LAPPEENRANTA UNIVERSITY OF TECHNOLOGY FACULTY OF ELECTRICAL ENGINEERING
ACTIVE DAMPING AND ENERGY RECOVERY IN A HYDRAULIC BOOM CONTROLLED BY AN ELECTRIC DRIVE
MASTER’S THESIS
Examiners: Prof. Juha Pyrh¨onen, D.Sc. Lasse Laurila Lappeenranta, May 19, 2008
Alexander Smirnov Punkkerikatu 2 A 1 53850 Lappeenranta Alexander.Smirnov@lut.fi
ABSTRACT
Lappeenranta University of Technology Faculty of Electrical Engineering Alexander Smirnov
Active Damping and Energy Recovery in a Hydraulic Boom Controlled by an Electric Drive
Master’s Thesis 2008
55 pages, 24 figures, 5 tables, and 2 appendices
Examiners: Prof. Juha Pyrh¨onen, D.Sc. Lasse Laurila
Keywords: Active damping, hydraulic crane, vibrations, mobile machines, energy recover, drive control
The aim of the study is to obtain a mathematical description for an alternative variant of controlling a hydraulic circuit with an electrical drive. The electrical and hydraulic systems are described by basic mathematical equations. The flexibilities of the load and boom is modeled with assumed mode method. The model is achieved and proven with simulations. The controller is constructed and proven to decrease oscillations and improve the dynamic response of the system.
Acknowledgments
The work was carried out at Lappeenranta University of Technology (LUT) during the period from autumn of 2007 up to spring 2008. I would like to express my sincere appreciation to the people who made this work possible.
At first, I want to thank you, my supervisors Professor Juha Pyrh¨onen and D.Sc.
Lasse Laurila for the possibility to work under your leadership, valuable suggestions and your scientific guidance.
I want to thank Professor Victor Vtorov from St. Petersburg Electrotechnical Uni- versity for his contributions to the work.
I wish to express my thanks to Olli Heinikainen for the help in obtaining data for experimental setup.
Special thanks to Julia Vauterin who has made my live and study in Lappeenranta possible.
I am indebted to my friends and colleges for uncountable discussions and ideas.
Without you the work might not have been finished in time.
I am grateful to my parents for their love and support.
Last but not the least my sweetheart Svetlana – thank you for your understanding and love.
Lappeenranta, May 2008 Alexander Igorevich Smirnov
List of Figures
1 Basic structure of hydraulic system . . . 3
2 The structure of proposed system . . . 4
3 Aerial Venn diagram describes the fields of Mechatronics . . . 6
4 Photo of the experimental setup (by Olli Heinikainen) . . . 7
5 Boom with basement . . . 7
6 Double acting hydraulic actuator . . . 12
7 Friction forces in hydraulic cylinder . . . 14
8 Scheme of hydraulic circuit . . . 15
9 Vector diagram of the PMSM . . . 18
10 Transfer function for electrical drive . . . 19
11 Principle of virtual work . . . 21
12 Sketch of the boom . . . 22
13 Forces applied to the boom . . . 22
14 Assume mode principle . . . 24
15 Drive step response . . . 29
16 Drive state variables during step response . . . 30
17 Hydraulic system step response . . . 30
18 Step response of the rigid boom . . . 31
19 Step response of the flexible boom . . . 31
20 Step response of the system with rigid boom only . . . 34
21 Step response of the system with addition of flexible motion . . . . 36
22 Response of the system with PID regulator . . . 39
23 Response of the system with different PID regulators . . . 40
24 Response of the flexible system with different PID regulators . . . . 40
25 Diagram of state-space elements . . . 41
26 Response of the system with state-space regulator . . . 42
List of Tables
1 Hydraulic fluid parameters . . . 82 Hydraulic pump parameters . . . 9
3 Electrical motor parameters . . . 11
4 Parameters of flexible boom . . . 32
5 Parameters of the system with flexible motion . . . 36
Nomenclature
Symbols
Ap surface area of piston
A state matrix
AHP state matrix of hydraulic circuit
Afl state matrix for the system with flexible motion
AH,b state matrix for the system with rigid boom and hydraulics Abm state matrix for the boom with flexible motion
B matrix of inputs
BHP matrix of inputs for hydraulic circuit
Bfl matrix of inputs for the system with flexible motion
BH,b matrix of inputs for the system with rigid boom and hydraulics Bbm matrix of inputs for the boom with flexible motion
C matrix of measurments
Cfl matrix of measurments for the system with flexible motion
CH,b matrix of measurments for the system with rigid boom and hydraulics Cbm matrix of measurments for the boom with flexible motion
D damping matrix
E Young’s modulus
E0 bulk modulus of hydraulic fluid
F force
Ff friction force
f frequency
I area moment of inertia
I identity matrix
J moment of inertia
JHP moment of inertia of hydraulic pump KP proportional gain
KD derivative gain
KI integral gain
K stiffness matrix
L inductance
Ld, Lq components of the axis inductances
l length
M mass matrix
m mass
mt total mass
mp piston mass
mA,fl mass of hydraulic fluid in the fisrs chamber and pipes
p pressure
Q flow of liquid
QA flow in the first chamber of hydraulic cylinder QLi internal leakage
QLe external leakage
qi generalized displacement
R resistance
T torque
Tf,HP friction torque of hydraulic pump Tmover torque of the drive for hydraulic pump Taux auxilary torque
THP,th theoretical torque of hydraulic pump
t time
tPWM time constant of supplying power electronics tM electromechanical time constant
tEl electrical time constant
V volume
VA volume of the first chamber VB volume of the second chamber
VHP theoretical displacement of the pump
xn state variable
un input signal
W energy
ϕ angle of rotation for the pump/drive
η efficiency
ηHP volumetric efficiency ofthe pump
Ψ flux linkage
Ψd, Ψq components of the stator flux linkage θ angle of rotation of the boom
0 zero matrix
Abbreviations
AMM assumed mode method
DTC direct torque control
DC direct current
PMSM permanent magnet synchronous machine PWM pulse width modulation
VFC voltage frequency converter PID proportional integral derivative LQR linear quadratic regulator
Contents
1 INTRODUCTION 3
2 SYSTEM DESCRIPTION 6
2.1 Mechanical Subsystem . . . 7
2.2 Hydraulic Subsystem . . . 8
2.2.1 Pumps . . . 8
2.2.2 Cylinders . . . 9
2.3 Electrical Subsystem . . . 10
3 MODELING 12 3.1 Hydraulic model . . . 12
3.2 Drive model . . . 16
4 BOOM MODELING 20 4.1 Rigid model . . . 20
4.2 Flexibility model . . . 23
4.2.1 Description with particular shape modes . . . 26
5 SIMULATION 28 5.1 Subsystems . . . 28
5.2 Full system . . . 33
6 CONTROL DESIGN 37
6.1 PID-Control . . . 37 6.2 State-Space Regulator . . . 41
7 CONCLUSION 43
APPENDICES 47
1 INTRODUCTION
Mobile machines are a kind of machines that work either manned or independently in demanding environments. Time of autonomous work is very important factor which is mostly defined by the energy consumption. Energy consumption is also valuable in the sense of rising price for fuels and friendliness to the environment.
