PRECISION MOTION CONTROL OF ELECTRO-HYDRAULIC SYSTEMS WITH
ENERGY RECOVERY
A Thesis
Submitted to the Faculty
of
Purdue University
by
Ned A. Troxel
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science in Mechanical Engineering
August 2011
Purdue University
West Lafayette, Indiana ii
ACKNOWLEDGMENTS
I would like to thank my adviser, Dr. Bin Yao, for his patience and skill in teaching me principles of advanced control theory and for his advice during the past two years. I would also like to thank the other members of the examination committee for reading the thesis and for their helpful input. I also want to express my thanks to Yiyuan Chen, Alain Zoro and Yong Lin for help with pressure sensor calibration and again to Yong for help with gathering and processing experimental data for system identification. Many thanks are due to Sun Hydraulics Corp. for their generous donation of hydraulic valves and accessories to incorporate an accumulator energy recovery system into the cur- rent experimental setup for future studies. Thanks to the staff at Herrick Labs for their support and friendship throughout the past year of my graduate study. The work is supported in part by the US National Science Foundation (Grant No. CMMI-1052872). I want most of all to acknowledge the source of all good gifts, the God of creation. Anything good and anything useful in this thesis is there by His desire, and without His help it would never have been completed. iii
TABLE OF CONTENTS
Page LIST OF FIGURES ...... v NOMENCLATURE ...... vii ABSTRACT ...... x CHAPTER 1. INTRODUCTION ...... 1 1.1 Saving Energy in Hydraulic Systems ...... 1 1.2 Technologies to Increase System Efficiency ...... 2 1.2.1 Individual Pump Control ...... 2 1.2.2 Load-Sensing Pumps ...... 3 1.2.3 Independent Metering Valves ...... 3 1.2.4 Regeneration Flow ...... 5 1.2.5 Energy Recovery Accumulator ...... 7 1.3 Outline of Thesis ...... 10 CHAPTER 2. PROBLEM FORMULATION ...... 11 2.1 System Definition and Modeling ...... 11 2.2 System Component Information and Identification ...... 15 2.2.1 Mechanical Linkage ...... 15 2.2.2 Hydraulic Cylinder ...... 15 2.2.3 Servo Valve ...... 16 2.2.4 Cartridge Valves ...... 17 2.2.5 Hydraulic Pump ...... 17 2.2.6 Hoses and Pipes ...... 18 2.2.7 Hydraulic Fluid ...... 19 2.2.8 Sensors ...... 20 2.2.8.1 Incremental Encoder ...... 20 2.2.8.2 Magnetostrictive Piston Position Sensor ...... 20 2.2.8.3 Pressure Transducers ...... 21 2.3 Simulation Model ...... 21 CHAPTER 3. CONTROL DESIGN ...... 24 3.1 Adaptive Robust Controller Design ...... 24 3.1.1 Step 1 ...... 25 3.1.2 Step 2 ...... 28 3.2 Theoretical Performance ...... 32 iv
Page 3.3 Gain Selection ...... 32 3.3.1 Selection of Scaling Factors (σ Values) ...... 33 3.3.2 Selection of h Values ...... 33 3.3.3 Selection of k and ε Values ...... 33 3.3.4 Selection of Adaptation Gains ...... 34 3.4 Supervisory Control ...... 37 3.4.1 Servo Valve Supervisor ...... 39 3.4.2 Independent Metering Valves (4- and 5-Valve) Supervisor .... 40 3.4.3 Regeneration Accumulator Configuration Supervisor ...... 42 CHAPTER 4. SIMULATION AND EXPERIMENTAL RESULTS ...... 46 4.1 Comparative Simulation Results ...... 46 4.1.1 Model Parameters and Setup ...... 47 4.1.2 Selection of Controller Gains ...... 49 4.1.3 Trajectory Tracking Performance ...... 52 4.1.4 Energy Usage Results ...... 53 4.1.5 Comparison of 5- and 6-Valve Configurations ...... 55 4.2 Servo Valve Experimental Results ...... 59 CHAPTER 5. CONCLUSIONS ...... 64 LIST OF REFERENCES ...... 66 APPENDIX ...... 67 APPENDIX A. PROOF OF THEORETICAL PERFORMANCE ...... 67 A.1 Proof of Prescribed Transient Performance ...... 67 A.2 Proof of Asymptotic Tracking in the Absence of Disturbance ...... 68 v
LIST OF FIGURES
Figure Page 1.1 Independent Metering Valves Configuration...... 4 1.2 Five Valve Configuration with True Cross Port Flow...... 6 1.3 Hydraulic System Schematic...... 9 1.4 Energy Recovered by Reducing Throttling Losses...... 9 2.1 Hydraulic Excavator Arm...... 11 2.2 Swing Coordinate System (top view)...... 12 2.3 Illustration of Cylinder Variables...... 12 ∂x 2.4 Illustration of ∂q as a Lever Arm...... 13 2.5 Experimental Servo Valve Flow Characteristic...... 17 2.6 Example Flow Map and Inverse Flow Map...... 18 2.7 Linear Piston Position Sensor...... 20 2.8 Comparison of Simulation and Experimental Results: Case 1...... 22 2.9 Comparison of Simulation and Experimental Results: Case 2...... 23 3.1 Overall Controller Structure...... 24 3.2 5-Valve Desired Pressure State Plot...... 42 3.3 Pressure State Diagram Showing Desired State Trajectories...... 44 3.4 Desired Pressures vs. Desired Load Force...... 44 4.1 Eigenvalue Analysis for Selection of Adaptive Gains...... 51 4.2 Tracking Error Results...... 53 4.3 Power Consumption for Constant Pressure Source...... 54 4.4 Power Consumption for Load-Sensing Pressure Source...... 55 4.5 Comparison of 5-Valve and 6-Valve Configurations...... 58 4.6 Experimental Tracking Error for Servo Valve...... 59 vi
Figure Page 4.7 Experimental Load Force Tracking Error...... 60 4.8 Parameter Estimates...... 61 4.9 Experimental Tracking Error with Additional Mass...... 62 4.10 Load Force Tracking Error with Addtional Mass...... 62 4.11 Parameter Estimates with Additional Mass...... 63 vii
NOMENCLATURE
SYMBOL UNIT DESCRIPTION 2 AA m Head side piston area 2 AB m Rod side piston area α rad Variable angle created by rod and cylinder pin joints with swing pivot pin as vertex N·s b m Coefficient of viscous damping for piston βe Pa Effective bulk modulus of hydraulic fluid d N·m Disturbance torque
ε [-] Weighted combination of ε2 and ε3 N·m·rad ε2 s Final tracking error design parameter N·m2 ε3 s Final tracking error design parameter δ N·m Bounding function for disturbance torque
∆0 N·m Constant component of disturbance torque ∆˜ N·m Time-varying component of disturbance torque
∆pLS Pa Load-sensing pump pressure differential m3 ∆QA s Flow discrepancy from desired flow rate into head chamber m3 ∆QB s Flow discrepancy from desired flow rate into rod chamber FL N Load force (given by AA pA − AB pB)
FLd N Desired load force (virtual input) Γ Matrix of adaptation rates (diagonal) γ [-] Adaptation strength h2 N·m Robust feedback parameter m2 h3 s Robust feedback parameter J kg · m2 Inertia of swing mechanism k [-] Polytropic gas constant −1 k1 s Swing angle error proportional gain k N·s Sliding surface error proportional gain 2 kg·m·rad k Pa·m2 Load force error proportional gain 3 N·s viii
SYMBOL UNIT DESCRIPTION kv L√ Valve flow coefficient min·V· kPa κ [-] Constant used for robust feedback lA m Distance constant used to calculate lC lB m Distance constant used to calculate lC lC m Distance between rod and cylinder pin joints rad λ s Converging rate of Lyapunov function µ1 rad Angle constant used to calculate α µ2 rad Angle constant used to calculate α rad ωLS s Bandwidth for load-sensing pump model pA Pa Head chamber pressure
pAd Pa Head chamber pressure set point pac Pa Accumulator pressure pB Pa Rod chamber pressure
pBd Pa Rod chamber pressure set point ppr Pa Accumulator precharge pressure ps Pa Source pressure (a.k.a. pump pressure) pt Pa Return line pressure (a.k.a. tank pressure) q rad Swing angle qd rad Desired swing angle trajectory
ϕ1 System vector for swing angle dynamics ϕ2 System vector for load force dynamics φ2 Adaptation regressor for z2 error φ3 Adaptation regressor for z3 error m2 ψLd s Weighted combination of desired flow rates (virtual input) m3 QA s Volumetric flow rate into head chamber m3 QAac s Flow rate from head chamber to accumulator m3 QAB s Cross port flow rate from head chamber to rod chamber m3 QAd s Desired flow rate into head chamber (virtual input) m3 QB s Volumetric flow rate into rod chamber m3 QBac s Volumetric flow rate from rod chamber to accumulator m3 QBd s Desired flow rate into rod chamber (virtual input) m3 Qs s Total flow from pump into system m3 Qt s Total flow from system to tank ix
SYMBOL UNIT DESCRIPTION θ Vector of unknown parameters
θmax Vector of known upper bounds on parameters θmin Vector of known lower bounds on parameters θˆ Vector of parameter estimates θ˜ Vector of parameter estimation errors uv V Servo valve spool command 3 VA m Head chamber volume (includes VAmin) 3 VAmin m Hose and pipe volume from valve to head side port 3 VB m Rod chamber volume (includes VBmin) 3 VBmin m Hose and pipe volume from valve to rod side port 3 Vf m Volume of oil stored in accumulator 3 Vtot m Total accumulator capacity
V3 Lyapunov function used to quantify theoretical performance x m Position of piston in cylinder xmax m Cylinder stroke length z1 rad Swing angle tracking error m z2 s Sliding surface tracking error z3 N Load force tracking error x
ABSTRACT
Troxel, Ned A. M.S.M.E., Purdue University, August 2011. Precision Motion Control of Electro-Hydraulic Systems with Energy Recovery. Major Professor: Bin Yao, School of Mechanical Engineering.
Hydraulic systems will continue to play an important role in our society because of their large power to weight ratios and compact actuators. Hydraulic servo-systems will only become more ubiquitous as demands for increased performance, faster production, tighter tolerances and more efficient operation continue to become more stringent. There is a need for energy-efficient hydraulic systems which can deliver the high performance control required in today’s society. In this thesis, the technologies of independent metering valves, cross-port regenera- tion and accumulator-based energy recovery systems and the reasons why they are able to boost the efficiency of hydraulic systems are discussed. A control methodology including low-level adaptive robust control (ARC) of the flow rates and high-level decision logic is presented for simultaneously achieving high performance and energy-efficient operation for hydraulic systems with these features. An experimental setup consisting of a scaled-down hydraulic excavator arm is used as a case study. Model parameters are identified and a correlated system model for the swing motion is presented. This system is then used to illustrate the control design procedure. The ARC algorithm is quite flexible and is applied without major modification to systems using four different valve configurations. A systematic method is proposed for selecting the controller gains to reduce the amount of trial and error required for this task. The ARC algorithm guarantees high performance motion control even in the presence of uncertainties and disturbances by combining adjustable nonlinear model compensation with robust feedback. The model compensation is adjusted by controlled parameter adap- tation. The high-level logic selects the most efficient way to implement the flow rates de- termined by the ARC algorithm and also determines the pressure level at which the system can operate most efficiently. xi
Simulation results are presented which demonstrate the ability to achieve both precision motion control and increase energy efficiency using the proposed control strategy. These results clearly show the operation and benefits of independent metering, cross-port regen- eration and energy recovery. Experimental results were obtained for one of the valve configurations studied. The tracking performance achieved demonstrates the effectiveness of the control algorithm. Ad- ditional experimental results are still needed to confirm that the energy efficiency predicted by the simulations can be achieved without loss of performance in practice. The addition of an energy recovery accumulator to a system with an under-utilized constant pressure source can allow the system to approach the efficiency which would be obtained by using a load-sensing pump. There is potential for increased efficiency by using a central energy recovery accumulator on systems with multiple actuators, since the accumulator is able to recover energy normally lost to throttling for actuators with widely- varying pressure requirements. 1
1. INTRODUCTION
Hydraulic systems are well-suited to a variety of applications, especially where large forces are required, but little space is available for actuators. Hydraulic servo-systems are able to provide high performance force or motion control with the addition of sensors and elec- tronic control elements such as solenoid-operated valves and variable-displacement pumps. Unfortunately, hydraulic systems usually have rather poor energy utilization compared with electro-mechanical systems. Simulations of a Bobcat 435, a mini-excavator weighing around 5 tons, were performed by Zimmerman et al. in [1]. The operation of the hydraulic system, which included a load- sensing pump and 4 active actuators, was simulated on a 27 second digging cycle and showed that of the total energy used for the cycle (around 260 kJ), only about one third was consumed in the actuators, yielding an efficiency of around 33%. The power used by the system varied from 2 to 25 kW. The energy utilization would have been even poorer had a constant pressure pump been used. Naturally, a much greater amount of energy is consumed for heavier equipment. In [2], Lin et al. presented a plot of the output power for a 20-ton excavator during a typical digging condition. The maximum output power was around 120 kW, with a mean value between 50 and 80 kW. Any technology which can boost the efficiency of hydraulic machinery has potential for significant fuel savings. With the great variety of commercial hydraulic components currently available, enough design flexibility can built into the system to allow a hydraulic positioning system to achieve both high-precision tracking accuracy and to increase the system efficiency. This thesis will present various system configurations and control strategies with the dual objec- tive of high-precision motion control and energy efficient operation.
