SOFTWARE ENABLED VARIABLE DISPLACEMENT PUMPS

Transcription

1 Proceedings of ASME-IMECE 2 ASME International Mechanical Engineering Congress and RD&D Expo November -11, 2, Orlando, FL USA IMECE SOFTWARE ENABLED VARIABLE DISPLACEMENT PUMPS Perry Y. Li, Cassie Y. Li and Thomas R. Chase Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis, MN 4 {pli,yli,trchase}@me.umn.edu ABSTRACT Direct pump control of hydraulic systems is more energy efficient than throttle valve based methods to control hydraulic systems. This requires variable displacement pumps that are responsive and capable of electronic control. Such Electronic Displacement Controlled (EDC) pumps tend to be significantly larger, heavier and more expensive than fixed displacement counterparts. In addition, achievable control bandwidths are typically lower than throttle valve based control approaches. We have recently begun a project to achieve the functionality of a variable displacement pump by combining a fixed displacement pump, a pulse width modulated (PWM) on/off valve under closed loop feedback control, and an accumulator. The proposed topology is the hydro-mechanical analog of the DC-DC boost converter in power electronics. Since on/off valves have little loss in either the on or the off state, this approach is potentially more efficient than throttle valve based control approaches. It has the small size/weight and low cost advantages of a fixed displacement pump. Faster response can be expected by eliminating the intervening inertias of the swash plate control system. The pump s functionalities can also be easily programmed by controlling the PWM on/off valve in different manners. This paper presents some preliminary results from this ongoing research program. While the PWM valve based approach provides desirable features, it also introduces undesirable ripples to the system pressure and flow rate. It is shown that increasing the accumulator pre-charge pressure and the accumulator volume can decrease ripple size at the expense of response time. This apparent trade-off can be alleviated by feedback control to achieve fast response time while keeping ripple small. Feedback control using PWM control must be implemented with care since the conventional state-space model may not be valid when the PWM frequency is low. On the other hand, increasing PWM frequency reduces ripple size and enables the system to achieve high control bandwidths. 1 Introduction Every hydraulic system requires at least one hydraulic pump that converts mechanical power from a prime mover (such as an I.C. engine) into hydraulic power. The lack of compact and portable hydraulic power supplies is a major impediment to applying hydraulics to many emerging robotic applications, such as personal assistance robots, robotic tools for search and rescue etc. where large forces are often needed in small or difficult to assess areas. Inability to control pump output precisely and quickly also contributes to the inefficient use of hydraulic power. Our research aims is to develop a new class of energy efficient hydraulic pumps that have 1) improved power and force densities, and 2) improved control capabilities with which a myriad of performance enhancing features can be implemented by changing software/firmware. The proposed new pumps will make hydraulic power supplies more compact. By utilizing the inherent control functionality of the pump, additional control valvings can be minimized, thus reducing the complexity of the hydraulic system and the energy loss through these valves. Hydraulic pumps are usually positive displacement devices. By being able to vary the flow rate as required, energy loss 1 Copyright c 2 by ASME

2 Figure 1. Fixed displacement pump Reflief valve Flow control valve Control Load (a) A common simple open hydraulic circuit that uses a metering control valve to control the cylinder motion Variable displacement pump Control Load (b) An energy efficient direct pump control alternative Valve controlled and direct pump controlled circuits. through valve throttling can be minimized. For example, in the valve-controlled system in fig. 1(a), losses occur in both the relief valve and the metering valve that are partially open. If the pump flow can be controlled quickly and accurately, an alternate circuit in Fig. 1(b) can be used. In this case, since neither throttling or relief valves are used, theoretically, the efficiency will be 1%. Moreover, compared to Fig. 1(a), the hydraulic circuit in Fig. 1(b) is simpler and requires fewer components and hydraulic connections. Since prime movers are normally operated at constant speeds, varying flow necessitates