The care about environment is directly bound with the question of fuel utilization.
One of the main consumer is a hydraulic subsystem of the mobile machine.
Hydraulic systems in mobile machines are usually presented as the actuation of manipulator. These systems have high rigidity and are able to produce large forces.
Hydraulic actuators have one more important factor, especially in the field of mobile machines, that they are small. Although they have some disadvantages such as non-linear dynamic characteristics, limitation on operating temperature and high energy spending.
The high consumption of energy in hydraulic systems arises from the control tech- niques. The typical hydraulic system, see Figure 1, consists of a cylinder, valve, pump, electrical drive and pipes (Jelali and Kroll, 2004). Electrical drive constantly rotates the pump thus providing flow of liquid. The flow to the cylinder is con- trolled by the valve. That structure consumes energy even if it does not need to produce any work.
Figure 1: Basic structure of hydraulic system
In this research another approach is examined. Modern electrical drives can provide fast response, and hence, it is possible to eliminate the valve and control the flow of liquid with the drive. The scheme is presented in Figure 2. The main part is the electrical drive with a variable frequency converter. An electric motor can work in both directions of energy flow e.g. as a motor and as a generator. While in generator mode the drive is powered by the hydraulic motor/pump.
Figure 2: The structure of proposed system
In this way the system consumes energy only on its positive movement. While moving back the system can restore a part of energy, whose amount depends on load. Finally not only constant consumption of energy is eliminated but a recovery technology introduced. The disadvantage is that we can not work with double- acting asymmetric cylinders in that case two drives are needed. Although the symmetric double-acting cylinder suits fine.
A typical example of hydraulic system in mobile machines is hydraulic cranes.
Such cranes can experience structural flexibility. It is perceptible because cranes lift heavy loads with a mass more than the structure itself. What is more the considerable crane length plays its role in flexibility. (Linjama and Virvalo, 1999) The lack of stiffness causes oscillations in systems with high dynamics. These vibrations are the reason for extra wear and damage. It is evident that vibrations should be avoided.
This goal can be reached in several different ways. The most simple is to keep the system from causing vibrations itself. This way means not to achieve speeds and accelerations that are greater or even close to the lowest natural frequency of object.
The lowest frequency can be determined experimentally, on that basis a common spring and damping system could be built as in (Nissing, 2000). This approach can provide a very simple model, and according to the author the accuracy is sufficient.
The problem is that the method does not include the load in separate manner and it is difficult to predict what parameters are affected by it. Although author suggested some techniques for solving this problem neither of them are described in addition to the model, so more effort in load estimation is needed. The solution was proposed by (Kovanen, 2003), where using computer design lowest frequency for all range of loads and boom lengths have been estimated.
These methods work well but they limit the speed of operation thus increasing the on state of the system. On the other side it is possible to take into account all the frequencies as in (Mikkola, 1994), but it leads to a huge computation burden and difficulties in applying control law. Linjama in his work (Linjama and Virvalo, 1999) suggested a model based on several lowest frequencies and showed the way to choose them.
The last methods can be referred as active damping methods. In general, damping techniques can be classified in three groups: passive damping, active damping and semi active damping. The first type uses some additional structures to introduce damping effect. The disadvantage of this method is the necessity for additional space and increased weight of the system. The second based on implementing control technique to apply additional force that will calm the oscillations. The last approach has a control system but it implements switching between structures with different parameters thus providing needed effect. The active damping approach has one more important benefit that it can save energy. The energy of oscillations in passive and semi active cases is dissipated through friction. While using an active method, energy can be stored or returned back to the grid.
All the mentioned approaches based on controlling the hydraulic system with pro- portional or servo valves. In this work an attempt is made to solve issues with another approach based on the control of an electrical drive rotating the hydraulic pump.
2 SYSTEM DESCRIPTION
The system presented in Figure 2 and shortly described in the introduction from general view can be considered as a mechatronic object. Mechatronics is a relatively new science; the term itself appeared in the year of 1969. This science represents cohesion of several fields that are shown in Figure 3. Such unity helps to solve challenging engineering tasks in hybrid systems. That results in better reliability, faster development and economically profitable systems.