1.1 Saving Energy in Hydraulic Systems Increasing the efficiency of any system means reducing total input energy required to perform a given task. The energy provided to a hydraulic system on the interval [t0,t1] from a pump or other pressure or flow supply may be calculated by:
t1 E = p (τ)Q (τ)dτ (1.1) s ˆ s s t0 2
where ps (t) is the supply pressure and Qs (t) is the flow rate into the system. Note that Equation (1.1) gives the energy provided to the system via the hydraulic fluid pressure, and does not account for factors such as pump efficiency. It is clear that the energy used by the system can only be reduced if either a lower supply pressure is used or the flow into the system is reduced. One or both of these can be done if less energy is dissipated by the system. The main sources of energy dissipation in hydraulic systems are mechanical friction, throttling losses (often most significant in the valves), and leakage. Throttling loss is often the largest of these three components and can be most influenced by changing the control valve configuration and control algorithm. Throttling losses are caused by friction between the hydraulic fluid and the flow passage and by viscous shear forces within the fluid. Re- ducing the throttling loss in a hydraulic system may be thought of as making it easier for hydraulic fluid to flow, as would happen by straightening and widening pipes and by fully opening valves, for example. The result is that same amount of fluid can flow with a lower pressure drop. The flow path obstruction created by a valve results in permanent pressure (and energy) loss. The power dissipated by oil flow through a valve (i.e. due to throttling) is the product of the flow rate and the pressure drop (i.e. P = ∆p·Q). Thus, the most efficient way to supply a given flow is with the smallest pressure drop possible. For hydraulic systems controlled by valves, some throttling losses are unavoidable. A valve-controlled system regulates the flow to actuators by changing the valve position. Some pressure drop across the valves is necessary for precision control of the flow.
1.2 Technologies to Increase System Efficiency Five technologies which improve the energy utilization in hydraulic systems will be discussed in this section. They include individual pump control, load-sensing pumps, in- dependent metering valves, regeneration flow and an accumulator-based energy recovery system.
1.2.1 Individual Pump Control The only way to eliminate valve throttling losses is to eliminate valves. This may essentially be done if each actuator is supplied by its own variable-displacement pump. The flow into the actuator can be controlled by simply controlling the flow generated by the pump. Theoretically, only the flow required by the system is delivered by the pump and the 3 pump outlet pressure will only be as high as required by the load on the actuator. Any valves included in the system are on-off valves which only serve to change the direction of the flow into the actuator. Since the valves are either fully open or fully closed, valve throttling loss is essentially eliminated. The losses of the system are mainly due to internal pump leakage and mechanical friction. The drawbacks of this type of system include increased equipment cost since more than one pump is required and slower dynamic response. The bandwidth of variable-displacement pumps is lower than for control valves, the system cannot react as quickly to changes in demanded flow or load. This will manifest itself as increased tracking error. Pump-controlled systems will not be addressed in further detail in this thesis, but it is well to recognize the benefits of their operation and the reasons for their increased efficiency.
1.2.2 Load-Sensing Pumps Load-sensing pumps maintain a slightly higher supply pressure than the maximum cylinder chamber pressure. Thus, when the highest chamber pressure decreases, so does the supply pressure and less energy is input to the system as compared to a constant pres- sure supply. Because the pressure drop between the source and the chamber pressures is reduced, the throttling losses are also reduced and the same performance can be achieved with lower input energy. When a load-sensing pump powers a single actuator, the opera- tion is nearly as efficient as would be achieved in a pump-controlled system. When multiple actuators with different pressure requirements are supplied, some energy is wasted in throt- tling the source pressure down for all actuators except the one with the highest pressure requirement.
1.2.3 Independent Metering Valves In many traditional hydraulic applications, a single valve (e.g. a 3-position, 4-way proportional directional valve) controls the flow rates into both cylinder chambers. For such a system, it is not possible to independently specify the flow into each cylinder chamber. The two flow rates are coupled by the single spool position, making the independent control of both chamber pressures impossible. While the net force on the piston (the load force) can be controlled, no additional freedom is available to influence the pressures. 4
Figure 1.1: Independent Metering Valves Configuration.
Independent metering valve configurations have been studied to see how separate valves may be used to increase efficiency and performance. A detailed review of the state of the art for these types of systems is given by Eriksson and Palmburg [3]. Book and Goering [4] and Hu and Zhang [5] studied a configuration of five cartridge valves. A schematic of this system is shown in Figure 1.1. In this configuration, the flow from the pump to each chamber and the flow from each chamber to the tank are con- trolled by four separate valves. A fifth valve connects the pump directly to the tank. They showed that such a configuration can emulate the characteristics of open-center, closed- center, tandem-center and float-center spool valve geometries simply by changing the con- trol software, though these characteristics are traditionally created by various spool valve hardware designs. Hu and Zhang also demonstrated a regeneration function which in- creased the cylinder extension speed. Two variations of the configuration in Figure 1.1 are possible. First, the fifth valve may be replaced with a pressure relief valve which provides the same function. Theoretically, this valve could also be eliminated when a variable displacement pump is used, but for safety reasons, a pressure relief valve should be included to protect the pump against pos- sible pressure shocks. The second variation is that the check valve in the return line could be omitted. This valve is not necessary for independent metering, but is added to allow for regeneration flow, as studied by Shenouda [6]. Much study of independent metering valves focuses on the ability of the configuration to operate as a “variable geometry” spool valve (that is, able to recreate the flow charac- teristics associated with different geometries). Changes made in software allow the valve’s 5 characteristics to be significantly altered, giving rise to names such as Smart Valve, Multi- function Valve and Programmable Valve. From an efficiency standpoint however, the more important property of independent metering valves is the ability to independently control the flows into and out of both cylinder chambers. The practical significance of this is inde- pendent control of the cylinder pressures. Both Shenouda [6] and Liu and Yao [7] have shown the potential for energy savings when such a configuration is used with a load-sensing pump. Since the pressures can be independently specified, a given net force on the cylinder can be achieved in a variety of ways. By setting one chamber pressure to a low level, the other chamber pressure can be used to achieve the required load force with as low a pressure as possible. The result is a lower maximum chamber pressure. For a load-sensing pump, the supply pressure (and thus the input energy) will be significantly reduced.
1.2.4 Regeneration Flow If a constant pressure source is used, then no matter how the system pressures change, the energy supplied for two different situations will be equal if the flows are equal. The only way to save energy is to reduce the flow from the pump. This may be accomplished by recycling hydraulic fluid already delivered to the system. Regeneration is the process of directing the flow from one chamber into the other with- out the use of the pump. Shenouda studied the potential for energy savings using normal and regeneration operation modes on a 4-valve configuration [6] similar to that studied by Hu and Zhang. Regeneration flow can be used whenever the cylinder chamber supplied by the pump actually has a lower pressure than the other chamber. This commonly occurs during decel- eration periods or when a large force is acting in the direction of motion (e.g. lowering a heavy load). This type of load is called an overrunning load. Shenouda showed in [6] that for a 4-valve setup, flow could be regenerated from one chamber to the other through the return valves (flow through valves V3 and V4 in Figure 1.1). This is referred to as “low side regeneration.” It should be noted that Shenouda’s system did have a check valve in the return line (as shown) which supports a slight pressure drop before opening, thus maintaining a positive pressure. The flow could be in either direction, depending on the direction of the overrunning load. Energy is saved because the required supply flow is reduced. 6
A single rod cylinder has a significant difference in the piston areas. When only a small positive load force is required (e.g when extending the rod with nearly constant velocity), the pressure of the rod-end side can exceed the pressure of the head end. Thus, fluid can be regenerated from the rod chamber to the head chamber. This type of regeneration has been used with a four valve configuration by allowing the flow to pass through the supply valves (fluid flows first through valve V2 and then valve V1 in Figure 1.1). In order for this to be possible, the rod chamber pressure must exceed the source pressure. This type of flow is called “high side regeneration.” High side regeneration can boost the maximum extension speed, but usually requires significantly higher source pressure (and hence greater energy consumption for a load-sensing setup) than normal operation, as observed by both Shenouda and Hu and Zhang [5, 6].
Figure 1.2: Five Valve Configuration with True Cross Port Flow.
Liu and Yao presented a system with a 5-valve configuration which used a valve to control flow directly between the cylinder ports and demonstrated its ability to save energy [7]. This configuration is shown in Figure 1.2; Valve #3 is the regeneration valve. It should be noted that this configuration differs from that studied by Hu and Zhang since the extra valve connects the two cylinder ports directly and does not connect the pressure source and tank (though a relief valve is included which does provide this function). 7
When true cross port flow is possible, the restrictions on the pressures during regener- ation are less restrictive. Thus, when an overrunning load is present, the pressure in the regeneration line can exceed the tank pressure significantly. Similarly for constant velocity extension, the rod chamber pressure need not exceed the supply pressure to use regenera- tion. The only restriction is that the cylinder chamber supplied by the pump actually has a lower pressure than the other chamber. The flow from the high pressure chamber can be used to either reduce the pump flow (thus saving energy) or increase the maximum possible flow (thus increasing the speed).
1.2.5 Energy Recovery Accumulator In fluid power systems, an accumulator often acts as an energy storage device. A hydro- pneumatic accumulator contains a quantity of precharged gas (often nitrogen is used) and a port through which oil can enter or leave the accumulator. The gas is separated from the oil by a barrier such as a piston, bladder, or diaphragm. When the port is exposed to high pressure, oil enters the accumulator, compressing the gas and storing energy. When the port pressure falls, oil will flow from the accumulator under the force of the pressurized gas. Accumulators are able to integrate directly into hydraulic systems and have been used in many applications. Yang et al. proposed an energy recovery system for a hydraulic elevator which incorpo- rated a hydraulic accumulator and a four-quadrant motor [8]. As the elevator traveled both upward and downward, oil was directed through the hydraulic motor. This enabled both capture and reutilization of the potential energy. A variable-frequency drive electric motor connected to the hydraulic motor allowed the hydraulic motor to supplement the energy provided to the load or to recover excess energy. Liang and Virvalo have developed an energy recovery system for a hydraulic crane [9]. This system incorporates a pair of assistant cylinders mounted in a parallel force configu- ration with the main boom cylinder, thus sharing the load among all three cylinders. When the load is first raised, the assistant cylinders draw oil from the oil reservoir through a check valve. During this motion, all the force is provided by the main cylinder. When the cylinders retract while lowering the load, the force of the load forces oil from the assistant cylinders into an accumulator. The next time a load is lifted (and for all subsequent cycles), the pressure stored in the accumulator acts on the assistant cylinders. This significantly re- duces the load required from the main cylinder, allowing the load-sensing pump to supply flow at a lower pressure to the main cylinder. This results in significant energy savings. 8
This energy recovery system is essentially decoupled from the original system since there is no flow directly between the main cylinder and the assistant cylinders. The disadvantage of this type of system is the requirement for one or two additional hydraulic cylinders. Kr- navek has shown several variations of how this technique of using an assistant cylinder can be employed [10]. Margolis has proposed that by combining an energy recovery system with an actuator, system efficiency can be boosted tremendously [11]. He presents a simulation of an appli- cation of the concept to a pneumatic positioning system by using a pneumatic accumulator in which the required energy for a step response is reduced from a conventional system by 40 times. The distinctive feature of the control strategy employed is that the servo valve is never open just a fraction, but switches very rapidly between the positive and negative limits (even when velocity is near zero). One effect of this is that very little energy is lost in throttling, since the orifice areas are either wide-open or fully-closed almost all the time. The rapid switching required makes such a control methodology impractical for hydraulic valves, but there is reason to believe that any strategy which significantly reduces throttling losses will also achieve significant energy savings. This thesis presents an energy recovery system for hydraulic cylinders. The system consists of an accumulator and two control valves, through which the flows to and from the cylinder chambers are controlled. A regeneration flow path is created when both valves are opened. A schematic of the system is shown in Figure 1.3. The effect of this modification is to decouple the regeneration flows, meaning the flow out of the high pressure cylinder need not equal the flow into the low pressure cylinder. This is the case for a simple regeneration path composed only of piping and valves, but when an accumulator is used, excess flow from one cylinder can be stored in the accumulator or extra flow may be supplied from the accumulator to the cylinder. The accumulator may be charged whenever there is flow out of a cylinder with pressure higher than that of the accumulator, and flow from the accumulator can replace pump flow whenever there is flow into a cylinder chamber at a lower pressure than the accumulator. The accumulator reduces throttling losses by acting as a second flow source or sink. High-pressure flow out of a cylinder can be directed to the accumulator rather than simply throttled to the tank. The throttling loss is lower because of the lower pressure drop. Figure
1.4 illustrates the situation where a flow Q from a cylinder at pressure pcyl is throttled to the reference pressure and the case when it flows into an accumulator with pressure pac. The shaded areas in the diagram represent the amounts of energy lost due to throttling and recovered by the accumulator. It is noted that although energy is stored in the accumulator, 9
Figure 1.3: Hydraulic System Schematic.
Figure 1.4: Energy Recovered by Reducing Throttling Losses.
it could only be reused immediately if another cylinder pressure is below the pressure
pac. Then, the accumulator could act as a secondary pressure source to supply flow to the cylinder at low pressure, replacing the flow from the pump. Of course, for a very slight pressure drop, the maximum achievable flow through a valve may be less than the required amount. In such cases, the accumulator can act as a source or sink in parallel with the main 10 pressure source or the tank, which still provides a portion of the benefits seen when the accumulator is used to provide the entire flow.
1.3 Outline of Thesis This chapter has discussed the factors affecting hydraulic system efficiency and re- viewed technologies which can increase it. These concepts provide a foundation for the methods which will be introduced later to optimizie the use of these technologies. Chapter 2 presents an analytical model for an example hydraulic system and outlines the control problems to be solved. Chapter 3 presents the control methodology used to achieve high performance tracking and efficient operation, gives a detailed derivation of the controller and proposes a method for choosing the various design parameters of the controller. Chap- ter 4 presents simulation and experimental results. A comparison of simulation results for four different system configurations shows the benefits of independent metering, regenera- tion flow and using an accumulator for energy recovery. Experimental results for a 4-way valve configuration are presented to demonstrate the effectiveness of the control algorithm in practice. Chapter 5 concludes the thesis. 11
2. PROBLEM FORMULATION
2.1 System Definition and Modeling The system to be considered is the swing mechanism of an excavator arm, driven by a single hydraulic cylinder whose motion is controlled by the flow of hydraulic fluid into the cylinder chambers. A photo of the actual system is shown in Figure 2.1. The primary goal is to have the swing angle q track a known desired angle trajectory qd (t) as accurately as possible. A secondary goal is to minimize the energy required to follow such a trajectory.
It is assumed that qd is sufficiently smooth to allow for its derivatives to be used in control design. The measurements of the swing angle q, swing velocityq ˙, head-end chamber pressure pA and rod-end chamber pressure pB are assumed to be available for the control design (i.e. full state feedback). These variables are illustrated in Figures 2.2 and 2.3. Additionally, pressure measurements from both sides of each valve used in the system are available as well. These are necessary to accurately calculate valve positions to achieve
Figure 2.1: Hydraulic Excavator Arm. 12 a particular flow rate. A controller which effectively controls the position could easily be adapted for the related problem of tracking an angular velocity trajectory.
