by the servo-valve, and accurately models the
case of a high actuator slew rate resulting in a load flow which
exceeds the flow capacity of the pump. In such cases the supply
pressure (P_{S}) falls, leading to a corresponding
reduction in control flow and loss of performance. The action of the
pressure relief valve may be modelled using a limited integrator to
clamp the system pressure to the nominal value.

Next in Part 3: Modelling a typical
servo actuator

To read Part 1 go to : "The
basics of electro-hydraulic servo actuator systems."

Richard Poley is Field Application Engineer at Texas Instruments with focus on digital control systems.

References:1. Richard Poley, DSP Control of Electro-Hydraulic Servo Actuators, TI Application Report SPRAA76

2. Herbert E. Merritt, Hydraulic Control Systems, Wiley, 1967

3. M.Jelali and A.Kroll, Hydraulic Servo Systems - Modelling, Identification & Control, Springer, 2003

4. Electrohydraulic Valves A Technical Look, Moog Technical Paper

5. Moog 760 Series Servovalves, product datasheet

6. R.H.Maskrey and W.J.Thayer, A Brief History of Electrohydraulic Servomechanisms, Moog Technical Bulletin 141, June 1978

7. T. P. Neal, Performance Estimation for Electrohydraulic Control Systems, Moog Technical Bulletin 126, November 1974

8. W.J.Thayer, Transfer Functions for Moog Servovalves, Moog Technical Bulletin 103, January 1965

9. J.C.Jones, Developments in Design of Electrohydraulic Control Valves, Moog Technical Paper, November 1997

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12. D.DeRose, The Expanding Proportional and Servo Valve Marketplace, Fluid Power Journal, March/April 2003

13. D.DeRose, Proportional and Servo Valve Technology, Fluid Power Journal, March/April 2003

14. D. Caputo, Digital Motion Control for Position and Force Loops NFPA Technical Paper I96-11.1, April 1996

15. B.C.Kuo and F.Golnaraghi, Automatic Control Systems, Wiley, 2003

16. J.Schwarzenbach and K.F.Gill, System Modelling & Control, Edward Arnold, 1992

17. J.B. Dabney and T.L. Harman, Mastering Simulink, Pearson Prentice Hall, 2004

18. Digital Control Applications With The TMS320 Family Selected Application Notes (SPRA019)

19. Implementation of PID and Deadbeat Controllers with the TMS320 Family (SPRA083)

20. Gene F. Franklin, J. David Powell & Michael L. Workman, Digital Control of Dynamic Systems,Addison-Wesley, 1998

21. TMS320F2812 Data Manual (SPRS174)

22. Signal Conditioning an LVDT using a TMS320F2812 DSP (SPRA946)

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