Figure 3: Aerial Venn diagram describes the fields of Mechatronics
Each part plays its own significant role. Mechanical part describes the physical structure of the system. Electrical part represents the source of power and also responsible for censors in this particular case. These two fields are tied together by the means of electromechanical conversion of energy in electrical drive. Both of them are an object for control system theory. Control theory is specialized in mod- eling the system and achieving desired output results with special mathematical methods. The last but not the least part of mechatronics - computers is character- ized by computer aided design, virtual prototyping, and model simulation.
The system used for current investigation is presented in Figure 4. It consists of a hydraulic motor, a hydraulic cylinder, an electrical drive, a basement, and a flexible boom. The boom represents the first part of a hydraulic crane to reduce the complexity of the model.
Figure 4: Photo of the experimental setup (by Olli Heinikainen)
2.1 Mechanical Subsystem
The mechanical part is presented with a boom and its basement. The boom, as it was mentioned, is assumed as the first part of the crane. This assumption is enough to validate the system structure and check the ability to eliminate vibrations.
Figure 5: Boom with basement
Cranes in mobile applications have some common features. The first one is the length relatively big for the machine itself. The length increases the ability to
operate in difficult environment and speeds up the work. The second is the hollow construction to reduce weight, the importance of that factor was mentioned earlier.
The boom is made of steel, with its relatively large length and hollow construction, it provides enough flexibility for this investigation. The density of the material is 7801 kg/m3, the modulus of elasticity or Young modulus is2,07·1011 N/m2. The dimensions of the mechanical subsystem are presented in Figure 5.
2.2 Hydraulic Subsystem
For the hydraulic subsystem a special fluid is used. It is of VG 32 class of viscos- ity according to ISO standard. In particular it is “ESSO UNIVIS N32” with the following parameters
Table 1: Hydraulic fluid parameters
Parameter Value Units
Density 844,4 kg/m3
Viscosity 15,7567 cSt Bulk modulus 1,29873·109 Pa
2.2.1 Pumps
Hydraulic pumps are used to move liquids. Usually pumps are driven rotationally.
By moving liquid they add energy to the system in order to overcome the difference in pressure. In that way the translation of mechanical energy to hydraulic energy happens.
Pumps can be separated in two general groups: rotodynamic pumps and positive displacement pumps. Positive displacement type is more common in industrial application. Its principle for moving fluid is trapping a fixed amount and then displacing that amount into the discharge pipe. The volume of fluid transported by a single rotation is called displacement or flow output.
Positive displacement pumps are subdivided into fixed and variable. In variable pumps such as piston pump it is possible to change the flow output. However in fixed displacement pumps the volume of displaced fluid is constant. The most popular types of fixed pumps are vane pumps and gear pumps.
In particular system there are some additional requirements to the pump. The pump should work not only in its traditional way but also in the reverse direction.
In the other words pump should act as a motor and convert the flow of liquid in rotational motion. This is to say the key principle of the system. The kind of pump that satisfies the requirements is an internal gear pump with constant displacement volume. Table 2 describes the main parameters of the pump.
Table 2: Hydraulic pump parameters
Parameter Value Units
Volume 13,3 cm3/rev
Continuous operating pressure 250 bar
Max speed 4000 min−1
Nominal speed 200−3600 min−1
Efficiency volume 94 %
2.2.2 Cylinders
Hydraulic cylinders are a type of hydraulic actuators. These devices translate hydraulic energy to mechanical. Cylinders (also called a linear hydraulic motor) in particular convert hydraulic energy and give a linear force or motion. The cylinders that convert force in one direction are known as single-acting.
Double-acting actuators can apply force in both directions. The type of these actuators with one rod is of the popular use and usually called asymmetric or differential cylinder. With one rod these type consume less space. The drawback is that when combined with symmetric valve the pressure jump exists around null velocity (Jelali and Kroll, 2004). Despite the disadvantage that makes the control more difficult, this type of actuators is good for closed loop control systems.
The hydraulic cylinder used in the setup is of the double-acting type. Although the structure for recovering energy needs a single-acting one, it is possible to use a double-acting. The cylinder’s first chamber is connected to the pump and performs all the work. While the second chamber is connected to the tank with atmosphere pressure. This connection provides only the necessary oil and prevents problems with leakage between chambers.
Cylinder in the installation has a rod with diameter 30 mm and length 300 mm.
The piston’s diameter is 50mm and mass is 3,913 kg.
2.3 Electrical Subsystem
As it was mentioned earlier pumps are driven rotationally and convert mechanical energy into hydraulic. The source of mechanical energy is electrical drive; it trans- lates electrical energy into mechanical one. The controllable electrical drive consists of power electronics, an electrical motor, a controller and measuring equipment.
Controller obtains values through measuring equipment, calculates the signal for power electronics which converts electrical energy in a suitable form for the motor.
The last one provides mechanical power.
Permanent Magnet Synchronous Machine (PMSM) became a very popular solution in modern drives. PMSM has a high efficiency, a high power density and a small volume these features make it so attractive. PMSM is a synchronous AC machine where the rotor winding is replaced with permanent magnets and stator winding is stayed the same, usually with 3 phases. Table 3 presents the parameters of the motor.