Figure 2.2: Swing Coordinate System (top view).
Figure 2.3: Illustration of Cylinder Variables.
The system is modeled as follows:
∂x Jq¨(t) = · (FL (t) − bx˙(t)) + d (t) ∂q FL = AA pA − AB pB VA (x) p˙A (t) = −AAx˙(t) + QA (t) (2.1) βe VB (x) p˙B (t) = ABx˙(t) + QB (t), βe
where the exact values of inertia J, viscous damping coefficient b and effective bulk mod- ∂x ulus βe are not known. The partial derivative ∂q is a known nonlinear function which is 13 equal to the distance between the swing pivot and the point where the axis of the cylinder crosses the center of the swing arm (i.e. the lever arm). This is more easily seen from the di- agram in Figure 2.4, which illustrates the relation between the lever arm and the arc length to angle ratio. The function d (t) represents the uncertain nonlinearities and disturbances acting on the system. Let d (t) be composed of a constant and time-varying portion such that d (t) = ∆0 + ∆˜ (t). The constant (or in practice, low frequency) component ∆0 may be estimated online. The signals x, VA and VB may be calculated from the angle measurement q as presented by DeBoer [12] and illustrated in Figure 2.2. The equations used for these calculations are as follows:
q 2 2 lc = lA + lB − 2lAlB cos(α) π α = − µ − µ − q 2 1 2 x = lC − x0 ∂x −lAlB sin(α) = q (2.2) ∂q 2 2 lA + lB − 2lAlB cos(α) ∂x x˙ = q˙ ∂q VA = VAmin + A1x(t)
VB = VBmin + A2 (xmax − x(t)),
where the lengths lA and lB, the angles µ1 and µ2 and the volumes VAmin and VBmin are known constants, whose values will be given in the following sections.
∂x Figure 2.4: Illustration of ∂q as a Lever Arm.
The flow rates QA and QB are treated as virtual inputs to the system. In reality, these flows are controlled by valves and the actual inputs are the valve commands. The valve dynamics will introduce a deviation of the actual flow rate from the desired flow rate. 14
Moreover, the calculation of valve commands from desired flow rates depends on accurate flow mappings. Even if highly accurate flow mappings are available, the flow characteris- tics of the valve may change with environmental factors such as oil temperature and wear, and so some calculation error will be present. It is thus practical to split the actual flow rates into the specified flow rate component and an error component, as follows:
QA (t) = QAd (t) + ∆QA
QB (t) = QBd (t) + ∆QB. (2.3)
The flow error components are assumed to be constant for the purposes of control design and will be estimated online to mitigate the effects of flow mapping error. The specified flow rates now become the virtual inputs to the system and may be freely specified. The sys- h iT 1 b ∆0 tem is linearly parametrized by the unknown parameter set θ = J J J βe βe∆QA βe∆QB as follows:
∂x ∂x q¨(t) = θ FL (t) − θ x˙(t) + θ + θ ∆˜ (t) 1 ∂q 2 ∂q 3 1 VA (x) p˙A (t) = θ4 (QAd (t) − AAx˙) + θ5 (2.4)
VB (x) p˙B (t) = θ4 (QBd (t) + ABx˙) + θ6.
The derivative of the load force, FL, is influenced by QAd and QBd as follows:
F˙L = AA p˙A − AB p˙B (2.5) AA ∂x θ5 AB ∂x θ6 = θ4 QAd (t) − AA q˙ + − θ4 QBd + AB q˙ + . VA (x) ∂q θ4 VB (x) ∂q θ4
It is assumed that vectors of minimum and maximum parameter values (denoted θmin th and θmax) are known. Thus, θmin,i < θi < θmax,i is always satisfied for each i component
of θ. Moreover, the uncertainty nonlinearity is bounded by a known function: ∆˜ ≤ δ (x,t). Strictly speaking, the inputs to the system are the control valve command signals. How- ever, when the valve response time is fast compared to the required flow response time, the dynamics of the valve position may be ignored during control design in order to simplify the design task. This assumes the flow rates are a static function of valve command and pressure drop. 15
2.2 System Component Information and Identification This section will describe the various components which make up the system, give their dimensions and relevant concerns. Offline system identification of the physical system parameters will also be discussed.
2.2.1 Mechanical Linkage In this thesis, up and down motion of the links of the excavator is not considered. This could be dealt with more rigorously by considering the dependence of the inertia of the linkage on the position of the links, with the effect of the constant mass of the links and the effect of an additional unknown load considered separately. When up and down motion of the links is not considered, the remaining motion of the swing mechanism is modeled as a rigid body rotation about the swing axis. The effect of flexible structural modes is neglected, and thus represents a source of neglected high-frequency dynamics. The inertia of the swing mechanism was estimated as 105 kg·m2 using Least Squares estimation on experimental data. The data used for this estimation will be presented later in Figures 2.8 and 2.9 as validation of the model. The relative locations of the pin joints determine the nonlinear kinematics given in
Equation (2.2). The lengths lA and lB have been measured to be 19.9 cm and 30.2 cm, respectively. The measures of angles µ1 and µ2 are given by DeBoer in [12] as 0.492 rad and 0.145 rad, respectively.
2.2.2 Hydraulic Cylinder A Parker hydraulic cylinder is used to control the swing motion (model number 02.00 D2HXTS23A / 11.00). It has a cylinder bore diameter of 2 inches and a rod diameter of
1.375 inches. These diameters allow calculation of the piston areas. The head side area, AA, 2 −3 2 2 −3 2 is 3.1416 in (2.027 × 10 m ) and the rod side area, AB, is 1.6567 in (1.069 × 10 m ). The cylinder stroke, xmax, is 11 in. (0.28 m). The interface of piston and cylinder is the primary source of mechanical friction in the system. This friction is modeled as linear, viscous damping. The same offline least-squares estimation mentioned in the previous section resulted in an estimated viscous damping N·s coefficient of 3700 m . 16
2.2.3 Servo Valve A Parker servo valve (model number BD760 AAAN10) was available to control the swing cylinder flow. For a 4-way spool valve, there are four distinct valve orifices which could potentially have different flow characteristics. Flow measurements were taken by moving the piston at constant velocity and using the velocity signal and known piston areas to calculate the flow. A regression analysis was used to determine the four flow coefficients using the following model:
p Qi = kvi |uv| ∆p, (2.6)
where Qi is the flow, uv is the command voltage, ∆p is the pressure drop and kvi is the flow coefficient. The pressure drops and the valve command were measured, while the flow was estimated as the product of the measured piston velocity and the associated piston area. Care was taken to use flow estimates where both the velocity and the pressures were approximately constant to reduce estimation error. This analysis was performed on each orifice separately, giving the 4 coefficients listed in Table 2.1. The resulting fit of the regression is plotted in Figure 2.5. Note that this figure does not show the pressure drop and this explains why the predicted flow is not a straight line when plotted against the input command alone. To simplify the control, it may be kv1+kv2 expedient to take an average of the two coefficients for QA and QB (i.e. use kv1eq = 2 , kv3+kv4 kv2eq = 2 ).
Table 2.1: Servo Valve Flow Coefficients.
Flow L 3 √ in√ Flow Path Coefficient min·V· kPa s·V· psi kv1 0.06288 0.1680 Pressure to Head Chamber kv2 0.07108 0.1899 Head Chamber to Tank kv3 0.06368 0.1701 Pressure to Rod Chamber kv4 0.05318 0.1421 Rod Chamber to Tank
No external measurement of the servo valve spool position is available, so for the pur- poses of identification and modeling, the valve command was regarded as a position. When the valve was modeled as a static function of the input command, the experimental pressure traces were found to lag those of the model. The discrepancy was eliminated by applying a second order linear filter with 8 Hz natural frequency and damping ratio of 0.7 to the 17
Figure 2.5: Experimental Servo Valve Flow Characteristic.
control input to model the spool dynamics. This is in agreement with the valve bandwidth of 8-10 Hz reported by Mohanty and Yao in [13].
2.2.4 Cartridge Valves The cartridge valves are Vickers proportional flow control valves (model number EPV10- L A-8H-12D-U-10). They have a maximum flow rate between 26 and 33 min. The valve is designed to provide a flow rate proportional to the voltage command at a wide range of pressures, but the flow actually remains a function of both the command voltage and pres- sure drop. To decrease the error in the specified flow rate, inverse valve mappings are used to calculate the voltage commands to give the desired flow rates. An example of such a mapping from the work of DeBoer [12] is shown in Figure 2.6. According to the manufacturer’s data sheet, the valves have a response time of 35 ms for a 50% stroke. The bandwidth of the valves is around 40 Hz.
2.2.5 Hydraulic Pump A variable displacement vane pump (Model Racine PSV PSS0 25ERM 13) provides L the flow to actuate the system. It has a rated flow of 76 min (20 gpm) at 69 bar (1000 18
Valve 2:HEPC Inverse Flow Mapping Valve 2:HEPC Flow Mapping
10 35
30 8
25 6 20
15 Volts (V) 4 Flow (LPM) 10 2 5
0 0 10 8000
8 8000 6000 40 6 30 6000 4000 4 4000 20 2000 2 2000 10 0 0 0 Volts (V) 0 PD (KPa) PD (KPa) Flow (LPM) Figure 2.6: Example Flow Map and Inverse Flow Map.
psi). The pump is pressure compensating so the flow is adjusted to meet the demand while maintaining a constant pressure around 1050 psi. A pressure relief valve and two 1 gallon accumulators are attached to the pump outlet to protect the pump from excessively high pressure and to reduce fluctuations in the source pressure. These components are included in the schematic in Figure 1.3.
2.2.6 Hoses and Pipes Oil flows from the control valves to the cylinder chamber through flexible high-pressure hoses (Weatherhead 3/4–H42512). These hoses have a double-layered, braided-steel con- struction and are rated for a working pressure of 172 bar (2500 psi) and a burst pressure of 620 bar (9000 psi). The hoses are important to consider for three reasons: they add inefficient volume to the chambers, they introduce compliance into the system, and they constitute a source of high-frequency dynamics. The inefficient volumes introduced by piping and hoses are given in Table 2.2. The volumes are larger for the cartridge valve block because the oil flows through an additional manifold composed of steel pipe. Both the change in volume and the increased compliance due to the hoses affect the pressure dynamics. While the inefficient volume introduced can be calculated quite accu- rately from the length and diameter of the hoses and piping, the effect on the bulk modulus is harder to characterize. Rather than modeling the compliance of the hoses explicitly, 19
Table 2.2: Inefficient Volumes.
Volume with Volume with Cartridge Valve Servo Valve Manifold Head Side, VAmin 0.5102 L 1.270 L Rod Side, VBmin 0.6377 L 1.208 L a more practical approach is to experimentally identify an effective bulk modulus, thus lumping the effects of mechanical compliance and the compressibility of the oil. For very long hoses, the pressure in the hose cannot be regarded as both uniform throughout the hose and identical to the cylinder chamber pressure, but the dynamics of the oil transmission line must be considered. For the relatively short lengths of hose used on the experimental system, the natural frequency of the effect is sufficiently high that it may be neglected.
2.2.7 Hydraulic Fluid The hydraulic oil used in the arm is Exxon NUTO H 68. Three parameters characterize hydraulic oil: density, viscosity, and bulk modulus. The pressure drop through a valve is increased for both higher density and higher viscosity. These values are treated as constant in the control design and are only accounted for indirectly with the valve coefficients and experimental flow mappings. Variations in temperature will affect density only slightly, but the viscosity will be greatly affected. The density is approximately 882 kg/m3 (at 15°C). Viscosity values of 68 cSt at 40°C and 8.5 cSt at 100°C are given by the manufacturer. The viscosity is estimated to be 168 cSt at 20◦C. If significant temperature variations occur, deviations from the commanded flow rate could result. Bulk modulus relates the change in pressure to the change in the density of the fluid. Theoretical values for bulk modulus of oil may be as high as 1.6 × 109 Pa, but are sig- nificantly reduced by entrained air and mechanical compliance according to Jelali and Kroll [14]. Entrained air, which is essentially the presence of air bubbles within the oil, only has an effect at low pressures as it is absorbed into the oil at high pressure. The best approach is to use experimental data to verify the accuracy of any value used for effective bulk modulus. The bulk modulus identified by Mohanty and Yao in [13] was 2.71×108 Pa. The simulation model used in this study uses a value of 2.5 × 108 Pa, which was selected to match the natural frequency of the system. 20
2.2.8 Sensors
2.2.8.1. Incremental Encoder
An incremental encoder measures the angular position. The encoder has 4096 counts, yielding an effective measurement resolution of 3.83 × 10−4 rad after quadrature. While an accurate angular velocity can always be obtained by filtering, the absolute position is unknown. The encoder only gives an accurate relative position from the position at start- up.
2.2.8.2. Magnetostrictive Piston Position Sensor
The hydraulic cylinder is equipped with a linear position sensor which uses magne- tostriction to sense the position of a magnet collocated with the piston. The sensor uses a rod contained inside the piston rod to send current pulses and detect the resulting strain. By measuring the time between the current pulse and when a strain pulse is detected, the distance may be determined. It should be noted that the sensor rod does not alter the area or volume of the head chamber significantly. A gap between the piston and the sensor rod allows hydraulic fluid to flow to the bottom of the hole in the piston rod, and the effective cross section is the same. The volume would be slightly increased, but the sensor rod dis- places nearly as much volume as is introduced by the hole in the rod. This may be seen more clearly from Figure 2.7. This position measurement from this sensor is used with the geometric formula in Equation (2.2) to calculate the initial angular position.
Figure 2.7: Linear Piston Position Sensor. 21
2.2.8.3. Pressure Transducers
Omega PX603 pressure transducers are used on the system. The sensors for the source pressure and return line are able to sense up to 207 bar (3000 psi). They have a sensitivity of 19×10−3 V (1.3 mV), a bias of 1 V and a rated accuracy of ±0.83 bar (12 psi). Sensors bar psi are mounted at the cylinder ports and are able to sense up to 138 bar (2000 psi). They have a sensitivity of 29×10−3 V (2 mV), a bias of 1 V and a rated accuracy of ±0.55 bar (8 bar psi psi).