varying the pump displacement. Of the various hydraulic pumps, internal and external gear pumps are easiest to manufacture, but they have fixed displacements (Fig.2). Variable displacement pumps are typically piston pumps and are occasionally vane pumps. Both are more complicated in their designs, heavier and bulkier. Displacement in a piston pump is varied by changing the swash plate angle. The size, weight, complexity and cost all increase significantly as we migrate from simple fixed displacement pumps, to manually controlled displacement pumps, to the so-called electronic displacement controlled (EDC) pumps. In this paper, we propose to combine a compact fixed displacement pump and a pulse width modulated (PWM) high speed on/off bleed-off valve to achieve a software controllable variable displacement pump so that the apparent pump displacement can be adjusted by varying PWM duty ratio. This approach is inspired by the advances and success of switched mode converters in power electronics to achieve compact, energy efficient, yet highly controllable devices. Just as pumps are used to convert mechanical power from the prime mover into the hydraulic power, power electronics is also concerned with the power transformation. Therefore, it is not surprising that ideas from one area can be used in the other. The advantages of the proposed approach are: Size By eliminating the swash plate control system, significant volume and weight savings, and hence power / force density improvement, can be obtained. Energy efficiency When the control valve is either fully on or fully off, the power loss in the bleed-off valve would be minimal (since either the pressure drop or flow is nominally zero) so that the pump operation will be efficient at all pump flows / effective displacements. Responsiveness By eliminating the cascade of mechanical control systems, and replacing it by a direct flow control, the relative degree of the control system is reduced, and the intervening inertias is not necessary. Thus, the responsiveness of the control system is expected to improve. Software enabled features By controlling the PWM signal, high bandwidth control of the pump output can be achieved. This in turn enables a plethora of efficiency and performance enhancing features to be implemented by programmable control software/firmware. Two important performance issues associated with the proposed device are ripple size and responsiveness. In this paper, these are investigated through a case study. Typically, ripple size can be reduced by increasing the capacity or the pre-charge of the accumulator, however, at the expense of slower response. It is shown that this apparent trade off can be mitigated by the use of feedback control. The rest of the paper is organized as follows. Section 2 discusses the rationale of the proposed approach based on the weight, size and performance of current pump designs. The concept of the hydraulic DC-DC converter is discussed in greater detail in section 3. Section 4 uses a design case study to illustrate the performance trade-offs of some design parameters. Fidelity of the averaged models for the PWM controlled systems will also be briefly discussed. Some concluding remarks are given in section. 2 Fixed and Variable Displacement Pumps Figs. 2-3 show a gear pump and a variable displacement piston pump respectively. Gear pumps are simple, light, compact and cost effective. Unfortunately, the pump flow rate cannot be changed unless the speed of the primer mover is changed. The variable displacement piston pumps are more complicated but the pump displacement can be varied by controlling the swash plate angle. 2 Copyright c 2 by ASME

3 Figure 2. cannot be varied. Gear pumps are simple and compact, but their displacements Computer Servo piston Prime mover EDC valve Swash plate Inport Output (a) Schematic of displacement control mechanism To illustrate the trade off between the weight, size and control options, a comparison of commercial pumps of various designs and features is shown table 1. It demonstrates that as control and mechatronic features are introduced, significant weight and size penalties are incurred. Notice that an EDC unit adds significantly to both the weight and size of the pump. For example, the weight and size of a fixed displacement gear pump is about % of those of a variable displacement piston pump. The EDC option adds an additional 3%-% in weight and size. In addition, the response times (neutral to full stroke) for the 2 EDC controlled pumps cited in table 1 are of the order of 1 sec, typical of other EDC pumps 1. This is clearly inadequate for high performance direct