Direct Displacement Control of Hydraulic Actuators ... - Me.umn.edu
Paper Number: 20.2
Direct Displacement Control of Hydraulic Actuators Based on a Self-spinning Rotary On/off Valve Meng Wang, Perry Y. Li, Haink Tu, Mike Rannow, Thomas R. Chase Center for Compact and Efficient Fluid Power University of Minnesota
ABSTRACT Throttling loss of hydraulic systems is a major loss that degrades the system efficiency. One approach to eliminate the throttling loss is displacement control. The flow rate going to the actuator is directly controlled from the power source (pump). In this way, the throttling valve is eliminated. This paper presents an open circuit to realize direct displacement control of a single actuator. The circuit includes one variable flow source, one directional valve, and one proportional valve. A hybrid nonlinear control strategy based on back-stepping technique and consideration of the pressure states was developed for actuator trajectory tracking while maintaining a low system operating pressure. The controller guarantees that neither of the two actuator chambers can cavitate nor go unbounded. The novel 3way valve based virtually variable displacement pump (VVDP) was selected as the power source to implement the circuit.
1. INTRODUCTION Hydraulic systems are typically controlled using two approaches: throttling valve control and displacement control. The first approach has the advantages of compactness, low cost, and high control bandwidth, and can be used to control multiple actuators from one single power supply. However, it is inefficient since significant energy is wasted through throttling. To address the high power consumption of valve control, load-sensing (LS) systems have been popular in the past decade. Since the system pressure is regulated to be only incrementally higher than the maximum required load pressure, LS system is more efficient than a typical constant pressure system. However, when one power source is used to power multiple actuators, and the pressure requirements have a large variation, only the actuator requiring the highest pressure can be controlled in an energy efficient manner. Another drawback with LS system is its challenge to maintain system stability [1]. In displacement control, a variable displacement pump (VDP) is used to generate
only the amount of flow required for each circuit. However, multiple VDPs are required in a multi-circuit system. This paper focuses on displacement control of a hydraulic actuator. From hydraulic configuration point of view, displacement control systems can be categorized into open circuits or closed circuits. In an open circuit, the pump inlet and the actuator return (via a directional valve) are connected to the hydraulic tank. In a closed circuit, the actuator return is directly connected to the pump inlet. Closed circuits eliminate the directional valves; however, when driving asymmetrical cylinders, the unequal volume of the cylinder should be compensated. In 1994, Hewett [1] developed a closed circuit with a charge pump, an accumulator, and a 3way 2position valve. The valve is actively controlled to connect a charge line to the low pressure side when volume compensation is required. In 2000, Rahmfeld and Ivantysynova [2] proposed a different closed circuit where the charging line and the low pressure side are connected via pilot-operated check valves. Heybroek [3] proposed an open circuit, in which four 2way valves are used to realize four-quadrant operation. If the pump can operate as a motor, energy regeneration can be achieved as well. Given the displacement control configurations, a lot of efforts have been spent on developing control strategy to manipulate the actuator in an energy efficient way. Instability can happen when the actuator switches operation mode, as reported by Williamson and Ivantysynova [4][5]. Cavitation is another issue which may be caused by over-running load. Control strategy has been developed to compensate this cavitation to the maximum flow capacity using the open circuit in reference [3]. In their works, the analysis was based on linearized dynamics. If the metering in and metering out pressures of the actuator cannot be controlled independently, one order of zero dynamics will be introduced. The stability of the zero dynamics has been investigated when the trajectory to be tracked has a constant velocity [6]. With an adaptive robust controller, the zero dynamics for tracking any nonzero constant velocity trajectory is shown to be
globally uniformly stable. However, the results cannot be easily adapted to an arbitrary trajectory. In this paper an open-loop circuit is developed to achieve direct displacement control using one pump, one directional valve, and one proportional valve. The proportional valve, placed on the return line, is to ensure that cavitation does not occur. A nonlinear control strategy based on back-stepping techniques was developed for a single actuator trajectory tracking. The hybrid control algorithm operates in various discrete modes based upon pressure states. A rigorous stability analysis of the actuator in all operating range was conducted. The controller guarantees accurate trajectory tracking in the presence of unknown load. It can lock the pressure on the charging line when it goes low to prevent cavitation. The control efforts are designed to maintain the throttling valve fully open whenever it is feasible, and use the variable flow source to achieve trajectory tracking. One variable displacement pump is required as the power source. Although VDPs have the advantage of high control precision, they are typically more expensive, bulkier and heavier than fixed displacement units with similar displacements. In addition, to achieve a high bandwidth, powerful actuators are needed to change the pump geometry (e.g. the swash plate angle). To take advantage of fixed displacement pumps, a virtually variable displacement pump (VVDP) was developed and implemented as the power source for displacement control. The VVDP consists of an on/off valve, a fixed displacement pump and an accumulator [7][8][9], as shown in Figure 1. By pulse-width-modulating (PWM) the on/off valve, mean partial flow is achieved with the virtual displacement set by the PWM duty ratio. Since the on/off valve has low loss in either on or off states, this approach reduces throttling loss. In addition, many types of fixed displacement pumps, including those that are low cost and compact but whose displacements cannot normally be varied, can be made variable.
Figure 1: Software enabled VVDP Circuit The key component to an efficient VVDP is the on/off valve. Ideally, the on/off valve should meet four requirements simultaneously, including: a large orifice, a fast transition, the ability to operate at high PWM frequencies, and a low actuation power. A novel type of high speed rotary self-spinning on/off valve has been developed, which can meet the four requirements at the same time [7]. The rest of the paper is organized as follows. In the next section, the working principle of the novel 3way valve based VVDP will be introduced. In section 3, the open circuit of displacement control using the VVDP will be described. The dynamic modeling of the system will be presented. An advanced nonlinear control strategy developed for actuator trajectory tracking in an energy efficient manner will be presented in section 4. Simulation results will be presented in section 5. Finally, some concluding remarks and the future research plan will be presented in section 6.