2.3 Simulation Model A Simulink model of the swing motion with the hydraulic cylinder and the servo valve was constructed using the parameters and modeling equations given throughout this chap- ter. Experimental data was gathered by manually controlling the servo valve input voltage with a joystick. The model was then correlated using this data. Two test cases are shown in Figures 2.8 and 2.9, where the experimentally applied valve commands are input to the model. There is some discrepancy in the pressure traces, due to a combined effect of the inaccuracy in the servo valve flow characteristic, the rough approximation of servo valve spool position dynamics, the unmodeled leakage flow and other unknown factors. The bulk modulus and the natural frequency of the servo valve were adjusted to make the natural frequency of pressure and velocity oscillations match. This model is adequate for control design purposes. 22
Figure 2.8: Comparison of Simulation and Experimental Results: Case 1. 23
Figure 2.9: Comparison of Simulation and Experimental Results: Case 2. 24
3. CONTROL DESIGN
A two-level control strategy is adopted for this problem. An adaptive robust controller is used to calculate a constraint on the flow rates which must be satisfied for high preci- sion motion control. This information can be used with any control valve configuration to determine the required valve commands. A “supervisor” controller or set of logic rules governs how the individual flow rates are specified, including whether or not to use regen- eration flow (if a regeneration path exists) or to charge or discharge the accumulator. This supervisor controller is specific to the valve configuration. For a directional servo valve, its function is limited to a simple desired spool position calculation, while for a regener- ation accumulator configuration, its task includes specifying the flow rates for 6 valves to maximize the efficiency and ensure the accumulator stays within limits. For the cylinder position to track a desired trajectory, a certain load force on the piston is required to produce the desired motion. Satisfying a single constraint between the two cylinder flow rates assures that this load force will be maintained.
Figure 3.1: Overall Controller Structure.
3.1 Adaptive Robust Controller Design Let θˆ denote the online parameter estimates and define the estimation error as θ˜ = θˆ − θ. The parameter adaptation law used in this control design is of the discontinuous projection type defined as follows: 25
ˆ˙ θ = Projθˆ (Γτ), (3.1) p p×p where θˆ ∈ R and Γ ∈ R is a diagonal, positive definite adaptation rate matrix and p τ ∈ R is the adaptation function, which will be specified later in the design. For such a Γ, th th ˆ the i component of Projθˆ (Γτ) is defined in terms of the i component of θ as follows: ˆ 0 ifθi = θmin,i and•i < 0 ˆ Projθˆi (•i) = 0 ifθi = θmax,i and•i > 0 •i else. Such a projection mapping ensures that the parameter estimates will always lie between the known bounds, and hence the maximum parameter estimation error is uniformly bounded by a known value θ˜ ≤ |θmax − θmin|. This fact will be used to design a controller with guaranteed transient performance. The system model (2.1) may be rewritten in simpler form as follows:
∂x T q¨ = θ FL + ϕ θ + θ ∆˜ 1 ∂q 1 1 ˙ T FL = θ4ψLd + ϕ2 θ, (3.2)
T 2 h 2 2 iT ∂x AA AB AA AB ∂x AA AB where ϕ1 = 0, −q˙ , 1, 0, 0, 0 , ψLd = QAd − QBd and ϕ2 = 0, 0, 0, − + q˙, , − . ∂q VA VB VA VB ∂q VA VB The function ψLd gives the combined effect of the desired flow rates on the load force and will be treated as a virtual input.
There are unmatched uncertainties and disturbances since the inputs QAd and QBd only appear in the F˙L channel and so cannot directly counteract the effect of ∆˜ or the effect of ˜ T ˜ parameter estimation error in θ1 and ϕ1 θ (which will arise from imperfect model com- pensation) in theq ¨ channel. The backstepping technique presented by Krstic´ [15] will be applied to overcome this problem. Two design steps will be required.
3.1.1 Step 1
To simplify the design, the tracking error z1 is used to define a sliding surface z2 as follows: 26
z1 = q − qd
z2 = z˙1 + k1z1, (3.3) where k1 is a positive constant. It is noted that a stable dynamics exists between z1 and z2, so that by regulating z2 to a small value, z1 is kept to a small level as well. The problem of making q track qd is reduced to regulating z2 to zero or as small as possible. In the first step, FL is treated as a virtual input and a virtual control law will be designed to regulate z2. The virtual control law, FLd is given by:
FLd = FLa + FLs −1 1 ∂x T ˆ FLa = q¨d − ϕ1 θ − k1z˙1 (3.4) θˆ1 ∂q −1 ∂x k2 FLs = FLs1 + FLs2, FLs1 = − z2, ∂q θmin,1
where k2 is a positive gain and FLs2 is a robust control function. The function of FLs1 is to provide a nominally stable dynamics for z2, since in the absence of uncertainties, the system is stable with FLs2 = 0. The purpose of FLs2 is to maintain tracking performance in the presence of parameter estimation errors and the unknown disturbance ∆˜ . Its synthesis will be discussed later.
Denote the discrepancy between FLd and FL as z3 (i.e. z3 = FL − FLd). Noting (3.3) and (3.4), the dynamics of z2 become:
∂x z˙ = θ (F + z ) + ϕT θ + θ ∆˜ − q¨ + k z˙ 2 1 ∂q Ld 3 1 1 d 1 1 −1 ! ˆ ˜ ∂x 1 ∂x T ˆ T = θ1 − θ1 q¨d − ϕ1 θ − k1z˙1 + ϕ1 θ − q¨d ∂q θˆ1 ∂q −1 ! ∂x ∂x k2 +k1z˙1 + θ1 − z2 + FLs2 + z3 + θ1∆˜ ∂q ∂q θmin,1 ∂x ˜ T ˜ θ1 ∂x ˜ ∂x = − FLaθ1 − ϕ1 θ − k2z2 + θ1 FLs2 + θ1∆ + θ1 z3 ∂q θmin,1 ∂q ∂q θ1 ∂x T ˜ ˜ ∂x = − k2z2 + θ1 FLs2 − φ2 θ + θ1∆ + θ1 z3, (3.5) θmin,1 ∂q ∂q 27
where φ2 is a grouping of all terms which contain parametric uncertainties:
T h iT T ˆ 2 ∂x q¨d−ϕ1 θ−k1z˙1 ∂x φ2 = ϕ1 + ∂q FLa 0 0 0 0 0 = − q˙ 1 0 0 0 . θˆ1 ∂q (3.6)
If FL were the actual input to the system, then the adaptation function would be given by:
τ2 = φ2σ2z2. (3.7)
This function will be used later in defining the final parameter adaptation law. The robust control function FLs2 must satisfy the following 2 properties:
∂x z θ F − φ T θ˜ + θ ∆˜ ≤ ε (3.8) 2 1 ∂q Ls2 2 1 2 ∂x z θ F ≤ 0, (3.9) 2 1 ∂q Ls2 where ε2 is a positive design parameter which is related to the guaranteed final tracking error. The parameter ε2 can theoretically be arbitrarily small, but in practice is limited by the finite physical response speed of the control. The property in Eq (3.8) assures that T ˜ ˜ FLs2 will dominate the parametric uncertainties (φ2 θ) and uncertain nonlinearities (∆) to a prescribed level. The second property (3.9) assures that the robust feedback will not interfere with the adaptation process.
Many different choices of FLs2 could satisfy properties (3.8) and (3.9), but it is practical to choose a function whose equivalent global gain with respect to z2 is low. Otherwise, control saturation will soon become a problem for large errors which may occur in the case of unexpectedly large disturbances or mismatched initial conditions. Also, in order to use
FLs2 for synthesis of the actual control law, it should be differentiable. A choice of FLs2 which has all these desirable properties is
−1 ∂x κh2 FLs2 = − h2 · tanh z2 , (3.10) ∂q ε2
1 T where h2 could be any function of the states and time which always satisfies h2 ≥ θ1∆˜ − φ θ˜ θ1 2 and κ = 0.2785. An example which theoretically works is
1 h2 ≥ φ2 q¨d,θˆ,q˙,z˙1 |θmax − θmin| + ∆M (x,t), (3.11) θmin,1 28
where ∆M is a bounding function of the disturbance. The use of such an h2 (which could be a nonlinear function of the state and time) could result in a more complicated FLs2 than necessary. For the purpose of this paper, it will be assumed that a sufficiently large, constant
h2 will do. This assumption implicitly places a limit on the size ofq ¨d which can be applied and still guarantee the theoretical results.
3.1.2 Step 2
The next step is to use the backstepping design technique to keep z3 as small as possible. The dynamics of z3 are given by:
z˙3 = F˙L − F˙Ld. (3.12)
To perfectly regulate z3, F˙Ld must be considered. For clarity, the terms containing the −1 ∂x nonlinear linkage compensation (the function ∂q ) may be separated from those which do not and FLd may be rewritten in the following form:
2 −1 ˆ ∂x ˆ x q¨d + θ2 ∂q q˙ − θ3 − k1z˙1 k h ∂ 2 κ 2 FLd = − z2 − h2 · tanh z2 ∂q θˆ1 θmin,1 ε2
−1 θˆ2 ∂x ∂x = q˙ + FLd, (3.13) θˆ1 ∂q ∂q
where FLd is given by:
q¨d − θˆ3 − k1z˙1 k2 κh2 FLd = − z2 − h2 · tanh z2 . θˆ1 θmin,1 ε2 The product rule may then be used to express the derivative as follows: 29
ˆ ! ˆ ∂x d θ2 ∂ ∂x θ2 2 F˙Ld = · q˙ + q˙ ∂q dt θˆ1 ∂q ∂q θˆ1 ! ∂x−1 d ∂ ∂x−1 + F + q˙ · F ∂q dt Ld ∂q ∂q Ld ˆ ˆ ˆ˙ ˆ˙ ˆ ! ∂x θ1 θ2q¨ + θ2q˙ − θ1θ2q˙ 1 ∂x2 θˆ = + l l cos(α) − 2 q˙2 q ˆ 2 l A B q ˆ ∂ θ1 C ∂ θ1 ... −1 ˆ q ˆ˙ ˆ˙ ˆ ∂x θ1 d − θ3 − k1 (q¨ − q¨d) − θ1 q¨d − θ3 − k1z˙1 + q ˆ 2 ∂ θ1 2 k2 κh2 2 κh2 − + · sech z2 z˙2 θmin,1 ε2 ε2 ! 1 1 + + q˙ · F . (3.14) x Ld lC ∂ π ∂q tan 2 − δ − γ − q
It is noted that the following simplifications have been made from (2.2):
∂ ∂x−1 ∂ l = C ∂q ∂q ∂q −lAlB sin(α (q)) ∂x −lAlB sin(α (q)) − lAlBlC cos(α (q)) = ∂q 2 2 2 lAlB sin (α (q)) 1 cos(α (q)) = + x lC ∂ ∂q sin(α (q))
∂ ∂x ∂ −l l sin(α (q)) = A B ∂q ∂q ∂q lC ∂x lAlBlC cos(α (q)) − lAlB sin(α (q)) ∂q = 2 lC ! 1 ∂x2 = lAlB cos(α (q)) − . lC ∂q 30
Unfortunately, the derivative of the load force as given in (2.5) is not fully calculable due to the unknown parameters and disturbances contained in terms such asq ¨ andz ˙2. The calculable portions (i.e. the best available estimates) ofq ¨ andz ˙2 are given by:
∂x T q¨ˆ = θˆ FL + ϕ θˆ (3.15) 1 ∂q 1 z˙ˆ2 = q¨ˆ − q¨d + k1z˙1. (3.16)
ˆ ˜ ∂x T ˜ ˜ ˆ ˜ ∂x T ˜ ˜ It should be noted thatq ¨ = q¨ − θ1 ∂q FL − ϕ1 θ + θ1∆ andz ˙2 = z˙2 − θ1 ∂q FL − ϕ1 θ + θ1∆. Now F˙Ld may be divided into calculable and incalculable components, denoted F˙Ldc and
F˙Ldu, respectively. The dynamics of z3 may thus be expressed as:
T ˙ ˙ z˙3 = θ4ψL + ϕ2 θ − FLdc − FLdu, (3.17) where F˙Ldc and F˙Ldu are given by:
∂FLd ∂FLd ∂FLd ∂FLd ∂FLd ˙ F˙Ldc = q˙ + q¨ˆ + z˙ˆ2 + + θˆ (3.18) ∂q ∂q˙ ∂z2 ∂t ∂θˆ ! 1 1 = + F q˙ x Ld lC ∂ ∂q tan(α (q)) 2! ˆ 1 ∂x θ2 2 + lAlB cos(α (q)) − q˙ lC ∂q θˆ1 ˆ −1 ! −1 2 ∂x θ2 ∂x k1 ∂x k2 κh 2 κh + − q¨ˆ − + · sech z2 z˙ˆ2 ∂q θˆ1 ∂q θˆ1 ∂q θmin,1 ε2 ε2 −1 ... ∂x q d + k1q¨d −1 ˙ + + FLa ∂x q˙ ∂x 1 θˆ − ˆ ∂q ˆ − ∂q ˆ 0 0 0 ∂q θˆ1 θ1 θ1 θ1 F F x ˙ ∂ Ld ∂ Ld ˜ ∂ T ˜ ˜ FLdu = + −θ1 FL − ϕ1 θ + θ1∆ , (3.19) ∂q˙ ∂z2 ∂q where
ˆ −1 2 ∂FLd ∂FLd ∂x θ2 ∂x k1 k2 κh2 2 κh2 + = − − − · sech z2 . (3.20) ∂q˙ ∂z2 ∂q θˆ1 ∂q θˆ1 θmin,1 ε2 ε2
Then, treating ψLd from (3.2) as a virtual input, the calculable part of F˙Ld will be included in the model compensation term ψLa, and a robust feedback term, ψLs2, will be used to mitigate the effect of the incalculable portion. The virtual control law for ψLd is: 31
ψLd = ψLa + ψLs x 1 ˆ ∂ σ2 T ˆ ˙ ψLa = −θ1 z2 − ϕ2 θ + FLdc (3.21) θˆ4 ∂q σ3 k3 ψLs = ψLs1 + ψLs2, ψLs1 = − z3, θmin,4
where σ2 and σ3 are scaling factors which will be discussed in Section 3.3.1. The term −1 ∂x σ2 −θˆ z is included in ψLa to compensate for the cross-coupling effect between 1 ∂q σ3 2 z2 and z3. The resulting dynamics for z3 are:
x ˆ ˜ 1 ˆ ∂ σ2 T ˆ ˙ z˙3 = θ4 − θ4 −θ1 z2 − ϕ2 θ + FLdc θˆ4 ∂q σ3 T ˙ θ4 ˙ +ϕ2 θ − FLdc − k3z3 + θ4ψLs2 − FLdu θmin,4 ∂x σ2 θ4 = − θ1 + θ˜1 z2 − k3z3 + θ4ψLs2 ∂q σ3 θmin,4 ˜ T ˜ ∂FLd ∂FLd ˜ ∂x T ˜ ˜ −θ4ψLa − ϕ2 θ − + −θ1 FL − ϕ1 θ + θ1∆ ∂q˙ ∂z2 ∂q ∂x σ2 θ4 T ˜ ∂FLd ∂FLd ˜ = −θ1 z2 − k3z3 + θ4ψLs2 − φ3 θ − θ1 + ∆, (3.22) ∂q σ3 θmin,4 ∂q˙ ∂z2
All terms containing parametric uncertainties have been grouped according to:
T ˜ ˜ ∂x σ2 ∂x ∂FLd ∂FLd ˜ ∂FLd ∂FLd T ˜ T ˜ φ3 θ = θ1 z2 − FL + + θ4ψLa − + ϕ1 θ + ϕ2 θ, ∂q σ3 ∂q ∂q˙ ∂z2 ∂q˙ ∂z2 so φ3 is defined as: ∂x σ2 ∂x ∂FLd ∂FLd ∂q σ z2 − ∂q FL ∂q˙ + ∂z 3 2 ∂FLd + ∂FLd ∂x q˙ ∂q˙ ∂z2 ∂q − ∂FLd + ∂FLd φ = ∂q˙ ∂z2 3 2 2 AA AB ∂x ψLa − + q˙ VA VB ∂q AA VA − AB VB 32
The function ψLs2 must satisfy robust conditions similar to (3.22) and (3.9):
T ˜ ∂FLd ∂FLd ˜ z3 θ4ψLs2 − φ3 θ − θ1 + ∆ ≤ ε3 (3.23) ∂q˙ ∂z2 z3θ4ψLs2 ≤ 0 (3.24)
where ε3 is a positive design parameter. A similar function as in (3.10) could be chosen κh3 (e.g. ψLs = h tanh z will do for properly chosen h ). The final adaptation function 2 3 ε3 3 3 is given by:
τ = τ2 + φ3σ3z3. (3.25)
3.2 Theoretical Performance Theorem: Using any flow distribution scheme which exactly satisfies the virtual control law (3.21) and using the parameter adaptation (3.1) with τ as in (3.25), the tracking error
z2 has a prescribed transient which may be predictably adjusted by changing the gains k1, k2, k3 and the design parameters ε2, ε3. This transient is quantified by the relation:
ε V (t) ≤ V (0)exp(−2λt) + (1 − exp(−2λt)) (3.26) 3 3 2λ 1 2 1 2 where λ = min(k2,k3), ε = σ2ε2 + σ3ε3, and V3 (t) = 2 σ2z2 + 2 σ3z3 is a positive semi- definite function. Moreover, if after a certain time, only parametric uncertainties exist (i.e.