Variable frequency controller (VFC) is the part of the system responsible for sup- plying energy in needed form and converting energy back to the network. VFC usually transform alternating supply voltage into constant one. On the next step the power is modulated in the form of suitable frequency and magnitude for the motor. The modulation in modern devices is produced by pulse width modulation technique (PWM). PWM constructs sinusoidal output waveform with a sequence of narrow voltage pulses and controls the width of the pulse. (Mohan et al., 2003)
Table 3: Electrical motor parameters
Parameter Value Units
Rated speed 3000 min−1
Standstill torque 14,5 N·m
Dynamic limit torque 52,2 N·m
Standstill current 10,1 A
Maximum current 40 A
Mass moment of inertia of the rotor 22.3·10−4 kg·m2
Phase inductance 5.7 mH
Phase resistance 533 mΩ
In the setup one of the modern drives is used. It is a high performance machinery drive from a well-known manufacture. This drive can control both synchronous and asynchronous machines including PMSM with an implemented direct torque control technique. The ability of using two control variants: speed and torque or motion control; makes it a very flexible tool. In addition inbuilt software provides the ability to implement different control laws without external devices. The switching frequency of the drive varies from 2 up to 16 kHz. The value of 4 kHz is used by default.
3 MODELING
Modeling is the way of describing the system in the universal language of math.
The mathematical model not only helps to save time but also provides a perfect space for testing new ideas. Nowadays it is difficult to imagine a system that is developed without use of computer aided design. In this work modeling is the necessary step to obtain mathematical description for control synthesis.
3.1 Hydraulic model
The description of hydraulic systems in most sources is made from the point that system is controlled through the valve. (Jelali and Kroll, 2004), (de Silva, 2005) In this research a different approach is used. Despite that it is possible to partly adapt models from those sources for current needs.
The hydraulic cylinder is the main linear actuator in the system. It operates a boom and plays a role in definition of the system’s dynamic. The general model is presented in Figure 6.
Figure 6: Double acting hydraulic actuator
Using equations of flow for both chambers yields QA−QLi = ˙VA+ VA
E0(pA)p˙A (3.1)
QB+QLi−QLe = ˙VB+ VB
E0(pB)p˙B (3.2)
whereQ denotes to the flow and subscripts “A”, “B”, “Li”, “Le” describe first cham- ber, second chamber, internal and external leakage respectively. Ff is the friction force, V is the volume of corresponding chamber and E0 is the bulk modulus.
In this research a double-acting cylinder is used as a single-acting one. The de- scription in that case can be obtained just by using equation (3.1). The effect of the second chamber is neglected.
The equation of piston motion is
mtx¨p+Ff( ˙xp) =AppA−Fext (3.3) whereFf represents a friction, Fext is a force of load,mt is the total mass of piston itself mp and hydraulic fluid in pipes and chamber mA,fl.
mt =mp+mA,fl (3.4)
The mass of fluid is usually much less than the piston mass and can be often omitted.
It is familiar to describe the friction force as a function of velocity. With this statement in mind we can obtain
Ff( ˙xp) = Fv( ˙xp) +Fc( ˙xp) +Fs( ˙xs) (3.5) where Fv is a viscous friction, Fc is a Columb friction, and Fs is a static friction.
The friction force is always a very complicated object and plays an important role in the dynamics of the actuator. Only more difficulties occur in hydraulic cylinder when taking in concern the friction in fluids. Mainly the friction appears between the piston and cylinder and acts in the opposite direction of the movement. The type of friction between layers of liquid is called viscous friction. It appears when surfaces of cylinder and rod do not contact. In the case when there is tangency among seal and cylinder a more difficult event occurs. This one is called the Coulomb friction. One more case that can happen is static friction but it is rarer especially in systems with fluid. A detailed description of friction types is provided by Olsson et al. (1998). The method of identification of friction effect is described by Hsu et al. (7-10 May 1996). The movement, pressure, stand time, temperature and many other parameters influence the amount of each friction component in
final description. Friction is the parameter that cannot be described and estimated analytically with enough accuracy. In that case it is always better to get actual measured values. Figure 7 shows the friction forces, the picture is reprinted from the work of Mikkola (1994) with his kind permission. Vertical axes is a friction
Figure 7: Friction forces in hydraulic cylinder
force in [N], horizontal axes is a speed in [mm/s]. Solid line shows the actually measured value, and thin line shows the analytical nonlinear model.
This work does not concentrate on detailed friction explanation and uses the simple viscous model.
The hydraulic system is presented with a cylinder and a pump. Adapting non-linear state-space model from (Jelali and Kroll, 2004) with assumptions:
• The tank pressure is constant
• The inefficient volumes are modeled as additive inefficient strokes
• The pump delivers a constant supply pressure
• The leakage between chambers neglected
• The load and piston are rigid
gives the definitions of state and input variables as x1 ≡xp x2 ≡x˙p x3 ≡pA
u2 ≡Fext (3.6)
The model itself can be presented as
˙ x1 =x2
˙ x2 = m1
t [x3Ap−Ff(x2)−U2]
˙
x3 = EVA0(x3)
A(x1)[Q(x3,ϕ)˙ −Apx2]
(3.7)
In valve controlled systems the flow highly depends on the pressure in the cylin- der’s chamber. Considering that electrical drive can provide enough torque this dependence can be neglected. Finally the flow is described as
Q=ηHPVHP
2π ϕ˙ (3.8)
whereVHP is a theoretical displacement of the pump,ηHP - volumetric efficiency of the pump, and ϕ - rotation angle of the pump.
Figure 8: Scheme of hydraulic circuit
Further it is possible to consider that the bulk modulus and chamber volume are constants and get the following model, whereU2 ≡ϕ. Figure 8 shows the principle˙ scheme of simplified hydraulic circuit that is described by the following equations
˙ x1 =x2
˙ x2 = m1
t [x3Ap−Ff(x2)−U2]
˙
x3 = EV0A
A
ηHPVHP
2π ϕ˙ −Apx2
(3.9)
Omitting friction forces it is possible to represent the model in the following matrix form
˙
x=AHx+BHϕ˙ (3.10)
where matrices them self described below
AH=
0 1 0
0 0 Amp
t
0 −EVA0
AAp 0
BH=
0 0 ηHPV2πHPEV0A
A
(3.11)
3.2 Drive model
Control of PMSM is a big field of science and this topic is covered in many sources.