2. A NOVEL 3WAY VALVE BASED VVDP Figure 2 shows a novel 3-way on/off valve based VVDP. The valve is composed of a stationary sleeve and a moving spool. To reduce the fluid volume between the pump outlet and the valve inlet, so as to minimize the compressible loss, the valve sleeve is directly mounted onto a fixed displacement pump by modifying the pump housing. Inlet nozzles tangential to the bore are casted inside the sleeve. A pressure rail inside the sleeve connects the nozzles and the pump outlet on the pump housing. The rhombus shape of the valve nozzle provides a fast transition
Figure 2: A novel 3-way rotary self-spinning on/off valve based VVDP Figure 3 shows the structure of the valve spool. The valve spool consists of three sections: one center PWM section and two outlet turbines. Helical barriers are wrapped
around the center PWM section. These barriers partition the center PWM section into two parts. The internal axial path directs the flow from the center PWM section to the adjacent outlet turbine. Both the center PWM section and the outlet turbines are designed to capture fluid moment to achieve spool self-spinning. The center PWM section functions as an impulse turbine. Fluid is accelerated through the nozzles inside the sleeve. The outlet section is designed as a reaction turbine. The turbine blade guides the fluid from flowing axially to exit the spool tangentially, and therefore a reaction torque is generated. In this way, the spool rotary motion is realized via self-spinning. The unidirectional rotary motion of the valve spool minimizes the on/off motion power consumption (proportional to PWM frequency squared) compared with linear poppet or spool type of on/off valves (proportional to PWM frequency cubed) As the spool rotates, the inlet nozzles land on the red triangles and blue triangles alternately. The red branch directs the fluid to load, and the blue branch directs the fluid back to tank. In this way, the valve pulse width modulates the inlet flow. If there are N triangle sets around the spool, then the PWM frequency is N times the spool rotary frequency. The PWM duty ratio is determined by the fraction of the time when the nozzles lands on the red triangle over one revolution. The helical shape offers a linear relationship between the duty ratio and the spool axial position.
fluid enters the spool tangentially, and leaves the spool tangentially as well. There is no net moment in the axial position, which minimizes the power required to regulate the spool axial position. Sensing of the axial and rotary positions of the spool is realized using non-contact optical sensors to maintain a simple sealing structure.
Figure 4: Valve spool sensing and actuation
3. DIRECT DISPLACEMENT OPEN CIRCUIT CONTROL USING A VVDP HYDRAULIC CIRCUIT - Figure 5 shows the hydraulic circuit for direct displacement control. The VVDP replaces a VDP as the flow source. The average flow is controlled by the PWM duty ratio. Compared with the VVDP presented in Figure 1, the accumulator is eliminated. Based on the preliminary experimental test, it is revealed that given the ability to operate at a high PWM frequency (~90Hz), the compliance of hoses is sufficient for smoothing out the pressure and flow ripples. A 2-position directional valve is used to change the flow direction. The directional valve is always fully open to reduce throttling loss. A proportional throttling valve is added on the return line, which will be used to avoid actuator chamber cavitation. In most operations, the throttling valve is fully open to reduce throttling.
Figure 3: Valve spool structure The spool axial motion is controlled externally, as shown in Figure 4. The spool axial position is controlled using an electro-hydraulic gerotor pump powered by a DC motor in a hydrostatic configuration. The gerotor pump ports fluid into the axial position ports as shown in Figure 1. By pumping fluid from one axial chamber to the other, the spool axial position is varied. One feature to notice is that
Pt ,
Fig. 6 (a) and (b). Let the tank pressure be
Kv = Cd Amax 2 / ρ , with Amax being the maximum valve opening area, and u ∈ [0 , 1] be the normalized throttling valve command. Depending on the operation of the directional valve, the pressure dynamics are: •
When u d
P1 = Figure 5: Hydraulic configuration using VVDP to control one single actuator in displacement control open circuit SYSTEM DYNAMICS – The hydraulic cylinder is modeled as a mass acted upon by the pressure in the two cylinder chambers, the linear viscous friction force, and the load force.
mx = P1 A1 − P2 A2 − bx + FL m is
(1)
A1 and A2 are the areas of the cylinder cap end and the rod end. b is the the mass of the cylinder rod.
viscous friction coefficient, and FL summarizes the carrying load, leakage, and un-modeled dynamics. Depending on the position of the directional valve, the circuit has two configurations, as shown in Figure 6. x is the position of the cylinder rod. Q is the flowrate into the
V1 and V2 are the volumes in the respective chamber and hose when x = 0 . The throttling upstream chamber.
valve on the return line is modeled as an orifice.
β V1 + A1 x
P2 = •
= 1 (Fig. 6 (a)): (Q − A1 x )
β V2 − A2 x
When u d
V1 + A1 x
P2 =
( A2 x − uK v | P2 − Pt |sign ( P2 − Pt ))
= −1 (Fig. 6 (b)):
β
P1 =
(2)
( A1 x − uK v | P1 − Pt |sign ( P1 − Pt ))
β V2 − A2 x
(3)
( A2 x + Q)
4. CONTROL STRATEGY The control objective is for the actuator position track a trajectory
r (t )
x (t )
to
in the presence of unknown
load FL (t ) . The system pressure should remain low to improve efficiency. Cavitation should be avoided in both cylinder chambers during the operation and the chamber pressures should be bounded. SYSTEM DYNAMICS IN STATE SPACE FORM – Four states are defined to represent the system dynamics: actuator position x1 = x , actuator velocity x2 = x , and the
P1 , P2 . Define a new state d = FL / m . Eqs. (1)-(3) become:
cylinder chamber pressures
x3 = P1 A1 − P2 A2 , and x1 = x 2
b 1 x2 + x3 + d m m x3 = − K ( x1 ) x2 x2 = −
+ H ( x1 , ud ) ⋅ Q + G ( x1 , ud )Ψ (P1 , P2 , ud )⋅ u Figure 6: Hydraulic circuit for different directional valve position The 2 positions of the directional control valve are denoted by u d = − 1, 1 and the respective circuits are shown in
{
}
Utotal
where
(4)
K ( x1 ) =
βA12 V1 + A1 x1
+
α1 := −λ1λ2ei + (λ1 + λ2 )e1
βA22 V2 − A2 x1
βA1 ⎧ ⎪⎪ V + A x , 1 1 H ( x1 , u d ) = ⎨ 1 − βA2 ⎪ , ⎪⎩V2 − A2 x1
(6)
The closed loop dynamics becomes:
ud = 1
1 ⎞⎛ ei ⎞ ⎛ 0 ⎞ d ⎛ ei ⎞ ⎛ 0 ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎜ ⎟ + ⎜ ⎟(e2 − α 1 ) (7) dt ⎝ e1 ⎠ ⎝ λ1λ 2 − (λ1 + λ 2 ) ⎟⎠⎜⎝ e1 ⎟⎠ ⎜⎝ 1 ⎟⎠ e ⎡ P11 P12 ⎤ Acl Let P = ⎢ be the symmetric positive definite ⎥ ⎣ P21 P22 ⎦
u d = −1
⎧ βA2 ⎪⎪ V − A x , u d = 1 2 1 G ( x1 , u d ) = ⎨ 2 − β A1 ⎪ , u d = −1 ⎪⎩V1 + A1 x1
matrix
that
satisfies
T Acl P + PAcl = − I .
the
Lyapunov
equation:
Define the Lyapunov function for the
first step as:
V1 = eT Pe ⎧⎪ K | P2 − Pt |sign( P2 − Pt ), u d = 1 Ψ( P1 , P2 , u d ) = ⎨ v ⎪⎩K v | P1 − Pt |sign( P1 − Pt ), u d = −1 The pressure dynamics are the same as described in Eqns (2) and (3).