pump control purposes. The response speed is limited by the size of the swash-plate servo hydraulic systems - larger servo-pistons will improve response speeds, but would also increase size, weight and cost of the unit. In the current cascade control structure (Fig. 3(b)), we first control an electrohydraulic valve, which in turn controls a hydraulic actuator, the swash plate angle, and finally pump output. This increases the relative degree of the control system, and requires large forces to move the intervening inertia of the components. Both factors contribute to the low bandwidths of EDC pumps. It would therefore be advantageous to achieve an alternate method of varying the displacement of the pump, that has significantly better control performance, but yet without the penalty on weight, size and cost that the existing option based on swash plate control suffers. Figure 3. (b) A EDC control unit Piston pumps are more complex but they can vary their displacement by moving the swash plate. The EDC control unit has dimension: 21mm x 1mm x 3mm, which is fitted on the side of a Sauer- Danfoss M4 pump and constitutes 3% of the pump size. To control the displacement of a piston pump, the swash plate angle is controlled. This can be achieved manually by using a lever (Direct Displacement Control - DDC), or via a set of additional servo pistons (hydraulic actuators) that push and pull on the swash plate. To interface with a computer, the servo pistons are in turn controlled via electrohydraulic valves which can then be interfaced to the computer. The electronic displacement control (EDC) unit can be a separate unit that is attached to a piston pump (such as the one in Fig. 3(b)) or integrated in the pump casing. 3 Main concept: Hydraulic boost (DC-DC) converter An alternative means to vary the pump displacement is based on the hydro-mechanical analog of a DC-DC switched mode converter in power electronics [1] (Fig. 4(a)). A driving force in the advances in power electronics was the invention of power transistors that can be switched on and off very quickly. When the transistor is fully on or fully off, power loss is minimal because either there is no voltage drop across the transistor or there is no current passing through it. A DC-DC converter (Fig. 4(a)) transforms input DC power at one voltage (V d ) to the output port at another voltage (V out ). In a boost converter, V out > V d whereas in a buck converter, V out < V in. A boost converter is shown in Fig. 4(a) where the transistor switch is pulse width modulated (PWM) so that the control is the on/off duty cycle. The circuit s operation is as follows. When the transistor switch is on, current is stored in the inductor. The input is disconnected from the output circuit. The load is temporarily supplied by the charge stored in the capacitor. When the switch 1 The response time depends on the sizes of the control/supply orifices used. Also, it should be pointed out that pressure compensation and load sensing pumps use integrated hydro mechanical feedback on the servo piston which have better response time (. to.9s). However, the flexibility of the control is limited. 3 Copyright c 2 by ASME

4 (a) Eaton family model pump type Displacement [cm 3 ] W x H x L [mm 3 ] volume [1 3 m 3 ] weight L2 Gear x 17 x kg (est) 73 Piston, MDC x 14 x kg 724 Piston, EDC x 222 x kg (b) Sauer-Danfoss family model pump type Displacement [cm 3 ] W x H x L [mm 3 ] volume [1 3 m 3 ] weight GNP3 Gear 44 1x 121 x n/a M44 piston, DDC x 182 x kg M4 Piston, EDC 4 24 x 242 x kg Table 1. Comparison of two manufacturers gear pump and variable displacement piston pumps with various control options. The trend is typical of other manufacturers. DDC refers to direct displacement control where swash plate is directly controlled via manual lever. MDC refers to manual displacement control where the manual lever controls a valve that controls the hydraulically actuated swash plate. EDC is electronically controlled swash plate. EDC for the Sauer-valve is shown in Fig. 3(b). I out I in L Inductor C Load PWM Vout Vin Boost converter (a) DC-DC boost converter Accumulator P1 P2 Flywheel To application circuit switching signal (PWM) Engine (b) PWM variable displacement pump Figure 4. Electrical DC-DC boost converter and its hydraulic analog is turned off, current from the input and the current stored in the inductor are used to feed the output load and to replenish the capacitor. The operation of the hydro-mechanical analog in Figs. 4(b) is exactly similar, with the inductor replaced by the inertia of the fixed displacement pump and possibly an extra flywheel, the capacitor replaced by an