Thus in (Vas, 1998) the latest modern techniques are described in great detail. The main disadvantage of such sources is the complexity of the model that is adapted for advanced control of the drive itself. It makes difficult to incorporate that model in general system for synthesizing the control law.
An important thing is choosing VFC for the dive setup as it is the main source of power both consumed and recovered and also it realizes the applied control strategy. According to Mohan et al. (2003) there are three main considerations when combining power electronics with the motor:
• Current rating
• Voltage rating
• Switching frequency and the motor inductance
Current rating that should be available is proportional to the peak torque. The torque can be derived from well known equations. The second Newton’s law gives an equation for the torque balance
JHPϕ¨+Tf,HP( ˙ϕ) = Tmover−ηHPTHP,th−Taux (3.12) whereJHPis the moment of inertia of the pump, Taux is the auxiliary torque,Tmover is the torque of the drive for the pump, THP,th is the theoretical torque, and Tf,HP is the frictional torque. ϕ¨ and ϕ˙ represent the second and the first derivative of the angle ϕ with the respect to time.
The same equation is available for the motor itself
Tm=Jϕ¨+Tf,m( ˙ϕ) +Tload (3.13)
where Tm is an electromagnetic torque of the motor, J moment of inertia of the motor, Tf,m is a friction torque, and Tload is a torque of the load. Substituting equation (3.12) into (3.13) it is possible to get the torque for the motor
Tm =Jϕ¨+Tf,m( ˙ϕ) +JHPϕ¨+Tf,HP( ˙ϕ) +ηHPTHP,th+Taux (3.14)
The rating of supply voltage is primary dependent on the maximum rotational speed of the motor. In the task of eliminating oscillation the motor does not achieve the maximum speed. That parameter should be estimated with the desired maximum speed of lifting the load. Considering that the maximum speed of the load is υ the drive speed in radians can be calculated as follows
ω = 2π Ap ηHPVHP
(3.15)
As PMSM has a relatively small inductance, so the switching frequency should be high. It is important because the drive should have a smooth torque value to eliminate oscillations provided by the boom. Of course a compromise must be found between losses in converter and torque ripple.
The PMSM is usually described with two mathematical models. The first one is in stationary coordinate system attached to axes ofa and b phases of stator winding.
The second is rotating coordinate system attached to a rotor of a synchronous machine.
The stationary model provides the most full and precise description of the PMSM.
The model contains the real physical quantities and is suitable for investigation the inner processes of the drive. In contrast such a description is too complex with parameters that are depending nonlinearly of the rotation angle.
The model in rotating system is useful because it has constant parameters and in general close to the model of the DC drive. The disadvantage is that description contains some abstract parameters that are not related to real quantities.
In this work the second type model is used, it is often referenced as a model based ond–q frame theory. The description can be found in the works of Vas (1998) and many other sources. A short and clear description is provided by Bian et al. (18-21 Aug. 2007).
Figure 9: Vector diagram of the PMSM
A vector diagram of the PMSM is presented by Figure 9. Voltages of the stator would be
ud = dΨd
dt −Ψqω+rsid uq = dΨq
dt −Ψdω+rsiq
(3.16) where ud,uq are d and q components of stator winding voltage respectively. Vari- ables Ψq and Ψd denotes components of the stator flux linkage, id and iq – compo- nents of stator winding current. The symbolω describes electric angular speed and rs is stator winding resistance. The electromagnetic torque is obtained as follows
Te= 3
2p(Ψdiq−Ψqid), (3.17) where pis the number of pole pairs and flux linkage is presented by
( Ψd =Ldid+ΨPM
Ψq =Lqiq (3.18)
A variable ΨPM denotes permanent motor flux linkage and Ld, Lq are axis induc- tances.
A simple model of PMSM ind–qcoordinates is derived by Борцов and Соколовский (1992). It is sufficient for presenting the dynamic behavior of the drive in more general system. Below the main points from that source presented.
The simplified model of the drive with PMSM is presented in Fig. 10. The model is linearized at the working point with following assumptions:
Figure 10: Transfer function for electrical drive
• The magnetic circuit is in linear region
• The magnetic flux from permanent magnets is constant
• The reaction of rotor and residual electromagnetic moment are neglected
• The stator field is circular
These assumptions are very common and used widely in literature for example in (Vas, 1998).
With the additional assumption that the drive is working at relatively low speed the following equations are valid
W1(s) = 1
tPWMs+ 1 W2(s) = 1 W3(s) = R−1
tEls+ 1 (3.19) wheretPWM is a time constant of supplying power electronics. It can be calculated as f 1
PWM where fPWM is the switching frequency of pulse width modulation block, see VFC in section 2.3 on page 10. tEl is an electrical time constant of the stator winding and defined as RL, whereL=Lq=Ld is the inductance of stator winding and R =rs is the active resistance.
After substituting the equations from (3.19) into the structure presented in Fig. 10 with simple algebraic operations we get the following transfer function
W(s) = ks
s(tMs+ 1)(tPWMs+ 1)(tEls+ 1) (3.20) where ks = Ck
f is a speed transfer coefficient of the drive. tM = (CJ R
f)2 is an elec- tromechanical time constant. Cf is a flux linkage coefficient,J - inertia moment of the rotor. The variable s denotes to the Laplace transformation. The input is a control voltage and output is an actual angle of the rotor.
Furthermore the modern FVC use very high switching frequencies. The pole intro- duced by these devices does not affect the low frequency dynamics of the system.
Finally the system can be reduced to the 3rd order.