V1 = −ei2 − e12 + 2(e2 − α 1 )( P21ei + P22 e1 )
Step 2 – The Lyapunov function defined in Eqn (8) is augmented with the error between e2 and α 1 :
Since one control objective is cylinder position trajectory tracking, define three system tracking error states as: e1 = r − x1 , e2 = r − x2 , e3 = mr + br − x3 . Define also the augment integral error state:
ei = e1 e1 = e 2
ei = ∫ e1dt
we have:
1 V2 = e T Pe + (e2 − α 1 ) 2 2 V2 = −ei2 − e12 + (e2 − α 1 )[2( P21ei + P22 e1 ) + (e2 − α1 )] (9)
b 1 e 2 = − e 2 + e3 + d m m e3 = mr + br + K ( x1 )r − K ( x1 )e 2 − U total
From this, the control law with e3 in Eq. (5) being the virtual control input is defined as: (5)
where K ( x1 ), and U total are defined in Eqn (4). There are three control inputs to manipulate: flowrate of the variable flow source Q ≥ 0 , the orifice area of the throttling valve
u ≥ 0 , and the directional valve ud ∈ {0, 1}.
Step 1 – Since the first two states are linear, using
e2 as
the virtual control input, the first two states can be stabilized simultaneously. Assuming all the states are measurable, the control input is determined by doing a pole placement with full states feedback. The poles are placed at λ1 and λ 2 , λ1 < 0, λ2 < 0 . The pole-placement
e2 as the input is:
⎡
b
α 2 = m ⎢− λ3 (e2 − α1 ) + e2 − λ1λ2e1 m ⎣ + (λ1 + λ2 )e2 − 2( P21ei + P22e1 )]
> 0.
where λ3
(10)
The derivative of the Lyapunov function
becomes:
BACK-STEPPING CONTROLLER -The error dynamics represented in Eqn (5) is in a strict-feedback form, which allows a Lyapunov function based back-stepping control technique to be applied.
control law with
(8)
V2 = −ei2 − e12 − λ3 (e2 − α 1 ) 2 + (e2 − α 1 )d +
1 (e2 − α 1 )(e3 − α 2 ) m
(11)
Step 3 – the Lyapunov function defined in Eqn (9) is augmented with the error between the virtual control input α 2 and e3 :
1 1 V3 = e T Pe + (e2 − α 1 ) 2 + (e3 − α 2 ) 2 2 2
V3 = −ei2 − e12 − λ3 (e2 − α 1 ) 2 + (e2 − α 1 )d +
e3d
(12)
1 (e2 − α 1 )(e3 − α 2 ) + (e3 − α 2 )(e3 − α 2 ) m
The desired value for
P1 =
e3 is: for λ4 > 0 :
1 = α 2 − (e2 − α 1 ) − λ4 (e3 − α 2 ) m
(13)
Q, u,
P1 =
(14)
* U total (t ) := −e3d + mr + br + K ( x1 )r − K ( x1 )e2
(18c)
β ( A2 x 2 + Q) V2 − A2 x
(18d)
V3 = −ei2 − e12 − λ3 (e2 − α1 ) 2 + (e2 − α 1 )d − λ4 (e3 − α 2 ) 2
(15)
If the disturbance d = 0 , V2 will be negative definite, and the tracking errors will go to zero exponentially. In the presence of the disturbance d , the errors do not vanish choosing
λ3
large
enough,
the
but
by
term
− λ3 (e2 − α 1 ) 2 can dominate the disturbance
quadratic
− α1 ) so that the tracking error can be arbitrarily
CONTROL EFFORT DISTRIBUTION – The control objectives are to ensure: 1. The trajectory tracking performance e1 → 0 . From the backstepping control design, this is ensured by:
U total (t ) := H ( x1 , ud ) ⋅ Q + G( x1 , ud )ψ (P1 , P2 , ud )⋅ u * total
(t )
(16)
2. The pressure in both chambers should be higher than a threshold values P1 > Pt and P2 > Pt for each chamber to prevent cavitation:
P1 ≥ P1 > Pt
,
P2 ≥ P2 > Pt
(17)
The pressure dynamics Eqns. (2)-(3) can be written as:
ud = 1
H (x1 , u d ) and G(x1 , u d ) are the same as the sign of by u d . If the pressure in the both cylinder chambers satisfies: For i = {1,2} From (4), the signs of
With this control effort, Eqn (12) becomes:
When
u d = −1
β ( A1 x 2 − Ψ ( P1 , P2 , u d )u ) V1 + A1 x
P2 =
* U total (t ) = U total (t )
=U
(18b)
.The desired total control efforts should
therefore satisfy:
term d (e2 bounded.
β ( A2 x 2 − Ψ ( P1 , P2, u d )u ) V2 − A2 x
When
and u d appear in the dynamics of e3 via the total control
U total (t )
(18a)
P2 =
According to Eqns. (4)-(5), the control inputs
effort
β (Q − A1 x 2 ) V1 + A1 x
Pi (t ) ≥ Pi > Pt , , Ψ ( P1 , P2 , u d ) is also positive. Hence, to satisfy (16), since u ≥ 0 and Q ≥ 0 , u d can be determined by the sign Then
of
* : U total
* ⎧ 1 U total (t ) ≥ 0 ud (t ) = ⎨ * ⎩− 1 U total (t ) < 0
(19)
The usage of Q and u is based on the rule that: u is fully open whenever it is feasible to reduce throttling loss, and to maintain the return line pressure low. There are three operating modes depending on whether any of the pressure constraints is active or not. Mode 1:
P1 (t ) > P1 and P2 (t ) > P2 :
This mode applies to *
the cases when no pressure constraint is active. U total is solved in an energy efficient way, which minimizes the usage of the throttling valve. Since the flow source is of positive displacement, if a negative flow is required, the flow is set to zero, and the trajectory tracking is achieved by using the throttling valve.