accumulator (for storing fluid at high pressure), and the transistor switch replaced by a high speed on/off valve. The accumulator serves to filter out the flow ripple due to the on/off flow output. The flywheel serves to filter out the load on the engine. If the prime mover s speed is regulated, then the flywheel will not be necessary. Roughly speaking, in steady state operation, the pump flow or the effective pump displacement will be a fraction of the flow rate / displacement of the fixed displacement pump, as determined by the duty ratio of the on/off valve. Thus, flow rate modulation can be achieved by varying the duty ratio of the on/off valve, in either open loop or closed loop. Fig. shows that the concept is easily adaptable to situations where a single pump is driving multiple circuits. Each additional circuit is controlled by a PWM valve/ accumulator combination which serves the function of lossless variable transformer and low pass filter. Notice that hardware controlled variable displacement pumps cannot be easily adaptable to this situation. Assuming that the on/off transition times are small compared to the times when the valve is in either the fully on or fully off states, the proposed device in Figs. 4(b) or will be very efficient. It is because the throttling loss through the valve will be negligible when either the flow through the valve, or the pressure drop across it, is nearly zero. 4 Copyright c 2 by ASME

5 Figure. Fixed displacement pump Flywheel Engine Adaptation to a single variable displacement pump driving multiple circuits P1 PWM PWM On/Off valve Effective Variable Displacement Pump Check valve Check valve PWM Switching Signal Accumulator P Accumulator P Load circuit 1 Load circuit 2 given by: J f ω = u(t) D P+Γ(t) (1) 2π Ṗ = γ P [ u(t) D ] V 2π ω Q out(t) (2) }{{} V where u = corresponds to the valve being opened, and u = 1 corresponds to the valve being closed; J f is the inertia of the flywheel/pump, ω is the pump speed, D is the pump displacement, Γ is the engine torque, P is the accumulator/output pressure, and γ is the ratio of heat capacities at constant pressure and and at constant temperature. V is the compressed gas volume in the accumulator, given by: PV γ = P V γ (3) The hydraulic boost converter in Fig. 4(b) can achieve the desired weight and size saving because simple gear pumps, which are compact and light, can be used as the fixed displacement pump. As shown in table 1, the weight and volume of a fixed displacement gear pump are only 1/3 of those of an EDC pump. Even if a fixed displacement piston pump is used to take advantage of their lower leakage and noise properties, the extra swash plate control unit will no longer necessary, and a saving of 3-% will likely be achieved. Beside the topology in Fig., the concept of common pressure rail (CPR) from which each hydraulic device obtains energy via a throttle-less hydraulic transformer [3, 4] is another possibility for situations when multiple circuits are fed by the same pump. The switched mode pump control concept proposed in this paper can also be extended easily to the efficient and compact implementation of hydraulic transformers as well. Notice that the hydraulic pump is itself a mechanical-to-hydraulic transformer. where P and V are the pre-charge pressure and the initial gas volume of the accumulator. Q out is the load flow. We assume that the valve input signal u(t) will be pulse width modulated (PWM) so that for a duty ratio s(t) [,1], u(t) is determined by comparing the s(t) with a periodic saw-tooth function: u(t) = { 1 if s(t) (t/t)mod1 if s(t) < (t/t)mod1 where T is the PWM period. If the PWM switching frequency (1/T ) is sufficiently fast [], the states space average system [1], obtained by treating the state variables and inputs as constants during the time averaging procedure, is valid and is given by: (4) J f ω = s(t) D P+Γ(t) () 2π Ṗ = γ P [ s(t) D ] V 2π ω Q out(t) () }{{} V In the steady state, we have the equilibrium condition, 3.1 Dynamics We assume that the gas charged accumulator obeys the ideal gas law and operates under adiabatic conditions. Ignoring the on/off valve dynamics and assuming that the valve has a sufficiently large orifice so that the pressure drop across it is negligible when it is open, the dynamics of the circuit in Fig. 4(b) are Q out = s(t)d 2π ω; Γ = s(t)d 2π P (7) which is the usual ideal positive displacement pump relationship with s(t)d playing the role of the effective displacement of the pump which can be controlled by software via the duty cycle s(t). Copyright c 2 by ASME