Using common technique, described in every book about control theory (Franklin et al., 1998) for example, the system represented in state-space form
Ad=
0 1 0
0 0 CJf 0 −tCf
elR −t1
el
Bd=
0 0
0 −1
ks
telR 0
Cd=
0 0 1
,
(3.21)
where state variables are x1 ≡ ϕ, x2 ≡ ω, x3 ≡ iϕ, and inputs are u1 ≡ u(t) – control signal, u2 ≡TL(t) – torque of the load.
4 BOOM MODELING
4.1 Rigid model
Firstly the rigid model of the boom is considered for examination. The hydraulic cylinder itself carries elasticity and damping elements. The damping comes from the viscous and Coulomb friction. Elasticity is a property of fluid itself and also of mechanical compliance.
The common way to describe the motion of connected bodies is the use of Newton’s laws. In the same way the system can be described by the principle of virtual displacement (Craig, 1981). In addition this method already includes inertia forces that on the contrary should be brought in Newton’s mechanics. Furthermore virtual displacement approach is powerful in describing continuous bodies.
The simple example presented by Figure 11 explains the basis of the method. The letters V and θ specify the displacement and angle of rotation respectively. The symbolθ is used in order to make the clear difference between the angle of rotation ϕ of a pump/drive. The virtual displacement of the structure is denoted by δV or
Figure 11: Principle of virtual work δθ depending whether it is linear or rotational movement.
The main principle of virtual displacement states that: consolidated virtual work of real and inertia forces for random virtual displacement must be equal to zero or in the language of math
δW0 ≡δWreal+δWinertia = 0. (4.1)
The main goal is to describe the oscillations of the system. For this purpose the motion of model assumed near the equilibrium state and the general large rotational displacement is not included in the description. For that reason the angle of rotation supposed to be small.
The values in Figure 11 can be evaluated as follows
V(x, t) =xtanθ(t). (4.2)
It is considered that θ is small
V(x, t) = xθ(t)
δV(x, t) = xδθ(t) . (4.3)
Figure 12 shows the boom with idealized components that define the motion. Letter l denotes to the length of the boom, parameter a shows the point of forces appli- cation, k is the spring constant, c is a coefficient of viscous damping, and f(t) is the force that operates the system.
All the applied forces and their points of application that affects the motion of the boom are explained in Fig. 13. The origin of coordinates labeled by lettersy andx,
Figure 12: Sketch of the boom
Figure 13: Forces applied to the boom
letters a and b show the position of applied forces. Spring, damping, inertia, load and applied forces denoted respectively asfs, fd, fi, fLandfp. The bending moment of inertia is Ml.
According to the chosen math model the equation of virtual works is δW0 =−fs(aδθ)−fI(lδθ
2 )−MI(δθ) +fp(aδθ)−fD(aδθ)−fL(bδθ). (4.4)
Forces themselves can be evaluated as follows fs =kaθ fI= ml2 θ¨ MI =
ml2 12
θ¨ fp =f(t) fD=caθ˙ fL =mLb2θ¨
. (4.5)
Substitution (4.5) into (4.4) and simplification produces ml2
3 +mLb3
θ¨+ (ca2) ˙θ+ (ka2)θ−af(t)
δθ= 0. (4.6)
As δθ6= 0, the equation of motion is ml2
3 +mLb3
θ¨+ (ca2) ˙θ+ (ka2)θ=f(t)a. (4.7)
For the purpose of integrating this equation (4.7) into the general model it is converted to the matrix form. The sate variables are
x1 ≡θ x2 ≡θ˙ u(t) = f(t), (4.8) and matrices are
Ab =
"
0 1
−ml23ka+3m2Lb3 −ml23ca+3m2Lb3
#
Bb =
"
0
3a ml2+3mLb3
#
. (4.9)
4.2 Flexibility model
Describing the flexibility is an essential task to achieve good results in reducing vibrations. A huge variety of methods to specify the system exists, their overview is presented by Shabana (1997). These methods involve nonlinear equations for precise description and it makes them difficult to use for control of the system.
The most accurate model is achieved using finite element method. That method is less valuable for this work because of its huge computation demands. However it can be used to achieve necessary results for assumed mode method - the one that provides sufficient correctness and can be linearized. The full process of receiving the flexible model is presented by Linjama (1998).
It was mentioned that the observed structure consists of 2 links: the basement and the boom. It is sufficient to consider that only the last one is flexible. The basement has a much greater cross section and less length so it will not introduce great effect in the flexibility of the system.
Figure 14: Assume mode principle
The assumed mode method chosen for implementing flexibility defines a function u(t, x). This function describes the displacement of the particular point x of the body from its neutral axis according to the time t Craig (1981). To separate the time dependent part of the function it is presented in the form:
u(x, t) = φ(x)q(t), (4.10)
where φ(x) is a shape function or the function that satisfies kinematic constraints placed on a body and approximates the deformation of the body. In the theory of AMM it is also convenient to call such functions assumed modes. The second part q(t)is a time function also called generalized displacement. Figure 14 describes an example.