Case 1:
U
* total
pressure is higher than the threshold, and the operation goes back to Mode 1. The control law is:
G( x1 , ud )Ψ ( P1 , P2 , ud ) , > H ( x1 , ud )
u = 1,
Case 1:
* U total − G ( x1 , u d )Ψ ( P1 , P2 , u d ) Q= H ( x1 , u d )
Case 2:
U
* total
(20)
Case 2:
G ( x1 , ud )Ψ ( P1 , P2 , ud ) , ≤ H ( x1 , ud )
u
0 and P1 (t ) ≤ P1 ), or ( U
(21)
* total
< 0 and
A2 x 2 ψ ( P1 , P2 , u d ) * U total < 0, P1 ≤ P1 :
− A1 x 2 ψ ( P1 , P2 , u d )
In both cases, once u is chosen, * U total − G( x1 , u d )ψ ( P1 , P2 , u d )u Q= H ( x1 , u d )
(23)
In this mode, the pressure in the upstream
cylinder chamber connected with the flow source is low, and has the potential to cavitate. From Eqns (18a) and (18c), the variable flow source Q will be used to increase the pressure. The throttling valve u will be used for trajectory tracking according to the variable flow source Q to satisfy Eqn (16). The controller will stay in this mode to increase the chamber pressure until the pressure is higher than the threshold, and the operation goes back to Mode 1. The control law is: *
Case 1: U total
> 0, P1 ≤ P1 :
Q > A1 x 2 * Case 2: U total < 0, P2 ≤ P2 :
Similar to mode 2, this imposes an upper bound on u . If this bound is greater than 1, then we can set u = 1 , and (23) reduces to Eq. (20) in mode 1. In the above controller modes, the calculation of
ud
,
Q and u are based on the Lyapunov function as presented in Eqn. (11). The control inputs can guarantee that the Lyapunov function remains decreasing during modes switching. This guarantees the trajectory tracking performance. Since whenever the pressure reaches the pressure lower bound Pi , the controller will force the chamber pressure to increase (Mode 2 and Mode 3); this will guarantee that if the initial chamber pressures are above the threshold, they will continue to be above threshold Pi . Thus, cavitation cannot happen in the
Q > − A2 x 2
cylinder chamber.
In both cases,
u=
u
0, P2 ≤ P2 :
* U total − H ( x1 , u d )Q G ( x1 , u d )ψ ( P1 , P2 , u d )
5. SIMULATION RESULTS (22)
Notice that this imposes lower bound on Q (or an upper bound on u ). If the upper bound on u is greater than 1, then we can set u = 1 , and (22) reduces to Eq. (20) in mode 1. Mode 3 – This mode applies to the cases when the pressure in the return chamber is low. To prevent the pressure from going below the threshold, considering pressure dynamics in Eqns (18b) and (18d), the throttling valve u will be used to increase the chamber pressure when it reaches the threshold. The controller will stay in this mode to increase the chamber pressure until the
The control strategy is simulated using Simulink Matlab. Some of the system parameters of are based on an experimental set up that is being constructed currently; the other parameters are estimated from typical values, as shown in Table 1. Parameter Description
Symbol
Value
Cylinder mass
m
3kg
Cylinder cap area
A1
20.26cm 2
Cylinder rod area
A2
10.77 cm 2
Figure 7. Cylinder position trajectory tracking and the chamber pressures
Viscous friction coefficient
b
1.65 N /(m / sec)
Oil Bulk Modulus
β
1× 10 7 Pa
Throttling Valve Full Open Area
Avalve
0.2cm 2
The control efforts are shown in Figure 8. As designed, the directional control valve does not change direction very frequently, especially after the trajectory is well tracked. The throttling valve is kept fully open most of time, and it starts to close when the trajectory velocity decreases, comes to a stop and switches direction. The throttling valve is also used when the pressure constraint is active, so that the pressures inside both cylinder chambers stay above the pressure lower bound.
The constant bulk modulus of the oil is estimated by assuming the air volume content in the hydraulic oil at atmosphere pressure is 7% , and the load pressure is under 70bar [14]. The actuator was simulated to track a sinusoid trajectory with amplitude of 3cm , and a frequency of 1Hz . A constant load of 50 N was added as the disturbance load. The initial volume on the cap side of the cylinder is 1.2 L , and the initial volume on the rod side is 1.0 L . The control
λ4 = 30.The
= P2 = 2bar .
Cylinder Position [cm]
The trajectory tracking of the cylinder is shown in Figure 7. With the introduction of the integral control, the cylinder can track the trajectory accurately even with a load disturbance. The pressures inside both cylinder chambers start from an initial value that is higher than the lower pressure threshold, and the pressure remain higher than the threshold. The system pressure remains low.
Chamber Pressure [bar]
10
= −30 , λ2 = −30 , λ3 = 30 ,
pressure lower bound for both cylinder
chambers are set to be P1
20
trajectory reference cylinder position
10 8
4 2 0
1
2 time [sec]
3
14
4
Cap side Rod side
12
2 time [sec]
3
4
50
0 0
1
2 time [sec]
3
4
1 0.5 0 -0.5 -1 0
6
1
100
Throttling valve opening %
and
30
0 0
Directional valve
gains were selected as: λ1
40
Flow rate [lpm]
Table 1 Simulated system parameter
1
2 time [sec]
3
4
Figure 8. Control efforts from variable flow source, throttling valve and directional valve
6. CONCLUSION AND FUTURE WORK
10 8 6 4 2 0
0.5
1
1.5
2
time [sec]
2.5
3
3.5
4
In this paper, the displacement control of a single hydraulic actuator was investigated. An open circuit was proposed to realize direct displacement using one variable flow source, one directional valve, and one throttling valve. A multiple mode control strategy was developed to achieve accurate trajectory tracking with maintaining a low system operating pressure. The trajectory tracking was
based on a nonlinear back-stepping controller. This controller guarantees that neither of the cylinder chambers cavitates or goes unbounded. An experimental set was being developed. In the next step, we will implement this control strategy on the backhoe with the novel VVDP as the variable flow source.
ACKNOWLEDGMENTS This material is based upon work performed within the ERC for Compact and Efficient Fluid Power, supported by the National Science Foundation under Grant No. EEC0540834.
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CONTACT Meng Wang, Haink Tu, and Mike Rannow: graduate students with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota. Email: {wang134, tuxxx021, rann0018}@me.umn.edu Dr. Perry Y Li: professor with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota. Email: [email protected] Dr. Thomas R. Chase: professor with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota.