6 3.2 Software enabled features The PWM control input s(t) can be used to modify the behavior of the pump via various types of feedback control algorithms, and thus enabling many possible useful features. We list some of these below. Ideal pump: This can be achieved by designing an appropriate feedback control law so that the ideal pump relationship (7) is strongly stabilized at the desired duty cycle s. Pressure control: this can be achieved by controlling the duty cycle so that the output pressure P follows some desired trajectory. From ()-(), we see that this is a state stabilization problem. Flow control: this can be accomplished by placing a small orifice downstream to the PWM pump outlet, and then controlling s(t) so that the pressure drop across the orifice corresponds to the desired flow rate. Direct load control: if the load is modeled, then Q out will be a function of P and the properties of the load. s(t) can then be used to directly control the relevant variables. Compensation for engine behavior: in current systems, the engine is regulated at constant speed ω. Using the PWM pump, variation in ω can be accounted for (e.g. by maintaining s(t)ω(t) constant). Optimal engine operation: This is related to the above. Since variation in engine speed can be accounted for, the engine can be operated at the speed that is most efficient for the power demand according to the engine s own speed-torque curve. and others... Notice that these features can be switched from one type to another by simply changing the control software. For this reason, we refer to this type of pump as software enabled variable displacement pumps. Notice also that the above features are only practical if the pump displacement can be varied sufficiently fast. 4 A Design Case study In this section, we use a case study to consider the performance trade-offs due to various design parameters. The simple circuit in Fig. 4(b) is considered with the load being a simple orifice: Q out = C P (8) The system is sized according to an experimental setup that is under construction. The displacement of the pump is D = 4.42cc. The pre-charge gas volume in the accumulator is V = 1cc. The gas constant for N 2 is γ = 1.4. The orifice coefficient in (8) is C = m 3 /s/pa.. In order to compare with a conventional variable displacement pump, the prime mover is assumed to be operating at constant speed ω = 14.7rad/s (1 RPM). This gives a maximum flow rate of m 3 /s (1.2 gpm). The dynamics are reduced to: [ Ṗ = γ P1+1/γ P 1/γ u(t) D ] V 2π ω C P where u(t) = when the on/off valve is open, and u(t) = 1 when it is closed. The states-space averaged system is in turn reduced to: [ Ṗ = γ P1+1/γ P 1/γ s(t) D ] V 2π ω C P (9) (1) where s(t) [,1] is the duty ratio of the PWM input. We consider the scenarios when we need to control the load flow through the orifice, from the initial flow of Q =. D 2π ω = m 3 /s (. gpm) to the final flow of Q f =. D 2π ω = m 3 /s (.72gpm), corresponding to the duty cycles of s =. and s =. respectively. Since the orifice is fixed, these correspond to the desired pressures of P = Pa and P f =.77 1 Pa. 4.1 Open loop response For the open loop tests, the duty cycle is changed from s =. to s =. at t = s. The response when the accumulator pre-charge of P =. 1 Pa and the PWM period of T =.1s is shown in Fig.. By t = s, the system has settled down at a steady oscillatory mode around the desired pressure P. After t = s, the average pressure rises to the desired pressure P f. The response time for this transition is 1.741s as determined from the time required for the average value to reach 9% of the overall change. This may be satisfactory but for the large peak-to-peak pressure ripples, which are as large as 1.2% of the average pressure. The average and ripple pressure characteristics translate directly into those of the load flow according to the orifice equation (8). Accumulator sizing The flow ripple can be improved by increasing the pre-charge pressure of the accumulator. Figure 7 shows the case when it has been increased 8 folds to P = 4 1 Pa. While the flow ripple has been reduced to 3.7% of the mean pressure, the time response has been significantly increased to 7.s. From (9), it can be seen that the term γ P1+1/γ plays P 1/γ V the role of the volumetric stiffness (dp/dv ) of the accumulator. Indeed, for the same operating pressure, increases in either the Copyright c 2 by ASME