The system’s dynamics is described by Lagrange’s equations d
dt δT
δq˙i
− δT
δqi +δW
δqi =Qi, i= 1,2, . . . , N, (4.11) where Q1, Q2, . . . , QN are generalized forces. Variation of coordinates δqi(i = 1,2, . . . , N) must satisfy the restriction of independence. The total kinetic en- ergyT, potential energyV, and virtual work of nonconservative forcesδWnc of the system must have the following forms
T =T(q1, q2, . . . , qN,q˙1,q˙2, . . . ,q˙N, t) V =V(q1, q2, . . . , qN, t)
δWnc =Q1δq1+Q2δq2+. . .+QNδqN
(4.12)
When describing the system with more than one degree of freedom the function u(t, x) can be approximated by the finite series as following
u(x, t) =
N
X
i=1
φi(x)qi(t) = φTq. (4.13) The Lagrange’s equation in that case can also be used. Expressing T, V, and δW through (4.13) leads to equation of motion for the N-degree of freedom model
M(q)¨q+Kq=Q. (4.14)
The potential energy is expressed as W = 1
2
N
X
i=1 N
X
j=1
kijqiqj, (4.15)
where
kij = Z L
0
EIφ00i(x)φ00j(x)dx. (4.16) The parameter E denotes the Young’s modulus of boom material, while I is the area moment of inertia. The moment of inertia for the boom can be defined as
I = Z
x2dm= ml2
3 . (4.17)
The double apostrophe (00) denotes the second derivative with the respect to variable x. The kinetic energy is described by following equations
T = 1 2
N
X
i=1 N
X
j=1
mijq˙iq˙j, (4.18) where
mij = Z L
0
Sρ φi(x)φj(x)dx. (4.19) The material density in the above equation is denoted as ρ, and S is the cross section. The generalized forces are given by
pi = Z l
0
f(x, t)φi(x)dx. (4.20)
The additional damping can be included as a viscous damping for eliminating higher frequencies modes according to Linjama (1998). It is suggested that damping matrix can be chosen proportional to the stiffness.
It is important that assumed modes, as it was mentioned earlier, must satisfy boundary conditions. For the boom which is connected to the base through a rotational joint the boundary conditions are
u(0, t) = 0
EIu00(x, t)|x=0,x=L = 0. (4.21) The first condition states the structural integrity of the system. The absence or rotary inertia is introduced by the second one.
4.2.1 Description with particular shape modes
The selection of modes is an important factor that affects the accuracy of the model.
For this reason it is good to have a suggestion on this topic. Fortunately the needed information can be obtained from the solution of finite elements model. There are different engineering products that can handle such task ANSYS Multiphysics or SolidWorks just few of many. The results for this research were obtained by Mikkola (1994) where the same experimental setup was used.
The higher frequency modes can be truncated because they do not affect seriously the dynamics of the system and anyway used techniques are not able to operate with such speeds. For eliminating these higher modes another approach should be used, piezoelectric materials looks promising (Sun et al., 2004). Afterwards the three lower mode shapes have been chosen.
φ1 = sin(πxl) φ2 = sin(3πxl) φ2 = sin(6πxl)
(4.22)
Using these shapes the system showed in Fig. 13 is described. The equations for mass matrix is
mij = Z l
0
Sρ φi(x)φj(x)dx+mLφi(b)φj(b). (4.23) The functions sin is symmetric so the part of equation under integral sign will be zero for non diagonal elements. The matrix is symmetric and will have the
following look
Sρ2l +mLφ21(b) mLφ1(b)φ2(b) mLφ1(b)φ3(b) mLφ2(b)φ1(b) Sρ2l +mLφ22(b) mLφ2(b)φ3(b) mLφ3(b)φ1(b) mLφ3(b)φ2(b) Sρ2l +mLφ23(b)
(4.24)
The components of general stiffness matrix is described as kij =
Z L 0
EIφ00i(x)φ00j(x)dx+kφi(a)φj(a), (4.25) the matrix itself is
EI(πl)42l +kφ21(a) kφ1(a)φ2(a) kφ1(a)φ3(a) kφ2(a)φ1(a) EI(πl)42l +kφ22(a) kφ2(a)φ3(a) kφ3(a)φ1(a) kφ3(a)φ2(a) EI(πl)42l +kφ23(a)
(4.26)
The elements of damping matrix are
cij =cφi(a)φj(a). (4.27)
The forces that act on the system are
pi =f(t)φi(a). (4.28)
Finally the system was described using the Lagrange’s equation in the form suitable for implementing control law
M¨q+D ˙q+Kq =Q, (4.29)
In state space representation it has the following form
"
˙ x1
˙ x2
#
=
"
0 I
−M−1D −M−1K
# "
x1 x2
# +
"
0 M−1P
#
u(t) (4.30)
5 SIMULATION
Simulation is an important part of a design procedure. It helps to save effort and recourses in prototyping from one side and get deeper understanding of inner pro- cesses. In this research simulation is used to validate models to be sure that they produce values close to the real one. The second reason is that no prototype is avail- able for testing purposes. The last but not the least reason is that implementation of synthesized control laws can be rather time consuming.
5.1 Subsystems
The most common type of input signals for testing purposes is a step function. It is became a de facto standard in control engineering society. The function itself defined as a discontinuous function that has positive unit value when argument is nonnegative and zero when it is negative. This function is also known as a Heaviside step function. The math description is following
H(x) =
( 0, x <0
1, x≥0 (5.1)
Lots of efforts have been put in analyzing the response for such type of input and most systems nowadays are described by the values obtained from that impact. In this section the step function is used until the opposite is stated.
Modern electrical drives either have a measured position and/or speed of a rotor or estimate these parameters. Assuming that both the speed and position are available the position signal can be used as a feedback. Updated equation (3.21) will have the following look
Ad =
0 1 0
0 0 CJf
−tks
elR −tCf
elR −t1
el
Bd =
0 0
0 −J1
ks
telR 0
Cd=
1 0 0
,
(5.2)
Firstly the model of the drive is examined. The input signal is the value in radians,
the output is the angle of rotor also in radians. The response of the system is presented by Figure 15. It can be seen that steady time is about 0,3 s, and overshoot is less than 5%.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.2 0.4 0.6 0.8 1
time [s]
Position [rad]
Figure 15: Drive step response
The results are accurate, PMSM is designed to have fast operation and in combi- nation with VFC provide a stable system. It is useful to note that system operated in linear region so the limitations of available current and torque have not been shown with such small rotation angle. It can be seen in Figure 16 that speed and torque do not exceed their maximum values. There is no reason to examine the system in saturated regions because the target is the elimination of oscillations and it can be achieved in linear region.