Email: [email protected]
Direct Displacement Control of Hydraulic Actuators Based on a Self-spinning Rotary On/off Valve Meng Wang, Perry Y. Li, Haink Tu, Mike Rannow, Thomas R. Chase Center for Compact and Efficient Fluid Power University of Minnesota
ABSTRACT Throttling loss of hydraulic systems is a major loss that degrades the system efficiency. One approach to eliminate the throttling loss is displacement control. The flow rate going to the actuator is directly controlled from the power source (pump). In this way, the throttling valve is eliminated. This paper presents an open circuit to realize direct displacement control of a single actuator. The circuit includes one variable flow source, one directional valve, and one proportional valve. A hybrid nonlinear control strategy based on back-stepping technique and consideration of the pressure states was developed for actuator trajectory tracking while maintaining a low system operating pressure. The controller guarantees that neither of the two actuator chambers can cavitate nor go unbounded. The novel 3way valve based virtually variable displacement pump (VVDP) was selected as the power source to implement the circuit.
1. INTRODUCTION Hydraulic systems are typically controlled using two approaches: throttling valve control and displacement control. The first approach has the advantages of compactness, low cost, and high control bandwidth, and can be used to control multiple actuators from one single power supply. However, it is inefficient since significant energy is wasted through throttling. To address the high power consumption of valve control, load-sensing (LS) systems have been popular in the past decade. Since the system pressure is regulated to be only incrementally higher than the maximum required load pressure, LS system is more efficient than a typical constant pressure system. However, when one power source is used to power multiple actuators, and the pressure requirements have a large variation, only the actuator requiring the highest pressure can be controlled in an energy efficient manner. Another drawback with LS system is its challenge to maintain system stability [1]. In displacement control, a variable displacement pump (VDP) is used to generate
only the amount of flow required for each circuit. However, multiple VDPs are required in a multi-circuit system. This paper focuses on displacement control of a hydraulic actuator. From hydraulic configuration point of view, displacement control systems can be categorized into open circuits or closed circuits. In an open circuit, the pump inlet and the actuator return (via a directional valve) are connected to the hydraulic tank. In a closed circuit, the actuator return is directly connected to the pump inlet. Closed circuits eliminate the directional valves; however, when driving asymmetrical cylinders, the unequal volume of the cylinder should be compensated. In 1994, Hewett [1] developed a closed circuit with a charge pump, an accumulator, and a 3way 2position valve. The valve is actively controlled to connect a charge line to the low pressure side when volume compensation is required. In 2000, Rahmfeld and Ivantysynova [2] proposed a different closed circuit where the charging line and the low pressure side are connected via pilot-operated check valves. Heybroek [3] proposed an open circuit, in which four 2way valves are used to realize four-quadrant operation. If the pump can operate as a motor, energy regeneration can be achieved as well. Given the displacement control configurations, a lot of efforts have been spent on developing control strategy to manipulate the actuator in an energy efficient way. Instability can happen when the actuator switches operation mode, as reported by Williamson and Ivantysynova [4][5]. Cavitation is another issue which may be caused by over-running load. Control strategy has been developed to compensate this cavitation to the maximum flow capacity using the open circuit in reference [3]. In their works, the analysis was based on linearized dynamics. If the metering in and metering out pressures of the actuator cannot be controlled independently, one order of zero dynamics will be introduced. The stability of the zero dynamics has been investigated when the trajectory to be tracked has a constant velocity [6]. With an adaptive robust controller, the zero dynamics for tracking any nonzero constant velocity trajectory is shown to be
globally uniformly stable. However, the results cannot be easily adapted to an arbitrary trajectory. In this paper an open-loop circuit is developed to achieve direct displacement control using one pump, one directional valve, and one proportional valve. The proportional valve, placed on the return line, is to ensure that cavitation does not occur. A nonlinear control strategy based on back-stepping techniques was developed for a single actuator trajectory tracking. The hybrid control algorithm operates in various discrete modes based upon pressure states. A rigorous stability analysis of the actuator in all operating range was conducted. The controller guarantees accurate trajectory tracking in the presence of unknown load. It can lock the pressure on the charging line when it goes low to prevent cavitation. The control efforts are designed to maintain the throttling valve fully open whenever it is feasible, and use the variable flow source to achieve trajectory tracking. One variable displacement pump is required as the power source. Although VDPs have the advantage of high control precision, they are typically more expensive, bulkier and heavier than fixed displacement units with similar displacements. In addition, to achieve a high bandwidth, powerful actuators are needed to change the pump geometry (e.g. the swash plate angle). To take advantage of fixed displacement pumps, a virtually variable displacement pump (VVDP) was developed and implemented as the power source for displacement control. The VVDP consists of an on/off valve, a fixed displacement pump and an accumulator [7][8][9], as shown in Figure 1. By pulse-width-modulating (PWM) the on/off valve, mean partial flow is achieved with the virtual displacement set by the PWM duty ratio. Since the on/off valve has low loss in either on or off states, this approach reduces throttling loss. In addition, many types of fixed displacement pumps, including those that are low cost and compact but whose displacements cannot normally be varied, can be made variable.
Figure 1: Software enabled VVDP Circuit The key component to an efficient VVDP is the on/off valve. Ideally, the on/off valve should meet four requirements simultaneously, including: a large orifice, a fast transition, the ability to operate at high PWM frequencies, and a low actuation power. A novel type of high speed rotary self-spinning on/off valve has been developed, which can meet the four requirements at the same time [7]. The rest of the paper is organized as follows. In the next section, the working principle of the novel 3way valve based VVDP will be introduced. In section 3, the open circuit of displacement control using the VVDP will be described. The dynamic modeling of the system will be presented. An advanced nonlinear control strategy developed for actuator trajectory tracking in an energy efficient manner will be presented in section 4. Simulation results will be presented in section 5. Finally, some concluding remarks and the future research plan will be presented in section 6.