∆˜ = 0 and ∆QA and ∆QB are constant), then z2 → 0 as t → ∞, and it follows that q → qd and so asymptotic tracking is achieved. The proof of this theorem is given in Appendix A.
3.3 Gain Selection This section will discuss the practical concerns of the gain selection process. The design
parameters which must be chosen are the scaling factors σ2 and σ3; the stabilizing gains ki for i = 1, 2, 3; the robust parameters hi and εi for i = 2, 3; and the adaptation rates γi for i = 1, 2, ..., 6. 33
3.3.1 Selection of Scaling Factors (σ Values)
The positive scaling factors σ2 and σ3 are used to scale the error signals. The error rad signal z2 has units of s , and suppose the maximum expected angular velocity for the rad system is 1.5 s . Then a reasonable choice for σ2 is given by taking the reciprocal of 1 10% of the maximum value, that is: σ2 = 1.5×0.1 = 6.667. Similarly, since the units of z3 are N, if the maximum achievable load force is 7000N, then a reasonable σ3 is given by 1 −3 σ3 = 0.1×7000 = 1.43 × 10 .
3.3.2 Selection of h Values As mentioned above, the h values should theoretically be chosen to dominate all possi-
ble uncertainties. Using h2 as an example, a possible approach is to choose a constant h2 which will satisfy
1 h2 ≥ φ2 q¨d,θˆ,x˙,z˙1 |θmax − θmin| + δ (x,t) (3.27) θmin,1 for the worst possible combination of xd (t), θˆ, and x(t). In practice, if the parameter estimate range is large, the magnitude of the vector |θmax − θmin| may be so large that the
resulting h2 value (which is a component of the desired load force) is several times larger than the maximum torque the system can apply. As this is unachievable in practice, it may
be necessary to choose the h2 value to use an appropriate portion of the achievable load force. Reducing h2 impacts the theoretical performance in that either the set of parameter variations, the size of disturbances, or the magnitude ofq ¨d for which the performance is guaranteed is reduced.
3.3.3 Selection of k and ε Values It can be shown that implementing the virtual control law (3.21) guarantees the tran-
sient performance has a converging rate of −2λ, where λ = min(k2,k3). This is similar to a linear system with a pole located at −2λ rad/s. Theoretically, k2 and k3 could be made arbitrarily large (resulting in arbitrarily high closed-loop bandwidth), but in reality the bandwidth is reduced by factors such as finite valve response speed and other neglected
high frequency dynamics. The values for k1, k2, and k3 are limited by the achievable band- width. Moreover, consider the coupled closed loop dynamics of z2 and z3 from (3.5) and (3.22) with z2,z3 ≈ 0. For small z2 and z3, FLs2 and ψLs2 are equivalent to proportional gains 34
h2 h2 of θ κ 2 on z and θ κ 3 on z . Noting the definition of z in (3.3), the closed loop system 1 ε2 2 4 ε3 3 2 behavior will be comparable to the following linear system (in which the uncertainties are treated as inputs, and nonlinear functions are treated as constants):
−k 1 0 z˙1 1 z1 2 θ1 κh2 ∂x z˙ = 0 − k2 − θ1 θ1 z (3.28) 2 θmin,1 ε2 ∂q 2 σ x θ κh2 z˙3 0 − 2 ∂ θ − 4 k − θ 3 z3 σ3 ∂q 1 θmin,4 3 4 ε3 0 0 T + θ1 ∆˜ + −φ θ˜ . 2 ∂FLd ∂FLd T −θ1 + −φ θ˜ ∂x˙ ∂z2 3
∂x If ∂q θ1 is small compared to the diagonal terms of the system matrix (which is true for large load mass), then the eigenvalues are approximately equal to the diagonals. To obtain a good system response, set each diagonal term equal to the achievable bandwidth, and
then evaluate the eigenvalues of this matrix for typical values of θ1 and θ4 to check that no eigenvalues have large imaginary components, iterating as necessary. From this it is clear 2 2 θ1 κh2 θ4 κh3 that the equivalent gains k eq = k − θ and k eq = k + θ are limited by 2 θmin,1 2 1 ε2 3 θmin,4 3 4 ε3 the achievable bandwidth. These relations express a theoretical trade-off between the high speed of the response
achieved by making k2 and k3 large and the low steady state tracking error level which results from very small ε2 and ε3. In order to make the k values large, the ε values must in- crease as well to satisfy the constraints on the bandwidth. If the system’s transient response
is more important (e.g. for a system with frequent starts where z2 is large), then k2, k3 and the ε values should be made larger. If trajectory planning is used, z2 can initially be set to zero and will stay near zero unless the parameters or disturbance falls outside the bounds.
For such a case, making ε2 and ε3 very small could result in good tracking and the slower transient from the necessary reduction in the k gains may have little effect. For small z2 and z3, there is little difference between a controller with small ε values and one with large k gains.
3.3.4 Selection of Adaptation Gains The adaptation gains which make up the matrix Γ are by far the most difficult controller parameters to choose a priori. This section will present a method for obtaining suitable 35 gains by extending the linear system in (3.28) to include at least some of the effects of adaptation. First define 2 additional states:
t η2 = σ2 z2 (ξ)dξ ˆ0 t η3 = σ3 z3 (ξ)dξ. (3.29) ˆ0
If all the parameter estimates are within their known bounds, then parameter estimates change according to: ˙ θˆ = Γφ2σ2z2 + Γφ3σ3z3. (3.30)
If φ2 and φ3 are constant, Equation (3.30) is linear with respect to z2 and z3 and integrating both sides yields an estimate of θˆ as a function of the new states:
θˆ (t) ≈ θˆ (0) + Γφ2η2 + Γφ3η3. (3.31)
The validity of this estimate depends chiefly on the linearity of Equation (3.30). An examination of the virtual control laws (3.4) and (3.21) reveals that θˆ1 and θˆ4 are used in a radically different way than are the remaining parameters. Because FL and ψL affect the
system through θ1 and θ4, the inverses of their parameter estimates are used in the virtual control laws. Moreover, this situation also leads to the inclusion of z2 in the first term of φ2 and the inclusion of z2 and z3 in the first and fourth terms in φ3. The main point is that the effects of adapting θˆ1 and θˆ4 are highly nonlinear, and it cannot be expected that analyzing a linear system will yield any valuable insight into how these gains should be chosen. The effect of remaining parameter adaptation is well-represented by Equation
(3.30). In proceeding with the linear analysis, the adaptation gains γ1 and γ4 will be set to zero, and the initial estimates θˆ1 (0) and θˆ4 (0) will be used instead of the actual (time- varying) online estimates. This approach does risk neglecting coupling between θˆ1 and θˆ4 and the remaining parameters. By substituting Equation (3.31) into (3.4) and (3.21), the model compensation terms
FLa and ψLa may be expressed in terms of η2 and η3 as follows:
−1 −1 1 ∂x T ˆ 1 ∂x T FLa = q¨d − ϕ1 θ (0) − k1z˙1 − ϕ1 (Γφ2η2 + Γφ3η3) θˆ1 (0) ∂q θˆ1 (0) ∂q (3.32) 36
x 1 ˆ ∂ σ2 T ˆ ˙ 1 T ψLa = −θ1 (0) z2 − ϕ2 θ (0) + FLdc − ϕ2 (Γφ2η2 + Γφ3η3). (3.33) θˆ4 (0) ∂q σ3 θˆ4 (0)
If θˆ1 (0) and θˆ4 (0) are close to their actual values, then there will be near cancellations −1 1 ∂x 1 of the and factors and the dynamics of z2 and z3 are approximated as θˆ1(0) ∂q θˆ4(0) follows:
θ1 ∂x T ˜ ˜ ∂x T z˙2 ≈ − k2z2 + θ1 FLs2 − φ2 θ + θ1∆ + θ1 z3 − ϕ1 (Γφ2η2 + Γφ3η3) θmin,1 ∂q ∂q ∂x σ2 θ4 T ˜ z˙3 ≈ −θ1 z2 − k3z3 + θ4ψLs2 − φ3 θ ∂q σ3 θmin,4 ∂FLd ∂FLd ˜ T −θ1 + ∆ − ϕ2 (Γφ2η2 + Γφ3η3). (3.34) ∂q˙ ∂z2
By appending η2 and η3 as states to the linear system (3.35), a new augmented system results:
z˙1 −k1 1 0 0 0 z1 0 −k ∂x − T Γ − T Γ z˙2 2eq ∂q θ1 ϕ1 φ2 ϕ1 φ3 z2 z = − σ2 ∂x −k − T − T z (3.35) ˙3 0 σ ∂q θ1 3eq ϕ2 Γφ2 ϕ2 Γφ3 3 3 η˙ 0 σ 0 0 0 η 2 2 2 η˙3 0 0 σ3 0 0 η3 0 0 T θ1 −φ θ˜ 2 ∂FLd ∂FLd T + −θ1 + ∆˜ + −φ θ˜ . ∂x˙ ∂z2 3 0 0 0 0
The eigenvalues of the 5×5 system matrix above may be used to evaluate a given choice of ∂x Γ (with γ1 and γ4 set to zero). It is noted that defining this system matrix requires that ∂q , ϕ1, ϕ2, φ2 and φ3 be constant. It remains to find a method for choosing the actual Γ matrix. 37
A logical starting point is to make the final adaptation rate of each parameter propor- tional to the extent of the uncertainty. This may be done by using Γ of the form
Γ = γ ·Wθ , (3.36)
where γ is a scalar value and Wθ is a diagonal scaling matrix defined as:
−1 Wθ = diag{θmax − θmin}diag φ2 + φ3 . (3.37)
Wθ is thus a product of two other scaling matrices. The purpose of the scaling matrix −1 diag φ2 + φ3 is to equalize the magnitude differences arising from φ2 and φ3. φ2 and φ3 are composed of positive terms of the average magnitudes expected for φ2 and φ3. The matrix diag{θmax − θmin} finally makes the adaptations proportional to the range of the expected parameters, which is also the maximum estimation error possible. In Equation
(3.36), γ may be thought of as setting the overall adaptation strength, while Wθ ensures proper relative scaling of the adaptation. When Γ is created in this manner, it is a positive- definite, diagonal matrix. Because the nonlinear functions in (3.35) must be treated as constants, the system should be evaluated for a large number of possible combinations. For each of these com- binations, the system eigenvalues for different values of γ should be calculated. The mag- nitude of the eigenvalues (which correlate with the bandwidth) should be large enough to make a difference on the time scale required (an equivalent time constant is given by 1 ) and should not be lightly-damped or oscillations in the estimates may lead to |Real(λ)| input chattering. After determining a suitable value for γ, simulations or experiments must
be done to find suitable values for γ1 and γ4. Equation (3.36) gives a starting point, but some trial and error will be required. Once these gains have been selected, some additional tuning of the Γ matrix may improve performance.
3.4 Supervisory Control
For the two inputs QAd and QBd, the foregoing design places only one constraint, namely: A1 A2 ψLd = QAd − QBd. (3.38) VA VB 38
This constraint may be satisfied for various valve configurations. If the flows to both cylin- der ports are controlled by a single directional valve, then the solution to above constraint will usually have a unique solution. For valve configurations which decouple the flows into each side of the cylinder, a unique solution will not exist. This results in additional design freedom to specify the individual desired flow rates and provides an opportunity to achieve a secondary objective or to take more factors into account to make the control more robust. Some possible secondary objectives include:
1. Maintain the system at as low a pressure as possible by utilizing one flow rate to regulate the pressure of the corresponding cylinder chamber (the off-side chamber) to a specified low pressure. Taking the desired flow rate for the off-side into account, the remaining desired flow rate must be used to satisfy (3.38). This type of design was done by Liu and Yao in [7]. This scheme increases the efficiency if the system uses a load-sensing pump, since the supply pressure (and thus the total energy input) will be redduced as much as possible.