The hydraulic subsystem is modeled without load and includes both pump and cylinder. The input is rotational speed that represents the rotation of the pump.
The position of the piston acts as an output. The input value of 1 rad/s is and equivalent of the 1,99 cm3/s flow of liquid.
Figure 17 describes the response of the system. It can be seen that input signal is applied at time value of 1 s. Furthermore as the description of the system does not include dead volume and other nonlinearities the response is a straight line. Such model satisfies the needs of current work because vibrations often occur during work but not start up process so dead volumes of cylinder are already filled.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0
1 2 3 4 5 6 7
time [s]
Speed [rad/s]
(a) Speed
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
−1
−0.5 0 0.5 1 1.5 2
time [s]
Torque [Nm]
(b) Torque
Figure 16: Drive state variables during step response
0 0.5 1 1.5 2 2.5 3
−0.5 0 0.5 1 1.5 2 2.5x 10−3
time [s]
Piston position [m]
Figure 17: Hydraulic system step response
The input flow is assumed to be constant as the natural pressure feedback will be incorporated lately in combination with drive model. Taking that statement in account the position can be approximated as the flow divided by piston surface.
The value will be about 0,1 cm/s which correlates nicely with the figure.
At the next step the rigid boom is examined. The input for the system is a force and the output is a displacement of the boom at particular point in meters. Figure 18 shows both position and speed of the same point.
0 1 2 3 4 5 6 7 8 9 10 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 10−5
time, s
Position [m]
(a) Position
0 1 2 3 4 5 6 7 8 9 10
−4
−3
−2
−1 0 1 2 3 4x 10−5
time, [s]
Speed, [m/s]
(b) Speed
Figure 18: Step response of the rigid boom
It is seen that lightly damped oscillations occur during that input. The frequency is 3,24 rad/s and damping is7,27·10−3. Also it should be noted that input force of 1 N is very small and the displacement less them1 mm fits such input.
The same investigation is made for the flexible part of the boom. Results are presented by Figure 19
0 0.5 1 1.5 2 2.5 3 3.5 4
0 1 2 3 4 5 6 7 8x 10−8
time, [s]
Position, [m]
1 1.05 1.1
0 2 4 6 8x 10−8
(a) Position
0 0.5 1 1.5 2 2.5 3 3.5 4
−3
−2
−1 0 1 2 3x 10−5
time, [s]
Speed, [m/s]
1 1.05
−4
−2 0 2 4x 10−5
(b) Speed
Figure 19: Step response of the flexible boom
The first most noticeable thing is that frequencies are slightly higher. The second is that the oscillations are tending to damp quickly at least to some level because the
damping factor of the boom itself is taken in account. Comparing with the rigid part of the boom we can see that speeds are grater according to position value.
Table 4 shows values of frequencies and damping factors.
Table 4: Parameters of flexible boom
Number Frequency [rad/s] Frequency [Hz] Damping factor
1 255 40,58 4,36·10−3
2 677 61,75 3,88·10−3
3 1870 297,62 6,81·10−3
The next step is to combine rigid and flexible motion of the boom. For that purpose lets define a function
φ(x) = h
φ1(x) φ2(x) φ3(x) i
, (5.3)
that is the row of shape modes. The matrices of the rigid part is defined by subscript
“b” and of the flexible by “f”. The combination of equations (4.9) and (4.29) is
Abm =
0 I
0 0
Af(2,1) ... Af(2,2) ...
0 0
0 Ab(2,1) 0 Ab(2,2)
Bbm=
0 φT(a) Bb(2)
Cbm=
φ(a) 0 0 φ(a)
(5.4)
where A(j, k) corresponds to the element of the matrix at the row j and column k.
The simulation of this model will not provide any additional information to the one already presented. Below is shown how to combine models of different parts of the system. In the following text the model of rigid motion and model with both rigid and flexible motions are treated separately to show the difference.
5.2 Full system
The model described by equation (4.9) has the force as the input. The force can be expressed in terms of hydraulic circuit as F = pA, where p is pressure and A - the surface area (piston surface in the first chamber in our case). One more important thing is that according to the equation (4.3) the vertical displacement was considered as the angle of rotation. In fact at this step it is possible without lose of accuracy to replace the rotation angle with the position of hydraulic cylinder.
Furthermore it is convenient to move piston mass to the boom’s model and describe it as a point mass. With this additions, combining equations (4.9) and (3.11) it is possible to obtain
AH,b =
0 1 0
−ml2+3m3kaLb32+3mta3 −ml2+3m3caLb32+3mta3 Apml2+3mL3ab3+3mta3
0 −EVA0
AAp 0
BH,b =
0 0 ηHPV2πHPEVA0
A
. (5.5)
In model aggregation it is needed to introduce a natural pressure feedback. The torque that acts on the pump and also the drive can be calculated as ηHPV2πHPp. It is the load torque for the drive, so the input matrix of the drive model (5.2) will have the following look
Bd =
0 0
0 −ηHPV2πJHP
ks
TelR 0
. (5.6)
In the second place by adding a model of electrical drive the full mathematical description is obtained. The model introduces a rigid boom controlled by electro- hydraulic circuit. Finally combining equations (5.5) and (3.21) and assuming that connection between drive and pump is rigid the following model is obtained
˙ x=
"
AH,b 0 BH,b 0 0 0 Bd(2) Ad
# x+
"
0 Bd(1)
#
u, (5.7)
where Bd(1) and Bd(2) are the first and the second column of the Bd matrix respectively.