2. A NOVEL 3WAY VALVE BASED VVDP Figure 2 shows a novel 3-way on/off valve based VVDP. The valve is composed of a stationary sleeve and a moving spool. To reduce the fluid volume between the pump outlet and the valve inlet, so as to minimize the compressible loss, the valve sleeve is directly mounted onto a fixed displacement pump by modifying the pump housing. Inlet nozzles tangential to the bore are casted inside the sleeve. A pressure rail inside the sleeve connects the nozzles and the pump outlet on the pump housing. The rhombus shape of the valve nozzle provides a fast transition
Figure 2: A novel 3-way rotary self-spinning on/off valve based VVDP Figure 3 shows the structure of the valve spool. The valve spool consists of three sections: one center PWM section and two outlet turbines. Helical barriers are wrapped
around the center PWM section. These barriers partition the center PWM section into two parts. The internal axial path directs the flow from the center PWM section to the adjacent outlet turbine. Both the center PWM section and the outlet turbines are designed to capture fluid moment to achieve spool self-spinning. The center PWM section functions as an impulse turbine. Fluid is accelerated through the nozzles inside the sleeve. The outlet section is designed as a reaction turbine. The turbine blade guides the fluid from flowing axially to exit the spool tangentially, and therefore a reaction torque is generated. In this way, the spool rotary motion is realized via self-spinning. The unidirectional rotary motion of the valve spool minimizes the on/off motion power consumption (proportional to PWM frequency squared) compared with linear poppet or spool type of on/off valves (proportional to PWM frequency cubed) As the spool rotates, the inlet nozzles land on the red triangles and blue triangles alternately. The red branch directs the fluid to load, and the blue branch directs the fluid back to tank. In this way, the valve pulse width modulates the inlet flow. If there are N triangle sets around the spool, then the PWM frequency is N times the spool rotary frequency. The PWM duty ratio is determined by the fraction of the time when the nozzles lands on the red triangle over one revolution. The helical shape offers a linear relationship between the duty ratio and the spool axial position.
fluid enters the spool tangentially, and leaves the spool tangentially as well. There is no net moment in the axial position, which minimizes the power required to regulate the spool axial position. Sensing of the axial and rotary positions of the spool is realized using non-contact optical sensors to maintain a simple sealing structure.
Figure 4: Valve spool sensing and actuation
3. DIRECT DISPLACEMENT OPEN CIRCUIT CONTROL USING A VVDP HYDRAULIC CIRCUIT - Figure 5 shows the hydraulic circuit for direct displacement control. The VVDP replaces a VDP as the flow source. The average flow is controlled by the PWM duty ratio. Compared with the VVDP presented in Figure 1, the accumulator is eliminated. Based on the preliminary experimental test, it is revealed that given the ability to operate at a high PWM frequency (~90Hz), the compliance of hoses is sufficient for smoothing out the pressure and flow ripples. A 2-position directional valve is used to change the flow direction. The directional valve is always fully open to reduce throttling loss. A proportional throttling valve is added on the return line, which will be used to avoid actuator chamber cavitation. In most operations, the throttling valve is fully open to reduce throttling.
Figure 3: Valve spool structure The spool axial motion is controlled externally, as shown in Figure 4. The spool axial position is controlled using an electro-hydraulic gerotor pump powered by a DC motor in a hydrostatic configuration. The gerotor pump ports fluid into the axial position ports as shown in Figure 1. By pumping fluid from one axial chamber to the other, the spool axial position is varied. One feature to notice is that
Pt ,
Fig. 6 (a) and (b). Let the tank pressure be
Kv = Cd Amax 2 / ρ , with Amax being the maximum valve opening area, and u ∈ [0 , 1] be the normalized throttling valve command. Depending on the operation of the directional valve, the pressure dynamics are: •
When u d
P1 = Figure 5: Hydraulic configuration using VVDP to control one single actuator in displacement control open circuit SYSTEM DYNAMICS – The hydraulic cylinder is modeled as a mass acted upon by the pressure in the two cylinder chambers, the linear viscous friction force, and the load force.
mx = P1 A1 − P2 A2 − bx + FL m is
(1)
A1 and A2 are the areas of the cylinder cap end and the rod end. b is the the mass of the cylinder rod.
viscous friction coefficient, and FL summarizes the carrying load, leakage, and un-modeled dynamics. Depending on the position of the directional valve, the circuit has two configurations, as shown in Figure 6. x is the position of the cylinder rod. Q is the flowrate into the
V1 and V2 are the volumes in the respective chamber and hose when x = 0 . The throttling upstream chamber.
valve on the return line is modeled as an orifice.
β V1 + A1 x
P2 = •
= 1 (Fig. 6 (a)): (Q − A1 x )
β V2 − A2 x
When u d
V1 + A1 x
P2 =
( A2 x − uK v | P2 − Pt |sign ( P2 − Pt ))
= −1 (Fig. 6 (b)):
β
P1 =
(2)
( A1 x − uK v | P1 − Pt |sign ( P1 − Pt ))
β V2 − A2 x
(3)
( A2 x + Q)
4. CONTROL STRATEGY The control objective is for the actuator position track a trajectory
r (t )
x (t )
to
in the presence of unknown
load FL (t ) . The system pressure should remain low to improve efficiency. Cavitation should be avoided in both cylinder chambers during the operation and the chamber pressures should be bounded. SYSTEM DYNAMICS IN STATE SPACE FORM – Four states are defined to represent the system dynamics: actuator position x1 = x , actuator velocity x2 = x , and the
P1 , P2 . Define a new state d = FL / m . Eqs. (1)-(3) become:
cylinder chamber pressures
x3 = P1 A1 − P2 A2 , and x1 = x 2
b 1 x2 + x3 + d m m x3 = − K ( x1 ) x2 x2 = −
+ H ( x1 , ud ) ⋅ Q + G ( x1 , ud )Ψ (P1 , P2 , ud )⋅ u Figure 6: Hydraulic circuit for different directional valve position The 2 positions of the directional control valve are denoted by u d = − 1, 1 and the respective circuits are shown in
{
}
Utotal
where
(4)
K ( x1 ) =
βA12 V1 + A1 x1
+
α1 := −λ1λ2ei + (λ1 + λ2 )e1
βA22 V2 − A2 x1
βA1 ⎧ ⎪⎪ V + A x , 1 1 H ( x1 , u d ) = ⎨ 1 − βA2 ⎪ , ⎪⎩V2 − A2 x1
(6)
The closed loop dynamics becomes:
ud = 1
1 ⎞⎛ ei ⎞ ⎛ 0 ⎞ d ⎛ ei ⎞ ⎛ 0 ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎜ ⎟ + ⎜ ⎟(e2 − α 1 ) (7) dt ⎝ e1 ⎠ ⎝ λ1λ 2 − (λ1 + λ 2 ) ⎟⎠⎜⎝ e1 ⎟⎠ ⎜⎝ 1 ⎟⎠ e ⎡ P11 P12 ⎤ Acl Let P = ⎢ be the symmetric positive definite ⎥ ⎣ P21 P22 ⎦
u d = −1
⎧ βA2 ⎪⎪ V − A x , u d = 1 2 1 G ( x1 , u d ) = ⎨ 2 − β A1 ⎪ , u d = −1 ⎪⎩V1 + A1 x1
matrix
that
satisfies
T Acl P + PAcl = − I .
the
Lyapunov
equation:
Define the Lyapunov function for the
first step as:
V1 = eT Pe ⎧⎪ K | P2 − Pt |sign( P2 − Pt ), u d = 1 Ψ( P1 , P2 , u d ) = ⎨ v ⎪⎩K v | P1 − Pt |sign( P1 − Pt ), u d = −1 The pressure dynamics are the same as described in Eqns (2) and (3).