7 7. x 1 Open Loop system T =.1, P =.e Pa, lambda = x 1 Open Loop system T =., P = 4e Pa, lambda = averaged. averaged 4. Figure. 4 Open-loop pressure response: T =.1s, P =. 1 Pa. The response time is and the peak-to-peak ripple is 1.2% of the average pressure at the final operating condition. Figure Open-loop pressure response: T =.s, P = 4 1 Pa. The response time is 7. and the ripple is 1.8% of the average pressure at the final operating condition. 7 x 1.. Open Loop system T =.1, P = 4e Pa, lambda = 2.3 averaged 4.2 Closed loop response The open loop response shows that the ripple size can be reduced by increasing the PWM frequency (decreasing T ). T is limited by the responsiveness of the on/off valve. Since throttling occurs when the valve is in transition (partially open), energy efficiency dictates that the transition times should be small compared to the times when the valve is either fully open or fully closed. For a given PWM frequency, feedback control can be used to improve the response time without adverse effects on the ripple size. To investigate the usefulness of the feedback, the duty cycle s(t) [,1] is determined by feedback as follows: Figure Open-loop pressure response: T =.1s, P = 4 1 Pa. The response time is 7.s and the ripple is 3.7% of the average pressure at the final operating condition. pre-charge pressure P or the accumulator volume V decreases its stiffness, which decreases ripple and slows down response. Notice that P cannot be higher than the operating pressure for it to be functional. PWM frequency As PWM period T decreases to.s, the ripple size decreases to 1.8% without affecting the response time as illustrated in Fig. 8. The trade-off between ripples and open loop response times can be seen for each PWM period T in Fig. 11. s(t) = s f f + K(P P(t)) (11) where s f f is the feedforward term, P is the desired equilibrium pressure, P is the sensed instantaneous pressure at the accumulator outlet, and K is the feedback gain determined by placing the pole of the linearization of the states space averaged system (1) as negative as possible without saturation. The PWM control design procedure followed here is traditional and similar to the literature (e.g. [1,,7]) in that the control law is derived based on a nonlinear model obtained using the states space averaging method. Typically, the control bandwidth is recommended to be an order of magnitude lower than the PWM frequency. For example, [] uses a 2kHz PWM frequency on an electromechanical system and is successful in tracking a Hz trajectory. Authors in [7] successfully used a 2Hz PWM to track a 1.2Hz trajectory on a pneumatic system. Notice that the PWM frequency in these cases are at least 2 times higher than the frequency of the trajectory. 7 Copyright c 2 by ASME

8 7 x Closed Loop system T =.1, P = 4e Pa, lambda = x Closed Loop system T =., P = 4e Pa, lambda = averaged.. averaged Closed Loop system T =.1, P = 4e Pa, lambda = Closed Loop system T =., P = 4e Pa, lambda = Duty Ratio d.7 Duty Ratio d Closed Loop system T =.1, P = 4e Pa, lambda = Closed Loop system T =., P = 4e Pa, lambda = Signal to PWM Valve. Signal to PWM Valve Figure 9. Closed-loop response for T =.1s, P = 4 1 Pa. Top: pressure response, middle: duty ratio s(t) and bottom: valve inputs u(t). The response time is 1.s and the peak-to-peak ripple is 3.7% of the average pressure at the final operating condition. Figure 1. Closed-loop response for T =.s, P = 4 1 Pa. Top: pressure response, duty ratio s(t) and valve inputs u(t): The response time is 1.2s and the peak-to-peak ripple is 1.8% of the average pressure at the final operating condition. Figures 9 and 1 show the closed loop responses when the accumulator pre-charge pressure is P = 4 1 Pa and the PWM periods are T =.1s and T =.s respectively. Compared with Figs. 7 and 8, the response times have been significantly reduced (from 7.s to 1.s, and 7.s to 1.2s respectively) without increasing the ripple size. The large transient increase (right after t = s) in duty cycles s(t) is responsible of the quickened response. If the PWM duty cycle is allowed to saturate, response times can be improved further. The trade-offs between ripple size and response times for different T s and pre-charge pressures P for the closed loop case is shown in Fig. 11. Decrease in PWM period causes a slight but noticeable decrease in response time for all pre-charge pressures. 