2. Use the flow rates most-likely to be achievable. Errors in the flow rate from the specified flow rate may be more severe for some flow rates than others (e.g. a small, nonzero flow rate may be difficult to apply, and flow rates near the maximum rated flow may not always be achievable). Both flow rates can be used to satisfy (3.38) by using Q = VA ψ and Q = − VB ψ , resulting in required flows of lower Ad 2A1 Ld Bd 2A2 Ld magnitude. Because this strategy does not actively regulate the pressure level, it is possible that the pressures will drift toward the tank or source pressure. If such a case occurs, then the specified flow rates will no longer be achievable, and problems such as cavitation may arise.
3. Maximize the likelihood of flow recycling to save energy. If the system contains a regeneration path, utilizing this flow reduces the flow required from the pump, reducing the energy input. This objective will be adopted for the purposes of this thesis. One way to achieve it is to regulate the pressures to provide the required load force and to allow either the regeneration path or the accumulator to be used. Methods for doing this will be described in detail later.
For each of these objectives and especially where discontinuous logic or decision elements are included in the control, care must be taken to mitigate the effects of switching suddenly between different flow rates, as frequent switching phenomena could lead to a significant loss of performance since the finite response time of hydraulic valves makes sudden flow 39 changes impossible to execute. In the following sections, a supervisor control is developed for each valve configuration which will be studied.
3.4.1 Servo Valve Supervisor The control design for the servo valve differs from the other configurations in that it has only 1 input. The flow rates QA and QB are coupled by the valve spool and may be expressed as follows:
p QA = uvkv1sgn(∆pA) |∆pA| + ∆QA p QB = uvkv2sgn(∆pB) |∆pB| + ∆QB, (3.39)
where uv is the servo valve command, kv1 and kv2 are flow coefficients, and ∆pA and ∆pB are defined as:
ps − pA, uv > 0 ∆pA = pt − pA, uv < 0 pt − pB, uv > 0 ∆pB = . (3.40) ps − pB, uv < 0
A1 A2 The virtual input ψL is defined as ψL = Q − Q . Noting (3.39), it may be VA Ad VB Bd expressed as a function of the servo valve command uv and the cylinder pressures:
kv1A1 p kv2A2 p ψL = uv sgn(∆pA) |∆pA| − sgn(∆pB) |∆pB| . (3.41) VA VB
This equation may be simplified by defining the following function:
kv1A1 p G(ps, pt, pA, pB,uv) = sgn(∆pA) |∆pA| (3.42) VA kv2A2 p − sgn(∆pB) |∆pB|. VB 40
Substituting (3.42) into (3.41), it is clear that the only way to satisfy the ψL constraint (3.21) is to choose the command according to
ψ u = Ld . (3.43) v G
It should be noted that Equation (3.43) defines uv implicitly since G actually depends on the sign of uv. The task of the servo valve supervisor is to calculate the two possible values for the function G based on the pressure measurements and choose the correct sign
for uv. Then, the magnitude of uv may be chosen to satisfy Equation (3.43), if possible. This is practical when |G| ≥ ψLd where u is the maximum command. When this uv,max v,max is not the case, the valve command will be saturated. When |G| ≈ 0, there will be little flow regardless of the valve command chosen and it may be best to leave the spool at its current position by basing uv on previous values. This will increase the life of the valve and avoid chattering between extremes. It may also be necessary to prevent a vacuum condition from arising during periods of large deceleration or when an overrunning load is present. Substantial vacuum could lead to the collapse of the metal-braid hose connected to the cylinder. The possiblilties for achieving secondary objectives are very limited since there is almost no freedom in choosing uv once ψLd is specified.
3.4.2 Independent Metering Valves (4- and 5-Valve) Supervisor For independent metering configurations, it is possible to specify both cylinder chamber pressures. The important consequence of this fact is that more than one set of pA and pB is able to produce the desired load force FLd, and there will be more than one way to specify QAd and QBd to satisfy ψLd. The most sure way to keep the cylinder chamber pressures under control is to actively regulate one pressure using the associated flow rate while compensating with the other flow rate to achieve ψd. The supervisor’s objective then becomes to determine which chamber pressure should be regulated and to what value. For the 5-valve configuration, a major objective will be to regulate the pressures to make regeneration possible. Regeneration flow is also technically possible for the 4-valve configuration under certain pressure restrictions if a check valve is included in the tank line. This adds a throttling loss, however, and regeneration would only be possible under an overrunning load. This thesis will not analyze regeneration with the 4-valve configuration. Since regeneration flow is out of the question for the 4-valve system, a reasonable objective is to regulate one pressure to a low level p0 while using the other to achieve the 41
desired load force. If the desired load force is below the quantity (AA − AB) p0, then pA should be regulated to p0. Otherwise, pB should be regulated to p0 and pA should be used to achieve the load force. This strategy has the advantages of being able to achieve very high load force and works for both a constant pressure or a load-sensing setup. The flow distribution for the 4-valve configuration is somewhat trivial. Positive flow into a cylinder chamber is provided through the pump valve and negative flow rates are achieved by opening the tank valves. The 5-valve configuration uses regeneration for energy saving and to boost perfor- mance, since higher speeds and flows are achievable. While the chamber pressures are not entirely free to choose (the desired load pressure must be satisfied), there is significant freedom in choosing how the pressures will satisfy the load pressure. A visualization tool which can help identify the constraints involved is a pressure state plot. An example de- sired pressure state plot is shown in Figure 3.2. In this plot, the head chamber pressure
(pA) is plotted on the horizontal axis and the rod chamber pressure (pB) is plotted on the vertical axis. On this plot, the set of pressures which result in a constant load force appear as slanted lines with slope AA . Thus, the maximum achievable load force occurs in the AB bottom right corner while the minimum (most negative) load force occurs in the top left corner. As expected for a single-rod cylinder, the greatest positive load force has much larger magnitude than the largest negative load force. This plot is useful for visualizing how the pressures should be set to make regeneration
flow possible. If the cylinder rod is extending, then regeneration can be used if pB > pA. This corresponds to the area above the solid blue line in Figure 3.2. Energy can be saved by staying above this line as much as possible during extension. During retraction of the rod,
pA > pB must be satisfied in order for regeneration to be possible. This is true for every point on the normal 4-valve trajectory, and there is no reason to depart from this trajectory when the velocity is negative. To follow the 5-valve extension trajectory along the slanted line, pB should be regulated according to:
FL + AA∆p pB = . (3.44) AA − AB
This will result in pB remaining above pA by the pressure differential ∆p . The value of ∆p should be set near the minimum level at which regeneration flow can accurately be con- trolled. The horizontal and vertical lines in the trajectory are easily tracked by regulating the pressure which is constant. 42
Figure 3.2: 5-Valve Desired Pressure State Plot.
The flow distribution for the 5-valve configuration should maximize the use of regen- eration flow. Thus, whenever the regeneration valve can be used to supply even a portion of the flow, it should do so. The maximum possible regeneration flow should be calculated based on the pressure drop and then any remaining flow requirement must be supplied by the pump and tank valves.
3.4.3 Regeneration Accumulator Configuration Supervisor The accumulator is only able to increase the energy efficiency of the system when it is able to replace pump flow. Yet, the accumulator cannot continue to supply flow unless it is periodically recharged. The main goal is then to use flow from the accumulator as much as possible and to capture as much energy from the cylinder chambers as possible, subject to two constraints:
1. The accumulator must not empty, so the pressure must be kept above a minimum
level at all times (pac > pmin).
2. The accumulator must not be over-charged (pac < pmax).
Since both charging and discharging are to be maximized, a straightforward strategy for determining the flow distribution is to use a priority flow distribution, meaning that flow to and from the accumulator is used whenever the system pressures make it possible. Thus, when the desired flow rate into a chamber is positive, the corresponding accumulator valve supplies either the total flow or as much as is possible given the pressure drop (which could 43 be none at all). The remaining flow is supplied from the pump. Similarly, it is always better to charge the accumulator than to throttle flow to the tank, so when a flow rate out of a chamber is specified, all the flow or as much as is possible is directed to the accumulator and any remainder is throttled to the tank. This scheme reduces the required supply flow and charges the accumulator as much as possible for the given QAd and QBd. If pac falls outside its operating limits, care should be taken to bring it back inside the limits. For example, suppose pac = pmin. An emergency charging mode should be acti- vated, such that no flow from the accumulator is used until pac rises above some safe level. The priority flow distribution and emergency modes together make up the flow distribution logic for the configuration. This flow distribution gives a reasonable way to decide the indi- vidual valve flow rates based on the desired cylinder chamber flow rates and the pressures, but it does not give a way to show how the system pressures should be set to allow for the optimal charging. Separate logic is necessary for this task. Desired pressure state plots will again be used to illustrate how the pressures can be set in order to make charging and discharging the accumulator possible. Two pressure state plots for the energy recovery accumulator system are shown in Figure 3.3. By plotting ver- tical and horizontal lines where pA = pac and pB = pac, four regions are created. In each of these zones, a different combination of accumulator usage is possible. For example, when both pA > pac and pB > pac, then only charging is possible. The chamber which is able to charge the accumulator depends on the direction of velocity, because flow is nearly pro- portional to the cylinder velocity unless the pressure is changing rapidly. Figure 3.3 shows these zones for both positive and negative velocity. Finally, two pressure trajectories are shown on each plot. One is a charging trajectory, which specifies a trajectory which prefers charging to discharging, while the other is a discharging trajectory, preferring discharging to charging. 44
Figure 3.3: Pressure State Diagram Showing Desired State Trajectories.
Figure 3.4: Desired Pressures vs. Desired Load Force.
These trajectories can then be expressed as a function of the desired load force, FLd. The charging and discharging trajectories for extension plotted against desired load force are shown in Figure 3.4. The goal of the charging trajectory is to maximize the range of 45
desired load force for which pB > pac, so that as much charging can occur as possible. Similarly, the goal of the discharging trajectory is to maximize the range of desired load force for which pA < pac, so that the accumulator can replace the pump as much as possible. If the accumulator pressure rises above a certain threshold, discharging trajectories should be chosen. Similarly, if the accumulator dips below a low threshold, the supervisor should begin choosing the charging trajectories. These thresholds should be within the operating limits (pmin and pmax) of the accumulator. The algorithm for choosing desired pressure setpoint is as follows:
1. Reset the charging mode if the accumulator pressure has passed the low or high threshold.
2. Check the velocity to choose between extension and retraction trajectories.
3. Check the accumulator charging mode to select the proper trajectory.
4. Choose to regulate the pressure which is constant for the current load pressure, or if there is none, choose the lower pressure to prevent it from dropping too low.
Regulating the pressure may be done in a number of ways. For simplicity, a proportional feedback term may be used to keep the pressure error small. Suppose that pA is to be regulated to the setpoint pAd. The control laws for QAd and QBd are as follows:
VA θˆ5 QAd = AAx˙− kp (pA − pAd) − θmin,4 θˆ4 VB AA QBd = QAd − ψLd , (3.45) AB VA where kp is a positive gain. This strategy may be used for both constant pressure or load-sensing pumps. For a load-sensing pump, however, the accumulator precharge, operating limits, and pressure thresholds should be set quite low. Otherwise, the load-sensing pump will use a higher than necessary pressure and thus waste energy. 46
4. SIMULATION AND EXPERIMENTAL RESULTS
The purpose of this chapter is twofold. First, the effectiveness of the control design strate- gies presented in Chapter 3 will be demonstrated. Second, the benefits associated with the various energy-saving technologies presented in Section 1.2 will be shown. The simulation results are intended to demonstrate the relative effectiveness of the technologies used to boost energy efficiency. Experimental results for the swing motion controlled by the servo valve are presented, showing the effectiveness of the controller in practice.
4.1 Comparative Simulation Results Reliable simulation models for the experimental cartridge valve configuration and the actual accumulator have not yet been developed. For this reason, existing simulation mod- els were utilized to perform a comparative simulation study of the different valve config- urations which have been studied. These models do not incorporate the kinematics of the swing motion of the arm, but instead simulate a large mass driven by a hydraulic cylinder. In some respects, this simplification will better serve the dual purpose of demonstrating the performance of the controllers and showing the benefits of the various technologies because the tracking error and the force requirements on the cylinder are more straightforward, and therefore more readily grasped. The nonlinear kinematics of the swing mechanism make it so that when the cylinder rod is moving with constant velocity, the angular acceleration of the swing arm must be nonzero. For this same reason, low tracking error is not equally easy to achieve at all positions. It should be noted that this simplification does not reduce the dynamic order of the system, the number of unknown parameters, or the nonlinearities of the system other than that associated with the swing kinematics. The effect of the simplifi- ∂x cation is to make the function ∂q a constant value. The load mass and the cylinder friction parameters for the simulation will be chosen to make the situation dynamically similar to driving the swing motion of the arm near the zero angle position. Simulations of four different valve configurations will be presented. They include:
1. Servo Valve: a 3-position, 4-way spool valve
2. 4-Valve: an independent metering configuration (see Figure 1.1) 47
3. 5-Valve: an independent metering configuration with cross-port regeneration flow (see Figure 1.2)
4. 6-Valve: an independent metering configuration with an energy recovery accumula- tor able to act as a secondary source or sink for either or both chambers (See Figure 1.3)
These simulations will show the relative benefits of load-sensing technology, independent metering, regeneration flow and an energy recovery accumulator. Results for each configu- ration with a constant pressure source and with a load-sensing pump will be presented and explained.