V1 = −ei2 − e12 + 2(e2 − α 1 )( P21ei + P22 e1 )
Step 2 – The Lyapunov function defined in Eqn (8) is augmented with the error between e2 and α 1 :
Since one control objective is cylinder position trajectory tracking, define three system tracking error states as: e1 = r − x1 , e2 = r − x2 , e3 = mr + br − x3 . Define also the augment integral error state:
ei = e1 e1 = e 2
ei = ∫ e1dt
we have:
1 V2 = e T Pe + (e2 − α 1 ) 2 2 V2 = −ei2 − e12 + (e2 − α 1 )[2( P21ei + P22 e1 ) + (e2 − α1 )] (9)
b 1 e 2 = − e 2 + e3 + d m m e3 = mr + br + K ( x1 )r − K ( x1 )e 2 − U total
From this, the control law with e3 in Eq. (5) being the virtual control input is defined as: (5)
where K ( x1 ), and U total are defined in Eqn (4). There are three control inputs to manipulate: flowrate of the variable flow source Q ≥ 0 , the orifice area of the throttling valve
u ≥ 0 , and the directional valve ud ∈ {0, 1}.
Step 1 – Since the first two states are linear, using
e2 as
the virtual control input, the first two states can be stabilized simultaneously. Assuming all the states are measurable, the control input is determined by doing a pole placement with full states feedback. The poles are placed at λ1 and λ 2 , λ1 < 0, λ2 < 0 . The pole-placement
e2 as the input is:
⎡
b
α 2 = m ⎢− λ3 (e2 − α1 ) + e2 − λ1λ2e1 m ⎣ + (λ1 + λ2 )e2 − 2( P21ei + P22e1 )]
> 0.
where λ3
(10)
The derivative of the Lyapunov function
becomes:
BACK-STEPPING CONTROLLER -The error dynamics represented in Eqn (5) is in a strict-feedback form, which allows a Lyapunov function based back-stepping control technique to be applied.
control law with
(8)
V2 = −ei2 − e12 − λ3 (e2 − α 1 ) 2 + (e2 − α 1 )d +
1 (e2 − α 1 )(e3 − α 2 ) m
(11)
Step 3 – the Lyapunov function defined in Eqn (9) is augmented with the error between the virtual control input α 2 and e3 :
1 1 V3 = e T Pe + (e2 − α 1 ) 2 + (e3 − α 2 ) 2 2 2
V3 = −ei2 − e12 − λ3 (e2 − α 1 ) 2 + (e2 − α 1 )d +
e3d
(12)
1 (e2 − α 1 )(e3 − α 2 ) + (e3 − α 2 )(e3 − α 2 ) m
The desired value for
P1 =
e3 is: for λ4 > 0 :
1 = α 2 − (e2 − α 1 ) − λ4 (e3 − α 2 ) m
(13)
Q, u,
P1 =
(14)
* U total (t ) := −e3d + mr + br + K ( x1 )r − K ( x1 )e2
(18c)
β ( A2 x 2 + Q) V2 − A2 x
(18d)
V3 = −ei2 − e12 − λ3 (e2 − α1 ) 2 + (e2 − α 1 )d − λ4 (e3 − α 2 ) 2
(15)
If the disturbance d = 0 , V2 will be negative definite, and the tracking errors will go to zero exponentially. In the presence of the disturbance d , the errors do not vanish choosing
λ3
large
enough,
the
but
by
term
− λ3 (e2 − α 1 ) 2 can dominate the disturbance
quadratic
− α1 ) so that the tracking error can be arbitrarily
CONTROL EFFORT DISTRIBUTION – The control objectives are to ensure: 1. The trajectory tracking performance e1 → 0 . From the backstepping control design, this is ensured by:
U total (t ) := H ( x1 , ud ) ⋅ Q + G( x1 , ud )ψ (P1 , P2 , ud )⋅ u * total
(t )
(16)
2. The pressure in both chambers should be higher than a threshold values P1 > Pt and P2 > Pt for each chamber to prevent cavitation:
P1 ≥ P1 > Pt
,
P2 ≥ P2 > Pt
(17)
The pressure dynamics Eqns. (2)-(3) can be written as:
ud = 1
H (x1 , u d ) and G(x1 , u d ) are the same as the sign of by u d . If the pressure in the both cylinder chambers satisfies: For i = {1,2} From (4), the signs of
With this control effort, Eqn (12) becomes:
When
u d = −1
β ( A1 x 2 − Ψ ( P1 , P2 , u d )u ) V1 + A1 x
P2 =
* U total (t ) = U total (t )
=U
(18b)
.The desired total control efforts should
therefore satisfy:
term d (e2 bounded.
β ( A2 x 2 − Ψ ( P1 , P2, u d )u ) V2 − A2 x
When
and u d appear in the dynamics of e3 via the total control
U total (t )
(18a)
P2 =
According to Eqns. (4)-(5), the control inputs
effort
β (Q − A1 x 2 ) V1 + A1 x
Pi (t ) ≥ Pi > Pt , , Ψ ( P1 , P2 , u d ) is also positive. Hence, to satisfy (16), since u ≥ 0 and Q ≥ 0 , u d can be determined by the sign Then
of
* : U total
* ⎧ 1 U total (t ) ≥ 0 ud (t ) = ⎨ * ⎩− 1 U total (t ) < 0
(19)
The usage of Q and u is based on the rule that: u is fully open whenever it is feasible to reduce throttling loss, and to maintain the return line pressure low. There are three operating modes depending on whether any of the pressure constraints is active or not. Mode 1:
P1 (t ) > P1 and P2 (t ) > P2 :
This mode applies to *
the cases when no pressure constraint is active. U total is solved in an energy efficient way, which minimizes the usage of the throttling valve. Since the flow source is of positive displacement, if a negative flow is required, the flow is set to zero, and the trajectory tracking is achieved by using the throttling valve.