4.3 Frequency dependent averaging In the closed loop responses (Figs. 9 and 1), the trajectories of the states-space averaged model (1) under the linear control law (11), as well as the average response (computed by a zero-phase Boxcar filter with a length approximately equal to 2T ), are plotted. Notice that the state-space averaged trajectories reach the desired pressures, but the average pressure trajectories tend to be lower. Hence, the states space average model over-estimates the average pressure. Since the control design is based on the average model, the control scheme generates pressures and flows that are below the the desired values. In contrast, for the open loop cases, the average models predict the average responses accurately (Figs. -8). The discrepancies that exist in the closed loop cases are due to the implementation of the PWM control in (4) as predicted in [2, ]. Because of the inevitable ripple in the response, a ripple function is also imposed on the duty cycle s(t) in (11) and (4). The mean of this function is typically different from that obtained from (11) assuming that the mean pressure P is fedback. In [2], the steady state bias and qualitative differences are predicted, and a frequency dependent averaging that takes this effect into account is proposed. The discrepancy between the state- 8 Copyright c 2 by ASME

9 1 P=.ePa Closed Loop vs.open Loop PWM System T=.2 % of Ripple of the Nominal Pressure 1 P=.1ePa T=.1 P=.ePa P=1ePa open loop P=4ePa Figure 11. pressure P closed loop T=. Response Time t Open loop and closed loop trade-off between peak-to-peak ripple size and response time for each PWM period T and accumulator pre-charge space averaged system and the average system response becomes more significant as the PWM frequency decreases. Following [2], the state-spaces averaged model in (1) is modified to be: [ Ṗ = γ P1+1/γ P 1/γ τ s (t) D ] V 2π ω C P (12) where τ s (t) is the modified average duty cycle determined by solving the implicit equation: τ s (t) = s f f + K(P (P(t)+ψ(t))) where ψ(t) is an estimate of the ripple function which is in turn a function of τ s (t). The readers are referred to [2] for details. Figure 12 shows that the modified average model can much better predict the average behavior of the system. Conclusions A fixed displacement pump, an accumulator, and an on/off switching valve under PWM control have been proposed to achieve the function of a variable displacement pump. The device is potentially more compact and controllable than a hardware controlled variable displacement pump so that a variety of 7 x Closed Loop system T =.1, P = 4e Pa, lambda = traditional average improved average 4. Figure 12. Actual, zero-phase, states-space averaged, and frequency dependent averaged closed-loop pressure responses for T =.1s, P = 4 1 Pa. software enable features can be implemented. The proposed device is the hydro-mechanical equivalent of a switched mode DC- DC boost converter in power electronics. Two important functional characteristics of the new device are ripple size and response time. It is shown that increases in the accumulator pre- 9 Copyright c 2 by ASME

10 charge pressure and volume can decrease ripple size at the expense of response times. This apparent trade-off can be alleviated by feedback control to achieve fast response time without adverse effect on the ripple size. Feedback control using PWM control must be implemented with care since the conventional state-space model may not be valid. The frequency dependent averaging technique proposed in [2] has been shown to be effective in predicting the average behavior for the new device. Increasing PWM frequency reduces ripple size and undesirable PWM effects. These results point toward several research directions that are currently being pursued: 1) design of an on/off switching valve that is capable of very high PWM frequency; 2) integrated design of the accumulator, on/off valve and pump that maximizes the ripple/responsiveness trade off of the system; 3) control strategies that take into account relatively low PWM frequencies that can be achievable by mechanical systems. Acknowledgment This material is based upon work supported by the National Science Foundation under grant number ENG/CMS REFERENCES [1] N. Mohan, T. Undeland, and W. Robbons, Power electronics: converters, applications and design. John Wiley and Sons, 2nd ed., 199. [2] B. Lehman and R. Bass, Switching frequency dependent averaged models for pwm dc-dc converters, IEEE Transactions on Power Electronics, vol. 11, no. 1, pp , 199. [3] Innas Hydraulic Transformer [4] L. Gu, M. Qiu, B. Jin, and J. Cao, New hydraulic systems made up of hydraulic power bus and switch-mode hydraulic power supplies, Chinese Journal of Mechanical Engineering, vol. 39, no. 1, pp , 23. [] B. Lehman and R. Bass, Recent advances in averaging theory for pwm dc-dc converters, in Proceedings of the IEEE CDC, pp , 199. [] D. G. Taylor, Pulse width modulated control of electromechanical systems, IEEE Transactions on Automatic Control, vol. 37, no. 4, pp , [7] E. J. Barth, J. Zhang, and M. Goldfarb, Sliding mode approach to pwm-controlled pneumatic systems, in Proceedings of the 222 ACC, pp , Copyright c 2 by ASME