4.1.1 Model Parameters and Setup The model used for simulation is given by:
x¨ = θ1FL − θ2x˙+ θ3 + θ1∆˜ F (t) (4.1) 2 2 AA AB AA AB F˙L = θ4ψL − θ4 + x˙+ θ5 − θ6 , VA VB VA VA
where θ = 1 , θ = b and θ = ∆F0 . The torque disturbance terms ∆ and ∆˜ have been 1 meq 2 meq 3 meq 0 replaced by equivalent force terms ∆F0 and ∆˜ F (t). The remaining parameters are identical to those for the swing arm model. In order to make the dynamic situation between the linear acceleration of a mass and the angular acceleration of the swing inertia, it is necessary to express the cylinder rod acceleration as a function of the angular accelerationq ¨. This relation may be derived as follows:
d d ∂x x¨ = (x˙) = q˙ dt dt ∂q ∂x ∂x = q˙2 + q¨. (4.2) ∂q ∂q
∂x For low velocities, a good approximation isx ¨ = ∂q q¨. An equivalent mass may now be determined by dividing the load force FL by the acceleration of the piston and rod which 48
−1 ∂x it causes. The angular acceleration caused by the load force is J ∂q FL. Combining this with the approximation ofx ¨ gives:
F m = L eq x¨ ∂x−1 F = L ∂q q¨ ∂x−2 F = J L ∂q FL ∂x−2 = J . ∂q
∂x m The value of ∂q at q = 0 is approximately 0.2 rad , which yields an equivalent mass kg·m2 of 105 = 2668 kg. The viscous friction of the cylinder has already been derived (0.2m)2 in terms of the cylinder velocityx ˙ in the modeling of Chapter 2. The viscous friction Ns force remains bx˙, and the same value as used in the swing model (b = 3700 m ) is used for the linear simulations. Other simulation parameters held constant include the head and −3 2 −3 2 rod side piston areas AA = 2 × 10 m , AB = 1.1 × 10 m , the effective bulk modulus 8 βe = 2.5 × 10 Pa, the inefficient volumes: VAmin = 0.510 L and VBmin = 0.638 L, and the stroke: xmax = 0.28 m. Friction is modeled with Stribeck, Coulomb, and viscous components. The Coulomb friction magnitude is 200 N and the maximum Stribeck force is 20 N. The two unmodeled
friction components serve as the disturbance force ∆˜ F .
For the constant pressure cases, the source pressure pS was held constant at 69 bar. The load-sensing pump is modeled as an ideal source which maintains ps a fixed amount (∆FLS) higher than the maximum cylinder chamber pressure with a first order lag of bandwidth
ωLS. Explicitly, the source pressure changes according to:
p˙s = ωLS (max(pA, pB) + ∆FLS) − ωLS ps. (4.3)
rad The source pressure was saturated at 69 bar. For all cases, 100 s was used for ωLS.A value of 10 bar was used for ∆FLS, except for the directional valve which required 20 bar to generate sufficient flow to follow the trajectory. The tank pressure was held constant at 0 bar. The accumulator is modeled as: 49
V˙ f = QAac + QBac 0, pac < ppr Vf = 1 (4.4) V 1 − ppr k p ≥ p , tot pac ac pr
where Vf is the accumulator oil volume, Vtot is the capacity, ppr is the precharge pressure, pac is the accumulator oil pressure, and k is the polytropic gas constant. If the pressure pac drops below ppr, then its dynamics change and pac changes in response to the pressure to
which it’s connected (pA and/or pB). For the constant pressure supply, ppr was set to 15 bar, while for the load-sensing pump, a lower value of 4 bar was used. For both pressure
supplies Vtot was 1.0 L and 1.4 was used for k, which models an adiabatic condition in the accumulator. This is reasonable given the rather rapid pressure changes experienced. The same desired position command (shown in Figure 4.5a) is input to the controllers for all cases.
4.1.2 Selection of Controller Gains This section will demonstrate the method of selecting gains presented in Section 3.3. The emphasis in gain tuning should be in obtaining a practically acceptable controller, and as with all control design, some trial and error is expected.
First, proper values for σ2 and σ3 must be chosen. The maximum expected cylinder m rod velocity is 0.3 s and the maximum load force attainable in both directions is A2 · ps, or approximately 7000 N. Taking 10% of these values and inverting as in Section 3.3.1 gives −3 σ2 = 33.3 and σ3 = 1.43 × 10 . The values for h2 and h3 are chosen as a fraction of the available control authority. In order to leave some control authority for the model compensation and stabilizing feedback
terms, 40% of the maximum FL and ψL are used. The maximum value of ψL may be found L L by setting QA = 30 min and QB = −30 min and choosing a set of volumes. This gives a value −3 m2 for h3 of 1.283 × 10 s . The achievable bandwidth of the system is limited by the response speed of the valves. The cartridge valve model used for the independent metering configurations in the simula- L tion has a maximum flow rate of around 30 min and the dynamics are modeled as a linear rad second order system with ωn = 200 s (approximately 30 Hz) and ζ = 0.4. Saturation limits are placed on the valve position to prevent the valve position from becoming greater 50 than fully open or reaching negative values. The servo valve has a lower bandwidth of rad 62.8 s (8 Hz), but has a higher damping ratio (ζ = 0.7). Thus, a desired bandwidth of rad 20 s is used for the independent metering valve configurations while the desired band- rad width of the servo valve system is 16 s , a less conservative but still achievable bandwidth. Values for k1, k2, k3, ε2, and ε3 were chosen to give eigenvalues near -20 for the matrix:
−k1 1 0 κh2 0 − θ1 k − θ 2 θ . θmin,1 2 1 ε2 1 h2 0 − σ2 θ − θ4 k − θ κ 3 σ3 1 θmin,4 3 4 ε3 Ns 8 The lower bounds on the values of meq, b and βe were 2600 kg, 2000 m and 1.3 × 10 Ns Pa, respectively. The upper bounds on the same parameters were 4000 kg, 5000 m and 8 L 4 × 10 Pa, respectively. In addition, the maximum flow error bound was 3 min . These bounds result in the following values for θmin and θmax:
T h −4 8 5 5 i θmin = 2.5 × 10 0.5 −1.54 1.3 × 10 −1.2 × 10 −1.2 × 10 T h −4 8 5 5 i θmax = 3.8 × 10 1.92 1.54 4 × 10 1.2 × 10 1.2 × 10 . (4.5)
In order to determine the proper form for Γ, the expected magnitude of the vectors
φ2 and φ3 must be characterized. One way to quickly generate data over a wide range of operating conditions is to use random data to create a large number of test cases. These cases may then be averaged to find the typical magnitudes which may be expected in the terms of φ2 and φ3. Uniformly distributed random numbers are scaled and shifted to the appropriate range before being input to the control algorithm. For example, numbers used to represent various values of x should come from the range [0,0.28], and numbers which represent pressures 6 pA and pB should be drawn from the range 0,6.9 × 10 . The measurements x,x ˙, pA and ... pB and the desired trajectory terms xd,x ˙d,x ¨d, and x d all need to be generated, but these must not be defined independently from one another or large error signals will result and the linear system representation will not be valid. For example, the desired accelerationx ¨d should be defined as a small random offset from the value θ1 (AA pA − AB pB) − θ2x˙. In this way, the error z3 which will be calculated will not be excessively large. The advantage of this method is that the normal control algorithm is used to construct the ϕ1, ϕ2, φ2 and φ3 51
vectors. The expected magnitudes of φ2 and φ3 may then be determined and the matrix Wθ −1 can be formed. The matrix diag φ2 + φ3 was chosen as:
−1 n −8 −4 −5 o diag φ2 + φ3 = diag 1.52 × 10 4.09 × 10 6.09 × 10 6.98 0.374 0.729 . (4.6) From the same test cases, an extended linear system like that in Equation (3.35) may be formed and the eigenvalues may be evaluated for various values of γ. It is important that while choosing γ the first and fourth diagonal entries of the Γ matrix are set to zero.
Figure 4.1: Eigenvalue Analysis for Selection of Adaptive Gains.
The eigenvalues of the extended linear system for 100 random cases are plotted for 6 different adaptation strengths in Figure 4.1. It is remarkable and speaks to the mainly linear nature of the remaining adaptation that the eigenvalues lie neatly on one trajectory and only a limited spread exists among the cases with the same adaptation strength. From this analysis, γ = 0.15 was chosen for the adaptation strength for the independent metering
configurations. After some experimentation, suitable values for Γ11 and Γ44 were deter-
mined to be 0.1 and 3000 times the corresponding diagonal entry of the Wθ matrix. A similar process was followed for the servo valve, resulting in a slightly lower choice of
γ = 0.08, with the same factors applied to give Γ11 and Γ44. The final adaptation matrix for the independent metering cases is given by: 52
2.05 × 10−13 −5 8.74 × 10 2.81 × 10−5 Γ = . 12 5.65 × 10 2.24 × 103 4.37 × 103 (4.7)
4.1.3 Trajectory Tracking Performance All controllers exhibit excellent tracking performance, keeping the error around the level of 1mm or less. The tracking errors for the constant pressure cases for all four config- urations are shown in Figure 4.2. The low tracking error reflects both the highly accurate model compensation and the effect of the adaptation. The same data was used to make flow maps to calculate the flow during simulation and to make inverse flow maps for the con- troller. Thus, the resulting flow discrepancies are lower than may be expected in practice. As can be seen in Figure 4.2, the error is relatively low even without adaptation, but the parameter adaptation reduces the error by about 50%. 53
(a) Adaptation Turned On (b) Adaptation Turned Off.
Figure 4.2: Tracking Error Results.
4.1.4 Energy Usage Results This thesis will not consider the effect of pump or motor efficiency on power consump- tion, but restricts the focus to the energy supplied to the system as in Equation (1.1). The supplied power for all configurations is plotted in Figures 4.3 and 4.4 for the constant pres- sure source and load-sensing pressure source, respectively. The energy consumed, average efficiency, and absolute maximum error are shown in Tables 4.1 and 4.2 for the constant pressure and load-sensing pump, respectively. The average efficiency is calculated by di- viding the total energy supplied to the load by the total energy consumed:
t f |FL (t)v(t)|dt η = t0 . (4.8) ´ t f ps (t)QS (t)dt t0 ´ The low values for efficiency reflect the low (on average) force required to drive the system. A large percentage of the desired trajectory is constant velocity motion, requiring high flow 54 but a very low net force. The efficiencies would be higher for a trajectory with a larger percentage of acceleration and deceleration or in the presence of larger disturbance forces.
Figure 4.3: Power Consumption for Constant Pressure Source.
Table 4.1: Constant Pressure Supply Energy Usage.
maximum System Con- Net Energy Average tracking error figuration Consumed (kJ) Efficiency (mm) Servo valve 10.37 11.5% 1.213 4-valve 10.24 11.7% 1.163 5-valve 6.48 18.3% 0.915 6-valve 4.27 28.1% 0.653
For a load-sensing supply, the energy required is greatly reduced by using the lowest pressure possible, but for a constant pressure supply, reducing the system pressures gives no benefit in terms of efficiency. As may be seen from Table 4.1, the energy required for the PD valve and 4-valve configurations is nearly identical for the constant pressure case. These configurations do not use regeneration flows, and thus use the same amount of source flow. The 4-valve configuration lowers the back-pressure in the return chamber to a very low level, and the resulting chamber pressures are much lower than those of the PD valve. This results in a lower source pressure and significantly lower energy consumption than the PD valve for the load-sensing case, as shown in Table 4.2. 55
Figure 4.4: Power Consumption for Load-Sensing Pressure Source.
Table 4.2: Load-Sensing Pump Energy Usage
maximum System Con- Net Energy Average tracking error figuration Consumed (kJ) Efficiency (mm) Servo valve 6.61 18.2% 1.722 4 valves 3.85 31.1% 1.149 5 valves 3.47 34.8% 1.059 6 valves 3.13 37.2% 1.091
A comparison of the load-sensing and constant pressure supply results shows that the load-sensing pump reduced the energy required for each configuration by 30-60% from the corresponding constant pressure case. It may be seen that the benefit of the 6-valve system over the 5-valve system is much greater when a constant pressure source is used than when a load-sensing pump is used. This is natural, since one function of the accumulator is to save the excess energy which results when the supply pressure is higher than required (see Section 4.1.5 for how this is done). A load-sensing pump eliminates this excess to a large extent, reducing the opportunity for the accumulator to save energy.
4.1.5 Comparison of 5- and 6-Valve Configurations Figure 4.5 shows pressure and flow variables for the 5-valve and 6-valve configurations for the constant pressure source case. A detailed comparison of the results will illustrate 56 the difference between the two configurations and explain why the 6-valve configuration requires less pump energy. First, the flow terms shown in Figure 4.5d and Figure 4.5e will be explained and then the operation of the 5- and 6-valve systems will be analyzed.
Qs is the total flow from the pressure source. By checking the velocity (shown in Figure 4.5a), it can be seen whether the flow is into the head chamber (for v > 0) or rod chamber
(for v < 0). Similarly, Qt is the total flow into the tank. For the 5-valve configuration, a single regeneration valve controls the cross port flow (QAB) from the head chamber at pressure pA to the rod chamber at pB. In Figure 4.5d, QAB < 0 implies the flow is from pB to pA. In Figure 4.5e, the flows QAac and QBac represent flows into the accumulator from the head chamber (pA) and rod chamber (pB), respectively. If QAac < 0, then the flow is from the accumulator to the head chamber. Similarly, QBac < 0 indicates flow from the accumulator to the rod chamber. There are two situations when the 5-valve system is able to use regeneration flow: dur- ing extension of the rod (v > 0) and during deceleration periods. When the velocity is positive, regeneration flow will not be possible if a great amount of positive acceleration is required. However, if the acceleration is not too great and especially if the rod is extending with constant velocity, the net force required may be quite low. During such times, pB > pA even though the net force is still positive. This is possible because the head side area AA is so much greater than AB. When the velocity is positive, the 5-valve supervisor seeks to keep pB > pA as much as possible, which is done by increasing both pressures when more load force is required. As may be seen from Figure 4.5b, both pressures rise sharply when positive acceleration is required at times t=0 sec. and t=4 sec. It can be seen from Figure 4.5d that there is a large amount of flow from the rod to head chamber during these times. Some flow from the pump is required to supplement the regeneration flow because of the larger head chamber area. During deceleration, the flow out of one of the cylinders is restricted and the pressure rises, providing a braking force. During deceleration from negative velocity (i.e. when the rod is being retracted), pA is providing the braking force and may rise above pB, as is seen to occur in Figure 4.5b. During these times, flow is regenerated from the head chamber to the rod. Some flow must always be diverted to the tank during these times because of the small rod side area. The 6-valve system also saves energy during the extension periods, but while the 5- valve still requires significant pump flow, the 6-valve system provides the flow almost en- tirely from the accumulator. During these times, fluid flows into the accumulator from the rod chamber but the accumulator supplies a much greater amount of flow to the head 57 chamber. This causes the pressure in the accumulator to drop significantly (see Figure 4.5c, 0 < t < 2s). The 6-valve system is able to charge the accumulator intermittently during periods of constant velocity retraction. Because of the low net force required, pA can at times exceed pac so that flow from the head chamber can be directed to the accumulator rather than the tank. Regeneration flow is never possible for the 5-valve configuration during such times. During deceleration from negative velocity, pB is too high for the accumulator to
supply flow to it, but almost all the flow from pA can be used to charge the accumulator during these periods. Unlike the 5-valve configuration, the 6-valve system can theoretically recycle the entire flow from the head chamber and does not always divert a portion of the flow to the tank. It can be seen in Figure 4.5e that the tank flow does indeed drop to zero around t = 3.25 sec. and t = 7.5 sec. To summarize, the proposed configuration is able to recycle more flow than the 5-valve configuration. This can be explained in two ways. First, the accumulator may be said to decouple the regeneration flow. Thus, the accumulator can supply more flow than it takes in or take in more flow than it supplies. Second, the accumulator may be viewed as an additional flow source or sink, which can allow flow which would normally be throttled to the tank to instead be used to charge the accumulator. The accumulator acts as a low- pressure source to replace the pump flow for light loads. 58
Figure 4.5: Comparison of 5-Valve and 6-Valve Configurations. 59
4.2 Servo Valve Experimental Results A servo valve controller was implemented using gains according to the selection method presented in Section 3.3, and achieves good performance. To allow better comparison with the simulation results presented in the previous section, a desired cylinder postion trajectory was calculated from the desired angle trajectory. Both of these trajectories, the experimen- tal results, and the position tracking error are shown in Figure 4.6. Two filters with different bandwidth were used to generate a slow and a fast trajectory. Figure 4.6 shows both the slow and fast trajectories in order.