Case 1:
U
* total
pressure is higher than the threshold, and the operation goes back to Mode 1. The control law is:
G( x1 , ud )Ψ ( P1 , P2 , ud ) , > H ( x1 , ud )
u = 1,
Case 1:
* U total − G ( x1 , u d )Ψ ( P1 , P2 , u d ) Q= H ( x1 , u d )
Case 2:
U
* total
(20)
Case 2:
G ( x1 , ud )Ψ ( P1 , P2 , ud ) , ≤ H ( x1 , ud )
u
0 and P1 (t ) ≤ P1 ), or ( U
(21)
* total
< 0 and
A2 x 2 ψ ( P1 , P2 , u d ) * U total < 0, P1 ≤ P1 :
− A1 x 2 ψ ( P1 , P2 , u d )
In both cases, once u is chosen, * U total − G( x1 , u d )ψ ( P1 , P2 , u d )u Q= H ( x1 , u d )
(23)
In this mode, the pressure in the upstream
cylinder chamber connected with the flow source is low, and has the potential to cavitate. From Eqns (18a) and (18c), the variable flow source Q will be used to increase the pressure. The throttling valve u will be used for trajectory tracking according to the variable flow source Q to satisfy Eqn (16). The controller will stay in this mode to increase the chamber pressure until the pressure is higher than the threshold, and the operation goes back to Mode 1. The control law is: *
Case 1: U total
> 0, P1 ≤ P1 :
Q > A1 x 2 * Case 2: U total < 0, P2 ≤ P2 :
Similar to mode 2, this imposes an upper bound on u . If this bound is greater than 1, then we can set u = 1 , and (23) reduces to Eq. (20) in mode 1. In the above controller modes, the calculation of
ud
,
Q and u are based on the Lyapunov function as presented in Eqn. (11). The control inputs can guarantee that the Lyapunov function remains decreasing during modes switching. This guarantees the trajectory tracking performance. Since whenever the pressure reaches the pressure lower bound Pi , the controller will force the chamber pressure to increase (Mode 2 and Mode 3); this will guarantee that if the initial chamber pressures are above the threshold, they will continue to be above threshold Pi . Thus, cavitation cannot happen in the
Q > − A2 x 2
cylinder chamber.
In both cases,
u=
u
0, P2 ≤ P2 :
* U total − H ( x1 , u d )Q G ( x1 , u d )ψ ( P1 , P2 , u d )
5. SIMULATION RESULTS (22)
Notice that this imposes lower bound on Q (or an upper bound on u ). If the upper bound on u is greater than 1, then we can set u = 1 , and (22) reduces to Eq. (20) in mode 1. Mode 3 – This mode applies to the cases when the pressure in the return chamber is low. To prevent the pressure from going below the threshold, considering pressure dynamics in Eqns (18b) and (18d), the throttling valve u will be used to increase the chamber pressure when it reaches the threshold. The controller will stay in this mode to increase the chamber pressure until the
The control strategy is simulated using Simulink Matlab. Some of the system parameters of are based on an experimental set up that is being constructed currently; the other parameters are estimated from typical values, as shown in Table 1. Parameter Description
Symbol
Value
Cylinder mass
m
3kg
Cylinder cap area
A1
20.26cm 2
Cylinder rod area
A2
10.77 cm 2
Figure 7. Cylinder position trajectory tracking and the chamber pressures
Viscous friction coefficient
b
1.65 N /(m / sec)
Oil Bulk Modulus
β
1× 10 7 Pa
Throttling Valve Full Open Area
Avalve
0.2cm 2
The control efforts are shown in Figure 8. As designed, the directional control valve does not change direction very frequently, especially after the trajectory is well tracked. The throttling valve is kept fully open most of time, and it starts to close when the trajectory velocity decreases, comes to a stop and switches direction. The throttling valve is also used when the pressure constraint is active, so that the pressures inside both cylinder chambers stay above the pressure lower bound.
The constant bulk modulus of the oil is estimated by assuming the air volume content in the hydraulic oil at atmosphere pressure is 7% , and the load pressure is under 70bar [14]. The actuator was simulated to track a sinusoid trajectory with amplitude of 3cm , and a frequency of 1Hz . A constant load of 50 N was added as the disturbance load. The initial volume on the cap side of the cylinder is 1.2 L , and the initial volume on the rod side is 1.0 L . The control
λ4 = 30.The
= P2 = 2bar .
Cylinder Position [cm]
The trajectory tracking of the cylinder is shown in Figure 7. With the introduction of the integral control, the cylinder can track the trajectory accurately even with a load disturbance. The pressures inside both cylinder chambers start from an initial value that is higher than the lower pressure threshold, and the pressure remain higher than the threshold. The system pressure remains low.
Chamber Pressure [bar]
10
= −30 , λ2 = −30 , λ3 = 30 ,
pressure lower bound for both cylinder
chambers are set to be P1
20
trajectory reference cylinder position
10 8
4 2 0
1
2 time [sec]
3
14
4
Cap side Rod side
12
2 time [sec]
3
4
50
0 0
1
2 time [sec]
3
4
1 0.5 0 -0.5 -1 0
6
1
100
Throttling valve opening %
and
30
0 0
Directional valve
gains were selected as: λ1
40
Flow rate [lpm]
Table 1 Simulated system parameter
1
2 time [sec]
3
4
Figure 8. Control efforts from variable flow source, throttling valve and directional valve
6. CONCLUSION AND FUTURE WORK
10 8 6 4 2 0
0.5
1
1.5
2
time [sec]
2.5
3
3.5
4
In this paper, the displacement control of a single hydraulic actuator was investigated. An open circuit was proposed to realize direct displacement using one variable flow source, one directional valve, and one throttling valve. A multiple mode control strategy was developed to achieve accurate trajectory tracking with maintaining a low system operating pressure. The trajectory tracking was
based on a nonlinear back-stepping controller. This controller guarantees that neither of the cylinder chambers cavitates or goes unbounded. An experimental set was being developed. In the next step, we will implement this control strategy on the backhoe with the novel VVDP as the variable flow source.
ACKNOWLEDGMENTS This material is based upon work performed within the ERC for Compact and Efficient Fluid Power, supported by the National Science Foundation under Grant No. EEC0540834.
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CONTACT Meng Wang, Haink Tu, and Mike Rannow: graduate students with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota. Email: {wang134, tuxxx021, rann0018}@me.umn.edu Dr. Perry Y Li: professor with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota. Email: [email protected] Dr. Thomas R. Chase: professor with Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota.
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