Figure 4.6: Experimental Tracking Error for Servo Valve.
The load force was calculated from pressure measurements and is plotted with the de- sired value in Figure 4.7. It can be seen that the desired load force becomes unachievable during the periods of highest acceleration, but that the load force tracks the desired value closely for the most part. The parameter estimates are well behaved for the most part and is shown in Figure 4.8. The discontinuous projection is active during some periods when the tracking error is large and greater oscillation of the paramters results. A 22.7 kg (50 lb) mass was added to the end of the hydraulic arm and the experiment was repeated. The position and force tracking results are shown in Figures 4.9 and 4.10, 60
Figure 4.7: Experimental Load Force Tracking Error.
respectively. Mohanty and Yao calculated the inertia for this case to be 217 kg·m2 in [13]. The parameter estimates for this case are shown in Figure 4.11. These experimental results show that the controller performs quite well, though some additional tuning of the adaptation gains may allow the steady state error to be reduced.
It may be possible to use larger adaptation rates for θ3, θ5, and θ6 without degrading the performance since the corresponding terms in φ2 and φ3 are approximately constant. 61
Figure 4.8: Parameter Estimates. 62
Figure 4.9: Experimental Tracking Error with Additional Mass.
Figure 4.10: Load Force Tracking Error with Addtional Mass. 63
Figure 4.11: Parameter Estimates with Additional Mass. 64
5. CONCLUSIONS
The principles behind current energy-saving technologies in hydraulic systems have been presented. The working principle of an energy recovery accumulator system was also pre- sented. A correlated system model of the swing motion of a hydraulic excavator system was presented to validate the use of the theoretical model in control design. A two-level controller structure was proposed which involves a low-level adaptive robust control algo- rithm for cylinder flow rate calculation and high level logic to optimize the system opera- tion for energy efficiency. The adaptive robust controller works by combining adjustable nonlinear model compensation with robust feedback to determine the desired flow rates. While accurate parameter estimates are by no means guaranteed, the gradient-type adap- tation functions as nonlinear integration of the error and effectively reduces the tracking error. A method of selecting the various controller parameters was proposed and applied with success. This controller structure was successfully applied to four systems with differ- ent control valve configurations. Simulation results demonstrated the ability of the control approach to achieve combined goals of precision motion control and energy efficiency. Experimental results for a 4-way servo valve demonstrated the controller performance in practice. The simulations for a typical motion and a constant pressure pump showed an efficiency of 28.1% for an independent metering valve configuration with an energy recovery accu- mulator. This was 2.4 times the efficiency for a conventional 4-way valve (11.5%). Using a load-sensing pump, the efficiency with the accumulator was 37.2%, more than double the 18.2% obtained for the 4-way valve. The system with cross port regeneration was 34.8% efficient, however, and its reduced complexity may make cross port regeneration the most attractive configuration for single-actuator, load-sensing systems. For multi-actuator sys- tems, an energy recovery accumulator would reduce the throttling losses normally induced by widely varying load levels for the different actuators. It would also be able to store and recycle more potential energy during lowering of loads than is possible for regeneration. Thus, an energy recovery accumulator promises increased energy savings for both constant pressure systems and multi-actuator load-sensing systems. There is still much work to be done in this area. Experimental results must be obtained to verify the simulation results and to show whether the gains in efficiency can be obtained 65 without a significant increase in tracking error. In order to do this, simulation models for the hydraulic arm controlled by independent metering valve configurations must be developed and correlated with actual experimental data to enable accurate testing of controllers in simulation before use in experiments. Finally, techniques similar to the presented approach should be applied to multi-actuator systems to study the benefits of a common energy re- covery accumulator. The main contributions of this thesis are the proposed gain selection method, the high- level logic used to optimize use of cross port regeneration and the energy recovery accumu- lator and the comparative simulation results. The theoretical development of the adaptive robust flow rate controller does not differ greatly from the approach taken by Liu and Yao in [7]. However, the method proposed for selecting the controller gains provides a sys- tematic approach to obtain reasonable adaptation rates without relying entirely on trial and error. The utility of the method was demonstrated by using it to select the gains for the simulations and experiments presented in this thesis. This method could also be applied to other systems. The high level logic for the cross port regeneration and energy recovery ac- cumulator configurations is clear and straightforward, yet effective. The strategy combines a priority flow distribution with a plan for regulating the pressures based on the desired load force. This strategy integrates with the ARC flow rate controller to utilize the full potential of these technologies. Finally, simulation results have illustrated the operation of various energy-saving technologies and shown their relative benefits. LIST OF REFERENCES 66
LIST OF REFERENCES
1. J.D. Zimmerman, M. Pelosi, C.A. Williamson, and M. Ivantysynova. Energy con- sumption of an ls excavator hydraulic system. ASME, 2007. 2. T. Lin, Q. Wang, B. Hu, and W. Gong. Development of hybrid powered hydraulic construction machinery. Automation in construction, 19(1):11–19, 2010. 3. B. Eriksson and J.-O. Palmberg. Individual metering fluid power systems: challenges and opportunities. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 225(3):196–211, 2011. 4. R. Book and C.E. Goering. Programmable electrohydraulic valve. SAE transactions, 108(2):346–352, 1999. 5. H. Hu and Q. Zhang. Multi-function realization using an integrated programmable e/h control valve. Applied Engineering in Agriculture, 19(3):283–290, 2003. 6. A. Shenouda. Quasi-static hydraulic control systems and energy savings potential using independent metering four-valve assembly configuration. PhD thesis, Georgia Institute of Technology, 2006. 7. S. Liu and B. Yao. Coordinate Control of Energy Saving Programmable Valves. Con- trol Systems Technology, IEEE Transactions on, 16(1):34–45, 2007. 8. H. Yang, W. Sun, and B. Xu. New Investigation in Energy Regeneration of Hydraulic Elevators. Mechatronics, IEEE/ASME Transactions on, 12(5):519–526, 2007. 9. X. Liang and T. Virvalo. An energy recovery system for a hydraulic crane. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 215(6):737–744, 2001. 10. F. Krnavek. Energy recovery device, December 26 1995. US Patent 5,477,677. 11. D. Margolis. Energy regenerative actuator for motion control with application to fluid power systems. Journal of dynamic systems, measurement, and control, 127:33, 2005. 12. Christopher C. DeBoer. Energy-Saving Control of Electrohydraulic Systems with Over- Redundant Programmable Valves. Master’s thesis, Purdue University, August 2001. 13. A. Mohanty and B. Yao. Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates. Control Systems Technology, IEEE Transactions on, (99):1–9, 2010. 14. M. Jelali and A. Kroll. Hydraulic servo-systems: modelling, identification, and control. Springer Verlag, 2003. 15. M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, et al. Nonlinear and adaptive control design. John Wiley & Sons New York, 1995. APPENDIX 67
APPENDIX A. PROOF OF THEORETICAL PERFORMANCE
Theorem: Using any flow distribution scheme which exactly satisfies the virtual control law (3.21) and using the parameter adaptation (3.1) with τ as in (3.25), the tracking error z2 has a prescribed transient which may be predictably adjusted by changing the gains k1, k2, k3 and the design parameters ε2, ε3. This transient is quantified by the relation:
ε V (t) ≤ V (0)exp(−2λt) + (1 − exp(−2λt)) (A.1) 3 3 2λ 1 2 1 2 where λ = min(k2,k3), ε = σ2ε2 + σ3ε3, and V3 (t) = 2 σ2z2 + 2 σ3z3 is a positive semi- definite function. Moreover, if after a certain time, only parametric uncertainties exist (i.e.
∆˜ = 0 and ∆QA and ∆QB are constant), then z2 → 0 as t → ∞, and it follows that q → qd and so asymptotic tracking is achieved.
A.1 Proof of Prescribed Transient Performance
Differentiate V3, noting (3.5) and (3.22):
x x ˙ θ1 ∂ ˜ T ˜ ∂ V3 = σ2z2 − k2z2 + θ1 FLs2 + θ1∆ − φ2 θ + θ1 z3 (A.2) θmin,1 ∂q ∂q ∂x σ2 θ4 T ˜ ∂FLd ∂FLd ˜ +σ3z3 −θ1 z2 − k3z3 + θ4ψLs2 − φ3 θ − θ1 + ∆ ∂q σ3 θmin,4 ∂q˙ ∂z2 θ1 2 θ4 2 ∂x T ˜ ˜ = − k2σ2z2 − k3σ3z3 + σ2z2 θ1 FLs2 − φ2 θ + θ1∆ θmin,1 θmin,4 ∂q ∂FLd ∂FLd +σ3z3 θ4ψLs2 − φ3θ˜ − θ1 + ∆˜ ∂x˙ ∂z2 2 2 ≤ −k2σ2z2 − k3σ3z3 + σ2ε2 + σ3ε3
≤ −2λV3 (t) + ε,
where λ = min(k2,k3) and ε = σ2ε2 + σ3ε3. The following steps are now taken:
1. Add 2λV3 (t) to both sides: V˙3 (t) + 2λV3 (t) ≤ ε 68
2. Multiply the functions on each side of the inequality in (A.2) by the positive function exp(2λt) (the inequality is preserved):
V˙3 (t) + 2λV3 (t) exp(2λt) ≤ ε · exp(2λt)
3. Rearrange the left side, using the product rule of differentiation:
d (exp(2λt) ·V (t)) ≤ ε · exp(2λt) dt 3
4. Integrate both sides with respect to time starting at t = 0:
ε V (t)exp(2λt) −V (0) ≤ (exp(2λt) − 1) 3 3 2λ
5. Multiply both sides by the positive function exp(−2λt), and rearrange: ε V (t) ≤ V (0)exp(−2λt) + (1 − exp(−2λt)) 3 3 2λ
A.2 Proof of Asymptotic Tracking in the Absence of Disturbance Assume the uncertainty in the system is due only to parametric uncertainties (i.e. ∆˜ = 0 and ∆QA and ∆QB are constant). Define the augmented, positive definite function Va as follows:
1 V (t) = V (t) + θ˜ (t)T Γ−1θ˜ (t) (A.3) a 3 2 ˙ ˙ Taking the derivative of Va while noting (3.1), (3.7), (3.25), (A.2) and the fact that θ˜ = θˆ since θ is constant yields:
T −1 ˙ V˙a = V˙3 + θ˜ Γ θ˜ (A.4) 69
˙ θ1 2 θ4 2 T ˜ Va = − k2σ2z2 − k3σ3z3 + σ2z2 θ1FLs2 − φ2 θ θmin,1 θmin,4 T −1 ˙ +σ3z3 θ4ψLs2 − φ3θ˜ + θ˜ Γ θˆ 2 2 ∂x T V˙a ≤ −k σ z − k σ z + σ z θ F − θ˜ φ 2 2 2 3 3 3 2 2 1 ∂q Ls2 2 ˜ T ˜ T −1 +σ3z3 θ4ψLs2 − θ φ3 + θ Γ Projθˆ (Γτ) (A.5)
The adaptation rate matrix Γ has the form Γ = diag(γ1,γ2,...,γp). Since Γ is diagonal, V˙a may be easily expressed as a sum of the individual components. By applying the robust properties (3.9) and (3.24) and grouping terms:
˙ ˜ T −1 Va ≤ −2λV3 (t) + θ −τ + Γ Projθˆ (Γτ) p ˜ 1 = −2λV3 (t) + ∑ θi −τi + Projθˆi (γiτi) i=1 γi
ˆ For the case when either θi = θmax,i and τi > 0 Projθˆi (γiτi) = 0. But for this case, θ˜i ≥ 0 since the estimate is at its the maximum bound and −θ˜iτi < 0. For this case, when ˆ ˜ ˜ θi = θmin,i and τi < 0, Projθˆi (γiτi) = 0 as well, but now θi ≤ 0, so that −θiτi < 0 again. Finally, for all other cases, Projθˆi (γiτi) = γiτi, causing the term to be zero. Thus,
V˙a ≤ −2λV3 (t) (A.6) and hence Va decreases. Moreover, Va has 0 as a lower bound and thus has a limit as t → ∞. Dividing by −2λ and integrating both sides gives:
1 1 t − Va (t) + Va (0) ≥ V3 (ξ)dξ. (A.7) 2λ 2λ ˆ0
From (A.7), it can be concluded that both z2 ∈ L2 and z3 ∈ L2. It may be proved thatz ˙1is bounded and so z1 is uniformly continuous. By Barbalat’s lemma, z1 → 0 as t → ∞. From (A.7), it can be concluded that both z2 ∈ L2 and z3 ∈ L2. It may be proved that z˙1is bounded and so z1 is uniformly continuous. By Barbalat’s lemma, z1 → 0 as t → ∞.